HP 50g manuel d'utilisation

HP  g gr aphing calc ulator user ’s guide H Ed i ti on 1 HP part number F2 2 2 9AA-9 0006[...]

Notice REG ISTER Y OUR PRODU CT A T: ww w .regis ter .hp.com TH IS MANUAL AND ANY E XAMPLE S CONT AINE D HEREIN ARE PR O VID E D “ AS IS” AND ARE SUB JECT T O CHANGE WITHOUT NOT ICE . HEWLET T -P ACKARD COMP ANY MAKE S NO W ARR ANTY OF ANY KIND WI TH REG ARD T O TH IS MANU AL , INCL UD ING, BUT NOT LIMITED T O, THE IMPLI ED W ARR ANTIE S OF MER[...]

Pref ace Y ou ha ve in y our hands a compact s ymboli c and numer ical computer that w ill fac ilitate calc ulati on and mathematical anal ysis o f pr oblems in a var iety of disc iplines, f r om elementary mathematic s to adv anced engineer ing and sc ience subjec ts. Although r ef err e d to as a calc ulator , because of its compact fo rmat r ese[...]

F or s ymboli c oper ati ons the calc ulator inc ludes a po werf ul Co mputer A lgebrai c S y ste m (CAS) that lets y ou select diff er ent modes o f oper ation , e .g ., complex number s vs . r eal numbers , or e x act (s y mbolic) v s . appr o x imate (numer ical) mode . T he displa y can be adju sted to pr ov ide te xtbook - type e xp r essi ons[...]

Pa g e TO C - 1 T abl e o f contents Chapter 1 - Getting started ,1-1 Basic Operations ,1-1 Batteries ,1-1 Turning the calculator on an d off ,1-2 Adjusting the display contrast ,1-2 Contents of the calculator’s display ,1-2 Menus ,1-3 SOFT menus vs. CHOOSE boxes ,1-4 Selecting SOFT menus or CHOOSE boxes ,1-5 The TOOL menu ,1-7 Setting time and d[...]

Pa g e TO C - 2 Chapter 2 - Introducing the calculator ,2-1 Calculator objects ,2-1 Editing expressions on the screen ,2-3 Creating arithmetic exp ressions ,2-3 Editing arithmetic expressions ,2-6 Creating algebraic expressions ,2-7 Editing algebraic expressions ,2-8 Using the Equation Writer (EQW ) to create expressions ,2-10 Creating arithmetic e[...]

Pa g e TO C - 3 Other flags of interest ,2-66 CHOOSE boxes vs. Soft MENU ,2-67 Selected CHOOSE boxes ,2-69 Chapter 3 - Calculation with real numbers ,3-1 Checking calculato rs settings ,3-1 Checking calculator mode ,3-2 Real number calculations ,3-2 Changing sign of a number, var iable, or expression ,3-3 The inverse function ,3-3 Addition, subtrac[...]

Pa g e TO C - 4 Physical constants in the calc ulator ,3-29 Special physical functions ,3-32 Function ZFACTOR ,3-32 Function F0 λ ,3-33 Function SIDENS ,3-33 Function TDELTA ,3-33 Function TINC ,3-34 Defining and using functions ,3-34 Functions defined by more than one expression ,3-36 The IFTE function ,3-36 Combined IFTE functions ,3-37 Chapter [...]

Pa g e TO C - 5 FACTOR ,5 -5 LNCOLLECT ,5-5 LIN ,5-5 PARTFRAC ,5-5 SOLVE ,5-5 SUBST ,5-5 TEXPAND ,5-5 Other forms of substitution in algebraic expressions ,5-6 Operations with transcendental functions ,5-7 Expansion and factoring using log-exp functions ,5-7 Expansion and factoring using trigonometric functions ,5-8 Functions in the ARITHMETIC menu[...]

Pa g e TO C - 6 The PROOT function ,5-21 The PTAYL function ,5-21 The QUOT and REMAINDER functions ,5-21 The EPSX0 function and the CAS variable EPS ,5-22 The PEVAL function ,5-22 The TCHEBYCHEFF function ,5-22 Fractions ,5-23 The SIMP2 function ,5-23 The PROPFRAC function ,5-23 The PARTFRAC func tion ,5-23 The FCOEF function ,5-24 The FROOTS funct[...]

Pa g e TO C - 7 Variable EQ ,6-26 The SOLVR sub-menu ,6-26 The DIFFE sub-menu ,6-29 The POLY sub-menu ,6-29 The SYS su b-menu ,6-30 The TVM sub-menu ,6-30 Chapter 7 - Solving multiple equations ,7-1 Rational equation systems ,7-1 Example 1 – Projectile motion ,7-1 Example 2 – Stresses in a thick wall cylinder ,7-2 Example 3 - System of polynomi[...]

Pa g e TO C - 8 List size ,8-10 Extracting and inserting elements in a list ,8-10 Element position in the list ,8-11 HEAD and TAIL functions ,8-11 The SEQ function ,8-11 The MAP function ,8-12 Defining functions that use lists ,8-13 Applications of lists ,8-15 Harmonic mean of a list ,8-15 Geometric mean of a list ,8-16 Weighted average ,8-17 Stati[...]

Pa g e TO C - 9 Changing coordi nate system ,9-12 Application of vector operations ,9-15 Resultant of forces ,9-15 Angle between vectors ,9-15 Moment of a force ,9-16 Equation of a plane in space ,9-17 Row vectors, column vector s, and lists ,9-18 Function OBJ  ,9-19 Function  LIST ,9-20 Function DROP ,9-20 Transforming a row vector into a co[...]

Pa g e TO C - 1 0 Function VANDERMONDE ,10-13 Function HILBERT ,10-14 A program to build a matrix out of a nu mber of lists ,10-14 Lists represent columns of the matrix ,10-15 Lists represent rows of the matrix ,10-17 Manipulating matrices by columns ,10-17 Function  COL ,10-18 Function COL  ,10-19 Function COL+ ,10-19 Function COL- ,10-20 Fu[...]

Pa g e TO C - 1 1 Function TRAN ,11-15 Additional matrix operations (The matri x OPER menu) ,11-15 Function AXL ,11-16 Function AXM ,11-16 Function LCXM ,11-16 Solution of linear systems ,11-17 Using the numerical solver for linear systems ,11-18 Least-square solution (function LSQ) ,11-24 Solution with the inverse matrix ,11-27 Solution by “divi[...]

Pa g e TO C - 1 2 Function QXA ,11-53 Function SYLVESTER ,11-54 Function GAUSS ,11-54 Linear Applications ,11-54 Function IMAGE ,11-55 Function ISOM ,11- 55 Function KER ,11-56 Function MKISOM ,11-56 Chapter 12 - Graphics ,12-1 Graphs optio ns in the calculator ,12-1 Plotting an expression of the form y = f(x) ,12-2 Some useful PLOT operations for [...]

Pa g e TO C - 1 3 Fast 3D plots ,12-34 Wireframe plots ,12-36 Ps-Contour plots , 12-38 Y-Slice plots ,12-39 Gridmap plots ,12-40 Pr-Surface plots ,12- 41 The VPAR variable ,12-42 Interactive drawing ,12-43 DOT+ and DOT- ,12-44 MARK ,12-44 LINE ,12-44 TLINE ,12-45 BOX ,12-45 CIRCL ,12-45 LABEL ,12-45 DEL ,12- 46 ERASE ,12-46 MENU ,12-46 SUB ,12-46 R[...]

Pa g e TO C - 1 4 The SYMBOLIC menu and graphs ,12-49 The SYMB/GRAPH menu ,12-50 Function DRAW3DMATRIX ,12-52 Chapter 13 - Calculus Applications ,13-1 The CALC (Calculus) menu ,13-1 Limits and derivatives ,13-1 Function lim ,13-2 Derivative s ,13-3 Functions DERIV and DERVX ,13-3 The DERIV&INTEG menu ,13-4 Calculating derivatives with ∂ ,13-4[...]

Pa g e TO C - 1 5 Integration with units ,13-21 Infinite series ,13-22 Taylor and Maclaurin’s se ries ,13-23 Taylor polynomial and reminder ,13-23 Functions TAYLR, TAYLR0, and SERIES ,13-24 Chapter 14 - Multi-variate Calculus Applications ,14-1 Multi-variate functions ,14-1 Partial derivatives ,14-1 Higher-order derivatives ,14- 3 The chain rule [...]

Pa g e TO C - 1 6 Checking solutions in the calc ulator ,16-2 Slope field visualizati on of solutions ,16-3 The CALC/DIFF menu ,16-3 Solution to linear and non-linear equations ,16-4 Function LDEC ,16-4 Function DESOLVE ,16-7 The variable ODETYPE ,16-8 Laplace Transforms ,16-10 Definitions ,16-1 0 Laplace transform and inverses in the calculator ,1[...]

Pa g e TO C - 1 7 Numerical solution of first-order ODE ,16-57 Graphical solution of first-order ODE ,16-59 Numerical solution of second-order ODE ,16-61 Graphical solution for a second-order ODE ,16-63 Numerical solution for stiff first-order ODE ,16-65 Numerical solution to ODEs with the SOLVE/DIFF menu ,16-67 Function RK F ,16-67 Function RRK ,1[...]

Pa g e TO C - 1 8 Chapter 18 - Statistical Applications ,18-1 Pre-programmed statistical features ,18-1 Entering data ,18-1 Calculating single-variable statistics ,18-2 Obtaining frequency distributions ,18-5 Fitting data to a function y = f(x) ,18-10 Obtaining additional summary statistics ,18-13 Calculation of percentiles ,18-14 The STAT soft men[...]

Pa g e TO C - 1 9 Paired sample tests ,18-41 Inferences concerning one proportion ,18- 41 Testing the difference betw een two proportions ,18-42 Hypothesis testing using pre-programmed features ,18-43 Inferences concerning one variance ,18-47 Inferences concerning two variances ,18-48 Additional notes on linear regression ,18-50 The method of least[...]

Pa g e TO C - 2 0 Custom menus (MENU and TMENU functions) ,20-2 Menu specification and CST variable ,20-4 Customizing the keybo ard ,20-5 The PRG/MODES/KEYS sub-menu ,20-5 Recall current user-defined key list ,20- 6 Assign an object to a user-defined key ,20-6 Operating user-defined keys ,20-7 Un-assigning a user-defined key ,20-7 Assigning multipl[...]

Pa g e TO C - 2 1 “De-tagging” a tagged quantity ,21-33 Examples of tagged output ,21-34 Using a message box ,21-37 Relational and logical operators ,21-43 Relational operators ,21-43 Logical operators ,21-45 Program branching ,21-46 Branching with IF ,21-47 The IF…THEN…END construct ,21-47 The CASE construct ,21-51 Program loops ,21-53 The[...]

Pa g e TO C - 2 2 Examples of program-generated plots ,22-17 Drawing commands for use in programming ,22-19 PICT ,22-20 PDIM ,22-20 LINE ,22-20 TLINE ,22-20 BOX ,22-21 ARC ,22-21 PIX?, PIXON, and PIXOFF ,22-21 PVIEW ,22-22 PX  C ,22-22 C  PX ,22-22 Programming examples using drawing functions ,22-22 Pixel coordinates ,22-25 Animating graphics[...]

Pa g e TO C - 23 Chapter 24 - Calculator objects and flags ,24-1 Description of calculator objects ,24-1 Function TYPE ,24-2 Function VTYPE ,24-2 Calculator flags ,24-3 System flags ,24-3 Functions for setting and changing flags ,24-3 User flags ,24-4 Chapter 25 - Date and Time Functions ,25-1 The TIME menu ,25-1 Setting an alarm ,25-1 Browsing ala[...]

Pa g e TO C - 24 Storing objects on an SD ca rd ,26-9 Recalling an object from an SD card ,26-10 Evaluating an object on an SD card ,26-10 Purging an object from the SD card ,26-11 Purging all objects on the SD card (by reformatting) ,26-11 Specifying a directory on an SD card ,26- 11 Using libraries ,26-12 Installing and attaching a library ,26-12[...]

Pa g e TO C - 2 5 Appendix F - The Applications (APPS) menu ,F-1 Appendix G - Useful shortcuts ,G-1 Appendix H - The CAS help facility ,H-1 Appendix I - Command catalog list ,I-1 Appendix J - MATHS menu ,J-1 Appendix K - MAIN menu ,K-1 Appendix L - Line editor commands ,L-1 Appendix M - Table of Built-In Equations ,M-1 Appendix N - Index ,N-1 Limit[...]

Pa g e 1 - 1 Chapter 1 G e t ting started T his chapte r pr ov ides basi c inf ormatio n about the oper ation of y our calculator . It is desi gned to familiar i z e y ou w ith the basic oper ations and se ttings b e fo r e y ou perfor m a calc ulation . Basic Operations T he follo w ing secti ons ar e designed t o get y ou acquainted w ith the har[...]

Pa g e 1 - 2 b . Insert a ne w CR203 2 lithium batter y . Make sur e its positi v e (+) side is f aci ng up . c. R eplace the plate and p u sh it to the ori ginal place. After installi ng the bat ter i es, pr ess [ON] to turn the po wer on . Wa rn i n g : When the lo w battery icon is displa y ed, y ou need to r eplace the batteri es as soon as pos[...]

Pa g e 1 - 3 At the top o f the display y ou will ha v e two lines o f infor mation that de sc ribe the settings o f the calculator . The f irst line sho ws the c har acte rs: R D XYZ HE X R= 'X' F or details on the meaning of the se s y mbols see C hapter 2 . T he second line sho ws the c harac ter s: { HOME } indicating that the HOME di[...]

Pa g e 1 - 4 E ach gr oup of 6 entr i es is called a Menu page . The c ur r ent menu , know n as the T OOL menu (s ee belo w) , has e ight en tri es ar ranged in tw o pages. T he next page , containing the ne xt two entr ies o f the menu is av ailable b y pr essing the L (NeXT menu) k e y . This k ey is the thir d ke y fr om the left in the thir d [...]

Pa g e 1 - 5 T his CHOOSE bo x is labeled B ASE MENU and pr o v ide s a list of n umber ed fu nct ion s, from 1 . H EX x to 6. B  R. T his displa y wi ll constitute the f irs t page of this CHOOSE bo x menu sho w ing si x menu f uncti ons. Y ou can nav igat e thr ough the menu b y using the up and do w n arr o w k e y s, —˜ , located in the u[...]

Pa g e 1 - 6 If y ou no w pr es s ‚ã , instead of the CHOO SE bo x that y ou sa w earli er , the displa y w ill no w show six s oft menu la bels as the f irst page of the S T A CK menu: T o na vi gate thr ough the func tions of this me nu , pr ess the L k ey to mov e to the ne xt page , or „« (ass oc iated w ith the L k e y ) t o m o v e t o [...]

Pa g e 1 - 7 The T OOL m enu T he soft menu k e y s for the men u c ur r ent ly displ ay ed , kno w n as t he T OOL men u , ar e assoc iat ed with oper ations r elated to manipulation o f var iables (s ee pages for more in forma tion o n variabl es) : @EDIT A EDIT the conten ts of a var ia ble (see Chapter 2 and Appendi x L for mor e infor mation o[...]

Pa g e 1 - 8 9 k e y the T IME c hoose bo x is acti vated . T his operati on can also be r epr esented as ‚Ó . Th e TIM E ch oo se box i s sh o wn in th e figu re b el ow: As indicated abov e, the TIME men u pr o vi des f our differ ent options , number ed 1 thr ough 4. Of inter es t to us as this poin t is option 3 . Se t time , date .. . U sin[...]

Pa g e 1 - 9 Let ’s change the minute f ield to 2 5, by pr ess ing: 25 !!@@OK#@ . T he seco nds f ield is no w highli ghted . Suppose that y ou w ant to c hange the seconds fi eld to 4 5, u se: 45 !!@@OK #@ T he time for mat f ield is no w highlighted . T o c h a n g e t h i s f i e l d f r o m i t s c u r r e n t set ting y ou ca n either pr ess[...]

Pa g e 1 - 1 0 Setting th e date After s etting the time for mat option , the SET T IME AND D A TE input for m w ill look as f ollo w s: T o set the date , f irst set the date f ormat . The de fault f or mat is M/D/Y (month/ day/y ear). T o modif y this f or mat , pre ss the do w n arr o w ke y . T his w ill hi ghlight the date f or mat as sho wn b[...]

P age 1-11 Intr oduc ing the calc ulator ’s k e yboar d The f igur e below sh ow s a di agram of the calculator ’s k ey boar d w ith the number ing of its ro ws and columns. T h e f i g u r e s h o w s 1 0 r o w s o f k e y s c o m b i n e d w i t h 3 , 5 , o r 6 c o l u m n s . R o w 1 has 6 ke ys , r ow s 2 and 3 hav e 3 ke y s each , and ro [...]

P age 1-12 shift ke y , k e y (9 ,1 ) , and the ALPHA k e y , ke y ( 7 ,1) , can be combined w ith some of the other k e y s to acti vat e the alternati ve func tions sho w n in the k e yboar d . F or e x ample , the P key , key(4,4 ) , has the follo wing si x func tions as soc iated wi th i t: P Main functi on , to acti vate the S Y MBoli c menu ?[...]

Pa g e 1 - 1 3 Pr ess the !!@ @OK#@ soft men u k e y to r etur n to nor mal displa y . Example s of s electing diffe r ent calc ulator modes ar e show n next . Oper at ing Mode T he calculator o ffer s two oper a ting mode s: the Algebr aic mode , and the Re v ers e P olish Notati on ( RPN ) mode . The de fa ult mode is the A lgebr aic mode (as ind[...]

Pa g e 1 - 1 4 T o enter this e xpre ssion in the calc ulator w e w ill f irs t use the equation w r iter , ‚O . P lease identify the f ollo w ing k e y s in the k e yboar d , besi des the nume ri c k e y pad ke y s: !@.#*+-/R Q¸Ü‚Oš™˜—` T he equation w rite r is a displa y mode in whi ch y ou can build mathematical e xpre ssi ons using[...]

Pa g e 1 - 1 5 Change the oper ating mode to RPN by f irst pr es sing the H butt on . Sele ct th e RPN oper ating mode b y either u sing the ke y , or pr essing the @ CHOOS soft m e n u k e y . P r e s s t h e !!@ @OK#@ soft men u k ey t o complete the oper ation . The displa y , for the RPN mode looks as f ollo w s: Notice that the displa y sho [...]

Pa g e 1 - 1 6 3.` Ent er 3 in le v el 1 5.` Ent er 5 in le v el 1, 3 mov es to y 3.` Ent er 3 in le v el 1, 5 mov es to lev el 2 , 3 t o lev el 3 3.* P lace 3 and multipl y , 9 appears in le v el 1 Y 1/(3 × 3), la st value in le v . 1; 5 in lev el 2 ; 3 in lev el 3 - 5 - 1/(3 × 3) , occ up ies le v el 1 no w ; 3 in lev el 2 * 3 × (5 - 1/(3 × 3[...]

Pa g e 1 - 1 7 Notice ho w the e xp r essi on is placed in stac k le ve l 1 after pre ssing ` . Pr essing the EV AL ke y at this point w i ll ev aluate the numer ical value o f that e xpr es sion Note: In RPN mode , pre ssing ENTER when ther e is n o command line w ill e xec ut e the D UP f uncti on whi ch cop ies the cont ents of stac k le vel 1 o[...]

Pa g e 1 - 1 8 mor e about r e al s, see C hapter 2 . T o illustr ate this and other numbe r for mats try the f ollo w ing ex er c ises: Θ Standard f ormat : T his mode is the most us ed mode as it sho ws nu mbers in the mos t famili ar notation . Pr es s the !!@@ OK#@ soft menu k ey , with the Number f or mat set to St d , to re turn to the calc [...]

Pa g e 1 - 1 9 Notice that the Number F or mat mode is set t o Fix f ollo wed b y a z er o ( 0 ). T his number indicat es the number of dec imals to be sho w n af t er the dec imal point in the calc ulator’s displa y . Pr ess the !!@@OK#@ soft menu ke y to r eturn to the calc ulator displa y . T he number no w is sho wn as: T his setting wi ll fo[...]

Pa g e 1 - 2 0 Pres s the !!@@OK#@ soft menu k ey to complete the sel ec tion: Pr ess the !!@@OK#@ soft menu k e y r eturn to the calc ulator displa y . The number no w is s h ow n as: Notice ho w the number is r ounded, not tr uncated . Th us , the number 12 3 .4 5 6 7 8 9 012 3 4 5 6 , f or this s etting, is displa yed a s 12 3 .4 5 7 , and not a[...]

Pa g e 1 - 2 1 same fa shion that w e c hanged the Fixe d number o f dec imals in the exa mp l e ab ove ) . Pr es s the !!@@OK#@ soft menu k ey r eturn to the calc ulator displa y . The number no w is s h ow n as: T his re sult , 1.2 3E2 , is the calculat or’s v ersio n of po w ers-o f- ten notatio n, i. e. , 1. 2 3 5 x 10 2 . In this, s o -calle[...]

Pa g e 1 - 2 2 Pr es s the !!@@OK#@ soft menu k ey re turn to the calc ulator dis pla y . The n umber no w is s h ow n as: Becau se this number has thr ee fi gur es in the inte ger part, it is sho wn w ith fo ur signif icati v e fi gur es and a z ero po wer o f ten , while using the Engineer ing f ormat . F or e xample , the number 0.00 2 5 6, w il[...]

Pa g e 1 - 23 Θ Pr es s the !!@@OK#@ soft menu k ey re turn to the calc ulator dis pla y . The n umber 12 3 .4 5 6 7 8 9012 , enter ed earlier , no w is sho wn as: Angle M easur e T r igonometr i c func tions , for e xample , r equir e arguments r epr ese nting plane angles . T he calculat or pr ov ides thr ee differ ent Angle Measur e modes fo r [...]

Pa g e 1 - 24 k e y . If u sing the lat t er appr oach , use u p and dow n arr ow k ey s , — ˜ , to se lect the pr ef err ed mode , and pr ess the !!@@OK#@ soft menu k e y to complete the ope r ation . F or e xample , in the follo w ing scr een, the R adians mode is selec ted: Coor dinate S y stem The coo r di na te sy ste m sel ectio n a ffect [...]

Pa g e 1 - 25 fr om the positi v e z ax is to the r adial dis tance ρ . T he Rec tangular and Spher ical coor dinate sy stems ar e re lated by the fo llo wi ng quantities: T o c hange the coordinat e s ys tem in y our calculat or , follo w these s teps: Θ Pr es s the H button. Ne xt, u se the do wn ar ro w k ey , ˜ , three times. Select the Angl[...]

Pa g e 1 - 26 _L ast S tac k : K eeps the conten ts of the last st ack en tr y f or us e with the f unct ions UNDO and ANS (see C hapter 2). Th e _Beep option can be us ef ul to adv ise the user a bout err ors . Y ou may w ant to des elect this option if u sing yo ur calc ulator in a cla ssr oom or libr ary . Th e _K ey Cli ck opti on can be usef u[...]

Pa g e 1 - 27 Selec ting Displa y modes T he calculator dis play can be c ustomi z ed to y our pr ef er ence b y selecting dif f erent disp lay mod es . T o see the op tional di splay sett ings use the follow ing : Θ F i r st , pr es s the H button to acti v ate the CAL CULA T OR MODE S input f or m. W ithin the CAL CUL A T OR MODE S input fo rm ,[...]

Pa g e 1 - 2 8 Pr essing the @ CHOOS so ft menu k e y w ill pr o vi de a list of a v ailable s y ste m fonts , as sho w n belo w: T he options a vaila ble ar e thr ee standar d Sys t e m Fo n t s (si z es 8, 7 , and 6 ) and a Br o wse .. opti on. T he latter w ill let y ou br o w se the calc ulator memory f or additional f onts that y ou may ha v e[...]

Pa g e 1 - 2 9 displa y the DISPLA Y MODE S input f orm . Pr ess the do wn ar r ow k e y , ˜ , tw i ce , to get to the Stack line . This line show s two pr operties that can be modif ied . When thes e pr oper ti es ar e selec ted (chec k ed) the f ollo w ing eff ects ar e acti v ated: _Small Changes f ont si z e to small . T his max imi z ed the a[...]

Pa g e 1 - 3 0 times , to ge t to the EQW (E quati on W rit er ) line . This line sho w s tw o pr oper ti es that can be modif ied . When thes e properti es ar e select ed (chec k ed) the fo llo w ing eff ects ar e acti vated: _Small Changes f ont si z e to small w hile using the equati on edito r _Small S tac k Disp Sho w s small font in the s tac[...]

Pa g e 1 - 3 1 r ight arr ow k ey ( ™ ) to select the under line in fr ont of the opti ons _Clock or _Analog . T oggle the @  @CHK@@ soft men u k e y until the desir ed setting is ac hie v ed. If the _Cloc k option is se lected , the time of the da y and date w ill be sho wn in the upper ri ght corner of the display . If the _Analog opti on is[...]

Pa g e 2- 1 Chapter 2 Intr oducing th e calculator In this c hapter we present a n umb er of basi c oper ations of the calculator inc luding the use of the E quation W r iter and the manipulation of data obj ects in the calc ulator . Stud y the ex amples in this chapt er to get a good gr asp of the capab ilities of the calc ulator f or futur e appl[...]

Pa g e 2- 2 the CA S, it mi ght be a good i dea to s w itch dir ectl y into appr o x imate mode . R efe r to Appendi x C f or mor e details . Mi x ing integers and reals together or mi s takin g an integer for a r eal is a common occ urr ence . Th e calc ulator w ill det ect su ch mi x ing o f obj ects and ask y ou if y ou w ant to s w itch t o app[...]

Pa g e 2- 3 Binary integ ers , obje cts of t ype 10 , are used i n some computer sc ienc e applicati ons. Graphics objec ts , ob jec ts of type 11, st or e graphi cs pr oduced by the calc ulator . T agg ed objects , obj ects of t y pe 12 , ar e us ed in the output of man y pr ograms t o identify r esults . F or ex ample, in the tagged objec t: Mean[...]

Pa g e 2- 4 T he r esulting e xpr essi on is: 5.*(1.+1./7.5)/( √ 3.-2.^3). Press ` to get the expr essio n in the dis play as f ollow s: Notice that , if your CA S is s et to EXA CT (see Appe ndi x C) and you en ter y our e xpr es sion us ing integer number s for in teger v alues , the r esult is a s y mbolic quantity , e . g ., 5*„Ü1+1/7.5™[...]

Pa g e 2- 5 T o e valuat e the e xpr essi on w e can us e the EV AL functi on , as f ollo ws: μ„î` As in the pr ev ious e xample , yo u wi ll be ask ed to appr ov e c hanging the CAS setti ng to Appr o x . Once this is done , y ou w ill get the same r esult as bef or e . An alte rnati v e wa y to e valuat e the e xpr essi on enter ed earli er b[...]

Pa g e 2- 6 T his lat t er r esult is pur el y numer ical , so that the two r esults in the stac k, although r epr esenting the same e xpr essi on, seem diff er ent . T o ver ify that they ar e not, w e subtr act the tw o values and e v aluate this differ ence using f uncti on EV AL: - Subtr act le v el 1 fr om lev el 2 μ Evalua te usin g funct i [...]

Pa g e 2- 7 T he editing cur sor is sho wn a s a blinking left arr o w ov er the f irs t char acter in the line to be edited. Since the editing in this case consists of r emov ing some c har acte rs and r eplac ing them w ith others , w e w ill use the r i ght and left ar r o w keys, š™ , to mo ve the c urs or to the a ppr opri ate place f or ed[...]

Pa g e 2- 8 W e set the calc ulator oper ating mode to Algebr aic , the CA S to Exac t , and the displa y to T e xtbook . T o ente r this algebr aic e xpr es sion w e us e the foll ow ing keys tro kes : ³2*~l*R„Ü1+~„x/~r™/ „ Ü ~r+~„y™+2*~l/~„b Press ` to get the fo llo w ing re sult: Enter ing this e xpr essi on when the calc ulato[...]

Pa g e 2- 9 Θ Pr ess the r ight arr o w k e y , ™ , until the c ursor is to the r ight of the x Θ Ty p e Q2 to enter the po wer 2 f or the x Θ Pr ess the r ight arr o w k e y , ™ , until the c ursor is to the r ight of the y Θ Pr ess the de lete k ey , ƒ , once to era se the c har acter s y. Θ Ty p e ~„x to enter an x Θ Pr ess the r ig[...]

Pa g e 2- 1 0 Θ Pr es sing ` once more to r eturn to normal display . T o see the entir e e xpr essi on in the sc r een, w e can change the optio n _Small Stack Di sp in the DIS P L A Y MODE S input for m (see Chapter 1). After eff ecting this change , the display w ill look as follo ws: Using the Equation W riter (E QW ) to create e xpressions T [...]

Pa g e 2- 1 1 T he six s oft menu k ey s f or the E quation W rit er acti vat e the follo wing f uncti ons: @EDIT : lets the u ser edit an entry in the line editor (see e x amples abo ve) @CURS : hi ghlights e xpr essi on and adds a graphi cs c urs or to it @BIG : if se lected (se lecti on sho wn b y the char acter in the label) the f ont us ed in [...]

Pa g e 2- 1 2 T he r esult is the e xpr essi on T he c ursor is sho w n as a left-fac ing ke y . T he c urso r indicat es the c ur ren t edition location . T yp ing a char act er , functi on name , or oper ation w ill enter the cor re sponding char acter or c har acter s in the cur sor location . F or e xample , for the c ursor in the location indi[...]

Pa g e 2- 1 3 Suppos e that no w y ou w ant to add the fr ac tion 1/3 to this entir e expr ession , i .e ., y ou wan t to en ter the e xpr es sion: F i r st , w e need to hi ghlight the entir e f ir st ter m b y using ei ther the r ight ar r o w ( ™ ) or the upper ar r o w ( — ) k ey s, r epeatedl y , until the entir e e xpr essi on is highli g[...]

Pa g e 2- 1 4 Sho wing the expression in smaller -siz e T o sho w the expr es sion in a smaller -si z e font ( whi c h could be u sef ul if the e xpr essi on is long and con vo luted), simply pr ess the @BIG soft menu k ey . F or this case, the scr een lo oks as follo ws: T o r ecov er the larger -font displa y , pr ess the @BIG soft me nu k e y on[...]

Pa g e 2- 1 5 If y ou w ant a floating-po int (numer ical) e v aluation , us e the  NUM fu nct ion (i .e ., …ï ) . T he r esult is as follo ws: Use the function UNDO ( …¯ ) on c e m ore t o rec ov er t h e o ri g in a l ex p ress io n : Ev aluating a sub-e xpression Suppos e that y ou w ant to ev a luat e only the e xpre ssio n in pare nth[...]

Pa g e 2- 1 6 A s ymboli c ev aluation once mor e. Suppo se that , at this point , w e want to e valuate the left-hand side fr acti on onl y . Pr ess the upper ar r o w ke y ( — ) thr ee times to selec t that fr acti on, r esulting in: Then , pres s the @EVAL so f t menu k ey to obtain: Let ’s tr y a numer ical ev aluation o f this term at this[...]

Pa g e 2- 1 7 Editing arithmetic e xpr essions W e w ill show s ome of the editing featur es in the E quation W riter as an e x erc ise . W e start b y enter ing the follo wi ng expr essi on used in the pr e v iou s ex er c ises: And w ill use the editing f eatur es of the E quati on E ditor to tr ansfo rm it into the fo llo w ing expr essio n: In [...]

Pa g e 2- 1 8 Pr es s the do wn ar r o w k e y ( ˜ ) to tri gger the c lear editing cur sor . The sc r een no w looks like this: B y using the le f t ar r o w ke y ( š ) y ou can mov e the c ursor in the gener al left dir ecti on , but stopp ing at each indi vi dual component of the e xpr essi on . F or e xam ple , suppose that w e will f irst w [...]

Pa g e 2- 1 9 Ne xt , we ’ll conv ert the 2 in f r ont of the parenth eses in the denominator into a 2/3 by using: šƒƒ2/3 At this point the e xpr essi on looks as f ollo w s: T he final step is to r emo ve the 1/3 in the r i ght-hand side of the e xpr ession . T his is accomplished by u sing: —————™ƒƒƒƒƒ T he final v ersi on w[...]

Pa g e 2- 2 0 Use t he fo llow ing k ey str ok es: 2 / R3 ™™ * ~‚n+ „¸ ~‚m ™™ * ‚¹ ~„x + 2 * ~‚m * ~‚c ~„y ——— / ~‚t Q1/3 T his re sults in the output: In this e xample w e us ed se ve ral lo we r -case English lett ers , e .g., x ( ~„x ), se ver a l Gr eek letters, e .g., λ ( ~‚n ) , and e v en a combinati o[...]

Pa g e 2- 2 1 Editing algebr aic ex pressions T he editing o f algebrai c equati ons follo ws the same r ules as the editing of algebr aic equati ons. Namely : Θ Use the ar r o w k e y s ( š™—˜ ) to highli ght e xpr essi ons Θ Use the do wn ar r o w ke y ( ˜ ) , r epeatedly , to tr igger the c lear editing c ursor . In this mode , use the [...]

Pa g e 2- 22 2. θ 3. Δ y 4. μ 5. 2 6. x 7. μ in the e xponential f unction 8. λ 9. 3 i n t h e √ 3 ter m 10. the 2 in the 2/ √ 3 fr acti on At an y po int we can c hange the clear editing c urs or into the insertio n cur sor b y pr essing the dele te k e y ( ƒ ) . Let’s u se these two c ursor s (the clear editing c ursor and the inse r [...]

Pa g e 2- 23 Ev aluating a sub-e xpression Since w e alr eady ha ve the sub-e xpre ssi on highli ghted , let ’s pr ess the @EVAL soft menu k e y to ev aluate this sub-e xpr ession . T he r esult is: Some algebr aic ex pre ssions cannot be simplif ied an ymor e. T r y the fo llow ing keys tro kes : —D . Y ou w ill notice that nothing happens , o[...]

Pa g e 2- 24 3 in the f irst te rm of the numer ator . Then , pr ess the r ight ar r o w k e y , ™ , to nav igate thr ough the expr essi on. Simplifying an e x pr ession Pr ess the @ BIG soft menu k e y to get the sc r een to look as in the pre vi ous f igur e (see abo ve). Now , pre ss the @SIMP s oft menu k ey , to see if it is pos sible to sim[...]

Pa g e 2- 2 5 Press ‚¯ to r ecov er the or iginal e xpre ssion . Ne xt , enter the f ollo w ing keys tro kes : ˜ ˜˜™™™™™™™———‚™ to sele c t the last two ter ms in the expr ession , i .e ., pr ess the @ FACTO soft menu k e y , to g e t Press ‚¯ to reco v er the ori ginal e xpre ssion . No w , let’s select the entir[...]

Pa g e 2- 26 Ne xt , select the command DER VX (the deri vati ve w ith r espec t to the v ari able X, the c urr ent CAS indepe ndent var iable) b y using: ~d˜˜˜ . Command DER VX w ill no w be sele c ted: Pr ess the @ @OK@@ soft me nu k e y to get: Ne xt , pr ess the L k e y to r eco ve r the ori ginal E quati on W r iter men u , and pr ess the @[...]

Pa g e 2- 27 Detailed e xplanation on the use of the help fac i lity f or the CA S is pr esented in Chapter 1. T o r eturn to the E quation W rite r , pre ss the @EXIT s oft menu k ey . Pr es s the ` k e y to e xit the E quation W rit er . Using the editing func tions BEGIN, END , COP Y , CUT and P ASTE T o f ac ilitate editing , w hether w ith the[...]

Pa g e 2- 28 Ne xt , we ’ll copy the f r actio n 2/ √ 3 from t he lef tm ost fa ctor in th e exp r es sion, and place it in the numerat or of the ar gument fo r the LN function . T r y the fo llo w ing k ey str ok es: ˜˜šš———‚¨˜˜ ‚™ššš‚¬ T he r esulting sc r een is as f ollo w s: T he functi ons BE GIN and END are no t [...]

Pa g e 2- 2 9 W e can no w cop y this expr essi on and place it in the denominator o f the LN ar gument , as follo ws: ‚¨™™ … ( 2 7 times ) … ™ ƒƒ … (9 times) … ƒ ‚¬ T he line editor n ow looks lik e this: Pr es sing ` sho w s the expr ession in the E quation W rit er (in small-font f ormat , pr ess the @ BIG soft menu k ey) [...]

Pa g e 2- 3 0 T o see the cor r esponding e xpr es sio n in the line editor , pr es s ‚— and the A soft menu k ey , to sho w : T his expr es sion sho w s the gener al form o f a summation typed dir ec tly in the stac k or line ed itor : Σ ( inde x = st ar ting_v alue , ending_value , summation e xpres sion ) Press ` to r eturn to the E quation[...]

Pa g e 2- 3 1 and the v ari able of diff er entiati on . T o f ill these input locati ons, use the f ollo w ing keys tro kes : ~„t™~‚a*~„tQ2 ™™+~‚b*~„t+~‚d The r esulting scr een is the follo w ing: T o see the cor r esponding e xpr es sio n in the line editor , pr es s ‚— and the A soft menu k ey , to sho w : T his indicates [...]

Pa g e 2- 32 Definite integr als W e w ill use the E quati on W r iter to ent er the follo w ing def inite inte gral: . Pr es s ‚O to ac ti vat e the E quation W rite r . Then pr ess ‚ Á to enter the integral sign . Notice that the si gn, w hen enter ed into the E quati on W rit er sc r een, pr ov ide s input locations f or the limits of integ[...]

Pa g e 2- 3 3 Double integr als ar e also pos sible . F or e x ample , w hich e v aluates to 3 6. P artial e valuati on is poss ible , for e x ample: T his integral e v aluates t o 3 6. Organi zing data in t he calculator Y ou can or gani z e data in yo ur calculator b y stor ing var iables in a dir ectory tr ee . T o unders tand the calc ulator ?[...]

Pa g e 2- 3 4 @CHDIR : Change to s elected direct or y @CANCL : Cancel action @@OK@ @ : Appr ov e a selec tion F or ex ample , to c hange dir ectory to the CA SDI R , pr ess the do w n -arr o w k ey , ˜ , and pr ess @CHDIR . T his acti on clo ses the Fi l e M a n a g e r w indow and r eturns us to normal calc ulator dis play . Y ou w ill notice th[...]

Pa g e 2- 3 5 T o mo ve betw een the differ ent so f t men u commands, y ou can u se not onl y the NEXT k e y ( L ), but also the PREV k ey ( „« ). T he user is in v ited to try these f uncti ons on his or her o w n. Their applicati ons ar e str aightf orwar d. T he HOME director y T he HOME dir ectory , as pointed out ear lier , is the base dir[...]

Pa g e 2- 36 T his time the CA SD IR is hi ghlighted in the scr een. T o s ee the contents of the dir ect or y pr ess the @@ OK@@ soft menu k e y or ` , to get the f ollo w ing sc r een: T he scr een sho w s a table des cr ibing the var iable s contained in the CA SD IR dir ect or y . T hese ar e v ar iable s pr e -def ined in the calc ulator memor[...]

Pa g e 2- 3 7 Pr essing the L k e y sho ws one mor e var iable st ored in this dir ectory: • T o see the contents o f the var ia ble EPS , f or e xam ple , use ‚ @EPS@ . T his sho w s the v alue of EP S to be .0000000 001 • T o see the v alue of a numeri cal v ari able , w e need t o pre ss onl y the soft menu k ey f or the v ar iable . F or [...]

Pa g e 2- 3 8 loc k the alphabetic k ey boar d tempor aril y and enter a f ull name bef or e unloc king it again. T he fo llo w ing combination s of k e y str ok es w ill lock the alphabeti c k e yboar d: ~~ locks the alpha betic k e y boar d in upper case . When lock ed in this fas hio n , press in g th e „ bef or e a letter k ey pr oduces a lo [...]

Pa g e 2- 3 9 Creating subdir ec tor ies Subdir ector i es can be cr eated by using the FI LE S env ironme nt or by u sing the c om ma nd C RD I R. Th e t wo ap proa ch es for cr e at i ng su b- di r e cto ries a r e pr esen ted next . Using the FI LE S menu Re gardles s of the mode of oper ation of the calc ulator (A lgebrai c or RPN) , w e can c [...]

Pa g e 2- 4 0 Th e Object input f i eld, the f irst input f ield in the f orm , is highlight ed by def ault . T his input fi eld can hold the conte nts of a ne w var ia ble that is being cr eated. Since w e hav e no contents f or the new sub-dir ectory at this po int , we simpl y skip this input f ield b y pr essing the do w n -ar r o w k ey , ˜ ,[...]

Pa g e 2- 4 1 T o mo v e into the MAN S dir ect ory , pr ess the co rr es ponding so ft menu k ey ( A in this case) , and ` if in algebr ai c mode. T he dir ectory tr ee w ill be sho wn in the second line o f the display as {HOME M N S} . Ho w e ver , ther e will be no labels as soc iat ed w ith the soft me nu k ey s , as sho w n belo w , becau se [...]

Pa g e 2- 42 Us e the do wn ar r o w k e y ( ˜ ) to selec t the option 2. M E M O RY … , or ju st press 2 . Then , pre ss @@OK@@ . T his will pr oduce the follo w ing pull-dow n menu: Us e the do wn ar r o w k ey ( ˜ ) t o select the 5 . DIRE CT OR Y option , or j ust press 5 . Then, pr ess @ @OK@@ . This w ill pr od u ce the follo w ing pull-d[...]

Pa g e 2- 4 3 Pr ess the @ @OK@ soft menu k ey to ac tiv ate the comm and , to cr eate the sub- dir ectory: Mov ing among subdirectories T o mo ve do wn the dir ectory tr ee , y ou need to pr ess the so ft menu k ey cor r esponding to the sub-dir ect or y y ou wan t to mo v e to . T he list o f var iable s in a sub-dir ecto r y can be pr oduced b y[...]

Pa g e 2- 4 4 T he ‘S2’ str ing in this f orm is the name of the sub-dir ectory that is being deleted . T he soft men u k ey s pro vi de the fo llo w ing options: @YES@ Pr oceed w ith deleting the sub-dir ectory (or var i able) @ALL@ Pr oceed w ith deleting all sub-dir ector ie s (or var iables) !ABORT Do not delete sub-dir ectory (or var ia bl[...]

Pa g e 2- 4 5 Us e the do wn ar r o w k e y ( ˜ ) to selec t the option 2. M E M O RY … T h e n , press @@OK@ @ . This w ill produ ce the fo llo w ing pull-do w n menu: Us e the dow n arr o w k e y ( ˜ ) to select the 5 . DIRE CT OR Y option . Then , press @@OK@ @ . This w ill produ ce the fo llo w ing pull-do w n menu: Us e the do wn arr ow k [...]

Pa g e 2- 4 6 Press @@OK@@ , to get: Then , pres s ) @@S3@@ to enter ‘S3 ’ as the ar gument to PGDI R . Press ` to delete the sub-dir ectory: Command PGDIR in RPN m o de T o us e the PGDIR in RPN mode y ou need to hav e the name o f the direc tory , between q uotes , alr eady a vaila ble in the stac k bef or e accessing the command . F or ex am[...]

Pa g e 2- 4 7 Using the PURGE command fr om the T OOL menu T he T OOL me nu is av ailable by pr essing the I ke y ( Algebr aic and RPN modes sho wn): T he PUR GE command is av ailable by pr essing the @PURGE s oft menu k e y . In the fo llo w ing e xample s w e want t o delete sub-dir ectory S1 : • Algebr aic mode: Enter @PURGE J ) @@S1@@ ` • R[...]

Pa g e 2- 4 8 Using the FI LE S menu W e w ill use the FILE S menu to enter the v ari able A. W e assume that w e are in the sub- dir ectory {HOME M NS IN TRO}. T o get t o this sub-dir ectory , use the f ollo w ing: „¡ and sel ect the INTR O sub-dir ectory as sho w n in this scr een : Press @@OK@@ to ent er the dir ectory . Y ou w ill get a f i[...]

Pa g e 2- 49 T o enter v ari able A (see table abov e) , w e fir st enter its contents , na me ly , the number 12 . 5, and then its name, A, as follo ws: 12.5 @@OK@@ ~a @@OK@@ . Resulting in the f ollo wing sc r een: Press @@OK@@ once more to c reate the v ari able. T he ne w var iable is show n in the fo llo w ing var ia ble listing: T he listing [...]

Pa g e 2- 5 0 Using the ST O  command A simpler w ay to cr eate a v ar ia ble is by us ing the S T O command (i .e ., the K k e y) . W e pro vi de e xample s in both the Algebr ai c and RPN modes, b y cr eating the r emaining of the v ar iable s suggested abo ve , namely : • Algebr aic mode Use the f ollo w ing k ey str ok es to s tor e the va[...]

Pa g e 2- 5 1 z1: 3+5*„¥ K~„z1` (if needed , accept c hange to Comple x mode) p1: ‚å‚é~„r³„ì* ~„rQ2™™™ K~„p1` . T he scr een , at this point , will look as follo ws: Y ou w ill see si x of the se ven v ari ables lis ted at the bottom of the sc r een: p1, z1, R, Q, A12 , α . • RPN mode Use the f ollo w ing k e ys tr ok [...]

Pa g e 2- 52 z1: ³3+5*„¥ ³~„z1 K (if needed , accept change to Comple x mode) p1: ‚å‚é~„r³„ì* ~„rQ2™™™ ³ ~„p1™` K . T he scr een , at this point , will look as follo ws: Y ou w ill see si x of the se v en var iables lis ted at the bottom of the sc reen: p1, z1, R, Q, A12 , α . Chec king v ariables contents As an ex [...]

Pa g e 2-53 Pr essing the soft me nu k e y cor r esponding t o p1 will pr o v ide an er r or messa ge (tr y L @@@p1@@ ` ): Note: By pre ss i n g @@@p1@@ ` we ar e trying t o acti vate (r un) the p1 progr am . Ho w ev er , this pr ogr am e xpects a numer ical input . T r y the fo llo w ing ex erc ise: $ @@@p1@ „Ü5` . Th e r esul t is: T he pr ogr[...]

Pa g e 2- 5 4 At this point , the scr een looks lik e this: T o see the contents o f A, use: L @@@A@@@ . To r u n p r o g r a m p1 w ith r = 5, use: L5 @@@ p1 @@@ . Notice that to run the pr ogram in RPN mo de , yo u only need to enter the in put (5) and pr es s the corr es ponding soft menu k ey . (In algebr aic mode , y ou need to place pare nth [...]

Pa ge 2- 55 Notice that this time the con tents of pr ogr am p1 are liste d in the scr ee n . T o see the r emaining v ari able s in this direc tory , pr ess L : Listing the con tents of all v ariables in the s c r een Use the k e y str ok e combinati on ‚˜ to list the cont ents of all v ar iable s in the sc r een . F or e xample: Press $ to r e[...]

Pa g e 2- 5 6 fo llow ed by the var iable ’s soft menu k e y . F or e xample , in RPN , if w e want to c ha nge the conten ts of var iable z1 to ‘ a+b ⋅ i ’, u s e : ³~„a+~„b*„¥` T his wil l place the algebrai c e xpr essi on ‘ a+b ⋅ i ’ in le v el 1: i n t h e st a ck . To en t e r this r esult into var iable z1 , us e: J„ [...]

Pa g e 2- 57 Use t he up ar r o w k ey — t o select the sub-dir ect or y MAN S and pres s @@O K@@ . If y ou no w press „§ , the scr een will sho w the contents of sub-direc tory MANS (notice that v ar iable A is show n in this list , as e xpected): Press $ @INTRO@ ` (A lgebrai c mode) , or $ @ INTRO@ (RPN mode) to r eturn t o the INTR O direc [...]

Pa g e 2- 5 8 Ne xt , use the delet e k ey thr ee times, to r emo ve the la st thr ee lines in the displa y : ƒ ƒ ƒ . At this po int , the stac k is r eady t o e xec ute the command ANS( 1)  z1. Pr es s ` to e xec ute this command . Then , use ‚ @@z1@ , to ve rify the contents of the v ar iable . Using the stac k in RPN mode T o demonstr a [...]

Pa g e 2- 59 Cop ying two or mor e v ariables using the stac k in RPN mode T he follo wing is an e xer cis e to de monstr ate ho w to copy tw o or mor e var iable s using the st ack w hen the calc ulator is in RPN mode. W e assume , again, that w e ar e wi thin sub-dir ectory {HOME MAN S INTRO} and that w e want to cop y the v ari able s R and Q in[...]

Pa g e 2- 6 0 T he sc r een no w sho w s the new o rde ring o f the var ia bles: RPN mode In RPN mode, the lis t of r e -or der ed var iables is list ed in the s tack be for e appl y ing the command ORDER. Su ppose that w e start fr om the same situati on as abo ve , but in RPN mode, i .e ., Th e re ord e red l i st i s c rea t ed by u si n g : „[...]

Pa g e 2- 6 1 Notice that v ar iable A12 is no longer ther e . If yo u no w pr ess 㤠, the sc r een w ill sho w the contents of sub-dir ectory MANS , including v ari able A12 : Deleting va riables V ar iables can be deleted using functi on P URGE . T his fu ncti on can be acc essed dir ectl y b y using the T OOLS men u ( I ), or by using the FI[...]

Pa g e 2- 6 2 va riab le p1 . Pr ess I @PURGE@ J @@p1@@ ` . The sc reen w ill no w s ho w va riab le p1 rem ove d : Y ou can us e the P URGE command to er as e mor e than one var iable b y plac ing their name s in a list in the ar gument of P URGE . F or e x ample , if no w we w anted to pur ge var iables R and Q , simult aneousl y , we can tr y th[...]

Pa g e 2- 6 3 the HIS T k ey : UNDO r esults f r om the k e ys tr ok e seq uence ‚¯ , w hile CMD r esults f r om the k e y str ok e seq uence „® . T o illus trat e the us e of UNDO , try the follo w ing ex er c ise in algebr aic (A L G) mode: 5*4/3` . T he UNDO command ( ‚¯ ) wi ll simply er ase the r esult . The same e xer c ise in RPN mo[...]

Pa g e 2- 6 4 As y ou can see , the number s 3, 2 , and 5, u sed in the fi rst calc ulation abo ve , ar e listed in the s electi on bo x , as w ell as the algebr aic ‘S IN(5x2)’ , but not the SIN f uncti on enter e d pr ev io us to the algebr aic . F lags A flag is a Boo lean value , that can be s et or clear ed (true or f alse) , that spec if [...]

Pa g e 2- 65 Ex ample of flag setting: general solutions v s. pr incipal value F or e xample , the def ault v alue f or s y ste m flag 01 is Gener al solu tions . What this means is that , if an equation has m ultiple soluti ons, all the s olutions w ill be r eturned b y the calculato r , most lik el y in a list . B y pr essing the @  @CHK@ @ so[...]

Pa g e 2- 6 6 ` (k eeping a s econd copy in the RPN stack) ³~ „t` Use the follo w ing k ey str oke sequence to enter the Q U AD command: ‚N~q (use the u p and dow n arr ow k ey s , —˜ , to se lect command QU AD) , pr ess @@OK@@ . The sc reen sho ws the pr inc ipal soluti on: No w , change the se t ting o f flag 01 to Ge ner al soluti ons : [...]

Pa g e 2- 6 7 CHOO SE bo x es vs . Soft MENU In some of the ex er c ises pr es ented in this chapter w e hav e seen menu lists of commands dis play ed in the scr een. T hes e menu lists ar e r ef err ed to as CHOO SE bo x es . F or ex ample, to us e the ORD ER command to r eorde r v ari ables in a dir ect or y , we u se , in alge br aic mode: „°[...]

Pa g e 2- 6 8 T he sc r een sho w s flag 117 not se t ( CHOO SE box es ) , as sho wn her e: Pr es s the @  @CHK @@ s oft menu k e y to set f lag 117 to soft MENU . The s cr een w ill r ef lect that c hange: Press @@OK@@ t w ice to retur n to normal calc ulator displa y . No w , we ’ll tr y to f i nd the ORDER command using similar k e y str ok[...]

Pa g e 2- 69 Note: mos t of the e xam ples in this user guide a ssume that the cur r ent s et ting o f flag 117 is its default setting (that is, not se t) . If y ou ha ve s et the flag but w ant to str i ctl y follo w the e xam ples in this guide , y ou should c lear the flag bef or e con tinuing . Selec ted CHOO SE bo x es Some men us w ill onl y [...]

Pa g e 2- 70 • T he CMDS (CoMmanD S) menu , acti v ated w ithin the Eq uation W rit er , i. e. , ‚O L @CMDS[...]

Pa g e 3 - 1 Chapter 3 Calculation with re al numbers T his chapte r demonstr ates the us e of the calc ulator f or oper ations and f uncti ons r elated to r eal numbers . Oper ations along the se lines ar e use ful f or mos t common calc ulati ons in the ph ysi cal sc iences and engineer ing. T he user should be acquaint ed w ith the ke yboar d t [...]

Pa g e 3 - 2 2 . Coordinate sy stem spe c ification (X Y Z , R ∠ Z, R ∠∠ ). T h e s y m b o l ∠ stands f or an angular coor dinate . XYZ: Carte sian or r ect angular (x,y ,z) R ∠ Z: cylindr ic a l P olar co or dinates (r , θ ,z ) R ∠∠ : Spher i cal coor dinates ( ρ,θ,φ ) 3 . Number base s pec ifi cati on (HEX, DEC , OCT , BIN) HEX[...]

Pa g e 3 - 3 R eal number calc ulations w ill be demonstr ated in both the Algebr ai c (AL G) and R ev er se P olish Notati on (RPN) modes . Changing sign of a number , var iabl e , or e xpression Use the k ey . In AL G mode , y ou can pr ess be fo re e nter ing the number , e.g ., 2.5` . Re sult = - 2 . 5 . In RPN mode , y ou need to enter at[...]

Pa g e 3 - 4 Alte rnati v el y , in RPN mode, y ou can separ ate the oper ands with a space ( # ) bef or e pr essing the oper ator k e y . Example s: 3.7#5.2 + 6.3#8.5 - 4.2#2.5 * 2.3#4.5 / Using parentheses P ar entheses can be used to gr oup operati ons, as w ell as to enclose ar guments of f unctions . T he par entheses ar e av ailable through t[...]

Pa g e 3 - 5 Squares and squar e roots T he squar e func tion , S Q, is av ailable thr ough the k e y str ok e combinati on: „º . When calc ulating in the st ack in AL G mode , e nter the fu ncti on bef or e the argument , e.g ., „º2.3` In RPN mode , enter the numbe r fir st , then the f uncti on, e .g., 2.3„º The s quar e r oot functi o[...]

Pa g e 3 - 6 Using po wers o f 10 in entering data P owe rs of te n, i.e. , nu mb e rs of t he fo rm - 4 .5 ´ 10 -2 , etc., ar e enter e d b y using the V k e y . F or e x ample , in AL G mode: 4.5V2` Or , in RPN mode: 4.5V2` Natural logar ithms and e xponential func tion Natur al logar ithms (i .e ., logarithms of base e = 2. 7 1 82 8 1 82 82[...]

Pa g e 3 - 7 the in ver se tr igonometr i c functi ons r e present angles , the ans w er fr om these func tions w ill be gi v en in the select ed angular measur e (DEG , R AD , GRD) . Some e xamples ar e show n ne xt: In AL G mode: „¼0.25` „¾0.85` „À1.35` In RPN mode: 0.25`„¼ 0.85`„¾ 1.35`„À All the func tions de sc ribed abo ve ,[...]

Pa g e 3 - 8 comb ination „´ . W ith the defa ult setting of CHOO SE bo xe s fo r syst em flag 117 (see C hapter 2) , the MTH menu is sho wn as the f ollo w ing menu list: As the y ar e a gr eat number of mathematic f uncti ons a vailable in the calc ulator , the MTH menu is s orted b y the t y pe of ob ject the f uncti ons appl y on . F or e x [...]

Pa g e 3 - 9 Hy perbolic functions and th eir in verses Selecting Option 4. HYP ERBOLIC.. , in the MTH men u , and pr es sing @@OK@@ , pr oduces the h yper bolic f unction men u: The h y perbolic f unctions ar e: Hy perbo lic sine , SINH , and its inv ers e , AS INH or sinh -1 Hy perbo lic cosine , CO SH, and its inv erse , A CO S H or cosh -1 Hy p[...]

Pa g e 3 - 1 0 T he r esult is: T he oper ations sho wn abo ve as sume that yo u are u sing the defa ult setting f or s y stem f lag 117 ( CHOO SE box es ). If y ou hav e changed the s etting of this flag (see Chapter 2) to SO FT m e nu , the MTH men u w ill sho w as labe ls of the s oft menu k ey s , as fo llo ws (l eft -hand si de in AL G mode , [...]

Pa g e 3 - 1 1 F or ex ample , to calculat e tanh( 2 . 5), in the AL G mode , when u sing SO FT m e nu s ove r CHOO SE bo xe s , f ollo w this pr ocedur e: „´ Sele c t MTH menu ) @@HYP@ Selec t the HYP ERBOLIC.. menu @@TANH@ Selec t the TA N H fun cti on 2.5` Ev aluate t anh(2 .5 ) In RPN mode , the same value is calc ulated using: 2.5` Ente r a[...]

Pa g e 3 - 1 2 Option 19 . MA TH.. r eturns the u ser to the MTH men u . The r emaining functi ons ar e gr ouped in to si x diffe r ent gr oups des cr ibed be low . If s y stem fl ag 117 is set to SO FT m e nu s , the REAL f uncti ons menu w ill look like this (A L G mode used , the same so ft menu k e y s w ill be a vailable in RPN mode) : The v e[...]

Pa g e 3 - 1 3 T he r esult is sho wn ne xt: In RPN mode , recall that ar gument y is located in the second le v el of the st ack , w hile argument x is located in the f i r st le vel o f the stac k . T his means , y ou should enter x f irst , and then , y , jus t as in AL G mode . Thus , the calculati on of %T(15, 4 5 ) , in RPN mode , and w ith s[...]

Pa g e 3 - 1 4 P lease notice that MOD is not a function , but r ather an operator , i . e ., in AL G mode , MOD sho uld be us ed as y MOD x , and not as MOD (y,x) . Th us, the oper ation o f MOD is similar to that of + , - , * , / . As an e x er c ise , v er ify that 15 M OD 4 = 15 mod 4 = r esidual o f 15/4 = 3 Absolute value , sign, mantissa, e [...]

Pa g e 3 - 1 5 G AMM A: T he G amma f unction Γ ( α ) P SI: N -th deri vati v e of the digamma f uncti on P si: Digamma f uncti on, de ri vati v e of the ln(Gamma) T he Gamma functi on is def ined b y . T his functi on has appli cations in applied mathematic s fo r sc ience and engineer ing , as well a s in pr obab ility and statis tic s. Th e PS[...]

Pa g e 3 - 1 6 Ex amples of thes e spec ial f unctions ar e sho w n her e using both the AL G and RPN modes. As an e x er c ise , v er if y that G AMMA(2 . 3) = 1.166 711…, PSI(1 . 5 , 3) = 1 .40909 .. , and P s i ( 1 .5) = 3. 6489 9 7 39 . . E- 2 . T hese calc ulations ar e sho w n in the fo llo w ing sc r een shot: Calculator constants T he fol[...]

Pa g e 3 - 1 7 Selec ting an y of thes e entr ies w ill place the value s elected , w hether a sy mbol (e .g ., e , i , π , MINR , or MAXR ) or a v alue ( 2 .71. ., (0,1) , 3 .14 .., 1E - 4 99 , 9. 9 9. . E 4 9 9 ) in the s tac k. P lease notice that e is a v ailable fr om the k e y board as exp ( 1) , i .e ., „¸1` , in AL G mode , or 1` „¸ [...]

Pa g e 3 - 1 8 T he user w ill recogni z e most o f these units (some , e.g ., dy ne , are not u sed v ery often no w aday s) fr om his or her ph ysi cs c lasse s: N = newto ns, dyn = dyne s, gf = gr ams – for ce (to distinguish f rom gr am-mass, or plainl y gr am, a unit of mas s) , kip = kilo -poundal (1000 pounds) , lbf = pound-f or ce (to dis[...]

Pa g e 3 - 1 9 A vailable units T he follo w ing is a l ist of the units av ailable in the UNI T S menu . T he unit sy mbol is sho wn f irs t follo wed b y the unit name in parenth eses: LENG TH m (meter ) , cm (centimeter ) , mm (millimeter ) , y d (yar d) , ft (feet) , in (inc h) , Mpc (Mega parsec) , pc (par sec) , ly r (light -y e ar ) , a u (a[...]

Pa g e 3 - 2 0 SPEED m/s (meter per s econd), cm/s (centimeter per second), f t/s (f eet per s econd) , kph (kilometer per ho ur ) , mph (mile per hour), knot (nautical mile s per hour), c (speed of light) , ga (accelerati on of gr av ity ) MA S S k g (kilogram), g (gr am) , Lb (av oir dupo is pound) , oz (ounce) , slug (slug) , lbt (T r o y pound)[...]

Pa g e 3 - 2 1 ANGLE (planar and soli d angle measur ements) o (se x agesimal degree), r (radi an) , gr ad (gr ade) , ar cmin (minute of ar c) , ar cs (second of ar c) , sr (ster adian) LIGHT (Illuminati on measur ements) fc (foot candle) , f lam (footlambe rt) , lx (lu x) , ph (phot), sb (stilb), lm (lumem) , cd (candela) , lam (lambert) RAD IA T [...]

Pa g e 3 - 2 2 Conv erting to base units T o conv er t an y of these units to the def ault units in the SI s yst em, u se the functi on UB A SE . F or e xample , to find out what is the v alue of 1 po ise (uni t of viscosit y) in the SI units , use the f ollo w ing: In AL G mode , s y ste m flag 117 se t to CHOOSE bo xes : ‚Û Select the UNIT S m[...]

Pa g e 3 - 23 ` Con vert the units In RPN mode , s y stem f lag 117 set to SO FT m e nu s : 1 Enter 1 (n o underline) ‚Û Select the UNIT S menu „« @ ) VISC Select the VIS C OS ITY opti on @@@P@@ Select the unit P (poise) ‚Û Select the UNIT S menu ) @TOOLS Select the T OOLS m en u @UBASE Select the UB A SE functi on Attac hing units to numb[...]

Pa g e 3 - 24 Notice that the under scor e is ente r ed automati call y when the RPN mode is acti v e . The r esult is the follo w ing sc r een: As indicated ear lier , if s ys tem flag 117 is s et to SOF T m en u s , then the UNI T S menu w ill sho w up as labels f or the soft menu k e ys . This se t up is very con veni ent f or extensi ve oper at[...]

Pa g e 3 - 25 Yy o t t a + 2 4 d d e c i - 1 Z z etta + 21 c c enti - 2 E e x a +18 m milli -3 P peta +15 μ mic r o -6 T ter a +12 n nano - 9 Gg i g a + 9 p p i c o - 1 2 Mm e g a + 6 f f e m t o - 1 5 k ,K kilo +3 a atto - 18 h,H h ecto +2 z z epto - 21 D(*) dek a +1 y yoc to - 2 4 ___________ _____________________ ___________________ (*) In the [...]

Pa g e 3 - 26 whi ch sho ws as 6 5_(m ⋅ yd). T o conv ert to units of the SI s y stem , use f uncti on UB A SE: T o calc ulate a di visi on , say , 3 2 50 mi / 5 0 h , ent er it as (3 2 50_mi)/(5 0_h) ` : w hich tr ansfor med to S I units , w ith func tion UB ASE , pr oduces: Additi on and subtr actio n can be perfor med, in AL G mode, w ithout u[...]

Pa g e 3 - 27 St ack calc ulations in the RPN mode , do not r equir e y ou to enc lose the diff er ent terms in par enth eses, e .g ., 12_m ` 1. 5_y d ` * 3 2 50_mi ` 5 0_h ` / T hese oper ati ons pr oduce the f ollo w ing output: Also , tr y the f ollo wing oper ations: 5_m ` 32 0 0 _ m m ` + 12_mm ` 1_cm^2 `* 2_s ` / T hese las t two ope rati ons[...]

Pa g e 3 - 28 UF A CT(x ,y) : fac tor s a unit y fr om unit obj ect x  UNI T(x ,y) : combines v alue of x w ith units o f y T he UB A SE func tion w as disc ussed in detail in an earli er sec tio n in this cha pter . T o access any o f these f unctions f ollow the e xamples pro vided ear lier f or UB A SE . Notice that , w hile func tion UV AL r[...]

Pa g e 3 - 2 9 Ex amples of  UNI T  UNIT( 2 5,1_m) `  UNI T(11. 3,1_mph) ` Ph y sical constants in t he calculator F ollow ing a l ong the treatment o f units, w e disc uss the u se of ph ysi cal constants that ar e av ailable in the calc ulato r’s memory . Thes e ph ysi cal cons tants ar e cont ained in a const ants libr ary acti vat ed[...]

Pa g e 3 - 3 0 T he soft menu k ey s cor r esponding t o this CONS T A NT S LIBRAR Y sc r een inc lude the f ollo w ing func tions: SI w hen selec ted , constants v alues ar e sho wn in S I units ENGL w hen se lected , constant s value s ar e sho w n in English units ( *) UNIT when s elect ed, cons tants ar e sho wn w ith units attac hed (*) V AL U[...]

Pa g e 3 - 3 1 T o see the v alues of the const ants in the English (or Imperi al) s ys tem , pre ss the @ENGL opti on: If w e de -select the UNIT S opti on (pr ess @UNITS ) onl y the v alues ar e show n (English units se lected in this case): T o cop y the value o f Vm to the s tack , select the var iable name , and pr ess ! , then , pr ess @QUIT@[...]

Pa g e 3 - 32 Special ph ysical functions Menu 117 , tr igge r ed by u sing MENU(117) in AL G mode, or 117 ` MENU in RPN mode , pr oduces the fo llo w ing menu (labels lis ted in the displa y b y using ‚˜ ): Th e fu nct ion s i ncl ud e: ZF A CT O R: gas compr essibilit y Z f actor function F AN NI NG : Fan ni ng fr ict ion fact or fo r fl uid f[...]

Pa g e 3 - 3 3 ZF A C T OR(x T , y P ) , w her e x T is the r educed t emper atur e , i .e ., the r atio of ac tual temper ature t o pseudo -c ri tical temper ature , and y P is the r educed pr essur e, i .e., the r atio of the ac tual pr essur e t o the pseudo -c r itical pr es sur e . The v alue of x T must be betw een 1. 05 and 3 . 0, while the [...]

Pa g e 3 - 3 4 Function T I NC F uncti on T INC(T 0 , Δ T) calculat es T 0 +D T . The ope rati on of this f uncti on is similar to that of f uncti on TDEL T A in the sense that it r eturns a r esult in the units of T 0 . Otherwise , it re turns a simple additi on of value s, e .g ., Defining and using functions Use rs can def ine the ir ow n funct[...]

Pa g e 3 - 3 5 Pr ess the J k ey , and y ou will noti ce that ther e is a new v ar iable in y our soft menu k ey ( @@@H@@ ) . T o see the contents of this v ar iable pr ess ‚ @ @@H@@ . The sc r een wi ll s how n o w: T hus , the var iable H contains a pr ogram de fined b y : <<  x ‘LN(x+1) + EXP(x)’ >> T his is a simple pr ogr [...]

Pa g e 3 - 3 6 T he contents of the v ar iable K ar e: <<  α β ‘ α+β ’ >>. Functions defined b y mor e than one e xpression In this secti on w e disc us s the treatme nt of f uncti ons that are de fi ned by tw o or mor e e xpre ssio ns. An e x ample o f such f uncti ons wo uld be The fun ct ion IFT E ( I F- Th en -E lse ) d e[...]

Pa g e 3 - 37 Combined IFTE functions T o pr ogr am a mor e compli cated f uncti on such as y ou can combine se v er al le ve ls of the IFTE func tion , i .e ., ‘ g(x) = IFTE(x<- 2 , - x, IF TE(x<0, x+1, IFTE(x<2 , x -1, x^2)))’ , Def ine this func tion b y an y of the means pr esent ed abo ve , and c hec k that g(-3) = 3, g(-1) = 0, g[...]

Pa g e 4 - 1 Chapter 4 Calculations with compl e x numbers T his chapte r show s e xam ples of calc ulations and a pplication o f functi ons to comp lex n umbers . Definitions A comple x number z is a number w r itten as z = x + iy , wher e x and y ar e real numbers , and i is the imaginary unit defined b y i 2 = - 1. The comple x number x+iy has a[...]

Pa g e 4 - 2 Press @@OK@@ , t w ice , to r eturn to the stack . Enterin g comple x numbers Comple x numbers in the calc ulator can be enter ed in either of the tw o Car tesian repr esenta tions, nam el y , x+iy , or (x ,y) . T he r esults in t he calc ulator w ill be show n in the or der ed-pair format , i.e ., (x ,y) . F or e x ample , w ith the c[...]

Pa g e 4 - 3 Notice that the las t entr y sho ws a comple x number in the f orm x+iy . T his is so becaus e the number w as enter ed bet w een single quot es, w hic h r epr ese nts an algebr aic e xpr essi on . T o ev aluate this number u se the EV AL k e y( μ ). Once the algebr aic e xpr essi on is e val uated, y ou reco v er the comple x number [...]

Pa g e 4 - 4 On the other hand , if the coor dinate s yst em is set to c ylindr ical coor dinates (us e C YLIN) , ent ering a com plex n umber (x,y), wher e x and y are r eal numbers , will pr oduce a polar repr esentati on . F or e x ample , in c y lindr ical coor dinates , enter the number (3 .,2 .) . T he fi gur e belo w show s the RPN st ack , [...]

Pa g e 4 - 5 Changing sign of a complex number Changing the si gn of a comple x number can be accomplish ed by u sing the k e y , e .g ., -(5-3i) = -5 + 3i Entering the unit imaginary number T o ent er the unit imaginar y number ty pe : „¥ Notice that the n umber i is enter ed as the order ed pair (0,1) if the CA S is set to AP PR O X mode . I[...]

Pa g e 4 - 6 CMP LX menu through the MTH menu Assuming that s y st em flag 117 is se t to CHOOSE bo x es (s ee Chapter 2), the CMPLX sub-men u w ithin the MTH menu is acc essed by using: „´9 @@OK@ @ . The follo wing sequen ce of scr een shots illustr ates t hese steps: T he fir st menu (opti ons 1 through 6) sho w s the follo w ing functi ons: R[...]

Pa g e 4 - 7 T his fir st sc r een sho ws f uncti ons RE , IM, and C  R . Noti ce that the last f uncti on r eturns a list {3 . 5 .} re pre senting the r eal and imaginar y compone nts of the comp lex n umber : T he follo wing s cr een sho ws func tions R  C, AB S , and ARG . Notice that the AB S functi on gets tr anslated to |3 .+5 .·i|, th[...]

Pa g e 4 - 8 T he re sulting menu inc lude some of the f uncti ons alread y intr oduced in the pr e vi ou s secti on , namely , AR G , AB S, C ONJ , IM, NEG , RE , and SIGN . It also inc ludes fu nctio n i whi c h serve s the same pur pos e as the k e y str ok e comb ination „¥ , i .e ., to enter the unit imaginar y number i in an e xpre ssi on.[...]

Pa g e 4 - 9 Functions fr om th e MTH menu T he h yper bolic f uncti ons and their in v ers es , as w ell as the Gamma, P SI , and P si func tions (spec ial f uncti ons) we re introduced and appli ed to r eal numbers in Chapte r 3 . Thes e functi ons can also be appli ed to comple x numbers b y fo llo w ing the procedur es pr esented in Chapte r 3 [...]

Pa g e 4 - 1 0 F uncti on DROI TE is f ound in the command catalog ( ‚N ). Using E V AL(AN S(1)) simplif ies the r esult to:[...]

Pa g e 5 - 1 Chapter 5 Algebraic and ar it hmetic oper ations An algebr aic ob ject , o r simpl y , algebr aic , is an y number , var i able name or algebr aic e xpr essi on that can be operat ed upon , manipulated, and comb ined accor ding to the rule s of algebr a . Example s of algebr aic ob jec ts ar e the fo llo w ing: • A number : 12 .3, 15[...]

Pa g e 5 - 2 (e xponential , logar ithmic , tr igonometry , h yper bolic , etc .) , as y ou would an y r eal or comple x number . T o demons trat e basic oper ations w ith algebr aic obj ects , let’s cr eate a c o up le of objects , say ‘ π *R^2’ and ‘ g*t^2/4’ , and stor e them in var iables A1 and A2 (See C hapter 2 to learn ho w to c [...]

Pa g e 5 - 3 ‚¹ @@A1@ @ „¸ @@A2@ @ T he same r esults ar e obtained in RPN mode if using the fo llo w ing ke ys tr ok es: @@A1@ @ @@A2@ @ +μ @@A1@ @ @@A2@ @ -μ @@A1@ @ @@A2@ @ *μ @@A1@@ @@A2@ @ /μ @@A1@@ ʳ ‚¹ μ @@ A2@@ ʳ „¸ μ Functions in the AL G menu T he AL G ( Algebr aic) menu is av ailable b y using the k e ys tr ok e seq u[...]

Pa g e 5 - 4 W e notice that , at the bottom of the sc r een , the line See: EXP AND F A CT OR suggests links t o other help f ac ility entr ies , the f unctions E XP AND and F A CT OR. T o mo ve dir ectly t o those entr ie s, pr ess the soft men u k ey @SEE1! for E XP AND , and @SEE2! for F A CT OR. Pr essing @SEE1 ! , f or e xample , show s the f[...]

Pa g e 5 - 5 F A CT OR: LNCOLLE CT : LIN: P ARTFRA C: S OL VE: S UBS T : TEXP AND: Not e: R ecall that , to use these , or any othe r functi ons in the RPN mode, y ou mus t enter the ar gument f irst , and then the func tion . F or e x ample , the e x ample for TE XP AND , in RPN mode will be s et up as: ³„¸+~x+~y` At this point , select f unct[...]

Pa g e 5 - 6 Other forms o f substitution in alg ebr aic e xpressions F uncti ons SUB S T , sho wn abo v e , is used to subs titute a var ia ble in an expr ession . A second f orm of substituti on can b e accomplished b y using the ‚¦ (ass oc iated w ith the I k e y) . F or e xample , in AL G mode , the fol lo w ing entry wi ll subs titute the v[...]

Pa g e 5 - 7 A differ ent appr oach to subs titution consis ts in def ining the substituti on e xpr essi ons in calc ulator v ar iables and plac ing the name of the var iables in the or iginal e xpr essi on . F or e xample , in AL G mode , stor e the fo llow ing var ia bles: Then , enter the e xpre ssion A+B: T he last e xpr essi on enter e d is a [...]

Pa g e 5 - 8 LNCOLLE CT , and TEXP AND ar e also contained in the AL G menu pr es ented earli er . F uncti ons LNP1 and EXP M wer e intr oduced in menu HYP ERBOLIC, under the MTH men u (See Chapt er 2) . T he only rem ainin g fun ctio n i s EXPL N. Its des cr ipti on is sho w n in the left-hand side , the e x ample fr om the help f ac ility is sho [...]

Pa g e 5 - 9 Functions in the ARITHME T I C menu T he ARITHME T IC menu cont ains a number of sub-menu s for s pec ifi c appli cations in n umber theory (int egers , poly nomials , etc .) , as w ell as a nu mber of f uncti ons that apply to ge ner al arithme tic ope rati ons . The AR ITHME TI C menu is tr igge r ed through the k ey str ok e combina[...]

Pa g e 5 - 1 0 L GCD (Greatest C ommon Denominator): P ROPFRA C (pr oper fr action) SI MP 2 : T he functi ons ass oci ated w ith the ARITHME T IC submenus: INTE GER , P OL YNOMIAL , MODUL O , and PERMUT A TION , are the fo llow ing: INT EG ER me nu EU LE R N u mb e r of in te g er s < n, c o - p rim e w i th n IABCUV Sol v es au + b v = c , w it[...]

Pa g e 5 - 1 1 F A CT OR F act ori z es an integer n umber or a poly nomial FCOEF Gener ates f rac tio n giv en r oots and multipli c ity FR OO T S R eturns r oots and multipli c ity giv en a fr action GCD Gr eatest common di v isor of 2 numbers or pol y nomials HERMITE n -th degree Her mite pol yn omial HORNER Horner e v aluation o f a pol yno mia[...]

Pa g e 5 - 1 2 Applications of the ARI THME T I C menu T his s ectio n is intended to pr es ent some of the back ground neces sar y f or appli cation of the ARI THMET IC menu f unctions . Def initions ar e pr esen ted ne xt r egarding the su bj ects of pol ynomials , pol ynomi al fr acti ons and modular ar ithmetic . T he ex amples pr esented belo [...]

Pa g e 5 - 1 3 multipl y ing j times k in modulus n arithmeti c is, in essence , the integer r emainder o f j ⋅ k / n in inf inite arithmeti c , if j ⋅ k>n . F or e xample , in modulus 12 ar ithmetic w e hav e 7 ⋅ 3 = 21 = 12 + 9 , (or , 7 ⋅ 3/12 = 21/12 = 1 + 9/12 , i .e ., the int eger r eminder of 21/12 is 9). W e can no w wr ite 7 ?[...]

Pa g e 5 - 1 4 Notice that , whene v er a r esult in the ri ght -hand si de of the “ congruence ” s ymbol pr oduces a r esult that is lar ger than the modulo (in this case , n = 6), you can alw ay s subtr act a multiple of the modulo fr om that re sult and simplif y it to a number smaller than the modulo . Thu s, the r esults in the f irst case[...]

Pa g e 5 - 1 5 [SP C] entry , and then pr es s the corr esponding modular arithme tic f uncti on . F or e x ample , using a modulus o f 12 , tr y the f ollo wing oper ations: ADDTMOD e xamples 6+5 ≡ -1 (mod 12) 6+6 ≡ 0 (mod 12) 6+7 ≡ 1 (mod 12) 11+5 ≡ 4 (mod 12) 8+10 ≡ -6 (mod 12) SUB TMOD ex amples 5 - 7 ≡ - 2 (mod 12) 8 – 4 ≡ 4 (m[...]

Pa g e 5 - 1 6 oper ating on them. Y ou can also conv er t an y number into a r ing number b y using the f uncti on EXP ANDMOD . F or ex ample, EXP A NDMO D(1 2 5) ≡ 5 (mod 12) EXP A NDMO D(17 ) ≡ 5 (mod 12) EXP ANDMOD(6) ≡ 6 (mod 12) The modular inv erse of a numb er Let a number k belong to a f inite ar ithmetic r ing of modulu s n , then t[...]

Pa g e 5 - 1 7 P ol ynomials P oly nomials ar e algebrai c expr essi ons consisting of one or mor e ter ms cont aining decr easing po we rs of a gi v en v ari able . F or e xample , ‘X^3+2*X^2 - 3*X+2’ is a thir d-or der poly nomi al in X, while ‘S IN(X)^2 - 2’ is a second-or der poly nomial in SI N(X) . A listing o f poly nomi al-r elated [...]

Pa g e 5 - 1 8 number s (func tion ICHINREM) . T he input consis ts of tw o v ector s [e xpr essi on_1, modulo_1] and [e xpr es si on_2 , modulo_2] . The o utput is a v ector containing [e xpr essi on_3, modulo_3] , wher e modulo_3 i s r elated to the pr oduct (modulo_1) ⋅ (modulo_2) . Example: CHINREM([X+1, X^2 -1],[X+1,X^2]) = [X+1,-(X^4 -X^2)][...]

Pa g e 5 - 1 9 An alter nate def initi on of the Hermite pol yn omials is wher e d n /dx n = n- th der i vati ve w ith r espec t to x . This is the def inition u sed in the calc ulator . Ex amples: The Her mite pol ynomi als of or ders 3 and 5 ar e giv en b y: HERMITE( 3) = ‘8*X^3-12*X’ , And HER MI TE(5) = ‘3 2*x^5-160*X^3+120*X’ . T he HO[...]

Pa g e 5 - 2 0 F or ex ample , for n = 2 , w e w ill w rit e: Chec k this r esult w ith yo ur calculator : L A GR ANGE([[ x1,x2],[y1,y2] ]) = ‘((y1-y2)*X+(y2*x1-y1*x2))/(x1- x2)’ . Other e x ample s: LA GR ANGE([[1, 2 , 3][2 , 8 , 15]]) = ‘(X^2+9* X-6)/2’ L A GRANGE([[0.5,1. 5,2 .5 , 3 .5, 4.5][12 .2 ,13 . 5,19 .2 ,2 7 . 3, 3 2 .5]]) = ‘ [...]

Pa g e 5 - 2 1 T he PCOEF function Gi ven an ar r ay co ntaining the r oots of a pol y nomial , the fu nction PC OEF gener ates an ar r ay containing the coe ffi c ients o f the corr esponding poly nomial . T he coeffi c ients cor r espond t o decr easing or der o f the independent var ia ble. F or ex ample: PCOEF([- 2 ,–1, 0,1,1,2]) = [1. –1. [...]

Pa g e 5 - 2 2 T he EPSX0 function and t he CAS v ariable EPS Th e va riab le ε (epsilon) is typ icall y used in mathemati cal te xtbooks to r epr esen t a v ery small number . T he calc ulator’s CAS cr eate s a v ari able EP S , w ith def ault v alue 0. 000000000 1 = 10 -10 , when y ou us e the EPSX0 f unction . Y ou can change this v alue , on[...]

Pa g e 5 - 23 Fra c ti on s F r acti ons can be expanded and fact or ed b y using func tions EXP AND and F A CT OR, f r om the AL G menu (‚×) . F or ex ample: EXP AND(‘(1+X)^3/((X-1) *(X+3))’) = ‘(X^3+3*X^2+3*X+1)/(X^2+2*X-3)’ EXP AND(‘(X^2)*(X+Y)/( 2*X-X^2)^2)’) = ‘(X+Y )/(X^2 - 4*X+4)’ EXP AND(‘X*(X+Y )/(X^2 -1)’) = ‘(X^2[...]

Pa g e 5 - 24 If y ou hav e the C omple x mode acti v e , the r esult w ill be: ‘2*X+(1/2/(X+i)+1/2/(X- 2 )+5/(X -5)+1/2/X+1/2/(X- i))’ T he FCOEF func tion T he function FC OEF is used to obta in a r a ti onal fr action , giv en the r oots and poles of the f r action . T he input f or the func tion is a v ector lis ting the r oots fo llo w ed [...]

Pa g e 5 - 25 mode selec ted, then the r esults w ould be: [0 –2 . 1 –1. – ((1+i* √ 3)/2) –1. – ((1–i* √ 3)/2) –1. 3 1. 2 1.]. Step-b y-step operations w ith poly nomials and fractions B y setting the CA S modes to S tep/st ep the calc ulato r wil l sho w simplif icati ons of fr actions or oper ations w ith poly nomi als in a step[...]

Pa g e 5 - 26 T he CONVERT M enu and algebr aic oper ations T he CONVER T menu is acti vated b y u sing „Ú ke y (the 6 key ) . T hi s menu summar i z es all con ver sion men us in the calc ulator . The lis t of these men us is sho wn next: T he functi ons a vaila ble in each o f the sub-menu s ar e sho w n next . UNIT S con vert menu (Option 1) [...]

Pa g e 5 - 27 B ASE con vert menu (Option 2) T his menu is the same as the UNI T S menu obtained b y u sing ‚ã . The appli cations of this menu ar e discu sse d in det ail in Chapter 19 . TRIGONOMETRIC conv er t menu (Option 3) T his menu is the same as the TRIG men u obtained b y using ‚Ñ . The appli cations o f this menu ar e disc uss ed in[...]

Pa g e 5 - 28 Fu n c ti o n  NUM has the same eff ect as the k ey str ok e combination ‚ï (ass oc iated w ith the ` key) . Fun ct io n  NU M co nver ts a s ym bo lic res ul t i nt o its floating-po int value . Func tion  Q conv er ts a floating-po int v alue into a fr acti on . F uncti on  Q π conv erts a floating-point v alue into [...]

Pa g e 5 - 2 9 LIN LNCOLLE CT P O WEREXP AND S IMPLIF Y[...]

Pa g e 6 - 1 Chapter 6 Solution to single equations In this c hapter w e featur e thos e functi ons that the calc ulator pr o vi des f or sol v ing single equations o f the for m f(X) = 0. Assoc iat ed with the 7 k e y ther e ar e two men us o f equation-sol v ing func tions , the S y mbolic S O L V er ( „Î ) , and the NUMer ical S oL V er ( ‚[...]

Pa g e 6 - 2 Using the RPN mode, the s olution is accomplished b y enter ing the equation in the stac k , f ollo we d by the v ar ia ble , bef or e enter ing f uncti on IS OL. R ight bef or e the e xec ution of I SOL , the RPN st ack should look as in the f igur e to the left. After appl y ing IS OL , the r esult is sho w n in the f igur e to the r[...]

Pa g e 6 - 3 The s cr e e n shot sho wn abo v e displa ys tw o solutions . In the firs t one , β 4 -5 β =12 5, SOL VE produce s no solu tions { }. In the s econd one , β 4 - 5 β = 6, S O L VE pr oduces f our soluti ons, sho w n in the last output line . The v ery last so lutio n is not v isible because the r esult occ up ies mor e c har acter s[...]

Pa g e 6 - 4 In the f irst case S OL VEVX could not find a s olution . In the second case , S OL VEVX f ound a single solu tion , X = 2 . The fol low i ng scr e ens sh o w the RP N stack for solving th e t wo exam pl es s hown abo ve (be for e and after applicati on of S OL VEVX) : T he equation u sed as ar gument fo r functi on S OL VEVX must be r[...]

Pa g e 6 - 5 The S ymbolic S olv er functi ons pre sented abo ve pr oduce soluti ons to rati onal equati ons (mainly , poly nomial equations). If the equation to be s ol ved f or has all numer i cal coeffi c ients , a numer ical soluti on is pos sible thr ough the use of the Numer ical S olv er f eatur es of the calc ulator . Numerical sol v er men[...]

Pa g e 6 - 6 P ol ynomial Equations Using the Sol ve poly… option in the calc ulator’s SOL V E en vir onment y ou can: (1) f ind the solu tions to a pol yn omial equati on; (2) obtain the coeff ic ie nts of the pol y nomial ha v ing a number of gi ven r oots; (3) obtain an algebr aic e xpr essi on f or the p o ly nomial a s a functi on of X. F [...]

Pa g e 6 - 7 All the s olutions ar e complex n umbers: (0.4 3 2 ,-0. 38 9) , (0.4 3 2 , 0. 38 9) , (-0.7 6 6, 0.6 3 2) , (-0.7 66 , -0.6 3 2) . Gene r ating poly nomial coefficients gi ven the polyn omial's roots Suppos e y ou w ant to gener ate the poly nomial w hose r oots are the n umbers [1, 5, - 2 , 4]. T o us e the calculat or fo r this [...]

Pa g e 6 - 8 Press ˜ to tr igger the line editor to see all the coeff i c ients . Gene r ating an algebraic e xpression f or the polynomial Y ou can use the calc ulator to gener ate an algebr aic e xpr es sion f or a poly nomial giv en the coe ffi c ients or the r o o ts of the pol y nomial . T he r esulting e xpre ssi on w ill be giv en in ter ms[...]

Pa g e 6 - 9 T o e xpand the pr oducts , y ou can us e the EXP AND command . The r esulting e xpr es si on is: ' X^4+-3*X^3+ -3*X^2 +11*X-6' . A differ ent appr oach to obtaining an e xpr essi on f or the poly nomi al is to gener a te the coeff ic ients fir st , then gene rat e the algebrai c e xpr essi on w ith the coeff ic ients highli [...]

Pa g e 6 - 1 0 Ex ample 1 – Calculating pa yment on a loan If $2 milli on ar e borr o w ed at an annual int er est rat e of 6 . 5% to be r epaid in 60 monthly pa y ments , what should be the monthl y pay ment? F or the debt to be totall y r epaid in 6 0 months, the f utur e value s of the loan should be z er o. S o , for the purpo se of using the[...]

Pa g e 6 - 1 1 pay m ents . Suppo se that w e use 2 4 peri ods in the firs t line of the amorti z ation scr een, i.e ., 24 @@OK @@ . T hen , pr ess @@AMOR@@ . Y ou w ill get the f ollo w ing re su l t : T his scr een is interpr eted as indi cating that after 2 4 months of pa y ing back the debt , the borr ow er has paid up US $ 7 2 3,211.43 int o t[...]

Pa g e 6 - 1 2 ˜ Skip P MT , since w e w ill be sol v ing fo r it 0 @@OK@@ Enter FV = 0, the opti on End is highlight ed @@CHOOS ! — @@OK@@ Change pa yme nt option t o Begin — š @@S OLVE! Highl ight P MT and so lv e f or it T he scr een no w sho ws the v alue of P MT as –38 , 9 21.4 7 , i .e. , the borr o w er mu st pay the lender US $ 3 8,[...]

Pa g e 6 - 1 3 ™ ‚í Enter a comma ³ ‚ @@PYR@ @ Enter name o f var iable P YR ™ ‚í Enter a comma ³ ‚ @@FV@ @ . Enter name o f var iable FV ` Ex ec ute P URG E command T he fo llo w ing two s cr een shots sho w the P URGE co mmand for purging all the v ari ables in the dir ectory , and the r esul t af t er e x ecu ting the command. In[...]

Pa g e 6 - 1 4 ³„¸~„x™-S„ì *~„x/3™‚Å 0™ K~e~q` Press J to see the ne w ly c r eated E Q v ari able: Then , enter the SOL VE env ironment and select S olv e equation… , by using: ‚Ï @@OK@@ . The corr esponding sc r een w ill be sho w n as: T he equation w e s tor ed in v ari able E Q is alr eady loaded in the Eq field in the[...]

Pa g e 6 - 1 5 This , ho w ev er , is not the only po ssible soluti on fo r this equation . T o obtain a negati ve s olutio n, f or e xampl e, ent er a negati v e number in the X: f ield be for e sol ving the equati on. T r y 3 @@@OK@@ ˜ @SOLVE@ . T he soluti on is no w X: - 3.045. Solution procedur e for Equation Solve ... T he numer ical sol ve[...]

Pa g e 6 - 1 6 T he equation is her e e xx is the unit str ain in the x -directi on , σ xx , σ yy , and σ zz , are the nor mal str esses on the partic le in the dir ecti ons of the x -, y-, and z -ax es , E is Y oung’s modulus or modulus of elasti c ity of the mat er ial , n is the P ois son r ati o of the mater ial , α is the thermal e xpans[...]

Pa g e 6 - 1 7 W ith the ex : fi eld highli ghted , pr ess @SOLVE @ to solve f or ex : T he soluti on can be seen fr om within the S OL VE EQU A TION in put for m b y pr essing @EDI T wh il e t he ex: f ield is highli ghted. T he r esulting value is 2. 4 7 0 833333333 E- 3. P r es s @@@OK@@ to e xit the ED I T featur e. Suppos e that y ou no w , w [...]

Pa g e 6 - 1 8 Spec ifi c ener gy in an open c hannel is def ined as the ener g y per unit w eight measur ed w ith r es pect to the c hannel bottom . Let E = s pec if ic ener g y , y = c hannel depth , V = flo w v eloc ity , g = acceler ation of gra v ity , the n we w rite T he flo w v eloc it y , in tur n , is giv en by V = Q/A, wher e Q = w ater [...]

Pa g e 6 - 1 9 Θ Solv e for y . T he r esult is 0.14 9 8 3 6.., i .e., y = 0.14 9 8 3 6 . Θ It is kno w n, how ev er , that ther e are ac tuall y two s oluti ons av ailable f or y in the spec if ic ener gy equatio n. T he soluti on w e jus t found cor r esponds to a numer i cal soluti on w i th an initial v alue of 0 (the de faul t va lu e fo r y[...]

Pa g e 6 - 2 0 In the ne xt e x ample w e w ill use the D ARC Y functi on f or f inding fr ic tion fact ors in pipeline s. T hus , w e def ine the f unctio n in the follo wing f r ame . Special function for pipe flo w: D ARC Y ( ε /D ,Re) T he D ar cy- W eis bach equatio n is used to calc ulate the ener g y loss (per unit wei g ht ) , h f , in a p[...]

Pa g e 6 - 2 1 Ex ample 3 – F low in a pipe Y ou may w a nt t o cr eate a separ ate sub-dir ectory (PIPE S) to tr y this e x ample . T he main equation go v ernin g flo w in a p ipe is, o f course , the D ar cy- W eisbac h equati on. T hu s, type in the fo llo w ing equation into E Q : Also , e nter the follo w ing var iables (f , A, V , Re): In [...]

Pa g e 6 - 2 2 T he combined equation has pr imitiv e v ari ables: h f , Q , L, g, D, ε , and Nu . Lau nch t he num erical solver ( ‚Ï @@ OK@@ ) to s ee the primiti ve v ari ables listed in the S OL VE E QU A TION in put f orm: Suppo se that w e us e the value s hf = 2 m, ε = 0. 00001 m , Q = 0. 05 m 3 /s , Nu = 0. 000001 m 2 /s, L = 20 m , an[...]

Pa g e 6 - 23 Ex ample 4 – Uni versal gr av itation Ne wton ’s la w of uni v ersal gr av itati on indicat es that the magnitude of the attr acti v e f or ce betw een tw o bodi es o f mas ses m 1 and m 2 s eparated b y a distance r is gi ven b y the equati on Her e , G is the uni v ersal gr av itati onal constant , who se v alue can be obtained [...]

Pa g e 6 - 24 Sol v e for F , and pr ess to r eturn to norm al calculator dis play . T he sol ution is F : 6 .6 7 2 5 9E -15_N , or F = 6 .6 7 2 5 9 × 10 -15 N. Different w ay s to enter equations into EQ In all the e xample s sho wn abo v e we ha ve en ter ed the equati on to be sol ved dir ectl y into v ar iable E Q befo r e acti vating the n um[...]

Pa g e 6 - 2 5 T ype an equati on, sa y X^2 - 12 5 = 0, dir ectl y on the stac k, and pr es s @@@OK@@@ . At this point the equati on is r eady f or solu tion . Alte rnati v ely , y ou can acti vate the equati on wr iter after pr essing @E DIT to enter y our equation . Pr ess ` to retur n to th e numeri ca l sol ver sc reen . Another wa y to enter a[...]

Pa g e 6 - 26 The S OL VE soft menu The SOL V E soft menu a llow s a ccess to some of the numerical solv er fu nctions thr ough the soft men u k e ys . T o access this men u us e in RPN mode: 7 4 MENU , or in AL G mode: MENU(7 4) . Alter nativ ely , y ou can use ‚ (hold ) 7 to acti v ate the S OL VE soft men u . T he sub-menu s pr ov ided b y the[...]

Pa g e 6 - 27 Ex ample 1 - Sol v ing the equati on t 2 -5t = - 4 F or ex ample , if you s tor e the equati on ‘t^2 -5*t=- 4’ into E Q, and pr ess @) SOLVR , it w ill acti v ate the f ollo wing menu: T his result indi cates that y ou can sol v e for a v alue of t f or the equation lis ted at the top of the displa y . If y ou tr y , fo r ex ample[...]

Pa g e 6 - 2 8 Y ou can also sol ve mor e than one equati on by s olv ing one equation at a time , and r epeating the pr ocess until a soluti on is fo und . F or e xample , if y ou enter the f ollo w ing list of equati ons into var i able E Q: { ‘ a*X+b*Y = c’ , ‘k*X*Y=s ’}, the k e y str ok e sequ ence @) ROOT @ ) SOLVR , w ithin the S OL [...]

Pa g e 6 - 2 9 Using units with the SOL VR sub-menu T hese ar e some rule s on the us e of units w ith the S OL VR sub-men u: Θ Enter ing a guess w ith units fo r a gi ven v ar ia ble , w ill intr oduce the u se of thos e units in the soluti on. Θ If a ne w gues s is gi ven w ithout units, the units pr ev iousl y sa ved f or that partic ular v ar[...]

Pa g e 6 - 3 0 T his functi on pr oduces the coeff ic ients [a n , a n- 1 , … , a 2 , a 1 , a 0 ] of a poly nomial a n x n + a n- 1 x n- 1 + … + a 2 x 2 + a 1 x + a 0 , g ive n a ve ct or o f i t s ro o ts [r 1 , r 2 , …, r n ]. F or ex ample, a v ect or who se r oots ar e gi v en b y [-1, 2 , 2 , 1, 0], w ill p r oduce the f ollo w ing coeff[...]

Pa g e 6 - 3 1 Press J to e x it the S OL VR en vi r onment . F ind y our w ay bac k to the TVM su b- menu w i thin the S OL VE sub-menu to try the other functi ons a vailable . Function T VMROO T This fun c tion requires as argument t he na me of one of the v ariables in t he T VM pr oblem . The f uncti on r eturns the s oluti on fo r that var ia [...]

Pa g e 7- 1 Chapter 7 Solv ing multiple equations Man y pr oblems of sc i ence and engineer ing req uir e the simultaneous solu tions of mor e than one equation . The calc ulator pro v ides s ev er al pr ocedur es f or solv ing multiple equations as pr esented belo w . P lease notice that no discu ssion of solv ing sy stems of linear equation s is [...]

Pa g e 7- 2 Use co mmand S OL VE at this point (f r om the S . SL V menu: „Î ) A fter about 40 s econds, may be more , yo u get as re sult a list: { ‘t = (x -x0)/(C OS( θ 0)*v0)’ ‘ y0 = (2*C OS( θ 0)^2*v0^2*y+(g*x^2(2*x0*g+2*SIN( θ 0))*CO S( θ 0 )*v0^2)*x+ (x0^2*g+2*S IN( θ 0)*C OS( θ 0)*v0^2*x0)))/(2*CO S( θ 0)^2*v0^ 2)’]} Press [...]

Pa g e 7- 3 the cont ents of T1 and T2 to the stac k and adding and subtr acting them . Here is ho w to do it w ith the eq uation w r iter : Enter and s tor e ter m T1: Enter and st or e ter m T2: Notice that w e ar e using the RPN mode in this ex ample, ho we v er , the pr ocedur e in the AL G mode should be v ery simi lar . Cr eate the equation f[...]

Pa g e 7- 4 Notice that the r esult include s a vec tor [ ] contained w ithin a list { }. T o r emo ve the list s y mbol , use μ . F inally , to decompose the v ector , use f uncti on OB J  . T he r esult is: T hese tw o ex amples constitu te sy stems of linear equatio ns that can be handled equall y w ell w ith func tion LIN S OL VE (see Chap [...]

Pa g e 7- 5 Ex ample 1 - Ex ample from the help facilit y As w ith all functi on entr ie s in the help fac ility , ther e is an e x ample at t ached to the MSL V entr y as sho wn abo v e . Notice that f uncti on MSL V r equir es thr ee ar guments: 1. A v ector co ntaining the equati ons, i .e., ‘[S IN(X)+Y ,X+SIN(Y )=1]’ 2 . A v ector containin[...]

Pa g e 7- 6 disc har ge (m 3 /s or ft 3 /s) , A is the c ro ss-sec tional ar ea (m 2 or ft 2 ), C u is a coeff ic ient that depends on the s ys tem of units (C u = 1. 0 fo r the SI, C u = 1.4 8 6 fo r the English s ys tem o f units) , n is the Manning’s coe ff ic ient , a measure o f the c ha nnel surf ace r oughness (e . g ., f or conc r ete , n[...]

Pa g e 7- 7 μ @@@EQ1@@ μ @@@EQ2@ @ . T he equatio ns ar e listed in the s tack a s follo ws (small fo nt option s elected): W e can see that these eq uations ar e indeed giv en in ter ms of the pr imitiv e var iable s b, m , y , g, S o , n, C u , Q, and H o . In or der to solv e for y and Q we need to giv e v alues to the other v ar iables. Suppo[...]

Pa g e 7- 8 Ne xt , we ’ll enter var iable E QS: LL @@EQS@ , fo llow ed by v ector [y ,Q]: ‚í„Ô~„y‚í~q™ and b y t he in itial gu esses ‚í„Ô5‚í 10 . Bef or e pre ssing ` , the sc r een w ill look like this: Press ` to sol v e the sy stem of equatio ns. Y ou ma y , if your angular measur e is not set to r adians , get the fo l[...]

Pa g e 7- 9 T he re sult is a list of thr ee v ector s. T he fir st vec tor in the list w ill be the equations sol ved . The second v ector is the list of unkno wns. The thir d v ector r epres ents the solu tion . T o be able to see the se v ector s, pr es s the do wn-a r r o w k e y ˜ to acti v ate the line editor . T he soluti on w ill be sho w [...]

Pa g e 7- 1 0 T he cosine la w indicat es that: a 2 = b 2 + c 2 – 2 ⋅ b ⋅ c ⋅ cos α , b 2 = a 2 + c 2 – 2 ⋅ a ⋅ c ⋅ cos β , c 2 = a 2 + b 2 – 2 ⋅ a ⋅ b ⋅ co s γ . In or der to sol v e an y tr iangle , you need to kno w at least thr ee of the fo llo w ing si x v ar iable s: a, b, c, α, β, γ . T hen, y ou can use the equ[...]

Pa g e 7- 1 1 ‘SIN( α )/a = SIN( β )/b ’ ‘SIN( α )/a = S IN( γ )/c’ ‘SIN( β )/b = S IN( γ )/c’ ‘ c^2 = a^2+b^2 - 2*a*b*CO S( γ )’ ‘b^2 = a^2+c^2 - 2*a*c*CO S( β )’ ‘ a^2 = b^2+c^2 - 2*b*c*CO S( α )’ ‘ α+β+γ = 180 ’ ‘ s = (a+b+c)/2’ ‘A = √ (s*(s-a)*(s-b)*(s-c))’ Then , enter the number 9 , and cr eat[...]

Pa g e 7- 1 2 Press J , if needed , to get y our var i ables me nu . Y our men u should sho w the va riab le s @LVARI! ! @TITLE @@EQ@ @ . Preparing to run t he ME S T he next s tep is to acti vate the ME S and tr y one s ample soluti on. Be for e we do that , ho we v er , w e want to s et the angular units to DEGr ees, if the y ar e not alr eady s [...]

Pa g e 7- 1 3 Let ’s tr y a sim ple soluti on of Cas e I, using a = 5, b = 3, c = 5 . Us e the fo llo w ing entr ies: 5 [ a ] a:5 is listed in the top left corner of the displa y . 3 [ b ] b:3 is listed in the top left corner of the displa y . 5 [ c ] c:5 is listed in the top left corner of the display . T o so lv e f or the angles u se: „ [ α[...]

Pa g e 7- 1 4 Pr es sing „ @@ALL@@ will s olv e for all the v ar iables , tempor aril y sho w ing the intermediate r esults. Pr ess ‚ @@ALL @@ to see t he solu tions: When done , pres s $ to r eturn to the ME S env ironment . Pre ss J to ex it t he ME S en v ir onment and r eturn to the nor mal calc ulator displa y . Org anizing th e v ariabl e[...]

Pa g e 7- 1 5 Progr amming the MES triangle solution using User RPL T o fac ilitate acti vating the ME S for f utur e soluti ons , w e will c r eate a pr ogram that w ill load the ME S wi th a single ke y str oke . The pr ogr am should look lik e this: << DEG MINI T TI TLE L V ARI MI TM MS OL VR >>, and can be ty ped in by using : ‚å[...]

Pa g e 7- 1 6 Use a = 3, b = 4, c = 6 . The soluti on pr ocedure us ed her e consists of so lv ing fo r all var ia bles at once , and then r ecalling the soluti ons to the st ack: J @TRISO T o clear up data and r e -start ME S 3 [ a ] 4 [ b ] 6 [ c ] T o en ter data L T o mo ve t o the next v ar iable s menu . „ @ALL! S ol v e fo r all the unkno [...]

Pa g e 7- 1 7 Adding an INFO but ton to your dir ec tory An inf ormati on button can be us ef ul for y our dir ectory to help y ou r emember the oper ation o f the func tions in the dir ectory . In this dir ecto r y , all w e need to r emember is to pr ess @TRISO to get a tr iangle s olution s tarted. Y ou may w ant to type in the fo llo w ing pr o[...]

Pa g e 7- 1 8 An e xplanatio n of the v ari ables f ollo ws : SOL VEP = a progr am that tr iggers the multiple equati on sol v er fo r the partic ular s et of equations s tor ed in var iable PEQ ; NAME = a v ari able stor ing the name of the multiple equati on sol ve r , namely , "v el . & acc. polar coor d. " ; LIST = a list of the v[...]

Pa g e 7- 1 9 Notice that afte r y ou enter a partic ular value , the calc ulator displa y s the v ari able and its value in the upper le f t co rner o f the displa y . W e hav e no w enter ed the kno wn v a r iables . T o calc ulate the unkno w ns w e can pr oceed in two ways: a) . So lv e fo r indiv idual var iables , for e xample , „ [ vr ] gi[...]

Pa g e 7- 2 0[...]

Pa g e 8 - 1 Chapter 8 Operations w ith lists L ists ar e a type o f calculat or’s ob ject that can be u sef ul f or data pr oces sing and in pr ogr amming. T his Cha pter pr esents e x amples of oper ations w ith lists . Definitions A list , within the conte xt of the calculat or , is a seri es of ob jec ts enclo sed between br aces and se parat[...]

Pa g e 8 - 2 T he fi gur e belo w sho w s the RPN stac k befo r e pre ssing the K key: Composing and decomposing lists Compo sing and decompo sing lists mak es sense in RPN mode onl y . Under such oper ating mode , decomposing a list is ac hie v ed by u sing functi on OB J  . With this func tion , a list in the RPN stac k is decompos ed into its[...]

Pa g e 8 - 3 In RPN mode , the follo wi ng scr een show s the thr ee lists and their name s read y to be stor ed. T o stor e the lis ts in this case you need to pr ess K three times . Changing sign T he sign - change k e y ( ) , w hen applied to a list of number s, w i ll c hange the sign o f all elements in the list . F or e xam ple: Addition , [...]

Pa g e 8 - 4 Subtr actio n, multiplicati on, and di v ision o f lists of numbers o f the same length pr oduce a list of the s ame length with ter m-b y- te rm oper ations . Exam ples: T he div ision L4/L3 w ill pr oduce an infinity entry becaus e one of the e lements in L3 is z er o: If the lists in v ol ved in the oper ation ha ve diff er ent leng[...]

Pa g e 8 - 5 AB S EXP and LN L OG and ANTIL OG S Q and squar e root SIN, ASIN COS, ACOS T AN, A T AN INVER SE (1/x) Real number functions from the MTH menu F uncti ons of inter est fr om the MTH me nu include , fr om the HYPERB OLIC menu: S INH, A S INH, CO SH , A C OSH , T ANH, A T ANH, and fr om the REAL menu: %, %CH, %T , MIN, MAX, MOD , SIGN, M[...]

Pa g e 8 - 6 T ANH, A T ANH S IGN, MANT , XP ON IP , FP FL OOR, CEIL D  R, R  D Ex amples of functions t hat use tw o arguments T he scr een shots belo w show appli cations o f the functi on % to list ar guments . F unction % r e quir es t w o ar guments. The f irst tw o ex amples sho w cases in w hic h only one o f the t w o ar guments is a [...]

Pa g e 8 - 7 %({10,20, 30},{ 1,2 , 3}) = {%(10,1),%(20,2),%(3 0, 3)} T his desc r iption o f func tion % f or list ar guments sh o ws the gener al pattern of e valuati on of an y f uncti on w ith two ar guments when one or both ar guments ar e lists . Ex amples of appli cations o f func tion RND ar e sho wn ne xt: Lists o f comple x numbers T he fo[...]

Pa g e 8 - 8 T he follo w ing ex ampl e sho w s applicati ons of the f uncti ons RE(R eal part) , IM(imaginar y part) , AB S(magnitude) , and AR G(argument) o f comple x numbers . The r esults are lists o f real n umbers: Lists o f algebraic objects T he follo wing ar e ex amples o f lists of algebr aic obj ects w ith the func tion S IN appl ie d t[...]

Pa g e 8 - 9 T his menu cont ains the fo llo w ing func tio ns: Δ LIS T : C alc ulate incr ement among consec uti ve elements in list Σ LIS T : Ca lculat e summation o f elemen ts in the list Π LIS T : Calc ulate pr oduct of elements in the list S OR T : Sorts elemen ts in incr easing order REVLI S T : R e v erse s or der of list ADD : Oper ator[...]

Pa g e 8 - 1 0 M anipulating elements of a list T he PR G (pr ogr amming) menu inc ludes a LI S T sub-m enu w ith a n umber of func tions t o manipulate ele ments of a list . W ith s ys tem f lag 117 se t to CHOO SE bo x es: Item 1. ELEMENT S.. co ntains the fo llo w ing func tions that can be us ed for the manipulation o f elements in lists: List [...]

Pa g e 8 - 1 1 F uncti ons GET I and PUT I , also av ailable in sub-me nu PR G/ ELEMENT S/, ca n also be us ed to ext rac t and place elements in a list . Thes e t w o functi ons, ho w e ve r , are u se ful mainl y in pr ogr amming . F uncti on GET I us es the same ar guments as GE T and r eturns the lis t , the element locati on plus one , and the[...]

Pa g e 8 - 1 2 SE Q is use ful t o pr oduce a list of v alues gi ve n a par ti c ular expr essi on and is desc r ibed in mor e detail her e . T he SEQ f uncti on tak es as ar guments an e xpr essi on in ter ms of an index , the name of the inde x , and starting, ending , and incr ement values f or the inde x , and r eturns a lis t consisting of the[...]

Pa g e 8 - 1 3 In both case s, y ou can ei ther t y pe out the MAP command (as in the e x amples abo v e) or select the command fr om the CA T menu . T he follo w ing call to func tion MAP us es a pr ogr am instead of a f uncti on as second a r gument: Defining functions t hat use lists In Chapte r 3 w e intr oduced the use o f the D EFINE f unctio[...]

Pa g e 8 - 1 4 to r eplace the plus sign (+) w ith ADD: Ne xt , we s tor e the edited e xpres sion in to v ari able @@@G@@@ : Ev aluating G(L1,L2) now pr oduces the f ollo w ing r esult: As an alter nati ve , y ou can define the f uncti on w ith ADD rathe r than the plus sign (+), fr om the s tart, i .e ., use DEFINE(' G(X,Y)=(X DD 3)*Y')[...]

Pa g e 8 - 1 5 Applications of lists T his sectio n show s a couple of appli cations o f lists to the calc ulation o f statisti cs of a sa mple. B y a samp le w e u nderstand a list of v alu es , sa y , {s 1 , s 2 , …, s n }. Suppo se that the sample o f inter est is the list {1, 5, 3, 1, 2, 1, 3, 4, 2, 1} and that w e stor e it into a var iable [...]

Pa g e 8 - 1 6 3 . Di v ide the r esult abo ve by n = 10: 4. Appl y the INV() func tion to the lat est r esult: T hus , the harmonic mean o f list S is s h = 1.63 4 8… Geometric mean of a list T he geometri c mean of a sample is def ined as T o f ind the geometri c mean of the list stor ed in S , w e can use the f ollo w ing pr ocedur e: 1. Appl [...]

Pa g e 8 - 1 7 T hus , the geometri c mean of list S is s g = 1.00 3 203… W eighted aver age Suppo se that the data in list S , def ined abo ve , namel y : S = {1,5,3,1 ,2,1,3,4,2,1 } is affec ted b y the we ights , W = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} If w e def ine the we ight lis t as W = {w 1 ,w 2 ,…,w n }, w e noti ce that the k -t h elemen[...]

Pa g e 8 - 1 8 3. U se f un c t io n Σ LIS T , once mo r e , to calc ulate the denominat or of s w : 4. Use the e xpre ssi on ANS( 2)/ANS(1) to cal culat e the w ei ghted a v er age: Th us, the w ei ghted av er age of list S w ith w eights in lis t W is s w = 2 .2 . Statistics of gr ouped data Gr ouped dat a is t y p icall y gi v en b y a table sh[...]

Pa g e 8 - 1 9 T he clas s mar k data can be st ored in v ari able S , while the fr equency coun t can be stor ed in var iable W , as follo ws: Gi ven the list of class marks S = {s 1 , s 2 , …, s n }, and the lis t of fr equenc y counts W = {w 1 , w 2 , …, w n }, the w ei ghted a v er age of the data in S w ith w eights W r epr esents the mean[...]

Pa g e 8 - 2 0 T o calc ulate this las t r esult , w e can us e the fo llow ing: T he standar d dev iati on of the gr ouped data is the sq uar e r oot of the var iance: N s s w w s s w V n k k k n k k n k k k ∑ ∑ ∑ = = = − ⋅ = − ⋅ = 1 2 1 1 2 ) ( ) ([...]

Pa g e 9 - 1 Chapter 9 V ec tors T his Chapter pr o v ides e x amples o f enter ing and operating w ith vect ors , both mathematical v ector s of man y elements, as w ell as ph y sical v ectors of 2 and 3 components . Definitions F r om a mathematical po int of v ie w , a vec tor is an arr a y of 2 or mor e elements arr anged int o a r o w or a col[...]

Pa g e 9 - 2 wher e θ is the angle betw een the tw o vec tors . The c r os s pr oduct pr oduces a vec tor A × B who se magnitude is | A × B | = | A || B |sin( θ ) , and its dir ecti on is gi v en by the s o -ca lled r ight -hand r ule (consult a textbook on Math , Ph y sics , or Mechani cs to s ee this oper ation illu str ated gr aphicall y) . [...]

Pa g e 9 - 3 Stor ing vectors int o var iables V ector s can b e s tor ed into var iables . The sc r een shots belo w show the v ectors u 2 = [1, 2] , u 3 = [-3, 2, -2] , v 2 = [3 ,-1] , v 3 = [1, -5, 2 ] stor ed into v ariables @ @@u2@@ , @@@u3@@ , @@@v2@@ , and @@@v3@@ , r especti v el y . F irst , in AL G mode: T hen, in RPN mode (bef or e pr es[...]

Pa g e 9 - 4 Th e ← WID k ey is u sed to decr ease the w idth of the columns in the spr eadsheet . Pr ess this k ey a couple o f times to see the column w idth dec r ease in y our Matr i x W r iter . Th e @ W I D → k ey is used to incr ease the w idth of the columns in the spr eadsheet . Pr ess this k ey a couple o f times to see the column w i[...]

Pa g e 9 - 5 Th e @+ROW@ k e y w ill add a r o w full o f z er os at the location o f the select ed cell o f the spr eadsheet . Th e @-ROW k ey w ill delete the r o w corr esponding t o the selec ted cell of the spr eadsheet . Th e @+COL@ k e y w ill add a column full o f z er os at the location o f the selec ted cell of the spr eadsheet . Th e @-C[...]

Pa g e 9 - 6 Building a vector with  ARR Y Th e fu nct ion → ARR Y , av ailable in the func tion catalog ( ‚N‚é , us e —˜ to locat e the functi on), ca n als o be u sed to build a v ect or or ar r ay in the f ollo w ing wa y . In AL G mode , enter  ARR Y( v ector el ements, numb er of elements ) , e .g ., In RPN mode: (1) Enter the [...]

Pa g e 9 - 7 In RPN mode , the functi on [ → ARR Y] take s the obj ects fr om stac k lev els n+1, n, n- 1 , …, dow n to st ack le ve ls 3 and 2 , and con v erts them into a v ector of n elements . T he object ori ginally at s tack le v el n+1 becomes the f irst element , the obj ect or iginally at le v el n becomes the second element , and so o[...]

Pa g e 9 - 8 Highli ghting the entir e e xpr essio n and using the @EV AL@ soft men u ke y , w e get the re su l t : -1 5 . T o r eplace an e lement in an arr a y use f uncti on PUT (y ou can find it in the func tion cat alog ‚N , or i n the P RG/LI S T/ELEMENT S sub-men u – the later w as intr oduced in Chapter 8). In AL G mode , y ou need to [...]

Pa g e 9 - 9 Simple oper ations with vectors T o illus tr ate oper atio ns w ith vec tor s we w ill use the v ector s A, u2 , u3, v2 , and v3, sto r ed in an ear lier e xe r c ise . Changing sign T o change the si gn of a v ect or use the k e y , e .g., Addition , subtraction Additi on and subtrac tion o f vec tors r equir e that the t w o v ecto[...]

Pa g e 9 - 1 0 Absolute value func tion T he absolute v alue func tion ( ABS), when appli ed to a vec tor , pr oduces the magnitude of the v ector . F or a v ector A = [ A 1 ,A 2 ,…,A n ], the magnitude is def ined as . In the AL G mode , enter the functi on name f ollo we d by the v ector ar gument . F or e xample: BS([1,-2 ,6]) , BS( ) , BS(u3)[...]

Pa g e 9 - 1 1 Dot pr oduct F uncti on DO T is used to calc ula t e the dot pr oduct of tw o vec tors o f the same length. S ome e xample s of applicati on of f uncti on DO T , using the v ecto rs A, u2 , u3, v2 , and v3, stor ed ear lie r , ar e show n next in AL G mode . Attempts t o calc ulate the dot pr oduct o f two v ector s of differ ent len[...]

Pa g e 9 - 1 2 In the RPN mode , appli cation o f func tion V  w ill list the components o f a ve ctor in the st ack , e .g., V  (A ) will pr oduce the f ollo w ing outpu t in the RPN stack (vector A is li sted in stack lev el 6: ). Building a t w o -dimensional v ec t or Fu n c ti o n  V2 is used in the RPN mode to bu ild a vect or w ith [...]

Pa g e 9 - 1 3 When the r ect angular , or Cartesi an, coor dinate s yst em is select ed, the t op line of the displa y w ill show an XY Z fi eld , and any 2 -D or 3-D v ector ent er ed in the calc ula t or is r eproduced as the (x ,y ,z) components o f the vecto r . T hus , to enter the v ector A = 3 i +2 j -5 k , w e use [3,2 ,-5], and the v ecto[...]

Pa g e 9 - 1 4 T he fi gur e belo w sho w s the tr ansfor mation o f the v ector f r om spher ical to Cartesi an coor dinates , w ith x = ρ sin( φ ) cos( θ ), y = ρ sin ( φ ) cos ( θ ), z = ρ co s( φ ) . F or this case , x = 3 .204 , y = 1.4 9 4, and z = 3 . 5 3 6. If the C YLINdri cal s y stem is s elected , the top line of the dis play w [...]

Pa g e 9 - 1 5 equi vale nt (r , θ ,z) with r = ρ sin φ , θ = θ , z = ρ co s φ . F or e x ample , the f ollo w ing f igur e sho ws the v ector ent er ed in spher ical coor d i nates, and tr ansf ormed to polar coor dinates . F or this case , ρ = 5, θ = 2 5 o , and φ = 4 5 o , while the tr ansfor mation sho ws tha t r = 3 .5 6 3, and z = 3[...]

Pa g e 9 - 1 6 Suppos e that yo u want t o find the angle between v e c tors A = 3 i -5 j +6 k , B = 2 i + j -3 k , y ou could tr y the f ollo w ing oper ati on (angular measur e set to degr ees) in AL G mode: 1 - Enter vec tors [3,-5, 6], pr ess ` , [2 ,1,-3], pr ess ` . 2 - DO T(ANS(1),ANS(2)) calc ulates the dot pr oduc t 3 - ABS( ANS( 3))*ABS(([...]

Pa g e 9 - 1 7 Thu s, M = (10 i +2 6 j +2 5 k ) m ⋅ N. W e kno w that the magnitud e o f M is such that | M | = | r || F |sin( θ ), wher e θ is the angle between r and F . W e can find this angle as , θ = sin -1 (| M | /| r || F |) by the f ollo w ing oper ati ons: 1 – AB S(AN S(1))/(AB S( ANS( 2))*ABS( ANS( 3)) calc ulates sin( θ ) 2 – A[...]

Pa g e 9 - 1 8 Ne xt , we calc ulate v ector P 0 P = r as ANS(1) – AN S(2), i .e ., F inally , w e tak e the dot pr oduct o f ANS(1) and ANS( 4) and mak e it equal t o z er o to complete the oper ation N • r =0: W e can no w use f uncti on EXP AND (in the AL G menu) to e xpand this ex p ress io n : T hus , the equation of the plane thr ough poi[...]

Pa g e 9 - 1 9 In this secti on w e w ill show ing yo u wa y s to transf or m: a column vec tor into a r o w vec tor , a r o w ve ctor int o a column vec tor , a list into a vec tor , and a vec tor (or matr i x) into a list . W e f irst demons tr ate thes e transf ormations u sing the RPN mode. In this mode , w e w ill use f uncti ons OB J  , ?[...]

Pa g e 9 - 2 0 If w e no w appl y func tion OB J  once mor e , the list in st ack le vel 1:, {3 .}, w ill be decomposed as follo ws: Function  LIS T T his functi on is used to c r eate a list gi ven the eleme nts of the list and the list length or si z e. In RPN mode , the list si z e, s a y , n, should be placed in stac k le vel 1:. T he ele[...]

Pa g e 9 - 2 1 3 - Use f uncti on  ARR Y to build the column vec tor T hese thr ee steps can be put t ogether into a U serRP L pr ogr am, e nter ed as fo llo ws (in RPN mode , still) : ‚å„° @) TYPE! @ OBJ  @ 1 + !  ARRY@ `³~~rxc` K A ne w v ar iabl e , @@RXC@@ , will be av ailable in the soft menu labels after pr es sing J : Press ?[...]

Pa g e 9 - 2 2 2 - Use f uncti on OB J  to d ecompose the l ist in stac k le vel 1: 3 - Pr ess the de lete k e y ƒ (also kno wn a s functi on DROP) t o eliminate the number in st ack lev el 1: 4 - Use f uncti on  LIS T to cr eate a list 5 - Use f uncti on  ARR Y to cr eate the r o w v ecto r T hese f i v e steps can be put t ogether into [...]

Pa g e 9 - 23 Th is variab le, @@CXR@@ , can no w be used t o dir ectl y transf orm a column v ector to a r o w vec tor . In RPN mode , enter the column v ector , and then pre ss @@CXR@ @ . T ry , fo r ex ample: [[1] ,[2],[3]] ` @@CXR@@ . After hav ing def ined v ar iabl e @@CXR@@ , w e can use it in AL G mode to transf or m a r o w vec tor into a [...]

Pa g e 9 - 24 A ne w v ar iabl e , @@LXV@@ , will be av ailable in the soft menu labels after pr es sing J : Press ‚ @@LXV@@ to see the pr ogram co ntained in the v ari able LXV : << OBJ  1  LIST  RRY >> Th is va r i ab le, @@LXV@@ , can no w be used to dir ectly tr ansfor m a list into a v ector . In RPN mode , enter the lis[...]

Pa g e 1 0 - 1 Chapter 10 ! Cr eating and manipulating matrices T his chapte r show s a number of e xamples aimed at c reating matr ices in the calc ulator and demons tr ating manipulation of matr i x elements . Definitions A matr i x is simpl y a re ctangular arr ay of ob jec ts (e.g ., numbers , algebr aics) hav ing a n umber of r o w s and colum[...]

Pa g e 1 0 - 2 Entering matr ices in the stac k In this secti on w e pr esent tw o diffe r ent methods to enter matr ices in the calc ulator st ack: (1) using the Matr i x W rit er , and (2) typ ing the matri x dir ectl y i nto th e sta ck. Using the M atr ix W riter As w i th the case of vectors, discussed in Ch apter 9 , ma tr ices c an be en ter[...]

Pa g e 1 0 - 3 If y ou ha ve s elected the t extbook displa y option (using H @) DISP! and c hecking off  Textbook ) , the matr ix w ill look lik e the one sho w n abo ve . O the r w ise , the displa y w ill sho w: T he displa y in RPN mode w ill look very similar to these . T yping in the matri x directly into the stac k T he same r esult as a [...]

Pa g e 1 0 - 4 or in the MA TRICE S/CREA TE menu a v ailable thr ough „Ø : T he MTH/M A TRI X/MAKE sub menu (let’s call it the MAKE menu) contains the fo llo w ing fu ncti ons: w hile the MA TRICE S/CREA TE sub-menu (let’s call it the CREA TE menu) has the fo llo w ing fu ncti ons:[...]

Pa g e 1 0 - 5 As y ou can see f r om e xploring the se menu s (MAKE and CREA TE) , the y both hav e the same f uncti ons GET , GET I, P UT , P U T I , SUB , REP L , RDM, R ANM, HILBERT , V ANDERMONDE , IDN, CON , → DIA G , and DI A G → . The CREA TE menu inc ludes the C OL UM N and RO W sub-menu s, that ar e also a vaila ble under the MTH/MA T[...]

Pa g e 1 0 - 6 Functions GET and P UT F uncti ons GET , GE TI , P UT , and P UT I, oper ate w ith matri ces in a similar manner as w ith lists or v ectors , i .e ., y ou need to pr o vi de the location o f the element that y ou want to GE T or PUT . Ho we ver , w hile in lists and v ector s only one inde x is r equired to identify an element , in m[...]

Pa g e 1 0 - 7 Notice that the s cr een is prepar ed fo r a subseq uent appli cation o f GET I or GE T , b y incr easing the column index o f the ori ginal re fer ence b y 1, (i .e., f r om {2 ,2} to {2 , 3}) , w hile sho w ing the ex trac ted v alue , namel y A(2 ,2) = 1.9 , in stac k lev el 1. No w , suppose that y ou w ant to inser t the v alue [...]

Pa g e 1 0 - 8 If the ar gument is a r eal matri x, TRN simpl y pr oduces the tr anspos e of the r eal matr i x. T r y , f or e xample , TRN(A ) , and compar e it w ith TRAN(A ) . In RPN mode , the tr ansconjugat e of matri x A is calc ulated by using @@@A@@@ TRN . Function CON T he functi on tak es as ar gument a list of tw o elements, cor r espon[...]

Pa g e 1 0 - 9 In RPN mode this is accomplished b y using {4 ,3} ` 1.5 ` CON . Function IDN F uncti on ID N (IDeNtity matri x) cr eates an identity matri x giv en its si z e. R ecall that an identity matr i x has to be a squar e matri x , ther ef or e , only one v alue is r equir ed to des cr i be it completel y . F or ex ample, t o cr eate a 4 ?[...]

Pa g e 1 0 - 1 0 v ector ’s dimensi on, in the latte r the number of r o ws and columns of the matr ix . T he follo wing e x amples illus tr ate the use o f functi on RDM: Re -dimensioning a vector into a matr ix T he follo w ing ex ample sho ws ho w to r e -dimension a v e c tor of 6 ele ments into a matr i x of 2 r o w s and 3 columns in AL G m[...]

Pa g e 1 0 - 1 1 If using RPN mode , we a ssume that the matr i x is in the stac k and us e {6} ` RDM . Function R ANM F uncti on RANM (RANdom Matr i x) w ill gener ate a matr i x w ith r andom integer elements gi ven a list w ith the number of r ow s and columns (i .e ., the dimensions of the matr i x) . F or e xample , in AL G mode , t w o diff e[...]

Pa g e 1 0 - 1 2 In RPN mode , assuming that the or iginal 2 × 3 matr i x is alread y in the stac k, u se {1,2} ` {2,3} ` SUB . Function REP L F uncti on REPL r e place s or inserts a sub-matr i x into a lar ger one . The input f or this func tio n is the matri x wh ere the r eplacement w ill tak e place, the location w here the r e placeme nt beg[...]

Pa g e 1 0 - 1 3 In RPN mode , wi th the 3 × 3 matri x in the stac k, w e simply hav e to acti v ate fu nct ion  DI G to obtain the same r esult as abo ve . Function DIA G → Fu n c ti o n D I A G → tak es a v ector and a lis t of matr i x dimensions {r o ws , columns}, and c r eates a diago nal matr ix w ith the main diagonal r eplaced w it[...]

Pa g e 1 0 - 1 4 F or ex ample, the f ollo wing command in AL G mode for the list {1,2 , 3, 4}: In RPN mode, ente r {1,2,3,4} ` V NDER MONDE . Function HIL BER T F uncti on HI LBER T cr eates the Hilbert matri x cor re sponding to a dimensi on n. B y def inition , the n × n Hilbert matr i x is H n = [h jk ] n × n , so that T he Hilbert matri x ha[...]

Pa g e 1 0 - 1 5 enter ed in the displa y as you perf or m those k ey str ok es . F irs t , we pr esent the steps ne cessar y to produce p r og r am C RMC. Lists r epr esent columns of the matri x Th e p r o gra m @CRMC allo w s yo u to put together a p × n matr i x (i .e., p r o w s, n columns) out of n lists of p elements each . T o cr eate the [...]

Pa g e 1 0 - 1 6 ~„n # n „´ @) MATRX! @ ) COL! @COL!  COL  ` Pr ogr am is display ed in le v el 1 To s a v e t h e p r o g r a m : ! ³~~crmc~ K T o see the contents o f the pr ogram u se J ‚ @CRMC . T he pr ogr am listing is the f ollo w ing: « DUP → n « 1 SWAP FOR j OBJ →→ RRY IF j n < TH EN j 1 + ROLL END NEX T IF n 1 >[...]

Pa g e 1 0 - 1 7 Lists r epr esent ro w s of the matrix T he pre vi ous pr ogram can be easil y modif ied to c r eate a matri x when the input lists w ill become the r o ws o f the r esulting matr i x. T he only ch ange to be perfor med is to c h ange COL → for ROW → in the pr ogr am listing. T o per f or m this c hange use: ‚ @CRMC L ist pr [...]

Pa g e 1 0 - 1 8 Both appr oa c hes w ill show the same func tions: When s y stem f lag 117 is set to S OFT menus , the COL menu is accessible thr ough „´ !) MATRX ) !)@@COL@ , or thr ough „Ø !) @CREAT@ ! ) @ @COL@ . Both appr oache s w ill sho w the same s et of f uncti ons: The op er ation of these functions is presented b elo w . Function [...]

Pa g e 1 0 - 1 9 In this r esult , the fir st column occ upi es the highe st stac k lev el af t er decompositi on , and stac k lev el 1 is occu pi ed by the n umber of co lumns of the or iginal matr ix . T he matr i x does not surv i v e decompositi on, i .e ., it is no longe r av ailable in the s tack . Function COL → Fu n c ti o n CO L → has [...]

Pa g e 1 0 - 2 0 In RPN mode , enter the matr i x fir st , then the v ector , and the column n umber , bef or e appl y ing fu nction C OL+. T he fi gur e belo w sho w s the RPN stac k bef or e and after appl y ing functi on COL+. Function COL - F uncti on COL - tak es as ar gument a matr ix and an intege r number r epr esenting the positi on of a c[...]

Pa g e 1 0 - 2 1 In RPN mode , functi on CS WP lets y ou s wap the columns of a matri x list ed in stac k le vel 3, who se indi ces ar e listed in s tac k lev els 1 and 2 . F or e x ample , the fo llo w ing fi gur e sho ws the RPN st ack bef or e and after a pply ing functi on CS WP to matr i x A in or der to s wap columns 2 and 3: As y ou can see [...]

Pa g e 1 0 - 22 When s y st em flag 117 is set to S OFT menus , the R O W menu is acces sible thr ough „´ !) MATRX ! )@@ROW@ , or thr ough „Ø !) @CREAT@ ! ) @@ ROW@ . Both appr oache s w ill sho w the same s et of f uncti ons: The op er ation of these functions is presented b elo w . Function → RO W Fu n c ti o n → R O W tak es as ar gume[...]

Pa g e 1 0 - 2 3 matr i x does not survi ve decompo sition , i .e ., it is no longer a vailable in the stack. Function RO W → Fu n c ti o n ROW → has the opposite e ffec t of the f unctio n → R O W , i .e ., gi v en n v ector s of the s ame length , and the number n , func tion R O W  builds a matri x b y plac ing the input v ectors as r o[...]

Pa g e 1 0 - 24 Function RO W- F uncti on RO W - tak es as ar gument a matri x and an integer number r epr esenting the positi on of a r o w in the matri x . The f unctio n re turns the or iginal matr i x , minu s a ro w , as w ell as the e xtr acted r o w sho w n as a v ector . H er e is an e x ample in the AL G mode using the matr i x stor ed in [...]

Pa g e 1 0 - 2 5 As y ou can see , the r ow s that or iginally occ up ied po sitions 2 and 3 ha ve been s wa pped. Function RCI F uncti on R CI stands f or m ultiply ing R ow I by a C onst ant value and r eplace the r esulting r o w at the same location . The f ollo wing e xample , wr itten in AL G mode , tak es the matr ix s tor ed in A, and multi[...]

Pa g e 1 0 - 26 In RPN mode , enter the matr i x fir st , fo llo w ed by the constant v alue , then b y the r o w to be multiplied b y the co nstant v alue , and f inally ente r the ro w that will be r eplaced. T he fo llo w ing fi gur e sho w s the RPN stac k befor e and af t er apply ing func tion R CIJ under the same conditi ons as in the AL G e[...]

P age 11-1 Chapter 11 M atr ix Oper ations and Linear Algebra In Chapte r 10 we intr oduced the concept of a matri x and pr esen ted a number of f uncti ons f or enter ing, c r eating, o r manipulating matri ces . In this Chapt er w e pr esen t ex a m ples of matr i x oper ations and appli cations t o problems of linear algebr a. Operations w ith m[...]

P age 11-2 Addition and subtr ac tion Consi der a pair of matr ices A = [a ij ] m × n and B = [b ij ] m × n . Addition and subtr action of the se tw o matri ces is onl y pos sible if they ha v e the same number of r ow s and columns . The r esulting matr i x , C = A ± B = [c ij ] m × n has elements c ij = a ij ± b ij . Some e xam ples in AL G [...]

P age 11-3 B y combining addition and subtr acti on w ith multiplicati on b y a scalar w e can fo rm linear combinati ons of matr ices of the same dimensions , e .g., In a linear combinati on of matr i ces, w e can multipl y a matr i x by an imaginary number to obtain a matri x of comple x n umbers, e .g ., M atr ix -vector multipli cation Matr i x[...]

P age 11-4 Matrix multiplication Matri x multipli cation is def ined b y C m × n = A m × p ⋅ B p × n , wher e A = [a ij ] m × p , B = [b ij ] p × n , and C = [ c ij ] m × n . Noti ce that matr i x multiplicati on is onl y possible if the number of columns in the f ir st oper and is equal to the number o f r o ws of the second oper and . T h[...]

P age 11-5 (another r o w vect or). F or the calculator to identify a r o w vector , y ou must us e double br ack ets to enter it: T erm-b y-term multiplication T erm-b y- t erm multiplicati on of two matr ice s of the same dimensions is pos sible thr ough the us e of f unction HAD AMARD . T he r esult is, o f course , another matri x of the sa me [...]

P age 11-6 In algebr aic mode , the k e y str ok es ar e: [enter or se lect the matr i x] Q [enter the po w er] ` . In RPN mode , the k ey str ok es ar e: [ent er or select the matr i x] † [ent er the po we r] Q` . Matri ces can be r aised to negati ve po wer s. In this cas e , the r esult is equi valent to 1/[matr i x]^AB S(po we r). The identit[...]

P age 11-7 T o v er if y the pr operties of the in v erse matr ix , consider the f ollo w ing multiplicati ons: Characteri zing a matr ix (T he matri x NORM menu) T he matri x NORM (NORMALIZE) menu is accesse d thr ough the k e ys tr ok e sequen ce „´ (s ys tem f lag 117 set t o CHOOSE bo xes): T his menu cont ains the fo llo w ing func tio ns: [...]

P age 11-8 Function ABS F uncti on ABS calc ulate s what is kno w n as the F r obenius nor m of a matr i x. F or a matr i x A = [a ij ] m × n , the F robeniu s norm of the matr ix is def ined as If the matr i x under consider ation in a r ow v ector or a column v ector , then the F r obe nius nor m , || A || F , is simply the v ector ’s magnitud[...]

P age 11-9 Functions RNRM and CNRM F uncti on RNRM r etur ns the Ro w NoRM o f a matr i x , whil e functi on CNRM r eturns the C olumn NoRM of a matr i x. Ex amples, Singular value decomposition T o unders tand the oper ation o f F uncti on SNRM, w e need to intr oduce the concept of matr i x decompositi on. Ba sicall y , matri x decompo sition in [...]

P age 11-10 Function SR AD F uncti on SRAD de termine s the Spectr al R ADius o f a matri x, de fined as the lar gest of the a bsolute v alues of its e igen v alues . F or e x ample , Function COND F uncti on COND deter mines the conditi on number of a matr ix : Definition of eigenv alues and eigenvectors of a matri x T he eigen v alues of a sq uar[...]

P age 11-11 T ry the follo wing ex erc ise f or matri x condition n umber on matr i x A3 3 . The conditi on number is COND( A3 3) , ro w norm , and column norm f or A3 3 ar e sho w n to the le ft . The cor r esponding numbers f or the in ver se matr i x, INV( A3 3) , ar e sho wn to the ri ght: Since RNRM(A3 3) > CNRM(A3 3) , then w e tak e ||A3 [...]

P age 11-12 F or ex ample , try finding the r ank for the matr i x: Y ou w ill f ind tha t the rank is 2 . That is becaus e the second r o w [2 , 4, 6] is equal to the f irs t r ow [1,2 , 3] multiplied b y 2 , thu s, r o w tw o is linearl y dependent of r o w 1 and the max imum number o f linearl y independent r o ws is 2 . Y ou can c heck that the[...]

P age 11-13 The determinant of a matr ix T he deter minant of a 2x2 and o r a 3x3 matr i x ar e r epr esen ted b y the same arr angement of elemen ts of the matr ices , but enc losed be t w een verti cal lines, i. e. , A 2 × 2 dete rminant is cal cul ated b y multiply ing the elemen ts in its diagonal and adding those pr oducts accompanied b y the[...]

P age 11-14 Function TR A CE F uncti on TRA CE calc ulates the tr ace of sq uare matr ix , def ined as the sum of the elements in its main diagonal , or . Ex amples: F or squar e matr i ces of hi gher or der de terminants can be calc ulated by u sing smaller or der deter minant called co fact ors . The gener al i dea is to "e xpand" a det[...]

P age 11-15 Function TR AN F uncti on TRAN re turns the tr anspo se of a r eal or the conj ugate tr anspo se of a comple x matri x. TRAN is equi v alent t o TRN. The oper ation of func tion TRN w as pr es ented in Cha pter 10. Additional matri x oper ations (T he matri x OPER menu) T he matri x OPER (OP ER A T IONS) is av ailable thr ough the k ey [...]

P age 11-16 MAD and RSD ar e relat ed to the soluti on of s y ste ms of linear equati ons and wil l be pr esen ted in a subseq uent secti on in this Chapt er . In this secti on w e ’ll disc us s only f uncti ons AXL and AXM. Function AXL F uncti on AXL con verts an arr a y (matri x) into a list , and v ice v ersa: Note : the latter oper ation is [...]

P age 11-17 T he implementati on of func tion L CXM fo r this case r equir es y ou to ente r: 2`3`‚ @@P1@@ LCXM ` T he follo w ing fi gur e show s the RPN s tack be fo r e and af t er appl y ing func tion LC X M : In AL G mode , this e x ample can be obtained b y using: T he progr am P1 must still ha ve been c r eated and stor ed in RPN mode . So[...]

P age 11-18 , , Using the num er ical solv er f or linear s ystems Ther e are man y w a ys to so lv e a s y stem of linear equatio ns w ith the calculator . One pos sibility is thr ough the numer i cal sol v er ‚Ï . F r om the numer ical sol v er scr een, sho w n belo w (left) , selec t the option 4. Sol v e lin s ys .., and pr ess @@@OK@@@ . Th[...]

P age 11-19 T his sy st em has the same number o f equations as o f unknow ns , and will be r efer r ed to as a sq uare s ys tem. In gener al, ther e should be a unique solu tion to the s y stem . T he soluti on w ill be the po int of inter secti on of the three planes in the coor dinate s ys tem (x 1 , x 2 , x 3 ) r epr esen ted b y the thr ee equ[...]

P age 11-20 T o chec k that the solu tion is cor r ect , enter the matr i x A and multiply times this solu tion v ector (e xample in algebr aic mode): Under-det ermined s ystem T he sy stem of li near equati ons 2x 1 + 3x 2 –5x 3 = -10, x 1 – 3x 2 + 8x 3 = 85, can be w ritten as the matri x equati on A ⋅ x = b , if T his sy stem has mor e unk[...]

P age 11-21 T o see the details of the so lutio n vect or , if needed, pr ess the @EDIT! butt on. T his w ill acti vat e the Matri x W r iter . W ithin this env ir onment , use the r ight- and left - arr o w k e y s to mo v e about the v ector : T hus , the solution is x = [15 . 3 7 3, 2 .46 2 6, 9 .6 2 6 8]. T o r eturn to the numer ical s olv er [...]

P age 11-2 2 Let ’ s store the latest r esult in a var iable X, and the matr i x into var iable A, as fo llo w s: Press K~x` to stor e the solution v ector into var iable X Press ƒ ƒ ƒ to clear thr ee lev els of the stac k Press K~a` to stor e the matri x into v ari able A No w , let’s v er ify the soluti on b y using: @@@A@@@ * @@@X@@@ ` , [...]

P age 11-2 3 can be w ritten as the matri x equati on A ⋅ x = b , if This s yst em has mor e equations than unkno w ns (an ov er -determined s yste m) . T he sy stem does not ha v e a single s oluti on. E ac h of the linear equations in the s y stem presented abo v e r epresen ts a s tr aight line in a two -dimensi onal Cartesi an coor dinate s y[...]

P age 11-2 4 Press ` to r eturn to the numer ical so lv er en v iro nment . T o check that the solu tion is corr ect , try the follo wing: • Press —— , to highlight the A: field . • Press L @CALC@ ` , to copy matri x A onto the stack . • Press @@@OK@@@ to r eturn to the n umer ical sol v er en vir onment . • Press ˜ ˜ @CALC@ ` , to co[...]

P age 11-2 5 • If A is a squar e matr i x and A is non- singul ar (i .e ., it’s in ver se matr i x e xis t , or its determinant is non - z er o) , LS Q r etur ns the ex act so lution to the linear s y stem . • If A has les s than full r ow r ank (u nde rde termined s y st em of equatio ns) , LS Q r eturns the solu tion w ith the minimum E ucl[...]

P age 11-2 6 Under-det ermined s ystem Consi der the s ys tem 2x 1 + 3x 2 –5x 3 = -10, x 1 – 3x 2 + 8x 3 = 85, wi th T he soluti on using LS Q is sho wn ne xt: Over-determin ed s ystem Consi der the s ys tem x 1 + 3x 2 = 15, 2x 1 – 5x 2 = 5, -x 1 + x 2 = 2 2 , wi th T he soluti on using LS Q is sho wn ne xt: . 85 10 , , 8 3 1 5 3 2 3 2 1 ⎥ [...]

P age 11-2 7 Compar e these thr ee solu tions w ith the ones calc ulated wi th the numer ical solv er . Solution with the in v erse matri x T he soluti on to the s ys tem A ⋅ x = b , wher e A is a squar e matri x is x = A -1 ⋅ b . T his re sults fr om multiply ing the f irst eq uation b y A -1 , i .e ., A -1 ⋅ A ⋅ x = A -1 ⋅ b . By def in[...]

Pa g e 1 1 - 2 8 T he pr ocedure f or the cas e of “ di v iding ” b by A is illustr ated belo w for the case 2x 1 + 3x 2 –5x 3 = 13, x 1 – 3x 2 + 8x 3 = -13, 2x 1 – 2x 2 + 4x 3 = -6 , The pr ocedu r e is sho wn in the follo wing s cr een shots: T he same solu tion as f ound abo ve w ith the in ver se matr i x . Solv ing multiple set of eq[...]

P age 11-29 [[14,9,- 2],[2,-5,2], [5,19,12]] ` [[1,2,3], [3,-2,1],[4,2 ,-1]] `/ T he re sult of this oper ation is: Gaussian and Gauss-Jordan elimination Gaus sian elimination is a pr ocedure b y w hic h the squar e matri x of coe ff ic ients belonging to a s y stem of n linear eq uations in n unkno wns is r educed to an upper - tr iangular matr i [...]

P age 11-30 T o start the pr ocess o f forw ar d elimination , w e di vi de the f irst equati on (E1) b y 2 , and s tor e it in E1, an d sho w the thr ee equ ation s again to pr oduce: Ne xt, w e r eplac e the s econd equation E2 b y (equ ati on 2 – 3 × equation 1, i . e ., E1-3 × E2) , and the thir d by (eq uation 3 – 4 × equation 1) , to g[...]

P age 11-31 an e xpre ssi on = 0. Thu s, the las t set of equati ons is interpr eted to be the fo llo w ing equiv alent set of equatio ns: X +2Y+3Z = 7 , Y+ Z = 3, - 7Z = -14. T he pr ocess o f back war d subs titution in Ga ussian e limination consis ts in finding the value s of the unkno wns , starting fr om the last equation and w or king up war[...]

P age 11-3 2 T o obtain a solution t o the sy stem matr i x equation us ing Gaussi an elimination , we f i r st c re a t e w h a t i s k n ow n a s t h e augmente d matr i x cor re sponding to A , i .e ., T he matri x A aug is the same as the or iginal matr i x A w ith a ne w r o w , cor re sponding to the elements o f the vec tor b , added (i.e .,[...]

P age 11-3 3 Multiply r o w 2 by –1/8: 8Y2 @RCI! Multiply r ow 2 b y 6 add it to ro w 3, r eplacing it: 6#2#3 @RCIJ! If y ou w er e perfor ming these oper ati ons by hand , y ou w ould wr ite the fo llo w ing: Th e symb ol ≅ ( “ is eq ui vale nt to ”) indicate s that what f ollo ws is equi valent to the pr e vi ou s matri x w ith so me r o[...]

P age 11-34 Multiply r o w 3 by –1/7 : 7Y 3 @ RCI! Multiply r ow 3 b y –1, add it to r o w 2 , r eplac ing it: 1 # 3 #2 @RCIJ! Multiply r ow 3 b y –3, add it to ro w 1, r eplacing it: 3#3#1 @RCIJ! Multiply r ow 2 b y –2 , add it to r ow 1, r e plac ing it: 2#2#1 @RCIJ! W r iting this pr ocess b y hand w ill r esult in the follo w ing s [...]

Pa g e 1 1 - 3 5 While perf orming p iv oting in a matr i x elimination pr ocedure , yo u can impro ve the numer i cal soluti on ev en mor e b y selecting a s the pi vo t the element w ith the lar gest ab solut e value in the column and r o w of inte r est . This oper ation ma y r equir e e xc h anging not onl y r o w s, but also columns , in some [...]

Pa g e 1 1 - 3 6 No w we are r eady to st ar t the Ga uss-Jor dan elimination w ith full p i vo ting . W e w ill need to k eep trac k of the permutati on matr ix b y hand , so tak e yo ur notebook and w r ite the P m a trix s h own ab ove. F i r st , w e chec k the pi vo t a 11 . W e notice that the element w ith the lar gest abs olute v alue in th[...]

P age 11-3 7 Hav ing f illed up w ith z er os the elements of column 1 below the p i v ot , now w e pr oceed to c heck the pi vot at po sition (2 ,2) . W e find that the number 3 in positi on (2 , 3) will be a better pi v ot , thus , w e ex change columns 2 and 3 b y using: 2#3 ‚N @ @@OK@ @ Chec king the pi v ot at positi on (2 ,2) , we no w find[...]

P age 11-38 2 Y #3#1 @RCIJ F i nall y , w e eliminate the –1/16 f r om positi on (1,2) by using: 16 Y # 2#1 @RCIJ W e no w hav e an identity matri x in the por ti on of the augmented matr i x cor re sponding to the or i ginal coeff ic ient matr i x A, thus w e can pr oceed to obtain the sol ution w hile accounting f or the ro w and column ex c h[...]

P age 11-3 9 T hen, f or this partic ular e x ample , in RPN mode , use: [2,-1,41] ` [[1,2,3 ],[2,0,3],[8 ,16,-1]] `/ T he calculat or sho ws an a ugmented matr i x consisting o f the coeff ic ients matr ix A and the iden tit y matr ix I , while , at the s ame time , sho w ing the ne xt pr ocedur e to ca lc ulate: L2 = L2 - 2 ⋅ L1 stands f or “[...]

P age 11-40 T o see the in ter mediate s teps in calc ulating and inv er se , j ust e nter the matr ix A fr om abov e, and pr ess Y , w hile keep ing the step-b y-st ep op ti on acti v e in the calc ulator’s CA S . Use the f ollo w ing: [[ 1,2,3],[ 3,-2,1],[4,2 ,-1]] `Y After go ing thr ough the diffe rent s teps , the soluti on r eturned is: Wha[...]

P age 11-41 T he r esult ( A -1 ) n × n = C n × n / det ( A n × n ) , is a gener al r esult that appli es to an y non -singular matr i x A . A gener al for m for the ele ments of C can be w r it te n based on the Gaus s-Jor dan algorithm . Based on the equation A -1 = C /det( A ) , sk etc hed abov e, the in v ers e matri x, A -1 , is not def ine[...]

P age 11-4 2 LINSOLVE([ X-2*Y+Z=-8,2 *X+Y-2*Z=6,5* X-2*Y+Z=-12], [X,Y,Z]) to pr oduce the s oluti on: [ X=-1,Y=2,Z = - 3]. F uncti on LINS OL VE w or ks wi th sy mb o lic e xpr es sions . F uncti ons REF , rr ef , and RREF , w ork w ith the au gmented matr i x in a Gaus sian eliminati on appr oach . Functions REF , rref , RREF T he upper tr iangula[...]

P age 11-4 3 T he diagonal matr i x that r esults f r om a Gaus s-Jor dan elimination is called a r o w-r educed echelon f or m. F unction RREF ( R ow-R educed E chelon F or m) The r esult of this functi on call is to pr oduce the r o w-r educed echelon f orm so that the matr i x of coeff ic ients is r educed to an identity matri x. T he e xtra col[...]

P age 11-44 T he re sult is the augmented matr i x corr esponding to the s yst em of equations: X+Y = 0 X- Y =2 Residual er rors in linear s ystem solutions (F unc tion RSD) F uncti on R SD calculate s the Re SiDuals or err ors in the soluti on of the matr i x equation A ⋅ x = b , repr esenting a s y stem o f n linear equations in n unkno w ns. W[...]

P age 11-45 Eigenv alues and eigenv ectors Gi v en a sq uar e matri x A , we can wr ite the eige nv alue equation A ⋅ x = λ⋅ x , w here the v alues of λ that satisfy the equation ar e know n as the ei gen value s of matr i x A . F or eac h value o f λ , we can f ind , fr om the same equation , values o f x that satisfy the e igen v alue equa[...]

Pa g e 1 1 - 4 6 Using the var iable λ to r epre sent e igen value s, this c har acter isti c pol ynomi al is to be interpr eted as λ 3 -2 λ 2 -2 2 λ +21=0. Function EG VL F uncti on E G VL (E iGenV aL ues) pr oduces the ei gen value s of a sq uar e matri x. F or e x ample , the ei gen value s of the matr ix sho wn belo w a r e calc ulated in A[...]

P age 11-4 7 of a matr i x , while the corr esponding ei gen values ar e the components of a vec tor . F or ex ample , in AL G mode , the ei gen vect ors and e igen v alues of the matr i x listed be lo w ar e found by a pply ing functi on E G V : T he re sult sho ws the e igen v alues as the columns of the matr i x in the re sult list . T o see the[...]

P age 11-48 • A list w ith the eigen v ect ors cor r es ponding to eac h ei gen v alue of matr i x A (stac k le ve l 2) • A v ector w ith the eige n vec tor s of matr i x A (st ack lev el 4) F or ex ample , try this ex erc ise in RPN mode: [[4,1,-2], [1,2,-1],[-2 ,-1,0]] JORD N T he output is the f ollo w ing: 4: ‘X^3+-6*x^2+2*X+8’ 3: ‘X^[...]

P age 11-4 9 Notice that the equati on ( x ⋅ I - A ) ⋅ p( x )=m ( x ) ⋅ I is similar , in f orm , to the ei gen value equati on A ⋅ x = λ⋅ x . As an e x ample , in RPN mode , try: [[4,1,-2] [ 1,2,-1][-2,- 1,0]] M D T he r esult is: 4: -8. 3: [[ 0.13 –0.2 5 –0.3 8][-0.25 0. 50 –0.2 5][-0.3 8 –0.2 5 –0.88]] 2: {[[1 0 0][0 1 0][0 [...]

P age 11-50 Function L U F uncti on L U tak es as input a s quar e matr ix A , and r eturns a lo wer - tr iangular matr i x L , an upper tr i angular matri x U , and a p e rmut ation matr i x P , in st ack le vels 3, 2 , and 1, re specti v el y . The r esult s L , U , and P , satisf y the equati on P ⋅ A = L ⋅ U . When y ou call t he L U func t[...]

P age 11-51 decompositi on, w hile the v ector s r epr esents the main diagonal of the matr i x S used earli er . F or ex ample, in RPN mode: [[ 5,4,-1],[2,- 3,5],[7,2,8] ] SVD 3: [[-0.2 7 0.81 –0. 5 3][-0. 3 7 –0. 5 9 –0.7 2][-0.8 9 3 . 09E -3 0.46]] 2 : [[ -0.68 –0.14 –0.7 2][ 0.4 2 0.7 3 –0. 54][-0.6 0 0.6 7 0.4 4]] 1: [ 12 .15 6 .88[...]

Pa g e 1 1 - 52 Function QR In RPN, f unction QR produces the QR fa ctoriz a tio n of a ma tr ix A n × m r eturning a Q n × n orthogonal matri x , a R n × m upper tr apez oi dal matri x, and a P m × m permu tation matr i x, in stac k le vels 3, 2 , and 1. The matr ices A , P , Q and R are re la t ed by A ⋅ P = Q ⋅ R . F or ex ample, [[ 1,-2[...]

Pa g e 1 1 - 5 3 T his menu includes f uncti ons AXQ, CHOLE SKY , G A US S, QX A, and S YL VE S TER. Function AX Q In RPN mode , f unction AXQ pr oduces the quadr ati c f orm cor r esponding to a matr i x A n × n in stac k le ve l 2 using the n var iable s in a vec tor placed in stac k le vel 1. F uncti on r eturns the quadr atic f orm in stac k l[...]

P age 11-54 suc h that x = P ⋅ y , b y using Q = x ⋅ A ⋅ x T = ( P ⋅ y ) ⋅ A ⋅ ( P ⋅ y ) T = y ⋅ ( P T ⋅ A ⋅ P ) ⋅ y T = y ⋅ D ⋅ y T . Function S YL VE STER F uncti on S YL V E S TER tak es as ar gument a s y mmetr ic s quar e matr ix A and r eturns a v ector cont aining the diagonal te rms of a diagonal matr i x D , and a[...]

Pa g e 1 1 - 5 5 Inf ormati on on the func tions list ed in this menu is pr esen ted belo w b y using the calc ulator ’s o w n help fac ility . The f igur es sho w the he lp fac ility entry and the attached e xamples . Function IMAGE Function ISOM[...]

P age 11-5 6 Function KER Function MKISOM[...]

Pa g e 1 2 - 1 Chapter 12 Gr a phi cs In this c hapter w e intr oduce some o f the gr aphic s capabiliti es of the calculat or . W e w ill pr esent gr aphics of f uncti ons in Cartesian coor dinates and polar coor dinates , par ametr ic plots , gr aphic s of coni cs , bar plots, scatter plots , and a v ari ety of thr ee -dimensi onal gr aphs . Grap[...]

Pa g e 1 2 - 2 T hese gr aph opti ons ar e desc ri bed bri ef ly ne xt . Fu n c ti o n : f or equations o f the for m y = f(x) in plane Cartesi an coordinates P olar : for equati ons of the fr om r = f( θ ) in polar coordinat es in the plane Pa r a m e t r i c : for plotting equati ons of the fo rm x = x(t) , y = y(t) in the plane Diff E q : f or [...]

Pa g e 1 2 - 3 Θ Ente r the PL O T en v ir onment b y pr es sing „ñ (pr ess them simultaneou sly if in RPN mode). Pr ess @ADD to get y ou into the equati on w riter . Y ou will be pr ompted to f ill the r ight-hand side of an equati on Y1(x) =  . T ype the f unction t o be plotted so that the E quatio n W rit er sho ws the follo wing: Θ Pre[...]

Pa g e 1 2 - 4 Θ Enter the P L O T WINDO W env ironme nt by enter ing „ò (pr ess them simultaneousl y if in RPN mode). Us e a range of –4 to 4 f or H- VI EW , then p r ess @AUT O to generate the V - VIEW automaticall y . The PL O T WINDO W sc r een looks as f ollo ws: Θ Pl ot t h e g ra ph : @ERASE @ DRAW ( wait till the calc ulator f inishe[...]

Pa g e 1 2 - 5 Some useful PL O T operations f or FUNCTION plots In or der to disc u ss these P L O T options , w e'll modif y the func tion to f or ce it to hav e some r eal r oots (Since the cur r ent curve is totall y contained abov e the x ax is, it has no r e al r oots.) Pr ess ‚ @@@Y1@@ to list the contents of the functi on Y1 on the s[...]

Pa g e 1 2 - 6 R OO T : 1.66 3 5 ... The calc ulator indicated , bef or e sho w ing the r oot , that it w as found thr ough SIGN REVER S A L . Press L to r ecov er the menu . Θ Pr es sing @ISE CT w ill gi ve y ou the int ersecti on of the c urve w ith the x -ax is, w hic h is esse ntiall y the roo t . Place the c urs or e xac tly at the r oot and [...]

Pa g e 1 2 - 7 Θ Enter the PL O T env ir onment by pr essing , simultaneously if in RPN mode , „ñ . Noti ce that the highli ghted fi eld in the PL O T en vir onment no w contain s the der i vati ve of Y1(X) . Pr ess L @@@OK@@@ to r eturn to return to nor mal calculat or displa y . Θ Press ‚ @@EQ@@ to chec k the contents of E Q. Y ou w ill no[...]

Pa g e 1 2 - 8 T o r etur n to nor mal calc ulator f uncti on , pr ess @) PICT @ CANCL . Graphics of tr anscendental functions In this secti on w e us e some of the gr aphics f eatur es of the calc ulator to sho w the typi cal beha vi or of the natur al log, e xponential , tri gonometr ic and h yper bolic func tions . Y ou w ill not see mor e gr ap[...]

Pa g e 1 2 - 9 10 by u si n g 1 @@@OK@@ 10 @@@OK@@@ . Ne xt , pr ess the s oft k e y labeled @AUTO to let the calc ulator det ermine the cor r esponding v ertical r ange . After a cou ple of seconds this r ange w ill be sho wn in the P L O T WINDOW -FUNCTION w indo w . At this po int w e ar e r eady to pr oduce the graph of ln(X) . Pre ss @ERASE @[...]

Pa g e 1 2 - 1 0 Graph of the e xponential function F irst , load the f uncti on e xp(X) , b y pr essing , simultaneous ly if in RPN mode , the left-shif t k e y „ and the ñ ( V ) k e y to acces s the PL O T -FUNCTI ON w indo w . Pres s @@DEL@ @ to r emo v e the func tion LN(X), if y ou didn ’t delete Y1 a s suggest ed in the pre vi ous n ote [...]

Pa g e 1 2 - 1 1 T he PP AR var iable Press J to r ecov er y our v ari ables menu , if needed. In y our v ari ables menu y ou should ha v e a v ar iable labe led PP AR . Pr ess ‚ @ PPAR to get the contents of this v ariable in the stac k . Pres s the do wn-arr o w k ey , , to launch the s tack editor , and u se the up- and do w n -ar ro w k e ys [...]

Pa g e 1 2 - 1 2 As indicated ear lier , the l n(x) and exp(x) f uncti ons ar e in v ers e of each othe r , i .e ., ln(e xp(x)) = x , and e xp(ln(x)) = x. T his can be v er if ied in the calc ulator b y typing and e v aluating the follo wi ng expr essi ons in the Eq uation W rit er: LN(EXP(X)) and EXP(LN( X)) . T he y should both ev aluate to X. Wh[...]

Pa g e 1 2 - 1 3 Summary of FUNCT I ON plot oper ation In this secti on w e pr esent inf ormati on r egar ding the PL O T SETUP , P L O T- FUNCT ION, and P L O T WINDO W sc r eens accessible thr ough the left-shif t k ey comb ined w ith the soft-menu k e y s A thr ough D . Based on the gr aphing e x amples pr esented abo ve , the procedur e to fo l[...]

Pa g e 1 2 - 1 4 Θ Use @CANCL t o cancel an y c hanges to the P L O T SE TUP w indo w and r eturn to nor mal calc ulator dis play . Θ Press @@@OK@@@ to sav e changes to the options in the P L O T SE TUP w indo w an d r etur n to normal calc ulator displa y . „ñ , simultaneousl y if in RP N mode: Ac cess to the PL O T windo w (in this case it w[...]

Pa g e 1 2 - 1 5 Θ Ente r lo w er and uppe r limits fo r hor i z ont al v ie w (H- V ie w) , and pr ess @AUTO , w hile the c urso r is in one of the V - Vi e w f ields , to ge ner ate the v ertical v i e w (V - Vie w) range automaticall y . Or , Θ Enter lo we r and upper limits f or v ertical v ie w (V- Vi ew), and pr ess @AUTO , w hile the c urs[...]

Pa g e 1 2 - 1 6 „ó , simult aneousl y if in RPN mode: Plots the gr aph based on the setting s stor ed in var ia ble PP AR and the cur r ent f unctions de fined in the PL O T – FUNCT ION scr een. I f a gr aph, diff er en t fr om the one y ou ar e plotting , alr eady e xis ts in the graphi c display s cr een, the ne w plot w ill be superimpo se[...]

Pa g e 1 2 - 1 7 Generating a table of v alues for a function T he combinati ons „õ ( E ) and „ö ( F ), pressed simultaneousl y if in RPN mode , let’s the us er pr oduce a table o f values o f functi ons . F or e x ample , w e wi ll pr oduce a table of the f unction Y(X) = X/(X+10), in the r ange -5 < X < 5 f ollo w ing thes e instruc[...]

Pa g e 1 2 - 1 8 the corr esponding value s of f(x) , listed as Y1 b y de fault . Y ou can use the up and do wn ar r o w k ey s to mov e about in the t able . Y ou w ill notice that w e did not ha ve to indicate an ending value f or the independent v ar iable x . Th us, the table co ntinues be y ond the max imum v alue fo r x suggested earl y , nam[...]

Pa g e 1 2 - 1 9 W e w ill tr y to plot the f uncti on f( θ ) = 2(1-sin( θ )), as follo w s: Θ F irs t , mak e sur e that y our calc ulator ’s angle measur e is set t o r adians. Θ Press „ô , simultaneousl y if in RPN mode , to access to the PL O T SE TUP wi ndo w . Θ Chan ge TYPE to Polar , by pr essing @CHOOS ˜ @@@OK@@@ . Θ Press ˜ a[...]

Pa g e 1 2 - 2 0 Θ Press L @ CANCL to ret urn to th e PL O T WI ND OW s cr e en. Press L @@@OK@@@ to r etur n to normal calc ulator displa y . In this e xe r c ise w e enter ed the eq uation to be plotted dir ectl y in the PL O T SETUP w indo w . W e can also enter equati ons f or plotting using the P L O T wi ndow , i .e., simultaneou sly if in R[...]

Pa g e 1 2 - 2 1 T he calculator ha s the ability of plotting one or more coni c c ur v es b y selecting Con ic as the functi on TYPE in the PL O T e nv ir onment . Mak e sure to dele te the var iables P P AR and E Q bef or e continuing . F or e x ample , let's sto r e the list o f equations { ‘(X-1)^2+(Y - 2)^2=3’ , ‘X^2/4+Y^2/3=1’ } [...]

Pa g e 1 2 - 2 2 Θ T o see labels: @EDI T L @) LABEL @MENU Θ T o r eco ver the men u: LL @) PICT Θ T o es timate the coor dinates of the po int of inter secti on, pr ess the @ ( X,Y ) @ menu k ey and mo v e the cur sor as c lose as po ssible to thos e points using the arr ow k ey s . The coor dinates of the c ursor ar e show n in the display . F[...]

Pa g e 1 2 - 23 whi ch in vol ve constant values x 0 , y 0 , v 0 , and θ 0 , we need to st ore the v alues of those par ameters in v ar iables . T o de velop this e xample , cr eate a sub-dir ect or y called ‘PR O JM’ fo r PR O Jectile Motion , and w ithin that sub-dir ectory stor e the fo llo w ing var iable s: X0 = 0, Y0 = 10, V0 = 10 , θ 0[...]

Pa g e 1 2 - 24 Θ Press @AUTO . This w ill gener ate autom ati c values of the H-V ie w and V - Vi e w r anges based on the v alues of the independent var iable t and the def initi ons of X(t) and Y(t) u sed . The r esult w ill be: Θ Press @ERASE @DR AW to dr a w the paramet ri c plot . Θ Press @EDIT L @LABEL @ MENU to s ee the gr aph w ith labe[...]

Pa g e 1 2 - 2 5 par ameters . The other v ar iables contain the v a lues o f constants us ed in the def initions o f X(t) and Y(t) . Y ou can stor e differ ent v alues in the v ari ables and pr oduce ne w par ametr ic plots of the pr o jectile eq uations us ed in this e xample . If you w ant to er as e the c urr en t pic tur e contents bef ore pr [...]

Pa g e 1 2 - 26 P lotting th e solution to simple differ ential equations T he plot of a simple differ ential equati on can be obtained by selec ting Diff Eq in the TYPE f ield o f the PL O T SETUP en v ir onment as f ollo ws: suppo se that w e w ant to plot x(t) fr om the diff er ential equati on dx/dt = exp(-t 2 ), w it h i ni ti al conditi ons: [...]

Pa g e 1 2 - 27 Θ Press L to reco v er th e menu . Pr ess L @) PICT to reco ver the or igin al gr aphics menu . Θ When w e obs erved the gr aph being plo tted, y ou'll noti ce that the gr aph is not v ery smooth . T hat is becaus e the plotter is using a time step that is too lar ge . T o r ef ine the gr aph and mak e it smoother , use a st [...]

Pa g e 1 2 - 28 T ruth plots T ruth plots ar e used to pr oduce two -dimensi onal plots of r egio ns that satisfy a certain mathemati cal condition that can be e ither true or f alse . F or ex a m ple , suppo se that y ou w ant to pl ot the regi on f or X^2/3 6 + Y^2/9 < 1, pr oceed as fo llo w s: Θ Press „ô , simultaneou sly if in RPN mode [...]

Pa g e 1 2 - 2 9 Θ Press „ô , simultaneou sly if in RPN mode , to access t o the PL O T SETUP wi n dow . Θ Press ˜ and ty pe ‘(X^2/3 6+Y^2/9 < 1) ⋅ (X^2/16+Y^2/9 > 1)’ @@@OK@@@ to def ine the conditions t o be plot t ed. Θ Press @ERASE @DRAW t o dra w the tr uth plot . Again , y ou ha v e to be patient w hile the calc ulato r pro[...]

Pa g e 1 2 - 3 0 [4. 5,5 .6, 4.4 ],[4.9 , 3.8 ,5 .5],[5 .2 ,2 .2 , 6.6]] ` to stor e it in Σ D A T , use the f uncti on S T O Σ (av ailable in the functi on catalog, ‚N ) . Pr ess V AR to reco v er y our var iable s menu . A soft menu k ey labeled Σ D A T should be a v ailable in the stac k. T he f igur e belo w sho w s the stor age of this ma[...]

Pa g e 1 2 - 3 1 accommodate the max imum v alue in column 1 of Σ D A T . Bar plots ar e usef ul when plotting categori cal (i .e ., non -numeri cal) data. Suppo se that y ou w ant to plot the data in co lumn 2 o f the Σ DA T m a t rix : Θ Press „ô , simultaneou sly if in RPN mode , to access t o the PL O T SETUP wi n dow . Θ Press ˜˜ to h[...]

Pa g e 1 2 - 32 Θ Press @ERASE @ DRAW to dr aw the bar plot . Pre ss @EDIT L @LA BEL @MENU to see the plot unenc umber ed b y the menu and w ith identify ing la bels (the c ursor w ill be in the middle of the plot , ho w e ver ) : Θ Press LL @) PICT to l eav e the EDI T en v ir onment . Θ Press @CANCL to r eturn to the PL O T W INDO W env ir onm[...]

Pa g e 1 2 - 3 3 Slope fields Slope fi elds ar e used to v isuali z e the solutio ns to a differ ential equati on of the fo rm y’ = f(x ,y) . Basi call y , what is pres ented in the plot ar e segmen ts tangenti al to the so lution c ur v es, since y’ = dy/dx , ev aluated at an y po int (x,y), repr esents the slope of the tangent line at point ([...]

Pa g e 1 2 - 3 4 of y(x ,y) = constant , for the soluti on of y ’ = f(x ,y) . Th us, slope f ie lds are u sef ul tools f or v isuali zing par ti c ularl y diffi cult equations t o sol v e . T ry als o a slope fi eld plot for the f uncti on y’ = f(x ,y) = - (y/x) 2 , by u sing: Θ Press „ô , simultaneou sly if in RPN mode , to access t o the [...]

Pa g e 1 2 - 3 5 Θ Press @ERASE @ DRAW to dr a w the thr ee -dimensional surf ace . The r esult is a w ir ef rame p ictur e of the surface w ith the re fer ence coor dinate sy stem sho w n at the lo w er left corner of the s cr e e n. B y using the arr ow k ey s ( š™— ˜ ) y ou can change the or ient ation of the surf ace. T he ori entati on [...]

Pa g e 1 2 - 3 6 Θ Press „ô , simul taneousl y if in RPN mode , to access the P L O T SETUP wi n dow . Θ Press ˜ and type ‘S IN(X^2+Y^2) ’ @@@OK@@@ . Θ Press @ERASE @DR AW to dr aw the plot . Θ When done , pres s @ EXIT . Θ Press @CANCL to r etur n to P L O T W INDO W . Θ Press $ , or L @@@OK@@@ , to retur n to normal calculator dis p[...]

Pa g e 1 2 - 37 Θ Press @EDIT L @LABEL @ MENU to s ee the gr aph w ith labels and r anges . This partic ular v ersi on of the gr aph is limited to the lo wer part of the dis play . W e can change the v ie wpoint to see a differ ent versi on of the gr aph. Θ Press LL @) PICT @CANCL to r eturn to the PL O T WINDO W env ir onment . Θ Change the e y[...]

Pa g e 1 2 - 3 8 T ry also a Wir ef r ame plot f or the surface z = f(x,y) = x 2 +y 2 Θ Press „ô , simul taneousl y if in RPN mode , to access the P L O T SETUP wi n dow . Θ Press ˜ and t y pe ‘X^2+Y^2’ @@@OK@@@ . Θ Press @ERASE @DRAW to dr aw the slope fie ld plot . Pr ess @ED IT L @) MENU @LABEL to see the plot unenc umb e red b y the [...]

Pa g e 1 2 - 3 9 Θ Press @EDIT ! L @ LABEL @MENU to see the gr aph w ith labels and r anges. Θ Press LL @) PICT@CANCL to retur n to the PL O T WINDOW en v ir onment . Θ Press $ , or L @@@OK@@@ , to retur n to normal calculator dis play . T ry als o a P s-Conto ur plot for the surf ace z = f(x,y) = sin x cos y . Θ Press „ô , simul taneousl y [...]

Pa g e 1 2 - 4 0 Θ Mak e sur e that ‘X’ is s elected as the Indep: and ‘ Y’ as the Depnd: variab le s. Θ Press L @@@OK@@@ to r eturn to nor mal calculat or displa y . Θ Press „ò , simultaneou sl y if in RPN mode , to acce ss the P L O T WINDO W sc r een. Θ Chan ge the def ault plot w indow r anges t o r ead: X-L eft:-1, X-Ri ght:1, Y[...]

Pa g e 1 2 - 4 1 Θ Press „ô , sim ultaneo usl y if in RPN mode , to acces s to the P L O T SETUP w indow . Θ Cha ng e TYPE to Gr idma p . Θ Press ˜ and t y pe ‘SIN(X+i*Y )’ @@@OK@@@ . Θ Mak e sur e that ‘X’ is s elected as the Indep: and ‘ Y’ as the Depnd: variab le s. Θ Press L @@@OK@@@ to r eturn to nor mal calculat or displa[...]

Pa g e 1 2 - 4 2 F or ex ample, t o pr oduce a Pr- Surface plot f or the surface x = x(X,Y ) = X sin Y , y = y(X,Y) = x cos Y , z=z(X,Y)=X, us e the fo llo w ing: Θ Press „ô , sim ultaneo usl y if in RPN mode , to acces s to the P L O T SETUP w indow . Θ Cha ng e TYPE to Pr - Surface . Θ Press ˜ and t y pe ‘{X*SIN( Y), X*COS( Y), X}’ @ @[...]

Pa g e 1 2 - 4 3 Inter ac ti ve dr a wing Whene v er w e pr oduce a tw o -dimensi onal gr aph , w e f ind in the gr aphic s sc r een a so ft men u k e y labe led @) EDIT . Pr essing @) EDIT pr oduces a menu that inc lude the fo llo w ing options (pr ess L to see additi onal func tions): T hro ugh the ex amples abo v e , yo u hav e the opportunit y [...]

Pa g e 1 2 - 4 4 Ne xt , we illus tr ate the use o f the differ ent dr a w ing functi ons on the r esulting gr aphi cs sc r een . The y req uir e use of the c ursor and the ar r o w k ey s ( š™— ˜ ) to mo v e the c ursor about the gr aphic s scr een. DO T+ and DO T - When DO T+ is selec ted , pi xels w ill be ac ti vat ed wher ev er the c urs[...]

Pa g e 1 2 - 4 5 should ha ve a s tr aight angle tr aced b y a hori z ontal and a v ertical segme nts. T he cur sor is still acti ve . T o deacti vat e it , w ithout mov ing it at all, pr ess @LINE . T he cu rsor r eturns to its n ormal sha pe (a cr o ss) and the LINE func tion is no longer acti ve . TLINE (T oggle LINE) Mo v e the cur sor to the s[...]

Pa g e 1 2 - 4 6 DEL T his command is u sed to r emov e parts of the gr aph betw een two MARK positi ons. Mo v e the cur sor to a po int in the gr aph, and pr ess @MARK . Mov e the c ursor t o a diff er ent point , pres s @M ARK again . Then , pr ess @@DEL@ . The s ection o f the gr aph bo x ed between the tw o marks w i ll be de leted. ERASE T he [...]

Pa g e 1 2 - 47 X,Y  T his command copies the coor dinates o f the cur r ent cur sor positi on, in us er coor dinates , in the stac k . Z ooming in and out in th e gr aphics display Whene v er y ou pr oduce a tw o -dimensi onal FUNCTION gr aphi c inter acti ve ly , the f irst s oft-menu k e y , labeled @) ZOOM , lets yo u access func tions that [...]

Pa g e 1 2 - 4 8 Y ou can alw a ys r etu r n to the v er y last z oom wi ndow b y u sing @ ZLAST . BO XZ Z ooming in and out of a gi v en gr aph can be pe rfor med by u sing the soft-menu k ey B O XZ . W ith BO XZ you s elect the re ctangular s ector (the “bo x”) that y ou want to z oom in into . Mo v e the curs or to one of the corners of the [...]

Pa g e 1 2 - 4 9 c ursor at the cent er of the sc reen , the w indo w gets z oomed so that the x -ax is e xtends fr om –64. 5 to 6 5 . 5 . ZSQR Z ooms the gra ph so that the plotting scale is maintained at 1:1 b y adjus ting the x scale , keep ing the y scale f i xe d, if the w indow is w ider than tall er . This f or ces a pr oportional z ooming[...]

Pa g e 1 2 - 5 0 S OL VER.. „Î (the 7 key ) C h. 6 TRIGONO ME TRIC. . ‚Ñ (the 8 key ) Ch. 5 EXP &LN.. „Ð (the 8 key ) C h. 5 T he S YMB/GRAPH menu T he GR AP H su b-menu w ithin the S YMB menu inc ludes the f ollo w ing f unctions: DEFINE: same as the k ey stro k e sequence „à (the 2 key ) GR OB ADD: paste s two GROB s fir st o v er[...]

Pa g e 1 2 - 5 1 T AB V AL(X^2 -1,{1, 3}) pr oduces a list of {min max} v alues o f the functi on in the interv al {1, 3}, w hile SIGNT AB(X^2 -1) show s the sign o f the func tion in the interv al (- ∞ ,+) , w ith f(x) > 0 in (- ∞ ,-1) , f(x) <0, in (-1,1) , and f(x) > 0 in (1,+ ∞ ). T AB V AR(LN(X)/X) pr oduces the f ollo w ing tab[...]

Pa g e 1 2 - 52 of F . T he question marks indicates uncer tainty or non-d ef inition. F or ex ample, fo r X<0, LN(X) is not defined , thu s the X lines sho ws a que stion mar k in that interv al. R ight at z er o (0+0) F is inf inite , for X = e , F = 1/e . F incr eas es bef or e r eaching this v alue , as indi cated by the u p war d ar r o w ,[...]

P age 13-1 Chapter 13 Calculus Applications In this Chapte r we dis cu ss appli cations of the calc ulator ’s functi ons to oper ations r elated to Calc ulus, e .g., limits , der i vati v es , integr als, po we r ser ies , etc. T he CAL C (Calc ulus) menu Man y of the func tions pr esented in this Chapte r ar e contained in the calc ulator ’s C[...]

P age 13-2 Function lim T he calculat or pr ov ides f uncti on lim t o c a l cu l at e l i m i t s of fu n c t io n s. Th i s f un c t io n use s as input an e xpre ssi on re pr esenting a func tion and the v alue wher e the limit is to be calc ulated. F unction lim is av ailable thr ough the command catalog ( ‚N~„l ) or thr ough opti on 2 . LI[...]

P age 13-3 T o calc ulate one -sided limits, add +0 or -0 t o the value to the v ari able . A “+0” means limit fr om the ri ght , w hile a “-0” means limit fr om the left . F or ex ampl e , the limit of as x appr oache s 1 fr om the le ft can be determined with the fo llo w ing k ey str ok es (AL G mode): ‚N~„l˜ $OK$ R!ÜX- 1™@íX@Å[...]

P age 13-4 in AL G mode . Re call that in RPN mode the ar guments must be e nter ed bef ore the func tion is appli ed. T he DERIV&INTEG menu T he functi ons a vailable in this sub-me nu ar e listed be low : Out of the se func tions DERIV and DER VX ar e used f or deri vati v es. The other func tions inc lude functi ons r elated to anti-der i va[...]

P age 13-5 be differ entiated . T hus , to calc ulate the deri vati v e d(sin(r ) ,r ) , us e , in AL G mode: ‚¿~„r„ÜS~„r` In RPN mode , this expr essi on must be enc los ed in quot es befo r e enter ing it into t he sta ck. Th e r e su lt in AL G mo de i s: In the E quati on W r iter , w hen y ou pr es s ‚¿ , the calc ulator pr ov ide[...]

P age 13-6 T o e valuate the der iv ati v e in the E quation W r iter , pr es s the up-arr o w k e y — , fo ur times, t o selec t the entir e e xpr essi on , then, pr ess @ EVAL . The der i vati ve w ill be e valuated in the E quation W riter as: T he c hain r ule T he chain rule f or der i vati ves appli es to der i vati ve s of composite f unct[...]

P age 13-7 Deri v ativ es of equations Y ou can use the calc ulator to calc ulate der i v ativ es o f equations , i .e ., e xpr essi ons in w hic h deri vati v es w ill ex ist in both sides o f the equal sign. S ome e xample s ar e sho wn belo w: Notice that in the e xpr es sions w her e the deri v ati ve si gn ( ∂ ) or function DERIV w as used ,[...]

P age 13-8 Analyzing gr aphics of func tions In Chapter 11 w e pre sented some f unctions that ar e av ailable in the graphic s sc r een f or anal yzing gr aphi cs of func tions of the f orm y = f(x). The se fu nctio ns inc lude (X,Y) and TR A CE f or determining po ints on the gr aph , as w ell as func tions in the Z OOM and FCN menu . The f uncti[...]

P age 13-9 Θ Press L @PICT @CANCL $ to r eturn to normal calc ulator displa y . Notice that the slope and tangent line that y ou r eques ted ar e listed in the stac k . Function DOMAIN F uncti on DOMAIN , av ailable thr ough the command catalog ( ‚N ), pr o v ides the domain of def inition of a func tion as a list of numbers and spec if icati on[...]

P age 13-10 T his re sult indicat es that the r ange of the f uncti on cor r esponding to the domain D = { -1,5 } is R = . Function SIGNT AB F uncti on SIGNT AB, a v ailable thr ough the command catalog ( ‚N ), pro v ides inf orma tion on th e sign of a function th r o ugh it s domai n . F or ex a mple , fo r the T AN(X) func tion , SIGNT AB indi[...]

P age 13-11 Θ Le v el 3: the f uncti on f(VX) Θ T w o lists, the f irs t one indicate s the var iati on of the f unction (i .e., w her e it inc reas es or dec reas es) in ter ms o f the independent var iable VX, the second one indicate s the var iati on of the f uncti on in term s of the dependent v a r iable . Θ A gr aphi c obj ect sh ow ing ho[...]

P age 13-12 The interpr etation of the v ariati on table show n abov e is as follo ws: the functi on F(X) incr eases f or X in the int erval (- ∞ , -1), reac hing a max imum equal to 3 6 at X = -1. Then , F(X) decr eas es until X = 11/3, r eaching a minimum of –4 00/2 7 . After that F(X) incr eases until r eac hing + ∞. Al so , at X = ±∞ ,[...]

P age 13-13 W e fi nd two c r itical po ints, one at x = 11/3 and one at x = -1. T o ev aluate the second der i vati ve at eac h point use: T he last s cr een show s that f”(11/3) = 14 , thus , x = 11/3 is a r elati v e minimum . F or x = -1, we ha ve the f ollow ing: T his r esult indi cate s that f ”(-1) = -14 , th us , x = -1 is a r elati v [...]

P age 13-14 Anti-deri v ati ves and integr als An anti-der iv ati ve o f a func tion f(x) is a func tion F(x) su ch that f(x) = dF/dx . F or e x ample , since d(x 3 ) /dx = 3x 2 , an anti-der i v ati ve o f f(x) = 3x 2 is F(x) = x 3 + C, w here C is a constant . One w ay to r ep r esent a n anti-der i vati ve is as a indefinite inte gr al , i .e .,[...]

P age 13-15 abo v e . The ir re sult is the so -called discr ete der i vati ve , i .e . , one de fined f or integer n umbers onl y . Definite integr als In a def inite integr al of a f uncti on, the r esulting anti-der i vati ve is e valuated at the upper and lo wer limit o f an int erval (a ,b) and the ev a l uated value s subtr acted . S y mbolic[...]

P age 13-16 T his is the gener al for mat for the de finit e integral w hen typed dir ectly into the stac k, i .e ., ∫ (lo w er limit , upper limit , in tegr and , var iable of in tegr ation) Pr es sing ` at this point w ill ev aluate the integral in the st ack: T he integral can be e valuated also in the E quation W rite r by s electing the enti[...]

P age 13-17 T he follo w ing ex ample sh o ws the e v aluation of a defi nite integr al in the E quation W riter , step-b y-step: ʳʳʳʳʳ Notice that the st ep-by-s tep pr ocess pr ov ide s infor mation on the inter mediate step s follo wed b y the CAS to solv e this integr al . F irst , CAS ide ntif ies a squar e r oot integr al, ne xt, a r ati[...]

P age 13-18 T ec hniques o f integr ation Se v er al techni ques of int egr ation can be im plemented in the calc ulators , as sho w n in the f ollo w ing e x amples . Substitution or chang e o f var iables Suppose w e want to calc ulate the integr al . If w e us e step-by- step calc ulatio n in the Eq uation W rit er , this is the sequence of v ar[...]

P age 13-19 Integration b y par ts and differentials A differ ential o f a functi on y = f(x) , is de fined a s dy = f’(x) dx , w her e f’(x) is the der i vati v e of f(x). Differ enti als ar e used to r epr esen t small incr ements in the var iables . The diff er ential o f a pr oduct of tw o functi ons , y = u(x)v(x) , is gi v en by dy = u(x)[...]

P age 13-20 Integration b y par tial fr actions F unction P A R TFRA C, pr esented in Chap te r 5, pr ov ides the decomposition of a fr action int o par ti al fr acti ons. T his techni que is us eful t o r educe a complicated fr action into a sum of simple f r actio ns that can then be integrated t erm b y ter m. F or ex ample , to integrate w e ca[...]

P age 13-21 Using the calc ulator , w e pr oceed as f ollo ws: Alternati ve ly , y ou can ev aluate the i n tegra l to inf inity fr om the start, e .g ., Integr ation with units An integr al can be calculated w ith units incorpor ated into the limits of integr ation , as in the e x ample sho w n belo w that uses AL G mode , w ith the CAS set to A p[...]

P age 13-2 2 Some n otes in the u se of units in the limits of int egrati ons: 1 – T he units of the low er limit of integr ation w i ll be the ones u sed in the f inal r esult , as illu str ated in the tw o e x amples belo w : 2 - Upper limit units mu st be consisten t w ith low er limit units. Otherw ise , the calc ulator sim ply r eturns the u[...]

Pa g e 1 3 - 23 T a ylor and Mac laur in’s series A fu nction f( x) can be expanded in to an inf inite ser ie s ar ound a point x=x 0 by using a T a y lor’s ser ie s, namel y , , wher e f (n) (x) repr esen ts the n - th der i vati ve of f(x) w ith respect to x , f (0) (x) = f(x) . If the v alue x 0 is z er o , the s er ies is r ef err ed to as [...]

P age 13-2 4 wher e ξ is a n umber near x = x 0 . Since ξ is ty pi cally unkn o wn , inst ead of an estimat e of the r esidual , w e pr ov ide an es timate of the or der of the r esi dual in re fe ren c e t o h, i. e. , we s ay t h a t R k (x) ha s an err or of orde r h n+1 , or R ≈ O(h k+1 ). If h is a small number , sa y , h<<1, then h [...]

P age 13-2 5 inc reme nt h. T he list r etur ned as the fir st output ob ject inc ludes the fo llo w ing items: 1 - Bi-dir ecti onal limit of the func tio n at point of e xpansion , i .e . , 2 - An eq uiv alent v alue of the f unctio n near x = a 3 - Expr essi on f or the T ay lor po ly nomi al 4 - Or der of the r esidual or r emainder Becau se of [...]

Pa g e 1 4 - 1 Chapter 14 M ulti-v ariate Calculus Applications Multi- v ar iate calculus r ef ers to functi ons of two or mor e v ar iables . In this Chapte r we dis c uss the basi c concepts of multi-v ari ate calc ulus including partial der i vati v es and multiple int egrals . Multi-var iate func tions A func tion of tw o or mor e var iables ca[...]

Pa g e 1 4 - 2 . Similarl y , . W e w ill use the multi-var i ate functi ons def ined earli er to calc ulate partial der i vati v es using thes e def initions . Her e ar e the der i vati ves o f f(x ,y) w ith r espec t to x and y , re specti vel y: Notice that the def inition of partial der i vati ve w ith r espec t to x, f or e xample , r equir es[...]

Pa g e 1 4 - 3 ther ef or e , w ith DERVX y ou can onl y calculat e deri vati v es w ith r espect to X . Some e xamples o f fir st-order partial der iv ati ve s are sho wn ne xt: ʳʳʳʳʳ Hi gh er -o rde r d erivat ives T he fo llo wing s econd-or der der i vati ves can be def ined T he last tw o e xpr essi ons r epr esen t cr oss-der i v ati ve [...]

Pa g e 1 4 - 4 T hir d-, fourth-, and higher or der der i vati ves ar e def ined in a similar manner . T o calc ulate higher o r der der i vati ves in the calculator , simply r e peat the der i vati v e functi on as man y times as needed . Some e xamples ar e show n belo w : T he c hain rule for partial deri vati ves Consi der the func tion z = f(x[...]

Pa g e 1 4 - 5 A diff er ent v ersi on of the c hain rule appli es to the cas e in whi ch z = f(x,y), x = x(u ,v), y = y(u, v) , so that z = f[x(u ,v) , y(u ,v)]. The f ollo wing f orm ulas r epre sent the c hain rule for this situati on: Determining e xtrema in functions of t w o v ariables In or der f or the functi on z = f(x ,y) to hav e an extr[...]

Pa g e 1 4 - 6 W e find c r itical points at (X,Y ) = (1, 0) , and (X,Y) = (-1, 0 ). T o c alc ulate the disc r iminant , we pr oceed t o calculate the second der i v ati ves , fXX(X,Y) = ∂ 2 f/ ∂ X 2 , fXY(X,Y) = ∂ 2 f/ ∂ X/ ∂ Y , and fYY(X,Y) = ∂ 2 f/ ∂ Y 2 . T he last r esult indi cates that the disc r iminant i s Δ = -12X, thus ,[...]

Pa g e 1 4 - 7 Appli cations of f uncti on HE S S ar e easier to v i suali z e in the RPN mode . Consi der as an ex ample the f uncti on φ (X,Y ,Z) = X 2 + XY + XZ , w e ’ll appl y fu nct ion H E S S to fu nct ion φ i n t h e f ol l owi n g e xa m p l e. T h e s cr e e n s h o t s s h ow t h e RPN stac k bef or e and after appl y ing func tion [...]

Pa g e 1 4 - 8 T he re sulting matri x has elements a 11 = ∂ 2 φ / ∂ X 2 = 6 ., a 22 = ∂ 2 φ / ∂ X 2 = - 2 ., and a 12 = a 21 = ∂ 2 φ / ∂ X ∂ Y = 0. T he disc r iminant , f or this cr itical point s2(1, 0) is Δ = ( ∂ 2 f/ ∂ x 2 ) ⋅ ( ∂ 2 f/ ∂ y 2 )- [ ∂ 2 f/ ∂ x ∂ y] 2 = (6.)(- 2 .) = -12 . 0 < 0, indicating a s[...]

Pa g e 1 4 - 9 Jacobian of coor dinate transf ormation Consi der the coordinat e tr ansfor mation x = x(u ,v) , y = y(u ,v) . T he Jacobi an of this tr ansf ormati on is def i ned as . When calc ulating an int egr al using suc h transf ormati on , the expr ession to u se is , w her e R’ is the r egi on R e xpre ssed in (u ,v ) coor dina te s. Dou[...]

Pa g e 1 4 - 1 0 w here the r egion R’ in polar coor dinates is R’ = { α < θ < β , f( θ ) < r < g( θ )}. Double integr als in polar coor dinates can be enter ed in the ca lc ulator , making sur e that the Jacobi an |J| = r is includ ed in the integr and . The f ollo w ing is an e x ample of a double in tegr al calc ulated in po[...]

P age 15-1 Chapter 15 V ec tor Anal y sis Applications In this Chapt er we pr esent a number of f unctio ns fr om the CAL C menu that appl y to the analy sis of scalar and ve ctor f iel ds. The CAL C menu w as pr esen ted in detail in Chapte r 13 . In partic ular , in the DERI V&INTE G menu w e identif ied a number of functi ons that hav e appl[...]

P age 15-2 At an y partic ular point , the maximum r a t e of change o f the functi on occ urs in the dir ecti on of the gr adien t , i .e ., along a unit vec tor u = ∇φ /| ∇φ |. The v alu e o f that dir ectional der iv ati ve is equal to the magnitude of the gr adient at an y point D max φ (x ,y ,z) = ∇φ •∇φ /| ∇φ | = | ∇φ | T[...]

P age 15-3 as the matri x H = [h ij ] = [ ∂φ / ∂ x i ∂ x j ], the gr adient o f the func tion w ith re spect t o the n -v ar ia bles , grad f = [ ∂φ / ∂ x 1 , ∂φ / ∂ x 2 , … ∂φ / ∂ x n ], and the list of va riab le s [ ‘ x 1 ’ ‘ x 2 ’…’x n ’]. Consider as an e xample the f unction φ (X,Y ,Z) = X 2 + XY + XZ , [...]

P age 15-4 not hav e a potential func tion assoc iated w ith it , since, ∂ f/ ∂ z ≠∂ h/ ∂ x. The cal c ula tor r espon se in th is case is sho wn bel o w: Di ver gence T he div er gence of a v ector f uncti on, F (x,y ,z ) = f(x ,y ,z) i +g(x,y ,z) j +h(x ,y ,z) k , is def ined b y taking a “ dot -pr oduct” o f the del oper ator w ith[...]

P age 15-5 Cur l The c url of a v ector fi eld F (x ,y ,z) = f(x, y ,z) i +g(x ,y ,z) j +h(x ,y ,z) k , is def ined b y a “ cr oss-pr oduct” of the del oper ator w ith the vec tor f ield, i .e ., T he cur l of v ect or fi eld can be calculat ed with f uncti on CURL . F or ex ample , f or the fu nction F (X,Y ,Z) = [XY ,X 2 +Y 2 +Z 2 ,Y Z], the [...]

P age 15-6 As an e xample , in an earlie r ex ample w e attempted to f ind a potenti al func tion for th e ve ctor f ie ld F (x,y ,z) = (x+y) i + (x-y+z) j + xz k , and got an e rr or message back f r om func tion P O TENT IAL. T o ve rify that this is a r otati onal f ield (i .e., ∇× F ≠ 0) , w e us e functi on CURL on this fi eld: On the oth[...]

P age 15-7 pr oduces the v e c tor potenti al func tion Φ 2 = [0, ZYX- 2YX, Y -( 2ZX-X)], w hic h is diffe r ent fr om Φ 1 . T he last command in the sc reen shot sho w s that indeed F = ∇× Φ 2 . Th us, a v ector potenti al functi on is not uniquel y determined . T he components of the gi ve n vect or fi eld, F (x ,y ,z) = f(x,y ,z) i +g(x,y [...]

Pa g e 1 6 - 1 Chapter 16 Differ ential Equations In this Chapte r we pr esent e xample s of so lv ing or dinar y diff er ential equati ons (ODE) using calc ulator f uncti ons. A differ ential equatio n is an equati on in vol v ing der i vati ves of the independen t var iable . In mo st cases , w e seek the dependent f uncti on that satisf ies the [...]

Pa g e 1 6 - 2 ( H @) DISP ) is not se lected . Pr ess ˜ to see the equati on in the E quati on Wr i t e r. An alter nati v e notatio n for der iv ati v es typed dir ectl y in the st ack is to u se ‘ d1’ f or the der i vati v e w ith r espect to the f irs t independent var ia ble , ‘ d2’ for the der i vati v e w ith r espec t to the seco n[...]

Pa g e 1 6 - 3 EV AL(AN S(1)) ` In RPN mode: ‘ ∂ t( ∂ t(u(t)))+ ω 0^2*u(t) = 0’ ` ‘ u(t)=A*SIN ( ω 0*t)’ ` SUBST EVAL The r esult is ‘0=0’ . F or this e xample , y ou could also us e: ‘ ∂ t( ∂ t(u(t))))+ ω 0^2*u (t) = 0’ to enter the diffe r ential equation . Slope field v isualiz ation of solutions Slope fi eld plots, in[...]

Pa g e 1 6 - 4 T hese f unctions ar e brie fl y desc r ibed next . T he y w ill be desc r ibed in mor e detail in later parts of this Chapte r . DE S OL VE: Differ enti al E quati on S OL VEr , pro vi des a solu tion if pos sible IL AP: In ver se L AP lac e tr ansf orm , L -1 [F(s)] = f(t) L AP: LAPl ace transf orm , L[f(t)]=F(s) LDE C: solv es Lin[...]

Pa g e 1 6 - 5 Both of thes e inputs must be gi ven in ter ms of the def ault independent v ar iable fo r the calculator ’s CAS (ty pi cally ‘X’) . T he output fr om the functi on is the gener a l soluti on of the ODE . The f unction LDE C is a v ailable thr ough in the CAL C/DI FF men u . The e x amples ar e sho wn in the RPN mode , ho w ev [...]

Pa g e 1 6 - 6 T he soluti on, sho w n par ti ally he re in the E quation W r iter , is: R eplac ing the combinatio n of constants accompan y ing the e xponenti al terms w ith simpler values , the e xpr essi on can be simplifi ed to y = K 1 ⋅ e –3x + K 2 ⋅ e 5x + K 3 ⋅ e 2x + ( 4 5 0 ⋅ x 2 +3 3 0 ⋅ x+2 41)/13 500. W e r ecogni z e the f[...]

Pa g e 1 6 - 7 2x 1 ’(t) + x 2 ’(t) = 0. In algebr aic f orm , this is wr itten as : A ⋅ x ’(t) = 0, wher e . T he s y stem can be s olv ed b y using func tion LDE C w ith argume nts [0, 0] and matri x A, as sho w n in the f ollo wing sc r een using AL G mode: T he soluti on is gi ve n as a vec tor containing the func tio ns [x 1 (t), x 2 ([...]

Pa g e 1 6 - 8 Ex ample 2 -- So lv e the second-o rde r ODE: d 2 y/dx 2 + x (dy/dx) = e xp(x) . In the calc ulator use: ‘ d1d1y(x)+x *d1y(x) = EXP( x) ’ ` ‘ y(x) ’ ` DESOLVE T he r esult is an e xpr essi on hav ing tw o impli c it integr ations , namel y , F or this parti cular eq uation , ho w ev er , w e r eali z e that the le ft -hand si[...]

Pa g e 1 6 - 9 P er f or ming the integr ation by hand, w e can only ge t it as far as: becaus e the integr al of exp(x)/x is no t av ailable in c losed f or m. Ex ample 3 – Sol v ing an equati on w ith initial co nditions . Sol ve d 2 y/dt 2 + 5y = 2 cos(t/2) , w ith initial conditi ons y(0) = 1.2 , y’(0) = -0. 5 . In the calculator , use: [?[...]

Pa g e 1 6 - 1 0 Press J @ODETY to get the str ing “ Linear w/ cst coeff ” for the ODE ty pe in this case . Laplace T r ansfor ms T he Laplace tr ansform o f a func tion f(t) pr oduces a f unction F(s) in the image domain that can be utili z ed to find the so lution o f a linear differ ential eq uation in vo lv ing f(t) thr ough algebr aic me t[...]

Pa g e 1 6 - 1 1 Laplace tr ansfor m and inv erses in the calc ulator T he calculat or pr o vi des the f uncti ons L AP and ILAP to calc ulate the L aplace tr ansfor m and the in v erse L aplace tr ansfor m, r especti v ely , of a func tion f(VX) , w here VX is the CA S def ault independent v ar iable , whi ch y ou should set t o ‘X’ . T hus , [...]

Pa g e 1 6 - 1 2 Ex ample 3 – Deter mine the in ve rse L aplace tr ansfor m of F(s) = sin(s) . Use: ‘SIN(X)’ ` IL AP . The calc ulator tak es a fe w seconds to r eturn the r esul t: ‘IL AP( SIN(X))’ , meaning that ther e is no c los ed-fo rm e xpr es sion f(t), such that f(t ) = L -1 {sin(s)}. Ex ample 4 – Determine the in ve rse L apla[...]

Pa g e 1 6 - 1 3 Θ Differ entiati on theor em for the n- th der iv ati v e . Let f (k) o = d k f/dx k | t = 0 , and f o = f(0) , then L{d n f/dt n } = s n ⋅ F(s) – s n-1 ⋅ f o − …– s ⋅ f (n - 2) o – f (n-1) o . Θ L inear it y theor em . L{af(t)+bg(t)} = a ⋅ L{f(t)} + b ⋅ L{g(t)}. Θ Differ entiati on theor em f or the image f [...]

Pa g e 1 6 - 1 4 Θ Shift theor em fo r a shif t t o the ri ght . Le t F(s) = L{f(t)}, then L{f(t-a)}=e –as ⋅ L{f(t)} = e –as ⋅ F(s) . Θ Shift theor em f or a shift to the left . Le t F(s) = L{f(t)}, and a >0, then Θ Similar ity theor em . L et F(s) = L{f(t)}, and a>0, then L{f(a ⋅ t)} = (1/a) ⋅ F(s/a) . Θ Damp ing theor em . L[...]

Pa g e 1 6 - 1 5 Dir ac’s d elta function and Heav isid e’s step function In the analy sis of contr ol s y stems it is cu stomary to utili z e a t y pe of f uncti ons that r epr esent certain ph y sical occ urr ences suc h as the sudden acti vati on of a s w itc h (Heav iside’s s tep func tion , H(t)) or a sudden, ins tantaneous , peak in an [...]

Pa g e 1 6 - 1 6 Y ou can pr o v e that L{H(t)} = 1/s , from wh ich it fol lows th a t L { U o ⋅ H(t)} = U o /s , wher e U o is a cons tant . Also , L -1 {1/s}=H(t) , and L -1 { U o /s}= U o ⋅ H(t) . Also , using the shift theor em f or a shift to the ri ght , L{f(t -a)}=e –a s ⋅ L{f(t)} = e –as ⋅ F ( s) , we c a n writ e L{ H ( t -k ) [...]

Pa g e 1 6 - 1 7 Applications of L aplace transf orm in the solution of linear ODEs At the beginning of the s ectio n on Laplace tr ansfor ms we indi cated that y ou could us e these tr ansfor ms to con v ert a linear ODE in the time do main into an algebr aic eq uation in the image domain . T he r esulting equati on is then sol v ed fo r a functi [...]

Pa g e 1 6 - 1 8 T he r esult is ‘H=( (X+1)*h0+a)/(X^2+(k +1)*X+k)’ . T o f ind the soluti on to the ODE , h(t) , w e need to us e the inv erse L aplace tr ansfor m, as f ollo w s: OB J  ƒ ƒ Isolat es ri ght -hand si de of las t expr essi on ILAP μ Obtains the in ver se L aplace tr ansfor m T he r esult is . R eplac ing X w ith t in this [...]

Pa g e 1 6 - 1 9 W ith Y(s) = L{y(t)}, and L{d 2 y/dt 2 } = s 2 ⋅ Y(s) - s ⋅ y o – y 1 , wher e y o = h(0) and y 1 = h ’(0) , the tr ansfor med equati on is s 2 ⋅ Y(s) – s ⋅ y o – y 1 + 2 ⋅ Y(s) = 3/(s 2 +9) . Use the c alc ulator to solv e for Y(s) , b y wr iting : ‘X^2*Y -X*y0 -y1+2*Y=3/(X^2+9)’ ` ‘Y’ I S O L T he r esul[...]

Pa g e 1 6 - 2 0 Ex ample 3 – Consider the equati on d 2 y/dt 2 +y = δ (t-3) , wher e δ (t) is Dir ac’s d e lta func tion . Using La place transf orms , w e can wr ite: L{d 2 y/dt 2 +y} = L{ δ (t- 3)}, L{d 2 y/dt 2 } + L{y(t)} = L{ δ (t-3)}. Wi th ‘ Delta(X-3) ’ ` L AP , the calc ulator pr oduces EXP(-3*X) , i .e., L{ δ (t -3)} = e –[...]

Pa g e 1 6 - 2 1 Chec k what the s olution t o the OD E w ould be if y ou us e the functi on LDEC: ‘Delta(X- 3)’ ` ‘X^2+1’ ` LDE C μ Note s : [1]. An alter nati ve w a y to obtain the in ver se L aplace tr ansfo rm of the e xpr es sion ‘(X*y0+(y1+E XP(-(3*X))))/(X^2+1)’ is b y separ ating the e xpr es sion in to partial f r actions , i[...]

Pa g e 1 6 - 22 T he re sult is: ‘S IN(X-3)*Heav isi de(X-3) + cC1*S IN(X) + cC0*CO S(X)’ . P lease notice that the v ari able X in this expr essi on actuall y r e p r esen ts the v ari able t in the or iginal ODE . Thu s, the tr anslation of the so lution in pape r may be w ritt en as: When compar ing this r esult w ith the pr ev i ous r esult[...]

Pa g e 1 6 - 2 3 Use o f the f unction H(X) w ith LD E C, L AP , or IL AP , is not allo wed in the calc ulator . Y ou hav e to us e the main results pr ov ided earlier w hen dealing w ith the Heav iside step f uncti on , i .e ., L{H(t)} = 1/s, L -1 {1/s}=H(t) , L{H(t-k)}=e –ks ⋅ L{H(t)} = e –ks ⋅ (1/s) = ⋅ (1/s) ⋅ e –ks and L -1 {e ?[...]

Pa g e 1 6 - 24 w here H(t) is Hea v iside ’s step f uncti on. Us ing Laplace tr ansfor ms, w e can wri te : L {d 2 y/dt 2 +y} = L{H(t- 3)}, L{d 2 y/dt 2 } + L{y(t)} = L{H(t- 3)} . The la st ter m in this e xpr essi on is: L{H(t -3)} = (1/s) ⋅ e –3s . With Y(s) = L{y(t)}, and L{d 2 y/dt 2 } = s 2 ⋅ Y(s) - s ⋅ y o – y 1 , w here y o = h([...]

Pa g e 1 6 - 2 5 Ex ample 4 – P lot the so lution to Ex ample 3 using the same v alues of y o and y 1 used in the plot of Ex ample 1, abov e . W e now plot the f unction y(t) = 0. 5 cos t –0.2 5 sin t + ( 1+sin(t -3)) ⋅ H(t-3) . In the r ange 0 < t < 20, and c hanging the vertical r ange to (-1, 3) , the gr aph should look lik e this: A[...]

Pa g e 1 6 - 26 f(t) = U o ⋅ [1-(t-a)/(b-1)] ⋅ [H(t-a) -H(t -b)]. Ex amples of the plots gener ated by the se func tions , fo r Uo = 1, a = 2 , b = 3, c = 4 , hori z ontal r ange = (0,5 ) , and v ertical r ange = (-1, 1.5 ) , ar e sho wn in the fig ure s b el ow: F ourier ser ies F ouri er ser ie s are s er ies in v olv ing sine and cosine func[...]

Pa g e 1 6 - 2 7 T he follo w ing ex erc ises ar e in AL G mode , with CA S mode s et to Ex act . ( W hen y ou pr oduce a gr aph , the CAS mode w ill be re set to Appr o x. Mak e sur e to se t it back t o Exact afte r pr oduc ing the gra ph.) Suppo se , f or ex ample , that the func tion f(t) = t 2 +t is per iodi c w ith per iod T = 2 . T o determi[...]

Pa g e 1 6 - 2 8 Function FOURIER An alter nati ve w a y to def ine a F our ier ser ies is by using comple x number s as fo llo w s: wh ere F uncti on FOURIER pr ov i des the coeff ic ient c n o f the complex -for m of the F ouri er ser i es giv en the functi on f(t) and the v alue of n. T he functi on F OURIER r equir es y ou to st or e the value [...]

Pa g e 1 6 - 2 9 Ne xt, w e mo ve to the CA SDI R sub-dir ector y under HOME to c hange the value of var iable PERIOD , e.g ., „ (hold ) §`J @) CASDI `2 K @PERIOD ` R eturn to the su b-dir ectory wher e y ou defined f uncti ons f and g, and calc ulate the coeff ic ients (A ccept change to C omple x mode w hen req uested): Th us, c 0 = 1/3, c 1 =[...]

Pa g e 1 6 - 3 0 T he fitting is some what accepta ble for 0<t<2 , although not as good as in the pr ev ious e xample . A general e xpression for c n T he functi on F OURIER can pro v ide a gener al e xpr essi on f or the coeff ic ient c n of the comple x F our ier ser ies e xpansion . F or ex ample , using the same f unction g(t) as befor e,[...]

Pa g e 1 6 - 3 1 The r esult is c n = (i ⋅ n ⋅π +2)/(n 2 ⋅π 2 ). P utting t ogether the comple x F ouri er ser ies Hav ing deter mined the gener al expr ession for c n , we can put toge ther a finite comple x F our ier se ri es b y using the summati on f unction ( Σ ) in the calculator as fo llo w s: Θ F irst , def ine a f uncti on c(n) r[...]

Pa g e 1 6 - 32 Or , in the calculator entry line as: DEFINE(‘F(X,k,c0) = c0+ Σ (n=1,k ,c(n)*EXP(2*i* π *n*X/T)+ c(-n)*EXP(-( 2*i* π *n*X/T))’) , w here T is the per iod , T = 2 . The fo llo w ing s cr een shots show the def i niti on of func tion F and the st orin g of T = 2 : Th e fu nct ion @@@F@@@ can be us ed to gener ate the e xpre ssi[...]

Pa g e 1 6 - 33 Accept c hange to Approx mode if r eques ted . The r esult is the v alue –0.40 46 7…. The ac tual value of the f uncti on g(0. 5) is g(0. 5) = -0.2 5 . Th e fo llo w ing calc ulations sh ow ho w w ell the F our ier ser ie s appr o x imate s this v alue as the number of componen ts in the ser ie s, gi v en b y k, inc r eases: F ([...]

Pa g e 1 6 - 3 4 per iodi c ity in the gr aph o f the ser ies . This per i odic it y is eas y to v isuali z e by e xpa nding the hor i z ontal range of the plot to (-0.5, 4) : F ourier series f or a triangular w av e Consi der the functi on w hich w e assume to be per i odic w ith peri od T = 2 . This f uncti on can be def ined in the calc ulator ,[...]

Pa g e 1 6 - 3 5 T he calculat or r eturns an int egr al that cannot be e valuat ed numer icall y becaus e it depends on the par ameter n . The coeff ic ient can s till be calc ulated by typing its de finiti on in the calc ulator , i .e ., w here T = 2 is the per i od. T he value of T can be st or ed using: T yp ing the firs t integr al abo ve in t[...]

Pa g e 1 6 - 3 6 Press `` to cop y this re sult to the scr een. T hen , r eacti vat e the E quation W r iter to calc ulate the second integr al defi ning the coeffi c ie nt c n , namel y , Once again, r eplac ing e in π = (-1) n , and using e 2in π = 1, we get: Press `` to cop y this second r esult to the sc r een . No w , add ANS(1) and ANS( 2) [...]

Pa g e 1 6 - 37 T his re sult is used to de fine the f unction c(n) as f ollo ws: DEFINE(‘ c(n) = - (((-1)^n-1)/(n^2* π ^2*(-1)^n)’) i. e. , Ne xt, w e def ine function F(X,k ,c0) to calc ulate the F our ier seri es (if you completed e x ample 1, y ou alr eady ha v e this functi on stor ed) : DEFINE(‘F(X,k,c0) = c0+ Σ (n=1,k ,c(n)*EXP(2*i* [...]

Pa g e 1 6 - 3 8 F r om the plot it is very diffi c ult to distinguish the or iginal functi on fr om the F ourier s eri es appr o ximati on. U sing k = 2 , or 5 ter ms in the ser ies, sho ws not so good a f itting: T he F our ier s eri es can be us ed to gener ate a per i odic tr iangular w a ve (or sa w tooth w av e ) by c hanging the hor iz ontal[...]

Pa g e 1 6 - 3 9 In th is case , th e per iod T , is 4. Mak e s ur e to chang e the value of v ari abl e @@@T@@@ to 4 (use: 4K @@@T@@ ` ) . F unction g(X) can be de fined in the calc ulator by us in g DEFINE(‘ g(X) = IFTE((X>1) AND (X<3) ,1, 0)’) The function plot ted as follo ws (hori z ontal r ange : 0 to 4 , v ert i cal r a nge: 0 to 1[...]

Pa g e 1 6 - 4 0 Th e si m pl i fic at io n of th e rig h t -h a nd s id e of c (n ) , a bove, i s ea si er d on e on p ap e r (i .e ., b y hand) . T hen , r et y pe the expr es sion f or c(n) as sho wn in the f igur e to the left abo v e , to def ine func tion c(n). T he F our ier s er ies is calc ulated w ith F(X,k ,c0) , as in e x amples 1 and 2[...]

Pa g e 1 6 - 4 1 W e can use this r esult as the f irs t input to the f uncti on LD E C w hen us ed to obtain a soluti on to the s y ste m d 2 y/dX 2 + 0.2 5y = S W(X) , w her e S W(X) stands f or Squar e W av e f uncti on of X. T he second inpu t item w ill be the char acter isti c equati on corr es ponding to the homogeneous ODE sho wn abo ve , i[...]

Pa g e 1 6 - 42 T he soluti on is sho wn belo w: F ourier T r ansf orms Befor e pr esen ting the concept of F our ier tr ansf orms , we ’ll d i scus s the gener al def initio n of an integr al tr ansf orm . In gener al , an integr al tr ansf orm is a tr ansfor mation that r elate s a functi on f(t) to a new f uncti on F(s) by an integr ation of t[...]

Pa g e 1 6 - 4 3 T he amplitudes A n w ill be r ef er red t o as the spectr um of the f uncti on and w ill be a measur e of the magnitude of the component of f(x) w ith fr equency f n = n/T . T he basic or f undamental fr equency in the F ouri er ser ies is f 0 = 1/T , thu s, all other fr equenc ies ar e multiple s of this basi c f req uency , i .e[...]

Pa g e 1 6 - 4 4 and The continuous spectrum is giv en by Th e fu nct ion s C ( ω ), S ( ω ), and A( ω ) ar e continuous functi ons of a v ari able ω , w hich beco mes the tr ansfor m v ari able fo r the F our ier tr ansfor ms def ined belo w . Ex ample 1 – D eter min e the coeffic ients C( ω ), S ( ω ) , and the continu ous spectr um A( ω[...]

Pa g e 1 6 - 4 5 Def ine this e xpr essio n as a f unction by u sing func tion DEFINE ( „à ) . Then , plot the continuo us spectr um, in the r ange 0 < ω < 10 , as: Definition o f Four ier transf orms Diffe r ent t y p e s of F ourie r transf or ms can be defined . T he fo llo wing ar e the def initio ns of the sine , cosine , and full F [...]

Pa g e 1 6 - 4 6 The continuous spect r um, F( ω ) , is calculated w ith the integral: T his re sult can be r ationali z ed b y multipl y ing numer ator and denominator b y the conjugat e of the denominator , namel y , 1-i ω . T he r esult is now : which is a co mp lex fu nct ion. T he absolute v alue of the r eal and imaginar y parts of the func[...]

Pa g e 1 6 - 4 7 Pr oper ties o f th e F ourier transfor m L inearity : If a and b are co nstants , and f and g functi ons, then F{a ⋅ f + b ⋅ g} = a F{f }+ b F{g}. T r ansfor mati on of partial deri vati v es . Let u = u(x ,t) . If the F ouri er tr ansfor m tr ansfor ms the var i able x , then F{ ∂ u/ ∂ x} = i ω F{u}, F{ ∂ 2 u/ ∂ x 2 [...]

Pa g e 1 6 - 4 8 the number o f oper ations u sing the FFT is r e du ced by a f act or of 10000/6 64 ≈ 15 . The FFT op er ates on t he sequenc e {x j } b y par titi oning it int o a number o f shorter seque nces . The DFT ’s of the shorter seq uences ar e calc ulated and later comb ined together in a highl y eff ic ient manner . F or details on[...]

Pa g e 1 6 - 49 T he fi gur e belo w is a box plot o f the data pr oduced. T o obtain the gra ph, f irs t cop y the ar r ay j ust c r eated, then tr ansfor m it into a column v ector b y using: OB J  1 +  ARR Y (F uncti ons OB J  and  ARR Y ar e av ailable in the command cat alog, ‚N ) . S tor e the arr ay into var ia ble Σ DA T by u[...]

Pa g e 1 6 - 50 Ex ample 2 – T o pr oduce the signal gi ven the s pectr um, w e modif y the pr ogr am GD A T A to inc lude an abso lute v alue , so that it r eads: <<  m a b << ‘2^m ’ EV AL  n << ‘(b-a)/(n+1)’ EV AL  Dx << 1 n F OR j ‘ a+(j-1 )*Dx ’ EV AL f AB S NEXT n  ARR Y > > >> >[...]

Pa g e 1 6 - 5 1 Ex cept for a lar ge peak at t = 0, the signal is mo stl y nois e . A smaller v er ti cal scale (-0. 5 to 0. 5) sho ws the si gnal as f ollo ws: Solution to specific second-or der differential equations In this secti on w e pr esent and so lv e spec ifi c t y pes of or dinar y differ ential equati ons who se solu tions ar e def ine[...]

Pa g e 1 6 - 52 w here M = n/2 or (n-1)/2 , whi che v er is an integer . Legendr e’s pol y nomials ar e pr e -pr ogr ammed in the calculator and can be r ecalled by u sing the func tion LE GENDRE gi v en the or der of the pol ynomi al , n. T he functi on LE GENDR E can be obtained fr om the command catalog ( ‚N ) or thr ough the menu ARITHME T [...]

Pa g e 1 6 - 5 3 wher e ν is not an integer , and the func tion Gamma Γ ( α ) is def ined in Chapter 3. If ν = n , an integer , the Bessel f uncti ons of the f ir st kind for n = intege r ar e def ined b y Regar dless of whether w e use ν (n on -int eger ) or n (integer ) in the calc ulato r , we can def ine the Bess el f unctions o f the fir [...]

Pa g e 1 6 - 5 4 Y ν (x) = [J ν (x) cos νπ – J −ν ( x)]/sin νπ , fo r non -int eger ν , and f or n integer , w ith n > 0, by wher e γ is the Euler cons tant , def ined by and h m r epre sents the har monic se r ies F or the case n = 0, the Bes sel f uncti on of the seco nd kind is def ined as With these def i niti ons, a gener al so[...]

Pa g e 1 6 - 5 5 T he modifi ed Bessel f unctions o f the second kind , K ν (x) = ( π /2) ⋅ [I - ν (x) − I ν (x)]/sin νπ , ar e also so lutions o f this OD E . Y ou can implement f uncti ons r epr esenting Bes sel’s f unctions in the calc ulator in a similar ma nn er to that used to def ine Bess el’s func tions of the f irst kind, but[...]

Pa g e 1 6 - 5 6 Laguerr e’s equation Lague rr e ’s equation is the s econd-orde r , linear OD E of the f orm x ⋅ (d 2 y/dx 2 ) +(1 − x) ⋅ (d y/dx) + n ⋅ y = 0. L aguerr e pol ynomi als, de fined as , ar e soluti ons to L aguerr e ’s equation . Laguer r e ’s pol ynomi als can also be calc ulated w ith: Th e te rm is the m-th coeff i[...]

Pa g e 1 6 - 57 L 2 (x) = 1- 2x+ 0.5x 2 L 3 (x) = 1-3x+1. 5x 2 - 0 . 16 666… x 3 . W eber ’s equation and H er mite poly nomials W eber’s eq uation is def ined as d 2 y/dx 2 +(n+1/2 - x 2 /4)y = 0, f or n = 0, 1, 2 , … A partic ular so lutio n of this eq uation is gi ven b y the functi on , y(x) = ex p (-x 2 /4)H * (x/ √ 2) , w her e the [...]

Pa g e 1 6 - 5 8 F i r st , c r eate the e xpr es sion de fining the de ri vati v e and stor e it into var i able E Q. T he fi gur e to the left sho ws the AL G mode command, w hile the ri ght-hand side f igur e sho ws the RPN s tack be for e pre ssing K . T hen, enter the NUMERICAL S OL VER en vir onment and select the differ ential equation s olv[...]

Pa g e 1 6 - 59 @@OK@ @ @INIT+ — .7 5 @@OK@@ ™™ @SOLVE (wai t) @EDIT (Changes initial v alue of t t o 0.5, and f inal v alue of t to 0.7 5, sol v e f or v(0.7 5) = 2 . 066…) @@OK@ @ @INIT+ — 1 @@OK@@ ™ ™ @SOLVE (wa it ) @EDIT (Changes initi al value o f t to 0.7 5, and final v alue of t to 1, s olv e for v(1) = 1. 5 6 2…) R epeat fo[...]

Pa g e 1 6 - 6 0 Θ „ô (simultaneousl y , if in RPN mode) to enter P L O T env i r onment Θ Hi ghligh t the f ield in f r ont o f TYPE , using the —˜ k ey s. T hen , pres s @CHOOS , and highlight Diff Eq , u sing the —˜ k ey s. Pr ess @@OK@@ . Θ Chan ge fi eld F: to ‘EXP(- t^2)’ Θ Mak e sur e that the f ollow ing paramet ers ar e se[...]

Pa g e 1 6 - 6 1 LL @) PICT T o re c over m e nu a n d re t u rn to PI C T envi ro n me n t. @ ( X,Y ) @ T o determine coor dina t es of an y point on the gr aph . Use the š™ k ey s to mo ve the cursor ar oun d the plot a r ea . At th e bottom of the sc r een y ou w ill see the coor dinates of the c urs or as (X,Y) , i .e., the calc ulator use s[...]

Pa g e 1 6 - 62 time t = 2 , the input for m fo r the differ ential equation s olv er should look a s fo llo w s (notice that the Init: v alue f or the Soln: is a v ect or [0, 6]) : Press @SOLVE (wai t) @EDIT to s ol ve f or w(t=2) . The so lution r eads [.16 716… - .6 2 71…], i .e ., x(2 ) = 0.16 716 , and x'( 2) = v(2) = -0.6 2 71. Pre s[...]

Pa g e 1 6 - 6 3 (Changes initi al value of t to 0.7 5, and f inal value o f t to 1, sol ve again f or w(1) = [-0.4 6 9 -0.6 0 7]) R epeat for t = 1.2 5, 1.5 0, 1.7 5, 2 .0 0. Pre ss @@OK@@ after v ie w ing the last r esult in @EDIT . T o r eturn to nor mal calculator displa y , pr ess $ or L @@OK@@ . T he diffe r ent soluti ons w ill be sho w n in[...]

Pa g e 1 6 - 6 4 Notice that the opti on V - V ar : is set to 1, indicating that the f irst ele ment in the v ector s oluti on, namel y , x ’ , is to be plotted against the independent v ar iable t . Accept c hanges to P L O T SETUP b y pr essing L @@OK@@ . Press „ò (simultaneousl y , if in RPN mode) to enter the P L O T WINDO W en vi r onment[...]

Pa g e 1 6 - 65 Press LL @PICT @CANCL $ to r etur n to nor mal calc ulator dis play . Numerical solution for stiff first-or d er ODE Consi der the ODE: d y/dt = -100y+100t+ 101, sub jec t to the initial conditi on y(0) = 1. Ex ac t solution T his equation can be w ri t t en as dy/dt + 100 y = 100 t + 101, and so lv ed using an integr ating fact or [...]

Pa g e 1 6 - 6 6 Her e w e are try ing to obtain the v alue of y( 2) giv en y(0) = 1. W ith the Soln: Final f ield highli ghted, pr ess @SOLVE . Y ou can chec k that a soluti on tak es abo ut 6 sec on ds, wh il e i n t he previou s fi rst - orde r exa mp le th e s ol ut ion was alm os t instantaneou s. Pr ess $ to cancel the calc ulati on. T his is[...]

Pa g e 1 6 - 67 Note: T he opti on Stiff is also a vailable f or gr aphical s oluti ons of differ ential equati ons. Numerical solution to ODEs w it h th e S O L VE/DIFF menu T he S OL VE soft men u is acti va ted b y using 7 4 MENU in RPN mode . T his menu is pr esent ed in detail in Cha pter 6 . One of the sub-menu s, DIFF , contains func tions f[...]

Pa g e 1 6 - 6 8 T he value of the so lution , y fin a l , w ill be av ailable in v ar iable @@@y@@@ . This f uncti on is appr opr iate f or pr ogramming since it lea v es the diff er ential eq uation spec if icati ons and the toler ance in the st ack r eady f or a new s olution . Notice that the soluti on use s the initial conditions x = 0 at y = [...]

Pa g e 1 6 - 69 contain only the v alue of ε , and the step Δ x w ill be tak en as a small default value . After running f unction @@RKF@ @ , the s tack w ill show the lines: 2 : {‘ x ’ , ‘ y’ , ‘f(x ,y)’ ‘ ∂ f/ ∂ x’ ‘ ∂ f/vy’ } 1: { εΔ x } T he value o f the soluti on , y fi nal , w ill be a vaila ble in var iable @@@[...]

Pa g e 1 6 - 70 T hese r esults indi cate that ( Δ x) ne xt = 0. 34 04 9… Function RRKS TEP T his f uncti on use s an input list similar to that of func tion RRK , as well as the toler ance for the so lution , a po ssible st ep Δ x , and a n umber (L A ST) spec ifying the last me thod used in the solu tion (1, if RKF w as used , or 2 , if R RK [...]

Pa g e 1 6 - 7 1 T hese r esults indi cate that ( Δ x) ne xt = 0. 005 5 8… and that the RKF method (CURRENT = 1) should be used. Function RKFERR T his functi on r etur ns the abso lute er r or estimate f or a gi ven s tep whe n sol v ing a pr oblem as that des cr ibed f or func tion RKF . T he input st ack looks as f ollo ws: 2: ʳʳʳ {‘ x ?[...]

Pa g e 1 6 - 72 T he follo w ing scr een shots sho w the RPN st ack bef ore and af t er applicati on of func tion R SBERR: T hese r esults indi cate that Δ y = 4.1514… and err or = 2 .7 6 2 ..., fo r Dx = 0.1. Chec k that , if Dx is redu ced to 0. 01, Δ y = -0. 003 0 7… and err or = 0. 0005 4 7 . Not e : As y ou e xec ute the commands in the [...]

Pa g e 1 7- 1 Chapter 17 Pr obability Applications In this Chapte r w e pr ov ide e xample s of applicati ons of calc ulator’s func tions to pr obab ility distr ibutions . T he MTH/PR OB ABILITY .. sub-m enu - part 1 T he MTH/PR OB ABILITY .. sub-men u is accessible thr ough the ke ys tr ok e sequence „´ . W ith sy stem flag 117 se t to CHOOSE[...]

Pa g e 1 7- 2 T o simplify notation , use P(n ,r) f or per mutati ons, and C(n ,r) f or combinations . W e can calculat e combinations , perm utations , and factor i als with f uncti ons CO MB, P ERM, and ! fr om the MT H/P R OBA BILITY .. sub-menu . The oper ation of those f uncti ons is desc r ibed next: Θ CO MB(n,r ): Combinati ons of n items t[...]

Pa g e 1 7- 3 R andom number gener ators , in gener al, oper ate b y taking a v alue , called the “ seed” of the gener ator , and per f or ming some mathematical algor ithm on that “ seed” that gener ates a ne w (ps eudo)r andom number . If yo u wa nt to gener ate a sequence o f number and be able to r epeat the s ame sequence lat er , yo u[...]

Pa g e 1 7- 4 fu nct ion (pmf) is r epr esented by f (x) = P[X=x], i .e ., the pr obability that the ra nd om va riab le X ta kes th e val ue x. T he mass distr ibuti on functi on mus t satisf y the conditi ons that f(x) >0, f or all x , and A c umulati ve dis tributi on func tio n (cdf) is def ined as Ne xt, w e w ill define a number o f functi[...]

Pa g e 1 7- 5 P oisson distribution The probabilit y mass f unction of the P oisson di str ibut ion is giv en by . In this e xpre ssi on, if the r andom var i able X r epre sents the n umber of occ urr ences o f an e ven t or observati on per unit time , length , area , volume , etc., then the par a meter l r epres ents the a v er age number of occ[...]

Pa g e 1 7- 6 Continuous pr obabilit y distr ibutions T he proba bility distributi on f or a continuou s r andom var ia ble , X, is c harac ter i z e d b y a f uncti on f(x) know n as the pr obab ilit y density functi on (pdf) . T he pdf has the foll o wing pr operties: f(x) > 0, f or all x , and Pr obabiliti es ar e calc ulated using the c u m [...]

Pa g e 1 7- 7 , w hile its cdf is giv en b y F(x) = 1 - e xp(- x/ β ) , f or x>0, β >0. T he beta distribution T he pdf for the gamma dis tr ibution is gi v en b y As in the case of the gamma dis tribut ion , the corr esponding cdf for the bet a distr ibuti on is also gi v en b y an integr al w ith no c losed-f orm solu tion . T he W eibull[...]

Pa g e 1 7- 8 Exponential pdf: 'epdf(x) = EXP(-x/ β )/ β ' Exponential cdf: 'ecdf(x) = 1 - EXP(-x/ β )' W eibull pdf: 'Wpdf(x) = α * β *x^( β -1)*EXP(- α *x ^ β )' W eibull cdf: 'Wcdf(x) = 1 - EXP(- α *x^ β )' Use f uncti on DEFINE to def ine all these f unctions . Ne xt , enter the v alues of α an[...]

Pa g e 1 7- 9 Continuous distributions f or statistical infer ence In this sec tion w e disc uss f our contin uous pr obability distr ibutions that ar e commonl y used f or pr oblems r elated to statis tical inf er ence . The se distr ibuti ons ar e the normal dis tributi on , the Student’s t distr ibution , the Chi-s quar e ( χ 2 ) distr ibuti [...]

Pa g e 1 7- 1 0 wher e μ is the mean , and σ 2 is the v ari ance of the dis tributi on . T o calc ulate the val ue of f( μ , σ 2 ,x) fo r the normal distr ibution , us e functi on NDIS T with the fo llo w ing ar guments: the mean , μ , the var iance , σ 2 , and, the v alue x , i.e ., NDIS T( μ , σ 2 ,x) . F or e xample , chec k that for a n[...]

Pa g e 1 7- 1 1 wher e Γ ( α ) = ( α -1)! is the G AMMA functi on def ined in Chapter 3 . T he calculator pr ov ides f or values o f the upper - t ail (cumulati v e) distr ibution func tion f or the t-distr ibution , f uncti on UTPT , gi ve n the paramet er ν and the value of t , i .e ., UTPT( ν ,t) . T he def inition of this f unction is , th[...]

Pa g e 1 7- 1 2 T he calculator pr ov ides f or values o f the upper - t ail (cumulati v e) distr ibution fu nct ion fo r th e χ 2 -distr ibutio n using [UTP C] gi v en the v alue of x and the par ameter ν . T he def inition of this f uncti on is, ther ef or e , T o use this f uncti on , we need the degr ees of f reedo m, ν , and the value of th[...]

Pa g e 1 7- 1 3 T he calculator pr ov ides f or values o f the upper - t ail (cumulati v e) distr ibution func tion f or the F distr ibuti on, f uncti on UTPF , gi ven the par ameter s ν N and ν D, and the value of F . The definition of th is function is, theref ore , F or ex ample, to calc ulate UTPF(10,5, 2 .5 ) = 0.1618 3 4… Diffe r ent pr o[...]

Pa g e 1 7- 1 4 Exponential: W eibull: F or the Gamma and Beta distr ibuti ons the e xpr essi ons to s olv e w ill be mor e compli cated due to the pr esence o f integr als, i .e ., • Gamma, • Beta , A numer ical soluti on w ith the numer i cal sol ver w ill not be feasible beca use of the integr al sign in v olv ed in the e xpre ssi on. Ho w e[...]

Pa g e 1 7- 1 5 Ther e are tw o r oots of this functi on f ound by using f unction @ROOT w i thin the plo t en vi r onment . Because o f the integr al in the equatio n, the r oot is appro ximat ed and w ill not be sho wn in the plot s cr een . Y ou w ill only get the me ssage Cons tant? Sho wn in the sc r een. Ho we v er , if you pr ess ` at this p[...]

Pa g e 1 7- 1 6 Notice that the second par amet er in the UTPN functi on is σ 2, n o t σ 2 , r epr esenting the v ar iance of the distr ibuti on. A lso , the s ymbol ν (the low er-case Gr eek letter no) is not a v ailable in the calc ulator . Y ou can us e , for e xample , γ (gamma) instead o f ν . T he letter γ is a vailable thought the c ha[...]

Pa g e 1 7- 1 7 Th us, at this point , you w ill hav e the four equati ons av ailable for so lution . Y ou needs ju st load one of the equati ons into the E Q f ie ld in the nume ri cal solv er and pr oceed w ith sol v ing for one o f the var ia bles . Example s of the UTPT , UTP C, and UPTF ar e show n belo w: Notice that in all the e xample s sho[...]

Pa g e 1 7- 1 8 W ith these four equati ons, w henev er y ou launch the numer i cal s olv er y ou ha ve the f ollo w ing cho i ces: Ex amples of s olution o f equations E QNA, E QT A, E QCA, and E QF A ar e sho w n belo w : ʳʳʳʳʳ[...]

P age 18-1 Chapter 18 Statistical Applications In this Chapte r we intr oduce statisti cal applicati ons of the calc ulator including statis tic s of a sample , f r equency dis tributi on of data , simple r egre ssi on, conf i dence int ervals , and h ypothe sis te sting . Pre-progr amm ed statistical f eatures T he calculat or pr o vi des pr e -pr[...]

P age 18-2 St or e the pr ogram in a v ar iable called LX C. After st or ing this pr ogram in RPN mode y ou can also us e it in AL G mode. T o sto r e a column vec tor into v ar iable Σ D A T use functi on S T O Σ , av ailable thr ough the catalog ( ‚N ) , e .g., S T O Σ ( ANS(1)) in AL G mode . Ex ample 1 – Using the pr ogram LX C, def ined[...]

P age 18-3 Ex ample 1 -- F or the data st or ed in the pr ev ious e x ample , the single -var iable statis tic s r esults ar e the f ollo w ing: M e a n : 2. 1 3333333333 , S t d D e v: 0 . 9 6 42 0 7 9 49 4 0 6, Va r i a n c e : 0 . 9 2969696969 7 T otal: 2 5 .6, Max imum: 4. 5, Minimum: 1.1 Definition s Th e d efi ni ti on s u sed f or these quan[...]

P age 18-4 Ex amples of calc ulation of these measur es, using lis ts, ar e a vailable in C hapter 8. T he median is the value that s plits the data set in the mi ddle when the e lements ar e placed in incr easing orde r . If y ou hav e an odd number , n, of or dered elements , the median of this sam ple is the value located in positi on (n+1)/2 . [...]

P age 18-5 Th e ran g e of the sample is the differ ence betw een the maximum and minim um v alues of the sample . Since the calc ulator , thr ough the pr e -pr ogr ammed statis tical f uncti ons pr o v ides the max imum and minimum values o f the sample , y ou can easily calc ulate the range . Coefficient of variation T he coeffi c ient o f var ia[...]

P age 18-6 Definition s T o unders tand the meaning of thes e par ameters w e pr esent the follo w ing def initions : Gi ven a se t of n data values: {x 1 , x 2 , …, x n } lis ted in no partic ular or der , it is often r equir ed to gr ou p these data into a ser ies of c lass es by counting the f r eque ncy or number o f values cor r esponding to[...]

P age 18-7 Θ Gener ate the list o f 200 number b y using RDLIS T(200) in AL G mode , or 200 ` @ RDLIST@ in RPN mode . Θ Us e pr ogram LXC (s ee abo ve) to con vert the list th us gener ated into a column vec tor . Θ St ore the column v ector into Σ DA T , by u s i n g f un c t i o n ST O Σ . Θ Obtain single -v ari able inf ormation u sing: ?[...]

P age 18-8 to calc ulate for unif orm-si z e c lasses (or bins) , and the class mar k is just the a ver age of the clas s boundari es f or eac h cla ss. F inally , the c umulati ve fr equency is obtain ed by adding to eac h v alue in the last column , e x cept the f irst , the fr equenc y in the ne xt r o w , and r eplac ing the r esult in the la s[...]

P age 18-9 « DUP S I ZE 1 GET  fr eq k « {k 1} 0 CON  cfr eq « ‘fr eq(1,1)’ EV AL ‘ cfr eq(1,1)’ S T O 2 k FOR j ‘ cfr eq(j-1,1) +fr eq(j,1)’ EV AL ‘ cfr eq (j,1)’ S T O NEXT cfr e q » » » Sa ve it un der the name CFRE Q. Use this pr ogram t o gener ate the list o f c umulati ve f r equenc ies (pr ess @CFRE Q w ith the c[...]

P age 18-10 Θ Press @CAN CEL to r etur n to the pre vi ous s cr een. Change the V - v ie w and Bar W idth once mor e , now to r ead V- Vi e w: 0 3 0, Bar Wi dth: 10. The ne w histogr am, bas ed on the same data set , no w looks lik e this: A plot of f r equency count , f i , v s. c lass mar ks, xM i , is kno wn as a f r eque ncy poly gon. A plot o[...]

P age 18-11 Θ F i r st , enter the two r ow s of data into column in the var iable Σ DA T by u s i n g the matri x wr iter , and func tion S T O Σ . Θ T o access the pr ogr am 3. Fit data.. , us e the follo wi ng k ey str ok es: ‚Ù˜˜ @@@OK@@@ T he input fo rm w ill sho w the c urr ent Σ D A T , alread y loaded. If needed , change y our se[...]

P age 18-12 Wher e s x , s y ar e the standar d de v iati ons of x and y , re spec ti vel y , i .e . Th e va lu es s xy and r xy ar e the "Co var iance" and "Cor r elati on," r es pecti v ely , obtained b y using the "F it data" featur e of the calc ulator . Lineari zed relationships Man y curv ilinear r elatio nships [...]

P age 18-13 T he gener al fo rm of the r egr essi on equati on is η = A + B ξ . Best data fitting T he calculat or can deter mine whi ch one of its linear or linear i z ed r elatio nship off ers the bes t fitting f or a set of (x ,y) data points . W e w ill illustr ate the u se of this featur e w ith an e x ample . Suppos e y ou w ant to f ind w [...]

P age 18-14 X-Col, Y -Co l: these options a pply onl y w hen yo u hav e mor e than t w o columns in the matr ix Σ D A T . B y def ault , the x column is column 1, and the y col umn is column 2 . _ Σ X _ Σ Y… : summary statis tics that y ou can choo se as r esults of this pr ogr am b y chec king the appr opri ate f ield u sing [  CHK] w hen [...]

P age 18-15 B. I f n ⋅ p is an integer , say k , calc ulate the mean of the k - th and (k -1) th or der ed observ ations . T his algorithm can be implemented in the f ollo w ing pr ogr am typed in RPN mode (See C hapter 21 for pr ogr amming inf ormati on): « S ORT DUP S I ZE  p X n « n p *  k « IF k CEIL k FL OOR - NO T THEN X k GE T X k[...]

P age 18-16 T he D A T A sub-menu T he D A T A sub-menu cont ains functi ons us ed to manipulate the statis tic s matri x Σ DA TA : The ope rati on of thes e func tions is as f ollo w s: Σ + : add r o w in lev el 1 to bottom of Σ DA T A ma t rix. Σ - : r emo ve s last r o w in Σ D A T A matri x and place s it in le vel o f 1 of the s tack . Th[...]

P age 18-17 Σ P AR: show s statis tical par ameter s. RE SET : r eset par ameter s to default v alues INFO: sh o ws s tatist ical par ameter s The MODL sub-menu w ithin Σ PA R T his sub-menu con tains fu nctio ns that let y ou change the dat a -f itting model to LINFIT , L OGFIT , EXPFI T , P WRFIT or BE S TFIT b y pr essing the appr opri ate but[...]

P age 18-18 T he functi ons inc luded ar e: B ARP L: produce s a bar plot with data in Xcol column of the Σ D ATA m a t r i x . HIS TP: pr oduces his togr am of the data in Xcol co lumn in the Σ DA T A m a t rix, using the de fault w idth corr esponding to 13 bi ns unless the bin si z e is modifi ed using func tion BIN S in the 1V AR sub-menu (se[...]

P age 18-19 Σ X^2 : pr o v ides the sum of s quar es of v alues in Xcol column . Σ Y^2 : pr ov ides the sum of squar es of value s in Ycol column . Σ X*Y : pr ov ides the sum o f x ⋅ y , i .e., the pr oducts o f data in columns Xcol and Ycol. N Σ : pr o v ides the number of col umns in the Σ DA T A m a tr ix. Ex ample of S T A T menu oper at[...]

P age 18-20 @) STAT @ ) £PAR @ RESET r esets statis tical par ameters L @ ) STAT @PLOT @SCATR pr oduces s catter plot @STATL dr aw s data fit as a s trai ght line @CANCL r eturns t o main display Θ Deter mine the f itting equation and so me of its statis tic s: @) STAT @ ) FIT@ @£LINE pr oduces '1.5+2*X' @@@LR@@@ pr oduces Intercept: 1[...]

P age 18-21 Θ F it data using columns 1 (x) and 3 (y) using a logar ithmic f itting: L @ ) STAT @ ) £ PAR 3 @YCO L sel ect Y col = 3, and @) MODL @LOGFI sele ct Mod el = Log f it L @ ) STAT @PLOT @ SCATR pr oduce scatter gr am of y v s. x @STATL sho w line for log f itting Ob v iou sly , the log-f it is not a good ch oi ce . @CANCL r eturns to no[...]

P age 18-2 2 L @ ) STAT @PLOT @ SCATR pr oduce scatter gr am of y v s. x @STATL sho w line for log f itting Θ T o r eturn to S T A T menu use: L @) STAT Θ T o get y our var iable menu bac k use: J . Confidence inter v als St atistical inf er ence is the proce ss of making conc lusi ons about a populati on based on info rmati on fr om sample data.[...]

P age 18-2 3 Θ P oint es timation: w hen a single value of the par amet er θ is pro v ided . Θ Co nfi dence interval: a n umer ical interv al that contains the par ameter θ at a gi ven le v el of pr obability . Θ E stimator : rule o r method of estimati on of the par ameter θ . Θ E stimate: v alue that the estimator y ields in a partic ular [...]

P age 18-2 4 Θ The par ameter α is kno wn as the si gnif icance le v el . T y pi cal v alues of α ar e 0. 01, 0. 05, 0.1, cor re sponding to conf idence le v els of 0.9 9 , 0.9 5, and 0.90, r especti vely . Confidence inter v als f or th e population mean w hen t he population v ariance is kno wn Let ⎯ X be the mean o f a random s ample of si [...]

P age 18-2 5 Small samples and large sampl es T he behav i or of the Student’s t distr ibution is suc h that for n>3 0, the distr ibution is indistinguishable fr om the standar d nor mal distribu tion . Th us, f or samples lar ger than 30 elements w hen the populati on var iance is unkno w n, y ou can use the same conf idence interval as w hen[...]

P age 18-2 6 E stimator s for the mean and s tandar d dev iation o f the diff er ence and sum of the statis tics S 1 and S 2 ar e gi v en b y: In t hese expressions, ⎯ X 1 and ⎯ X 2 ar e the v alues of the statis tics S 1 and S 2 fr om samples tak en fr om the t w o populati ons, and σ S1 2 and σ S2 2 ar e the v ar iance s of the populati ons[...]

P age 18-2 7 In this case , the cente red conf idence intervals f or the sum and diff er ence of the mean v alues of the populations , i .e ., μ 1 ±μ 2 , are gi ven b y : wher e ν = n 1 +n 2 - 2 is the number of degrees o f fr eedom in the Student’s t distr ibuti on. In the last tw o options w e spec ify that the population v ari ances, altho[...]

P age 18-2 8 These options ar e to be i nterpr eted as follow s : 1. Z -INT : 1 μ .: Single sample conf idence in te r v al fo r the population mean , μ , w ith kno wn populati on var iance , or for lar ge s amples w ith unkno wn populatio n v ari ance . 2. Z - I N T: μ1−μ2 .: Conf ide nce interval f or the differ ence o f the population mean[...]

P age 18-29 Press @HELP to obtain a sc r een e xplaining the meaning of the confi dence interval in terms o f r andom numbers gener ated by a calc ulator . T o scr oll do wn the r esulting sc r een use the do w n -ar r o w k ey ˜ . Pr ess @@@OK@@@ whe n done with the help sc r een. T his w ill r eturn y ou to the sc r een sho wn abo v e. T o calcu[...]

P age 18-30 Ex ample 2 -- Data f r om two s amples (s amples 1 and 2) indicat e that ⎯ x 1 = 5 7 .8 and ⎯ x 2 = 60. 0. The sample si z es ar e n 1 = 4 5 and n 2 = 7 5 . If it is kno w n that the populations ’ standar d dev iati ons ar e σ 1 = 3 .2 , and σ 2 = 4. 5, deter mine the 90% co nfi dence interval f or the diff er ence of the popula[...]

P age 18-31 When done , pre ss @@@OK@@@ . The r esults, as te xt and gr aph, ar e sho wn be lo w: Ex ample 4 -- Determine a 90% conf idence inter v al f or the differ ence between two pr oportions if sample 1 sho ws 20 su ccess es out of 120 tr ials , and sample 2 s ho ws 15 s uccesses out of 1 00 trial s . Press ‚Ù— @@@OK@@@ to access the con[...]

P age 18-3 2 Ex ample 5 – Determine a 9 5% conf idence in terval f or the mean of the populatio n if a s ample of 50 elements has a mean of 15 . 5 and a st andard de vi atio n of 5 . The popul ation ’s standar d dev iation is unkno wn . Press ‚Ù— @@@OK@@@ to access the confi dence inter v al f eatur e in the calc ulator . Pr ess —— @@@[...]

P age 18-3 3 T hese r esults assume that the v alues s 1 and s 2 ar e the population st andar d de vi ations . If these v alues actuall y r epr esent the s amples ’ standar d de v iatio ns, y ou should enter the s ame values as be for e, bu t wi th the option _pooled selected . T he r esults no w become: Confidence inter v als f or th e v ariance[...]

P age 18-34 T he confi dence interv al fo r the population v ari ance σ 2 i s therefor e , [(n -1) ⋅ S 2 / χ 2 n-1 , α /2 ; (n-1) ⋅ S 2 / χ 2 n-1,1- α /2 ]. wher e χ 2 n-1 , α /2 , and χ 2 n -1,1- α /2 ar e the value s that a χ 2 va riab le, wi th ν = n-1 degr ees of fr eedom , e x ceeds with pr obabiliti es α /2 and 1- α /2 , r es[...]

P age 18-35 Hy pot hesis testing A h ypo thesis is a declar ation made about a populati on (for ins tance , w ith r espect to its mean) . A cceptance of the h y pothesis is based o n a statisti cal test on a sample tak en fr om the population . The consequent acti on and dec isi on - making ar e called h y pothesis te sting . T he proce ss of h ypo[...]

Pa g e 1 8 - 3 6 Err ors in h ypothesis testing In h ypothe sis testing w e use the ter ms err ors of T y pe I and T y pe II to def ine the case s in w hich a tr ue h ypothe sis is re jec ted or a fals e h ypothe sis is accepted (not r ejected) , respect i vel y . Let T = val ue of test sta tistic, R = re ject i on region, A = acceptance r egion , [...]

P age 18-3 7 Th e va lu e of β , i .e ., the pr obability of making an er r or of T ype II , depends on α , the sample si z e n, and on the tr ue value o f the paramet er tes ted . Th us, the val ue of β is deter mined af t er the hy pothesis testing is perf ormed . It is c ust omary to dr a w gra phs sho w ing β , or the pow er of the test (1-[...]

P age 18-38 T he cr ite ri a to use f or h y pothesis te sting is: Θ Rej ec t H o if P -value < α Θ Do not r ej ect H o if P -value > α . T he P -v alue fo r a two -si ded tes t can be calc ulated using the pr obability f unctio ns in the calc ulator as f ollo w s: Θ If using z , P -value = 2 ⋅ UTPN(0,1,|z o |) Θ If using t , P -value[...]

P age 18-3 9 Ne xt, w e u se the P - v alue assoc iated w ith either z ο or t ο , and compar e it to α to dec ide w hether or no t to r ej ect the n ull hy pothesis. T he P - v alue f or a tw o -sided tes t is defined as e ither P -value = P(z > |z o |), or , P -value = P(t > |t o |) . T he cr ite ri a to use f or h y pothesis te sting is:[...]

P age 18-40 val ue s ⎯ x 1 and ⎯ x 2 , and st andard de vi ations s 1 and s 2 . If the populations standar d dev iati ons cor r esponding to the samples , σ 1 and σ 2 , ar e kno wn , or if n 1 > 30 and n 2 > 30 (la r ge sa mples) , th e test stati stic to be used is If n 1 < 30 o r n 2 < 30 (at least one small s ample) , u se the [...]

P age 18-41 T he cr ite ri a to use f or h y pothesis te sting is: Θ Rej ec t H o if P -value < α Θ Do not r ej ect H o if P -value > α . P aired sample tests When w e deal w ith tw o samples o f si z e n w ith pair ed data point s, ins tead of tes ting the null h y pothesis , H o : μ 1 - μ 2 = δ , using the mean v a l ues and st andar[...]

P age 18-4 2 wher e Φ (z) is the c umulativ e distributi on fu nctio n (CD F ) of the st andar d normal distr ibuti on (see Cha pter 17). R ejec t the null h ypothe sis, H 0 , if z 0 >z α /2 , or if z 0 < - z α /2 . In other w ords , the r ej ecti on r egi on is R = { |z 0 | > z α /2 }, whil e the acceptance r egion is A = {|z 0 | <[...]

P age 18-4 3 T wo - tail ed test If using a two -tailed test w e w ill find the v alue of z α /2 , fr om Pr[Z> z α /2 ] = 1- Φ (z α /2 ) = α /2 , or Φ (z α /2 ) = 1- α /2 , wher e Φ (z) is the c umulativ e distributi on fu nctio n (CD F ) of the st andar d normal distr ibuti on. R ejec t the null h ypothe sis, H 0 , if z 0 >z α /2 ,[...]

P age 18-44 1. Z - T est : 1 μ .: Single s ample hy pothesis testing f or the population mean , μ , w ith kno w n population v ar iance , or f or lar ge samples w ith unknow n populatio n v ari ance . 2. Z - Te s t : μ1−μ2 .: Hy pothesis tes ting for the diff er ence of the population means, μ 1 - μ 2 , with e ither kno wn populati on var i[...]

P age 18-45 Then , w e r ej ect H 0 : μ = 150 , against H 1 : μ ≠ 150 . The test z v alue is z 0 = 5. 656854 . T he P- va l u e i s 1. 54 × 10 -8 . Th e cri ti ca l va lu es of ± z α /2 = ± 1.9 5 9 9 64 , corr esponding to cr iti cal ⎯ x r ange of {14 7 .2 15 2 .8}. T his infor mation can be obse r v ed gra phicall y b y pre ssing the sof[...]

P age 18-46 W e r ej ect the null h ypothe sis, H 0 : μ 0 = 15 0, against the alter nati v e h ypo thesis , H 1 : μ > 15 0. The t est t v alue is t 0 = 5. 6 5 68 5 4, w ith a P -v alue = 0. 000000 3 9 3 5 2 5 . The c r itical v alue of t is t α = 1.6 7 6 5 51, cor r esponding to a cri tic al ⎯ x = 15 2 . 3 71. Press @GRAPH to see the re sul[...]

P age 18-4 7 T hus , w e accept (mor e acc urat ely , we do no t r ejec t) the h y pothesis: H 0 : μ 1 −μ 2 = 0 , or H 0 : μ 1 =μ 2 , against the alternati ve h y pothesis H 1 : μ 1 −μ 2 < 0 , or H 1 : μ 1 =μ 2 . The test t value is t 0 = -1. 3417 7 6 , w ith a P - v alue = 0. 0 91309 61, and cr itical t is –t α = -1.6 5 9 7 8 2 .[...]

P age 18-48 T he test c r iter ia ar e the same as in h y pothesis te sting of means , namely , Θ Rej ec t H o if P -value < α Θ Do not r ej ect H o if P -value > α . P lease noti ce that this pr ocedur e is valid onl y if the populati on fr om w hic h the sample w as tak en is a Normal populati on . Ex ample 1 -- Co nsider the case in w [...]

P age 18-4 9 T he follo w ing table sho ws h ow to select the nu merat or and denominator f or F o depending on the alter nati ve h ypothe sis cho sen: ___________ _____________________ _____________________ _______________ Alterna ti ve T est Degr e es h ypothe sis statis tic o f fr eedom ___________ _____________________ _____________________ ___[...]

P age 18-50 Ther efor e , the F test stati stics is F o = s M 2 /s m 2 =0. 3 6/0.2 5=1. 44 T he P -v alue is P -value = P(F>F o ) = P(F>1.44) = UTPF( ν N , ν D ,F o ) = UTPF( 20, 30,1.44) = 0.17 88 … Since 0.17 88… > 0 . 05, i .e ., P - v a l ue > α , ther ef or e , w e cannot re ject the null h ypothe sis that H o : σ 1 2 = σ [...]

P age 18-51 W e get the , so -called, nor mal equations: T his is a s y stem o f linear equati ons w ith a and b a s the unkno w ns, whi c h can be sol v ed using the linear equation f eature s of the calculator . T her e is, ho w ev er , no need to bother wi th these calc ulations becau se y ou can use the 3. Fit Data … option in the ‚Ù men u[...]

Pa g e 1 8 - 52 F r om w hic h it fo llow s that the standar d dev iations o f x and y , and the co var iance of x ,y ar e giv en , r espec tiv el y , by , , and Also , the sample corr elation coeff ic ient is In ter ms of ⎯ x, ⎯ y, S xx , S yy , and S xy , the soluti on to the no rmal equati ons is: , Prediction error T he r egr essi on c urve[...]

Pa g e 1 8 - 5 3 Θ Co nfi dence limits for r egres sion coeff i c ients: F or the slope ( Β ): b − (t n- 2 , α /2 ) ⋅ s e / √ S xx < Β < b + (t n- 2 , α /2 ) ⋅ s e / √ S xx , F or the inter cept ( Α ): a − (t n- 2 , α /2 ) ⋅ s e ⋅ [(1/n)+ ⎯ x 2 /S xx ] 1/2 < Α < a + (t n- 2 , α /2 ) ⋅ s e ⋅ [(1/n)+ ⎯ x 2[...]

P age 18-54 a+ b ⋅ x+(t n- 2 , α /2 ) ⋅ s e ⋅ [1+(1/n)+(x 0 - ⎯ x) 2 /S xx ] 1/2 . Pr ocedure f or inference statistics f or linear regression using the calculator 1) Ent er (x ,y) as columns of data in the st atistical matr ix Σ D AT. 2) Pr oduce a scatter plot f or the appr opri ate column s of Σ D A T , and use appr opri ate H- and V [...]

Pa g e 1 8 - 5 5 1: Covariance: 2.025 T hese r esults ar e interpr eted as a = -0.8 6 , b = 3 .2 4, r xy = 0.9 8 9 7 20 2 2 9 7 4 9 , and s xy = 2 . 0 2 5 . T he corr elati on coeff ic ient is c los e enough to 1. 0 to co nfir m the linear tr end obs erved in the gr aph . Fro m t he Single-var… option o f the ‚Ù menu w e f ind: ⎯ x = 3, s x [...]

P age 18-5 6 Ex ample 2 -- Suppos e that the y-data used in Ex ample 1 r e pr esent the elongation (in h undr edths of an inc h) of a me tal w ir e w hen sub jec ted to a f or ce x (in tens o f pounds) . The ph y sical phe nomenon is suc h that w e e xpect t he inter cept , A, to be z er o . T o chec k if that should be the ca se , w e test the nu [...]

P age 18-5 7 Multiple lin ear fitting Consi der a data set of the fo rm Suppo se that w e sear c h for a data f itting of the for m y = b 0 + b 1 ⋅ x 1 + b 2 ⋅ x 2 + b 3 ⋅ x 3 + … + b n ⋅ x n . Y ou can obtain the least-squar e appr ox imation to the values of the coeffi cients b = [b 0 b 1 b 2 b 3 … b n ], b y pu t ti ng together the m[...]

P age 18-5 8 W ith the calculat or , in RPN mode , yo u can pr oceed as fo llo ws: F irst , w ithin y our HOME dir ect ory , cr eate a sub-dir ect or y to be called MPFI T (Multiple linear and P oly nomial data FI Tting) , and ent er the M P FIT su b- dir ectory . Within the sub-dir ectory , t y pe this pr ogr am: «  X y « X TRAN X * INV X TRA[...]

P age 18-5 9 Compar e these f itted value s with the or iginal data as sho w n in the table belo w: P oly nomial fitting Consi der the x -y data set {(x 1 ,y 1 ), ( x 2 ,y 2 ), … , ( x n ,y n )}. Suppos e that we w ant to f it a poly nomial or or der p to this data set . In other w or ds, w e seek a f it ting of the f or m y = b 0 + b 1 ⋅ x + b[...]

P age 18-60 If p > n -1 , then add columns n+1, …, p-1, p+1 , to V n to f or m matri x X . In st ep 3 fr om this lis t , we hav e to be a war e that column i ( i = n+1, n+2 , …, p+1 ) is the v ector [x 1 i x 2 i … x n i ]. If w e w er e to use a list o f data values f or x r ather than a v ector , i .e ., x = { x 1 x 2 … x n }, w e can e[...]

P age 18-61 « Open pr ogram  x y p Enter l ists x and y , and p (le v els 3,2 ,1) « Open subpr ogram 1 x SI ZE  n Deter mine siz e of x list « Open subpr ogram 2 x V ANDERMOND E P lace x in stac k , obtain V n I F ‘ p<n -1’ THEN This IF implements st ep 3 in algorithm n P lace n in stac k p 2 + Calculate p+1 FOR j Start loop j = n -[...]

P age 18-6 2 Becau se w e w ill be using the same x -y data for f itting poly nomi als of diff er ent or ders , it is adv isable to s av e the lists of data v alues x and y into v ari ables xx and yy , re specti vel y . This w a y , we w ill not ha ve to t y pe them all o v er again in eac h applicati on of the pr ogr am P OL Y . Th us, pr oceed as[...]

P age 18-63 Θ T he corr elation coe ff ic ient , r . T h is value is constr ained to the r a nge –1 < r < 1. T he cl os er r is to +1 or –1, the better the data f itting. Θ T he sum of squar ed er ro rs, S SE . T his is the quantity tha t is to be minimi z ed by lea st-squar e appr oac h. Θ A plot of re siduals . T his is a plot of the[...]

P age 18-64 x V ANDERMOND E P lace x in stac k, obtain V n I F ‘ p<n -1’ THEN T his I F is s tep 3 in algor ithm n P lace n in stac k p 2 + Calc ulate p+1 FOR j S tar t loop , j = n-1 to p+1, step = -1 j C OL − D R OP R emo ve column , drop f r om stac k -1 S TEP Clos e FOR -S TEP loop ELSE I F ‘ p>n -1’ THEN n 1 + Calc ulate n+1 p [...]

P age 18-6 5 “SSE”  T A G T ag r esult as S SE » Close sub-progr am 4 » Clo se sub-pr ogram 3 » C lose su b-pr ogr am 2 » Clo se sub-pr ogr am 1 » Clo se main pr ogram Sa ve this pr ogr am under the name P OL YR , to emph asi z e calculati on of the correlation coeffic ient r . Using the POL YR progr am for v alues of p between 2 and 6 [...]

P age 19-1 Chapter 19 Numbers in Differ ent Bases In this Chapt er w e pre sent e x amples of calculati ons of number in bases other than the dec imal basis . Definitions T h e nu m b e r sys t e m u s e d fo r e ve r yd a y a ri t h m e t ic i s k n own a s t h e decimal syst em fo r it uses 10 (L atin , deca) digits , namely 0 -9 , to w r ite out[...]

P age 19-2 W ith sy st em flag 117 set to S OFT menus, the B A SE menu sho ws the f ollo w ing: W ith this for mat , it is ev ident that the L OGIC, BIT , and B YTE entri es w ithin the B ASE menu ar e th emselv es sub-menus. These menus are discussed later in this Chapter . Functions HEX, DEC, OCT , and B IN Number s in non-dec imal s ys tems ar e[...]

P age 19-3 As the dec imal (D E C) sy stem has 10 digits (0,1,2 , 3, 4,5, 6, 7 , 8 , 9) , the he xadec imal (HEX) sy stem has 16 digits (0, 1,2 , 3, 4 ,5,6 , 7 , 8 , 9 ,A,B ,C,D ,E ,F) , the octal (OCT) sy stem has 8 digits (0,1,2 , 3, 4,5, 6, 7) , and the binar y (BIN) s ys tem has only 2 di gits (0,1) . Conv ersion between number s ystems Whate v[...]

P age 19-4 T he only e ffec t of selecting the DE C imal s y stem is that dec imal numbers , whe n started w ith the s ymbol #, ar e wr itten with the suff ix d . W ordsi ze T he wor dsi z e is the number of b its in a b inar y obj ect . B y defa ult , the w ordsi z e is 64 bites . F uncti on RCW S (R eCall W ordSi z e) show s the c urr ent wor d s[...]

P age 19-5 The L OGIC m enu T he L OGIC menu , av ailable thr ough the B A SE ( ‚ã ) pr ov ides the f ollo wing fu nct ions : T he functi ons AND , OR, X OR (e x c lusi v e OR) , and NO T ar e logical f uncti ons. T he input to these f uncti ons ar e t w o v alu e s or e xpre ssi ons (one in the cas e of NO T) that can be e xpr esse d as binar y[...]

P age 19-6 AND (BIN) OR (BIN) X OR (BIN) NO T (HEX) T he B I T menu T he BIT men u , av ailable thr ough the B ASE ( ‚ã ) pr ov ide s the follo w ing fu nct ions : F uncti ons RL, SL , A SR, SR, RR , contained in the BIT menu , ar e used to manipulate b its in a binary integer . T he def initi on of thes e func tions are sho wn belo w : RL: R ot[...]

P age 19-7 T he B Y TE menu T he B YTE menu , av ailable thr ough the B A SE ( ‚ã ) pr o v ides the f ollo w ing fu nct ions : F uncti ons RLB, SLB , SRB , RRB, cont ained in the BIT menu , ar e used to manipulate b its in a binary integer . T he def initi on of thes e func tions are sho wn belo w : RLB: R otate Left one byte , e.g ., #110 0b ?[...]

Pa g e 2 0 - 1 Chapter 20 Customi zing menus and k e yboar d T hro ugh the use of the man y calc ulator menus y ou hav e become famili ar w ith the oper ati on of men us f or a v ar iety of appli catio ns. A lso , y ou ar e f amiliar w ith the man y func tions a vaila ble by u sing the k ey s in the ke yboar d , whether thr ough the ir main functi [...]

Pa g e 2 0 - 2 M enu numbers (R CLMENU and MENU func tions) E ac h pre-defined men u has a number attac hed to it . F or ex ample, su ppose that y ou acti vate the MTH menu ( „´ ). Then , using the f uncti on catalog ( ‚N ) f ind functi on R CLMENU and ac ti vate it. In AL G mode simple pres s ` after RCLMEN U() sho ws u p in the s cr een . Th[...]

Pa g e 2 0 - 3 T o acti vate an y of those functi ons y ou simply need to enter the f unction ar gument (a number ) , and then pr es s the corr es ponding soft menu k ey . In AL G mode , the list to be ent er ed as ar gument o f functi on TMENU or MENU is mor e complicated: {{“ exp ” , ”EXP(“},{“ln” , ”LN( “},{“Gamma ” , ”G AM[...]

Pa g e 2 0 - 4 Y ou can try using this list w ith TMENU or MENU in RPN mode to ve rify that y ou get the same menu a s obtained ear lier in AL G mode . M enu specification and CST v a r iable F r om the tw o e xer c ises sho wn abo v e w e notice that the most general men u spec if icati on list include a n umber of sub-lists equal to the number of[...]

Pa g e 2 0 - 5 Customizing the k e y board E ach k ey in the k e yboar d can be i dentif ied by tw o numbers r e pr esenting their r o w and column. F or e xam ple , the V AR ke y ( J ) is located in r o w 3 of column 1, and w ill be r ef er red t o as k ey 31. No w , since each k e y has up t o ten func tions as soc iated w ith it , eac h func tio[...]

Pa g e 2 0 - 6 T he functi ons av ailable ar e: A SN: Assigns an obj ect to a k e y spec ified b y XY .Z S T O KE Y S : Stor es user -d ef ined k e y l ist RC LK EY S : Ret urn s curren t use r-de fine d key l ist DELKEY S: Un-assigns one or mor e ke y s in the cu rr ent us er -def ined k ey lis t , the ar guments ar e either 0, to un -assi gn all [...]

Pa g e 2 0 - 7 Operating user-defined ke ys T o oper a t e this user -def ined k e y , enter „Ì be fo re pre ssing the C key . Notice that afte r pre ssing „Ì the sc r een sho w s the spec ifi cation 1US R in the second displa y line . Pr essing f or „Ì C for this e xample , y ou should r eco v er the P L O T menu as follo ws: If y ou hav [...]

Pa g e 2 0 - 8 T o un -assign all user -defined k ey s use: AL G mode: DELKE YS(0) RPN mode: 0 DELKEYS Chec k that the use r -k e y def initions w er e r emov ed b y using f unction R C LKE Y S.[...]

P age 21-1 Chapter 21 Pr ogr amming in User RP L language Use r RP L language is the pr ogramming language mo st commonl y used to pr ogr am the calculator . The pr ogram components can be put t ogether in the line editor by inc luding them betw een pr ogram container s « » in the appr opr iat e orde r . Because ther e is more e xperi ence among [...]

P age 21-2 „´ @LIST @ADD@ AD D Calc ulate (1+x 2 ), / / then di v ide ['] ~„x™ 'x' „° @) @MEM@@ @ ) @DIR@@ @ PURGE PURGE P u rge varia b le x ` Pr ogr am in lev el 1 ___________ ____________ ________ __ _____________________ T o sa v e the pr ogra m use: ['] ~„gK Press J to r ecov er your v ar iable menu , and ev alu[...]

P age 21-3 use a local v ar iable w ithin the pr ogram that is only de fi ned for that pr ogr am and w ill not be a v ailable fo r use afte r pr ogr am e xec ution . The pr e v iou s pr ogr am could be modifi ed to r ead: « → x « x SINH 1 x SQ ADD / »» T he arr ow s ymbol ( → ) is obtained b y combining the r i ght-shift k e y ‚ w ith the[...]

P age 21-4 Global V ariable Scope An y var iable that y ou def i ne in the HO ME dir ectory or an y other dir ecto r y or sub-dir ectory w ill be consider ed a global var iable fr om the point o f vi ew of pr ogr am dev elopment . Ho we v er , the sco pe of suc h v ari able , i .e ., the locati on in the dir ecto r y tr ee w her e the var iable is [...]

P age 21-5 Local V ariable Scope L ocal var iable s are ac tiv e only w ithin a pr ogr am or sub-pr ogram . The r ef or e , their scope is limited t o the pr ogr am or sub-pr ogram w her e the y’r e def ined . An e x ample of a local v ari able is the inde x in a F OR loop (des cr ibed late r in this chapter ) , f or e x ample « → n x « 1 n F[...]

P age 21-6 S T ART : S T AR T -NEXT -S TEP constru ct f or br anching FOR: F O R - NE XT- S TEP constr uct f or loops DO: DO-UNT IL -END constr uct f or loops WHILE: WHILE-REP EA T-END co nstru ct f or loops TE S T : Compar iso n operator s, logi cal oper ators , flag t esting f unctio ns TYPE: F unctions f or conv erting objec t types , splitting [...]

P age 21-7 Functions listed b y sub-menu T he follo wing is a lis ting of the func tions w ithin the P RG sub-menus lis ted b y sub- menu . ST A CK MEM/DIR BR CH/IF BRCH/WHILE TYP E DUP P URGE IF WHILE OB J  SW A P RC L T H E N R E PE A T  ARR Y DR OP S T O EL SE END  LIS T O VER P A TH END  ST R RO T CRD IR TE ST  TAG UNRO T PGDIR B[...]

P age 21-8 LIS T/ELEM GROB CHARS MODES/FLAG MO DES/MISC GE T  GROB S UB SF BEEP GE TI BL ANK REP L CF CLK PU T GO R POS F S ? S Y M PU TI G X O R SIZE F C ? S T K S IZE SUB NUM F S?C ARG P O S REPL CH R F S?C CMD HEAD  LC D O B J  FC?C INF O TA I L L C D  ST R STO F SIZE H EA D RC LF IN LIS T/PR OC ANIMA TE T AIL RE SET INF ORM DOLIS[...]

P age 21-9 Shortc uts in the PR G menu Man y of the func tions lis ted abo ve f or the P RG menu ar e r eadily a v ailable thr ough other means: Θ Compar ison oper ators ( ≠ , ≤ , <, ≥ , >) ar e a vailable in the k ey boar d. Θ Man y func tions and settings in th e MODE S sub-me nu can be acti v ated b y using the input f uncti ons pr[...]

P age 21-10 „ @ ) @IF@ @ „ @CASE@ „ @ ) @IF@ @ „ @CASE@ „ @ ) START „ @) @ FOR@ „ @ ) START „ @) @ FOR@ „ @ ) @@DO@@ „ @ WHILE Notice that the ins ert pr ompt (  ) is av ailabl e after the k e y w or d fo r each constr uct s o yo u can start t y ping at the r ight locatio n. K e y strok e sequence for commonl y used commands [...]

P age 21-11 @) STACK DUP „° @) STACK @ @DUP@@ SW A P „° @) STACK @SWAP@ DR OP „° @) STACK @DROP@ @) @MEM@@ @ ) @DIR@@ PU RG E „° @) @MEM@@ @ ) @ DIR@@ @PUR GE ORDER „° @) @MEM@@ @ ) @DIR@@ @ORDER @) @BRCH@ @ )@IF@@ IF „° @) @BRCH@ @ ) @IF@@ @@@IF@@@ THEN „° @) @BRCH@ @ ) @IF@@ @THEN@ ELSE „° @) @B RCH@ @ ) @ IF@@ @ELSE @ END[...]

P age 21-12 @) @BRCH@ @ ) WHILE@ WHILE „° @) @B RCH@ @ ) WHILE @ @WHILE REP EA T „° ) @BRCH@ @ ) WHILE@ @REPEA END „° ) @BRCH@ @ ) WHILE@ @ @END@ @ ) TEST@ == „° @ ) TEST@ @ @@ ≠ @@@ AND „° @ ) TEST@ L @@AND @ OR „° @ ) TEST@ L @@@OR@@ XO R „° @ ) TE ST@ L @@XOR@ NO T „° @ ) TEST@ L @@NOT@ SA M E „° @ ) TEST@ L @SAME SF[...]

P age 21-13 @) LIST@ @ ) PROC@ REVLI S T „° @) LIST@ @ ) PROC@ @REVLI@ SO RT „° @) LIST@ @ ) PROC@ L @SORT@ SE Q „° @) LIST@ @ ) P ROC@ L @@SEQ@@ @) MODES @ ) ANG LE@ DE G „°L @) MODES @ ) A NGLE@ @@DE G@@ RAD „°L @) MODES @ ) ANGLE@ @ @RAD@@ @) MODES @ ) MEN U@ CS T „°L @) MODES @ ) MENU@ @@CST@ @ MENU „°L @) MODES @ ) M ENU@ [...]

P age 21-14 fu nctio ns from th e M TH m enu . Spe c ifica lly , you ca n use fun ction s for li st oper ations suc h as S ORT , Σ LIS T , etc ., a vail able thr ough the MTH/LI S T menu . As additional pr ogramming e xer cis es, and to try the ke ystr ok e seque nces listed abo v e , we pr esent her ein thr ee pr og r ams for c r eating or manipu[...]

P age 21-15 Ex amples of sequential pr ogramming In gener al , a pr ogr am is an y sequence o f calc ulato r instruc tions enc lo sed between the pr ogram container s and ». Subpr ograms can be inc luded as part o f a pr ogr am. The e xamples pr esented pr e v iou sly in this guide (e .g., in Chapt ers 3 and 8) 6 can be cla ssif ied ba sicall y in[...]

P age 21-16 wher e C u is a constant that depends on the sy st em of units used [C u = 1. 0 for units of the Internati onal S ys tem (S.I .) , and C u = 1.4 8 6 f or units o f the English S y ste m (E . S .)], n is the Manning’s r esist ance coeff ic ient , whi ch depends on the type of c hannel lining and other f actor s, y 0 is the flo w depth,[...]

P age 21-17 Y ou can also separ ate the in put data w ith spaces in a single stac k line r ather than using ` . Pr ograms that simulate a sequence of stack operations In this case , the terms to be in v olv ed in the sequence o f oper ations ar e as sumed to be pr es ent in the stac k . The pr ogram is ty ped in by f ir st opening the pr ogr am con[...]

P age 21-18 As y ou can see , y is used f i r st , then w e us e b, g , a n d Q, in that order . Ther efor e, for the pur pose of this calculatio n we need to enter the v ar iables in the in ve rse or der , i .e. , (do not t y pe the f ollo w ing) : Q ` g ` b ` y ` F or the spec if ic v alues under consider ation w e use: 23 ` 32. 2 ` 3 ` 2 ` T he [...]

P age 21-19 Sa ve the pr ogram int o a var iable called hv: ³~„h~„v K A ne w var iable @@@hv @@@ should be av ailable in y our soft k e y menu . (Pr ess J to see y our v ar iable lis t .) The pr ogram le ft in the stac k can be e valuat ed by u sing func tion EV AL. T he r esult should be 0.2 2 8 17 4…, as befor e. Als o , the progr am is av[...]

P age 21-20 it is al wa y s pos sible to r ecall the pr ogr am def inition int o the stac k ( ‚ @@@q@@@ ) to see the or der in w hic h the v ari ables mu st be ent er ed , namely , → Cu n y0 S0 . Ho w ev er , f or the case of the pr ogram @@hv@@ , its def inition « * SQ * 2 * S W AP SQ S W AP / » does not pr o v ide a c lue of the or der in w[...]

P age 21-21 w hich indi cates the positi on of the diff er ent stac k input le vels in the fo rmula . B y compar ing this r esult w ith the or iginal f ormula that w e pr ogr ammed , i .e ., w e find that w e mu st enter y in s tack le vel 1 (S1), b in stac k lev el 2 (S2), g in stac k le v el 3 (S3) , and Q in st ack le vel 4 (S4). Pr ompt with an[...]

P age 21-2 2 T he re sult is a stac k pr ompting the user f or the value o f a and plac ing the cu rsor r ight in fr on t of the prompt :a: Ent er a value f or a , sa y 3 5, then pre ss ` . T he r esult is the input s tring :a:35 in stac k lev el 1. A function with an input string If y ou w er e to use this p iece o f code to calculate the functi o[...]

P age 21-2 3 @SST ↓ @ R esult: em pt y s tack , e x ec uting → a @SST ↓ @ R esult: empty stac k, ente ring subpr ogr am « @SST ↓ @ R esult: ‘2*a^2+3’ @SST ↓ @ R esult: ‘2*a^2+3’ , leav ing subpr ogram » @SST ↓ @ R esult: ‘2*a^2+3’ , leav ing main pr ogr am» F urther pr essing the @SST ↓ @ s oft menu k e y pr oduces no m[...]

P age 21-2 4 F ixi ng th e pr ogram T he only pos sible explanati on f or the failur e of the pr ogr am to pr oduce a numer ical r esult seems to be the lac k of the command  NUM after the algebr aic e xpr essi on ‘2*a^2+3’ . Let ’s edit the progr am by adding the mis sing EV AL functi on . T he progr am, after editing , should read as f o[...]

P age 21-2 5 Input string progr am for two input v alues T he input str ing pr ogr am fo r t w o input values , say a and b , looks as f ollo ws: « “ Enter a and b: “ { “  :a:  :b: “ {2 0} V } INPUT OBJ → » T his progr am can be ea sily c r eated b y modif y ing the contents o f INPT a. St or e this pr ogr am into v ar iable INP T[...]

P age 21-2 6 ` . The r esult is 4 9 88 7 . 06_J /m^3 . The units of J/m^3 ar e equiv alent to P ascals (P a) , the pr ef err ed pres sur e unit in the S .I. s y stem . In pu t st ring prog ram for th ree i npu t val ues T he input str ing pr ogr am f or thr ee input value s, sa y a ,b , and c, loo ks as fo llo w s: « “ Enter a, b and c: “ { ?[...]

P age 21-2 7 Enter v alues o f V = 0. 01_m^3, T = 300_K , and n = 0.8_mol . Bef or e pr es sing ` , the stac k w ill look like this: Press ` to get the re sult 199 5 4 8.2 4_J/m^3, or 199 5 48.2 4_ P a = 199 .5 5 kP a. Input through input f orms F uncti on INFORM ( „°L @) @@IN@ @ @INFOR@ .) can be used to c r eate detailed input f orms f or a pr[...]

Pa g e 2 1 - 2 8 T he lists in items 4 and 5 can be em pty lists. Also , if no v alue is to be select ed for these opti ons y ou can use the NO V AL command ( „°L @) @@IN@ @ @NOVAL@ ). After f unction INFORM is acti vated y ou will get as a r esult either a z er o , in case the @CANCEL option is en ter ed , or a list w ith the v alues ente r ed [...]

P age 21-29 3 . F ield f ormat info rmation: { } (an empty lis t , thus , defa ult value s used) 4. L ist of r eset val ues: { 120 1 .0001} 5 . L ist of initial v alues: { 110 1.5 .00001} Save th e prog ram i nto va riab le IN F P1 . P ress @INFP 1 to run the pr ogram . T he input f orm , w ith initial v alues loaded , is as follo ws: T o see the e[...]

P age 21-30 T hus , we demonstr ated the u se of f uncti on INFORM. T o see h o w to use the se input v alues in a calc ulation modify the pr ogr am as follo ws: « “ CHEZY’S EQN” { { “ C:” “Chezy’s coefficient” 0} { “R:” “Hydraulic radius” 0 } { “S:” “Channel bed slope” 0} } { } { 120 1 .0001} { 110 1.5 .000 01 } IN[...]

P age 21-31 « “ CHEZY’S EQN” { { “C:” “Chezy’s coefficient” 0} { “R:” “Hydraulic radius” 0 } { “S:” “Channel bed slope” 0} } { 2 1 } { 120 1 .0001} { 110 1.5 .00001 } INFORM IF THEN OBJ  DROP  C R S ‘C*(R*S)’  NUM “Q”  TAG ELSE “Operation cancelled” MSGBOX END » R unning pr ogr am @INFP2 pr [...]

P age 21-3 2 Ac tiv ati on of the CHOO SE func tion w ill re turn e ither a z er o, if a @CANCEL ac ti on is used , or , if a c hoi ce is made , the ch oi ce s elect ed (e .g., v) and the numbe r 1, i .e ., in the RPN stac k: Ex ample 1 – Manning ’s equation f or calc ulating the v eloc ity in an open ch an nel fl o w in clu de s a co ef fic ie[...]

P age 21-3 3 commands “Operation canc elled” MSGBOX w ill sho w a message bo x indicating that the oper ation w as cancelled. Identif y ing output in pr ograms T he simplest w ay to identify numer ical pr ogr am output is to “tag ” the pr ogr am r esults . A tag is simply a str ing attached to a numbe r , o r to a n y objec t . The str ing [...]

P age 21-34 Ex amples of tagged output Ex ample 1 – tagging output fr om function FUNC a Let ’s modif y the f uncti on FUNCa, de f ined earlier , to pr oduce a tagged output . Use ‚ @FUNCa to r ecall the contents of FUNCa to the s tack . T he ori ginal func tion pr ogram r eads « “ Enter a: “ { “  :a: “ {2 0} V } INPU T OBJ →→[...]

Pa g e 2 1 - 3 5 « “ Enter a: “ { “  :a: “ {2 0} V } INPUT OBJ →→ a « ‘ 2*a^2+3 ‘ EVAL ” F ” → TAG a SWAP »» (R ecall that the functi on S W AP is av ailable b y using „° @) STACK @SW AP@ ). Stor e the pr ogram bac k into FUNCa b y using „ @FUNCa . Ne xt , run the pr ogr am by pr essing @FUNCa . En ter a v alue of [...]

Pa g e 2 1 - 3 6 Ex ample 3 – tagging input and outpu t fr om f uncti on p(V ,T) In this e xample w e modify the pr ogr am @@@p@@@ so that the o utput tagged input v alues and t agged r esult . Use ‚ @@@p@@@ t o recall the conte nts of the pr ogr am to the st ack: « “ Enter V, T, and n: “ { “  :V:  :T:  :n: “ {2 0} V } INPUT O[...]

P age 21-3 7 Stor e the progr am back into var ia ble p by using „ @@@p@@@ . Ne xt , r un the pr ogr am by pr essing @@@p@@@ . Ent er value s of V = 0. 01_m^3, T = 30 0_K, and n = 0.8_mol , when pr ompted . Bef or e pre ssing ` for input , the stack w ill look lik e this: After e xec uti on of the pr ogr am , the stac k w ill look lik e this: Usi[...]

P age 21-38 T he r esult is the f ollo w ing message bo x: Press @@@OK@@@ to c ancel the mes sage bo x . Y ou could us e a message bo x for o utput fr om a progr am b y using a tagged output , con verted to a s tring , as the output str ing f or MS GBO X. T o con v ert any tagged r esult , or any algebr ai c or non- tagged v alue , to a str ing , u[...]

P age 21-3 9 Press @@@OK@@@ to cancel message b o x output . The stack w ill now look like this: Including input and output in a m essage bo x W e could modify the pr ogram so that not onl y the output , but also the input , is inc luded in a message bo x . F or the case of pr ogram @@@p@@@ , the modifi ed pr ogr am wi ll look lik e: « “ Enter V[...]

P age 21-40 Y ou w ill notice that after ty ping the k e ys tr ok e sequence ‚ë a ne w line is gener a t ed in the stac k. T he last modif icati on that needs to be included is to type in the plu s sign three times after the call t o the functi on at the v ery e nd of the sub-pr ogram . T o see the pr ogr am oper ating: Θ S tor e the pr ogr am [...]

P age 21-41 Incorpor ating units within a program As y ou ha ve bee n able to obse r v e fr om all the ex amples f or the diffe r ent vers ion s of pro gram @@@p@@@ pr es ented in this cha pter , attac hing units to input v alues may be a t ediou s pr ocess . Y ou could ha v e the pr ogr am itself attach those units to the input and output v alues [...]

P age 21-4 2 2. ‘ 1_m^3 ’ : T he S .I. un its corr espo nding to V ar e then placed in stac k lev el 1, the tagged input f or V is mo v ed to stack lev el 2 . 3 . * : B y multiply ing the contents of st ack le vels 1 and 2 , w e gener ate a number w ith units (e .g ., 0. 01_m^3) , but the ta g is lost . 4. T ‘ 1_K ’ * : Calc ulating v alue [...]

P age 21-4 3 Press @@@OK@@@ to cancel me ssage bo x output . Me s sag e bo x output without units Let ’s modify the progr a m @@@p@@@ once mor e to eliminate the us e of units thr oughout it . The unit-less pr ogram w ill look like this: « “ Enter V,T,n [S.I.]: “ { “  :V:  :T:  :n: “ {2 0} V } INPUT OBJ →→ V T n « V DTAG T [...]

P age 21-44 oper ators ar e used to mak e a statement r egarding the r elativ e position of t w o or mor e r eal numbers . Depending on the ac tual numbers us ed, su ch a st atement can be true (r epr es ented b y the numer i cal value o f 1. in the calc ulator ) , or fals e (r epr ese nted by the numer ical value of 0. in the calc ulator ) . T he [...]

P age 21-45 Logical oper ators L ogical oper ator s ar e logical partic les that ar e used to jo in or modify simple logical s tatements . The logical ope rat ors a vaila ble in the calculat or can be easily acc essed thr ough the ke ys trok e sequence: „° @ ) TEST@ L . T he av ailable logi cal oper ator s ar e: AND , OR, X OR (e xc lusi ve or )[...]

Pa g e 2 1 - 4 6 T he calculat or include s also the logi cal oper ator S AME . This is a non-standar d logical ope rat or used t o deter mine if two ob jec ts ar e identi cal . If they are identi cal , a value o f 1 (true) is r eturned , if no t, a value of 0 (f alse) is r etur ned. F or ex ample , the f ollo wing e xer cis e , in RPN mod e , re t[...]

P age 21-4 7 Br anc hing w ith I F In this secti on w e pr esen ts e xample s using the constr ucts IF…THEN…END and IF…THEN…ELSE…END . T he I F…THEN…END construct T he IF…THEN…END is the simplest of the IF pr ogr am constr ucts . The gener al fo rmat of this co nstruc t is: IF logical_statement THEN program_statements END . T he o[...]

P age 21-48 W ith the cur sor  in fr ont of the IF st atement pr ompting the user f or the logical stat ement that wi ll acti vate the I F cons truct when the pr ogr am is e xec ut ed. Ex ample : T y pe in the follo w ing pr ogr am: « → x « IF ‘ x<3 ’ THEN ‘ x^2 ‘ EVAL END ” Done ” MSGBOX » » and sa v e it under the name ‘[...]

P age 21-4 9 Ex ample : T y pe in the f ollo w i ng pr ogram: « → x « IF ‘ x<3 ’ THEN ‘ x^2 ‘ ELSE ‘ 1-x ’ END EVAL ” Done ” MSGBOX » » and sa v e it under the name ‘f2 ’ . Pre ss J and v er ify that var iable @@@f2@@@ is indeed av ailable in your var ia ble menu . V er ify the follo wing r esults: 0 @@@f2@@@ Result: 0 [...]

P age 21-50 IF x<3 THEN x 2 ELSE 1-x END While this simple cons truc t w orks f ine w hen y our f uncti on has onl y tw o br anche s, y ou ma y need to nes t IF…THEN…ELSE…END constru cts to deal w ith func tion w ith three or mor e branc hes . F or e xample , conside r the functi on Her e is a possible w a y to e valuate this f uncti on us[...]

P age 21-51 A comple x IF construc t like this is called a set o f n ested IF … THEN … EL SE … END constr ucts . A poss ible wa y to e valuate f3(x), based on the nested IF constr uct sho wn abo ve , is to w rite the pr ogr am: « → x « IF ‘ x<3 ‘ THEN ‘ x^2 ‘ ELSE IF ‘ x<5 ‘ THEN ‘ 1-x ‘ ELSE IF ‘ x<3* π ‘ TH[...]

Pa g e 2 1 - 52 pr ogr am_stateme nts , and pa sses pr ogram f lo w to the statement f ollow ing the END state ment. T he CASE , THEN, and END st atements ar e a vailable f or selecti ve typ ing by using „° @) @ BRCH@ @ ) CASE@ . If y ou ar e in the BRCH menu , i .e., ( „° @) @ BRCH@ ) y ou can use the f ollo w ing shortc uts to type in y our[...]

Pa g e 2 1 - 5 3 5. 6 @@ f3c@ Re su l t : -0.6 312 6 6… (i .e ., sin(x) , w ith x in r adians) 12 @@f3c@ Res ul t : 16 2 7 54.7 91419 (i.e ., e xp(x)) 23 @@f3c@ Res ul t - 2 . (i .e ., - 2) As yo u can see , f3c produces e xactl y the same r esults as f3 . The onl y diffe r ence in the pr ogr ams is the branc hing constr ucts u sed . F or the cas[...]

P age 21-54 Commands in v ol ved in the S T AR T constru ct ar e av ailable thr ough: „° @) @BRCH@ @ ) START @ST ART W ithin the BRCH men u ( „° @) @BRCH@ ) the follo wi ng ke ys tr ok es ar e a vailabl e to gener ate S T AR T construc ts (the s y mbol indicates c ur sor positi on) : Θ „ @START : S tarts the S T ART…NE XT constru ct: S T[...]

Pa g e 2 1 - 5 5 1. T his pr ogr am needs an integer numbe r as inpu t . Th us , bef or e e xec utio n, that number (n) is in st ack le v el 1. The pr ogram is the n ex ec uted . 2 . A z er o is ente r ed , mov ing n to stac k lev el 2 . 3 . T he command DUP , w hi ch can be typed in a s ~~dup~ , copi es the contents of s tack le v el 1, mo ves all[...]

P age 21-5 6 „°LL @) @RUN@ @@DBG@ Start the debugger . SL1 = 2 . @SST ↓ @ SL1 = 0., SL2 = 2 . @SST ↓ @ SL1 = 0., SL2 = 0. , SL3 = 2 . (DUP) @SST ↓ @ Empty stac k (-> n S k) @SST ↓ @ Empty stac k ( « - s tart subpr ogr am) @SST ↓ @ SL1 = 0., (s tart value of loop inde x) @SST ↓ @ SL1 = 2 .(n) , SL2 = 0. (end v alue of loop inde x)[...]

P age 21-5 7 @SST ↓ @ SL1 = 1. (S + k 2 ) [Stor es value o f SL2 = 2 , into SL1 = ‘k ’] @SST ↓ @ SL1 = ‘S’ , SL2 = 1. (S + k 2 ) @SST ↓ @ Empty st ack [S tor es value of SL2 = 1, int o SL1 = ‘S’] @SST ↓ @ Empty stac k (NEXT – end of loop) --- loop e xec ution n umber 3 f or k = 2 @SST ↓ @ SL1 = 2 . (k) @SST ↓ @ SL1 = 4. (S[...]

P age 21-5 8 3 @@@S1@@ Re su lt : S:14 4 @@@S1@@ Res ul t: S:30 5 @@@S1@@ Re su lt : S:55 8 @@@S1@@ Res ul t: S:204 10 @@@S1@@ Res ul t: S:385 20 @@@S1@@ Res u lt : S:2870 30 @@@S1@@ Res ul t: S:9455 100 @@@S1@@ Re su l t: S:338350 The ST ART…STEP construct T he gener al fo rm of this statemen t is: start_value end_value START program_statements [...]

P age 21-5 9 J 1 # 1. 5 # 0.5 ` Enter par ame ters 1 1. 5 0. 5 [ ‘ ] @GLIST ` En ter the pr ogr am name in lev el 1 „°LL @) @RUN@ @@DBG@ S tart the debugger . Use @SST ↓ @ t o step into the pr ogr am and see the detailed ope rati on of eac h command . T he FOR construct As in the case of the S T AR T command, the F OR command has tw o v ari [...]

P age 21-60 T o av oid an inf inite loop , mak e sur e that start_value < end_value . Ex ample – ca lc ulate the summation S using a F OR…NEXT construc t T he follo w ing pr ogram calc ulates the summation Using a FOR…NEXT loop : « 0 → n S « 0 n FOR k k SQ S + ‘ S ‘ STO NEXT S “ S ” → TAG » » Stor e this pr ogram in a v ar [...]

P age 21-61 Ex ample – gener ate a list of number s using a FOR…S TEP construc t T ype in the pr ogram: « → xs xe dx « xe xs – dx / ABS 1. + → n « xs xe FOR x x dx STEP n → LIST » » » and stor e it in var ia ble @GLIS2 . Θ Chec k out that the pr ogr am call 0. 5 ` 2. 5 ` 0.5 ` @ GLIS2 pr oduces the list {0. 5 1. 1.5 2 . 2 . 5}. ?[...]

P age 21-6 2 T he follo w ing pr ogram calc ulates the summation Using a DO…UNTIL…END loop: « 0. → n S « DO n SQ S + ‘ S ‘ STO n 1 – ‘ n ‘ STO UNTIL ‘ n<0 ‘ END S “ S ” → TAG » » Stor e this pr ogram in a v ar iable @@ S3@@ . V er ify the follo wing e xe r c ises: J 3 @@@S3@@ Re su lt : S:14 4 @@@S3@@ Res ul t: S:30[...]

Pa g e 2 1 - 6 3 T he WHILE construct T he gener al str uctur e of this command is: WHILE logical_statement REPEAT program_statements END T he WHILE stateme nt w ill r epeat the program_statements wh il e logical_statement is tr ue (non z er o) . If not , pr ogram contr ol is pa ssed to the stat ement r ight afte r END . The program_statements must[...]

P age 21-64 and stor e it in var ia ble @GLIS4 . Θ Chec k out that the pr ogr am call 0. 5 ` 2. 5 ` 0.5 ` @ GLIS4 pr oduces the list {0. 5 1. 1.5 2 . 2 . 5}. Θ T o see st ep-by-step oper ation u se the pr ogr am D B UG for a short list , for e xample: J 1 # 1. 5 # 0.5 ` Enter par ame ters 1 1. 5 0. 5 [‘] @GLIS4 ` Enter the pr ogram name in lev [...]

P age 21-6 5 If y ou enter “ TR Y A G AIN” ` @ DOERR , p r oduc e s the follow ing m essage: TR Y AGA I N F inally , 0` @DOERR , pr oduces the messa ge: I nter rupted ERRN T his functi on r etur ns a number r epr es enting the most r ecent err or . F or e xample , if y ou try 0Y$ @ERRN , y ou get the n umber #30 5h . This is the binary integer [...]

P age 21-66 T hese ar e the components of the IFERR … THEN … END construc t or of the IFERR … THEN … EL SE … END constr uct . Both logi cal constr ucts ar e used fo r tr appi ng err or s dur ing pr ogr am ex ec uti on . Within the @) ER ROR sub-men u , enter ing „ @) IFERR , or ‚ @) IFERR , w ill place the IFERR s truc tur e component[...]

P age 21-6 7 User RP L pr ogramming in algebraic mode While all the pr ogr ams pre sent ed earli er are pr oduced and run in RPN mode , y ou can al wa y s type a pr ogr am in Us er RPL w hen in algebrai c mode b y using func tion RP L>. T his functi on is a vaila ble thr ough the command catalog . As an e x ample , try cr eating the follo wing p[...]

P age 21-6 8 Wher eas , using RP L, ther e is no proble m when loading this pr ogram in algebr aic mode:[...]

Pa g e 22 - 1 Chapter 2 2 Pr ogr ams for gr aphic s manipulation T his chapt er include s a number of e x amples sho w ing ho w to use the calculat or’s func tions f or manipulating gr aphics int er acti v el y or thr ough the us e of pr ogr ams. As in Cha pter 21 w e r ecommend u sing RPN mode and setting s ys tem f lag 117 to S OFT menu labels.[...]

Pa g e 22 - 2 T o us er -def ine a k e y yo u need to add to this list a command or pr ogram fo llo w ed by a r efer ence to the k e y (see details in C hapter 20) . T y pe the list { S << 81.01 M ENU >> 13.0 } in the stac k and use f uncti on S T OKEY S ( „°L @) MODES @ ) @ KEYS@ @@ STOK@ ) to user -d ef ine k ey C as the acc e ss t[...]

Pa g e 22 - 3 LA BE L (10) T he functi on L ABEL is us ed to label the ax es in a plot including the v ar iable names and minimum and max imum value s of the axe s. T he var ia ble names ar e select ed fr om info rmatio n contained in the var ia ble PP AR. AU TO ( 1 1 ) T he func tion A UT O (A UT Oscale) calc ulates a dis play r ange for the y-ax [...]

Pa g e 22 - 4 EQ ( 3) T he var ia ble name EQ is r es er v ed by the calc ulator to stor e the c urr ent equatio n in plots or solut ion to eq uations (s ee chapt er …) . T he soft menu k ey la beled E Q in this menu can be us ed as it w ould be if y ou hav e y our v ar iable men u av ailable , e .g., if y ou pr es s [ E Q ] it w ill lis t the c [...]

Pa g e 22 - 5 T he follo w ing diagr am illu str ates the f uncti ons av ailable in the P P AR menu . T he letter s attached to eac h f unction in the di agr am ar e used f or r ef er ence purpos es in the desc ripti on of the func tions sho wn belo w . INFO (n) and PP AR (m) If y ou pr ess @INFO , or enter ‚ @PPAR , while in this menu , you w il[...]

Pa g e 22 - 6 INDEP (a) T he command IND EP spec ifi es the independent v ar iable and its plotting r ange . T hese spec ifi cations ar e stor ed as the thir d paramet er in the v ar ia ble PP AR. T he def ault v alue is 'X'. T he v alues that can be assigned t o the independent var iable spec if icati on ar e: Θ A v ari able name , e .g[...]

Pa g e 22 - 7 CENTR (g) T he command CENTR tak es as ar gument an or der e d pair (x ,y) or a value x , and adju sts the fi rst tw o elements in the v ari able P P AR, i .e., (x min , y min ) and (x max , y max ) , s o that the center o f the plot is (x ,y) or (x , 0) , r especti v el y . S CALE (h) T he SCALE command dete rmines the plotting scale[...]

Pa g e 22 - 8 A list o f two b inar y intege rs {#n #m}: sets the ti c k annotations in the x - and y- ax es to #n and #m pi xels , r espec tiv el y . AXE S (k) T he input value f or the axes command consis ts of e ither an order ed pair (x,y) or a list {(x ,y) atic k "x-ax is label" "y-ax is label"}. The par ameter atick s tand[...]

Pa g e 22 - 9 The PTYP E menu within 3D (IV) T he PTYP E menu under 3D cont ains the follo w ing functi ons: T hese f uncti ons corr espond to the gr aphi cs opti ons Slopef ield , Wir efr ame , Y - Slice , P s-Contour , Gri dmap and Pr -Sur f ace pre sented ear lie r in this chapt er . Pr essing one o f these s oft menu k e y s , while ty ping a p[...]

Pa ge 22- 1 0 XV OL (N) , YV OL (O) , and ZV OL (P) T hese f unctions t ake as input a minimum and maxi mum value and ar e used to spec ify the extent o f the parallelep iped wher e the gr aph w ill be gener ated (the v ie w ing par allelepiped). Thes e values ar e s tor ed in the v ar iable VP AR . The def ault values f or the r anges XV OL , YV O[...]

Pa ge 22- 1 1 The S T A T menu within PL O T T he S T A T menu pr o v ide s access to plots r elated to st atistical anal y sis. W ithin this menu w e find the f ollo wing men us: T he diagr am belo w sho ws the br anc hing of the S T A T me nu w ithin P L O T . The numbers and let t ers accompan ying eac h f unction or menu ar e us ed f or r ef er[...]

Pa ge 22- 1 2 The P T YP E m enu wi thin ST A T (I) The P TYP E menu pr o v ides the f ollo wing f uncti ons: The se ke ys cor res pond to the p lot ty pes Bar (A ) , H istogr am (B) , and Scatter (C) , pr esented ear lier . Pr essing one of these soft menu k ey s, w hile typing a pr ogr am, w ill place the corr esponding f uncti on call in the pr [...]

Pa ge 22- 1 3 X COL (H) T he command XC OL is used t o indicate w hi ch o f the columns of Σ D A T , if mor e than one , w ill be the x - column or independent var iable column . YC O L ( I ) T he command Y COL is used to indicate w hic h of the columns of Σ DA T , i f m o re than one , w ill be the y- column or dependent v ar iable column . MODL[...]

Pa ge 22- 1 4 Θ SIMU: w hen selec ted , and if more than one gr aph is to be plotted in the same set o f axe s, plots all the gr aphs simultaneousl y . Press @) PLOT to r eturn to the PL O T menu . Generating plots w ith progr ams Depending on w hether w e ar e dealing w ith a tw o -dimensional gr aph def ined by a fun ctio n, by d at a from Σ D [...]

Pa ge 22- 1 5 T hree -dimensional gr aphics T he thr ee -dimensional gr aphi cs a vaila ble , namel y , options Slopef ield , Wir efr ame , Y -Sli ce , P s-Contour , G r idmap and Pr -Surface , use the VP AR v ar iabl e w ith the fol low ing fo rmat: { x left , x right , y near , y far , z low , z high , x min , x max , y min , y max , x eye , y ey[...]

Pa ge 22- 1 6 @) PPAR Sho w plot par ameters ~„r` @INDEP Def ine ‘ r’ as the indep . v ari able ~„s` @DEPND De fine ‘ s ’ as the depende nt v ari able 1 # 10 @XRNG De f ine (-1, 10) as the x -r ange 1 # 5 @YRN G L Def ine (-1, 5 ) as t he y-r ange { (0, 0) {.4 .2} “Rs ” “Sr ”} ` Axes de finiti on list @AXES Def in e ax es cent[...]

Pa ge 22- 1 7 @) PPAR Sho w plot par ameters { θ 0 6 .2 9} ` @INDEP De f ine ‘ θ ’ as the indep . V ariable ~y` @DEPND De fine ‘ Y’ a s the dependent v ar iable 3 # 3 @XRNG De fine (-3, 3) as the x -r ange 0. 5 # 2. 5 @YRNG L Def ine (-0. 5,2 . 5) as the y-r ange { (0, 0) {. 5 .5} “ x ” “ y”} ` Ax es def inition lis t @AXES Defi[...]

Pa ge 22- 1 8 « S tart pr ogram {PPAR EQ} PURGE P u r ge c urr ent P P AR and E Q ‘ √ r’ STEQ Sto r e ‘ √ r’ i nto E Q ‘r’ INDEP Set independent v ari able to ‘ r’ ‘s’ DEPND Set dependent v ar iable t o ‘ s ’ FUNCTION Selec t FUNCTION as the plot type { (0.,0.) {.4 .2} “Rs” “Sr” } AXES Se t axe s inf or matio n [...]

Pa ge 22- 1 9 Ex ample 3 – A polar plot . Enter the follo wing pr ogr am: «S t a r t p r o g r a m RAD {PPAR EQ} PURGE Change to r adians , pur ge vars . ‘1+SIN( θ )’ STEQ St ore ‘ f( θ )’ into E Q { θ 0. 6.29} INDEP Set indep . var iable to ‘ θ ’ , w ith range ‘Y’ DEPND Set dependent v ar iable t o ‘Y ’ POLAR Selec t POL[...]

Pa ge 22- 2 0 P I CT T his soft k e y re fer s to a var iable called PICT that stor es the cur r ent conten ts of the gr aphi cs w indo w . This v ar iable name , ho w ev er , cannot be placed within quot es, an d ca n only stor e graph i cs object s. In tha t sens e , PIC T is like no oth er calc ulato r v ari ables . PDI M T he functi on P DIM ta[...]

Pa ge 22- 2 1 BO X T his command tak es as in put two or dered pair s (x 1 ,y 1 ) (x 2 , y 2 ) , or two pair s of pi xel coor dinates {#n 1 #m 1 } {#n 2 #m 2 }. It dr aw s the bo x wh ose di agonals ar e r epr esente d by the tw o pairs of coor dinates in the input . ARC T his command is u sed to dr aw an ar c. AR C tak es as in put the fol low ing[...]

Pa ge 22- 22 Θ PI X? Chec ks if pi xe l at location (x ,y) or {#n, #m} is on . Θ PI X OFF turns o ff pi x el at location (x ,y) or {#n , #m}. Θ PI X ON turns on p i xe l at location (x ,y) or {#n , #m}. PVIEW T his command take s as input the coor dinates of a po int as use r coor dinates (x ,y) or pi x els {#n, #m}, and place s the contents of [...]

Pa g e 22 - 23 (5 0., 5 0.) 12 . –180. 180. AR C Dr aw a c ir cle cen ter (5 0,5 0) , r= 12 . 1 8 FOR j Dr aw 8 lines w ithin the c ir cle (50., 5 0 .) DUP L ines ar e center ed as (5 0,5 0) ‘12*COS( 45 *(j-1))’  NUM Calc ulate x, other end at 5 0 + x ‘12*SIN( 4 5*(j-1))’  NUM Calc ulates y , other end at 5 0 + y R  C Con vert x [...]

Pa g e 22 - 24 It is suggest ed that you c r eate a separ a t e sub-dir ectory to sto r e the progr ams. Y ou could call the sub-dir ectory RIVER , since w e ar e dealing w ith irr egular open c han nel c r os s-secti ons , t y pi cal of r i ver s . T o see the pr ogram XSE CT in acti on, use the f ollo wi ng data sets . Enter the m as matr ices o [...]

Pa g e 22 - 2 5 P ix el coordinates T he fi gur e belo w sho w s the gr aphic coor dinate s fo r the t y pi cal (minimum) scr een of 13 1 × 64 pi xels . P i x els coor dinates ar e measured f r om the top left corner of the screen {# 0 h # 0h}, w hich corresponds to user-defined coor din ates Data set 1 Data set 2 xy x y 0.4 6 .3 0.7 4.8 1. 0 4.9 [...]

Pa ge 22- 26 (x min , y max ) . T he max imum coor dinates in terms of p i xels cor r espond to the lo w er ri ght corner of the sc r een {# 8 2h #3Fh}, w hic h in use r-coor d inate s is the point (x max , y min ). T h e coor dinates of the t w o other corner s both in p i xel as w ell as in user - def ine d coor di nates ar e show n i n the figur[...]

Pa g e 22 - 27 Animating a collec tion o f graphics T he calc ulato r pr o v ide s the f unction ANIMA TE to animate a n umber o f gr aphi cs that hav e been placed in the st ack . Y ou can gener ate a gr aph in the gr aphic s sc r een b y using the commands in the PL O T and PICT men us . T o place the gener ated gr aph in the stac k, u se PICT R [...]

Pa g e 22-2 8 ANIMA TE is av ailable b y us ing „°L @) GROB L @ ANIMA ) . T he animation w ill be r e -started. Pr ess $ to st op the animation once mor e. Noti ce that the number 11 w ill still be lis ted in stac k le v el 1. Pr ess ƒ to dr op it fr om the stack. Suppos e that yo u want t o keep the f igur es that compose this animation in a v[...]

Pa g e 22 - 2 9 Ex ample 2 - Animating the plotting of diff er ent po w er f uncti ons Suppos e that yo u want t o animate the plotting of the functi ons f(x) = x n , n = 0, 1, 2 , 3, 4, in the same set o f axe s. Y ou could use the f ollo w ing pr ogr am: «B e g i n p r o g r a m RA D Set angle units to r adians 131 R  B 64 R  B PD IM Se t [...]

Pa ge 22- 3 0 pr oduced in the calc ulator’s sc reen . T her ef or e , when an image is con v er ted into a GROB , it becomes a s equence of binary digits ( b inary dig its = bit s ), i . e . , 0’s and 1’s . T o illustr ate GR OBs and con ve rsi on of image s to GR OBS consider the f ollo w ing e xe r c ise . When w e pr oduce a gr aph in the[...]

Pa ge 22- 3 1 1` „°L @) GROB @  GRO B . Y ou w ill no w ha ve in le v el 1 the GROB desc r ibed as: As a gr aphic ob ject this eq uation can no w be placed in the gr aphi cs displa y . T o r ecov er the gr aphics dis play pr ess š . Then , mo ve the c urso r to an empt y sect or in the graph , and pr ess @) EDIT LL @ REPL . The equatio n ‘[...]

Pa g e 22 - 32 BLANK T he functi on BL ANK , w ith ar guments #n and #m, c r eates a blank gra phics obj ect of w i dth and height spec ifi ed by the v alues #n and #m, r es pecti v ely . T his is similar to the func tion P DIM in the GRAPH men u . GOR The fun ctio n GO R ( Grap hics OR ) ta k es as in put gr ob 2 (a target GROB) , a set of coor di[...]

Pa g e 22 - 3 3 An e xample o f a progr am using GROB T he follo w ing pr ogram pr oduces the gr aph of the sine f unctio n including a fr ame – dra w n w ith the func tion B O X – and a GROB t o label the gr aph. Here is the listing o f the progr am: «B e g i n p r o g r a m RA D Set angle units t o radi ans 131 R  B 64 R  B PD IM Se t [...]

Pa g e 22 - 3 4 sho w s the state o f str es ses w hen the element is r otated b y an angle φ . In this case, the normal str esses are σ ’ xx and σ ’ yy , while the shear str esses ar e τ ’ xy and τ ’ yx . The relationsh ip bet w een the origina l state of str esses ( σ xx , σ yy , τ xy , τ yx ) and the stat e of str ess w hen the [...]

Pa g e 22 - 35 The stress cond ition for whic h t he she ar stress , τ ’ xy , is z er o , ind i cated by segment D’E’ , produces the s o -called princ ipal str esses , σ P xx (at po int D’) and σ P yy (at point E’). T o obtain the princ ipal str esses y ou nee d to r otate the coor dinate s y stem x ’-y’ by an angle φ n , counter [...]

Pa g e 22-3 6 separ ate v ar iables in the calc ulator . Thes e sub-pr ogr ams are then link ed by a main pr ogr am, that w e w ill call MOHRCIRCL . W e will fir st c r eate a sub- dir ect or y called MOHR C w ithin the HOME dir ectory , and mov e into that dir ect or y t o type the pr ograms . T he next s tep is to c r eate the main pr ogr am and [...]

Pa g e 22 - 37 At this point the pr ogram MOHR CIRCL s tarts calling the su b-pr ograms t o pr oduce the fi gur e . Be pa ti ent . The r esulting Mohr ’s c ir cle w ill look as in the pic tur e to the le ft. Becau se this v ie w of P ICT is in vok ed through the f uncti on PVIEW , we cannot get an y other inf ormati on out of the plot beside s th[...]

Pa g e 22 - 3 8 inf ormatio n tell us is that some w here betw een φ = 5 8 o and φ = 5 9 o , the shear stress, τ ’ xy , becomes z er o . T o f ind the actual v alue of φ n, pr ess $ . Then type the list corr esponding to the v alues { σ x σ y τ xy}, for this case , it w ill be { 25 75 50 } [ENTER] Then , pres s @CC&r . T he last r esul[...]

Pa g e 22 - 3 9 necess ar y to plot the c irc le . It is suggest that w e r e -or der the var iable s in the sub-dir ectory , so that the pr ogr ams @MOHRC and @PRNST ar e the two f ir st v ari ables in the soft-menu k e y labels. T his can be accomplished b y cr eating the list { MOHRCIRCL PRNS T } using: J„ä @MOHRC @PRNST ` And then , order in[...]

Pa ge 22- 4 0 T o find the v alues o f the str ess es corr esponding to a r otatio n of 3 5 o in the angle of th e stressed p art i cle, w e use: $š Clea r sc reen, show PICT in graphics scr e en @TRACE @ ( x,y ) @ . T o mov e c ursor o v er the c irc le sho w ing φ and (x ,y) Ne xt , pr ess ™ until y ou r ead φ = 3 5 . T he corr esponding coo[...]

Pa ge 22- 4 1 Since pr ogr am IND A T is use d also f or pr ogram @PRNST (P R iNc ipal S T resses), running that partic ular pr ogr am w ill no w use an input f or m, f or e x ample , T he r esult , after pr es sing @@@OK@@@ , is the follo wing:[...]

Pa g e 23 - 1 Chapter 2 3 Character strings Char acter s tring s are calc ulator obj ects enc losed betw een double quotes . T hey ar e tr eated as te xt b y the calc ulator . F or e x ample , the str ing “SINE FUNCT ION” , can be transf or med into a GR OB (Gra phic s Objec t) , to la bel a gr aph , or can be us ed as output in a pr ogr am. Se[...]

Pa g e 23 - 2 String concatenation Str ing s can be concatenated (j oined together ) b y using the plu s sign +, f or exa mp l e: Concat enating str ings is a pr actical w a y to cr ea t e output in pr ogr ams. F or e x ample , concatenating "Y OU ARE " A GE + " YEAR OLD" cr eate s the string "Y OU ARE 2 5 YE AR OLD", [...]

Pa g e 23 - 3 T he operati on of NUM, CHR , OB J  , and  S TR w as pr esen ted ear lier in this Chapt er . W e hav e also s een the functi ons S UB and REP L in r elation t o gr aphic s earli er in this chapte r . Func tions S UB , REPL , P OS , S IZE , HEAD , and T AIL hav e similar eff e c ts as in lis ts, namel y : SI ZE: number o f a sub-[...]

Pa g e 23 - 4 sc r een the ke y str ok e sequence to get such c harac ter (  . fo r this case) and the numer ical code corr esponding to the c har acter (10 in this cas e) . Char acte rs that ar e not def ined appear a s a dark squar e in the c har acte rs list (  ) and sho w ( None ) at the bottom of the displa y , e ven t hough a numer ical[...]

Pa g e 24 - 1 Chapter 2 4 Calculator objec ts and flags Numbers , lists, v ec tors, matri ces, algebr ai cs, etc ., ar e calc ulator objec ts. T hey ar e classif ied accor ding to its nature into 30 diff er ent t y pes , whic h ar e desc r ibed belo w . F lags ar e var ia bles that can be us ed to contr ol the calculat or propert ies . F la gs w er[...]

Pa g e 24 - 2 Number T y pe Ex am ple ___________ _____________________ _____________________ _______________ 21 Ext ended Real Number Long Real 2 2 Extended C omple x Number Long Complex 2 3 Link ed Arr a y Linked rr ay 2 4 Char acter Ob ject Character 25 C o d e O b j e ct Code 2 6 L ibrary Data Library D ata 2 7 Exter nal Obj ect External 28 I n[...]

Pa g e 24 - 3 Calculator flags A flag is a v ar iable that can e ither be set or uns et . The st atus of a f lag affec ts the behav ior of the calc ulator , if the f lag is a sy stem f lag, or o f a pr ogr am, if it is a user f lag . The y ar e desc r ibed in mor e detail ne xt . S y stem flags S y ste m flags can be access ed by using H @) FLAGS! [...]

Pa g e 24 - 4 T he functi ons contained w ithin the FL A G menu ar e the f ollow ing: The ope rati on of thes e func tions is as f ollo w s: SF Set a f lag CF C lear a flag F S? R eturns 1 if flag is set , 0 if not set FC? R eturns 1 if flag is c lear (not set), 0 if f lag is set F S?C T ests flag as F S does, then c lears it FC?C T ests flag as FC[...]

Pa g e 2 5 - 1 Chapter 2 5 Date and T ime F unc tions In this Chapt er w e demonstr ate some o f the func tions and calc ulations using times and date s. T he TIME menu T he TIME men u , av ailable thr ough the ke ys trok e sequence ‚Ó (the 9 k ey) pr o v ides the f ollo w ing f unctio ns, w hic h ar e des cr ibed ne xt: Setting an alarm Option [...]

Pa g e 2 5 - 2 Br ow sing alarms Option 1. Br o ws e alarms ... in the T IME menu lets y ou r e v ie w y our cur r ent alarms . F or e x ample , after enter ing the alarm us ed in the ex ample a bov e, this option w ill show the f ollo w ing scr een: T his s cr een pro vi des four s oft menu k ey labe ls: EDIT : F or editing the selected alar m , p[...]

Pa g e 2 5 - 3 T he applicati on of these f uncti ons is demonstr ated belo w . D A TE: P laces c urr ent date in the stac k  D A TE: Set s y stem date to spec ifi ed value T IME: P laces c ur r ent time in 2 4 -hr HH.MM S S f ormat  T IME: Set sy stem time to spec if ied v alue in 2 4 -hr HH.MM. S S f ormat T ICK S: Pr o v ides s y stem time[...]

Pa g e 2 5 - 4 Calculating with tim es Th e fu nct ion s  HMS , HM S  , HMS+, and HM S - ar e us ed to manipulate value s in the HH.MM S S for mat . This is the same f ormat us ed to calc ulate w ith angle measur es in degr ees, min utes , and seconds. T hu s, thes e oper ations ar e usef ul not onl y fo r time calculati ons, but als o for an[...]

Pa g e 26 - 1 Chapter 2 6 M anaging memory In Chapte r 2 w e intr oduced the basic co ncepts of , a nd ope rati ons fo r , cr eating and managing var i ables and dir ec tor ies . In this Chapt er w e disc uss the management of the cal culat or’s memory , including the partition of memo r y and tec hniques f or backing u p data. Me mo ry S t r uct[...]

Pa g e 26 - 2 P or t 1 (ERAM ) can contain up to 12 8 KB of data . P ort 1, together with P ort 0 and the HOME dir ectory , cons titute the calc ulator’s RAM (R andom Acce ss Memory) segment of calc ulator ’s memory . T he R AM memory segment r equir es contin uous elec tr ic po w er suppl y f r om the calculat or bat t er ies t o operat e. T o[...]

Pa g e 26 - 3 Chec king objec ts in memor y T o see the ob jec ts stor ed in memor y y ou can use the FILE S func tio n ( „¡ ). Th e sc ree n b el ow sh ows t he H OM E d i rec to r y wi th five d i re cto ri es, n a m ely , TRIANG , MA TRX , MPFIT , GRPH S , and CASDIR . Additi onal dir ector ie s can be vi e wed b y mo v ing the c ursor do wn [...]

Pa g e 26 - 4 Bac k up objec ts Bac ku p obj ects ar e used t o copy data f r om y our home dir ect or y int o a memor y port. The pur pose of bac king up obj ects in memory port is to pr eserve the contents of the objects f or f utur e usage . Back up objec ts hav e the fo llow ing ch ara cte ris ti cs: Θ Bac k up obj ects can onl y e x ist in po[...]

Pa g e 26 - 5 Bac king up and r estoring HOME Y ou can back u p the cont ents of the c urr ent HOME dir ectory in a single bac k up obj ect . T his ob jec t w ill contain all var iables , k e y assi gnments , and alar ms c urr en tly def ined in the HO ME dir ectory . Y ou can also r esto r e the contents o f y our HOME dir ectory fr om a back u p [...]

Pa g e 26 - 6 Stor ing, deleting, and r estoring back up objects T o c r eate a bac k up obj ect us e one of the f ollow ing appr oaches: Θ Us e the F ile Manager ( „¡ ) t o c o p y t h e o b j e c t t o p o r t . U s i n g t h i s appr oach , the back up obj ect w ill hav e the same name as the o ri ginal object . Θ Us e the S T O co mmand to[...]

Pa g e 26 - 7 Using data in backup objects Although y ou cannot dir ectl y modify the contents o f back up objec ts, y ou can use tho se cont ents in calculat or oper ations. F or e x ample , y ou can r un pr ogr ams stor ed as back up objec ts or us e data fr om back up obj ects t o run pr ograms . T o run bac k up-obj ect pr ogr ams or use data f[...]

Pa g e 26 - 8 T o r emo ve an SD car d , turn o ff the HP 50 g, pr ess ge ntly on the e xposed edge of the car d and push in . The car d should spring out o f the slot a small distance , allo w ing it now to be easil y r emo ved f r om the calculator . F ormatting an SD card Most SD car ds will alr ead y be fo rmatted, but the y may be f or matted [...]

Pa g e 26 - 9 Accessing objects on an SD card Acce ssing an obj ect f r om the SD car d is similar to whe n an objec t is located in ports 0, 1, or 2 . How ev er , P ort 3 wi ll not appear in the menu when using the LIB fu ncti on ( ‚á ) . T he SD file s can only be managed u sing the F iler , or F i le Manager ( „¡ ). When st ar ting the F i[...]

Pa g e 26 - 1 0 Note that if the name of the object y ou intend to st ore on an SD car d is longer than ei ght c harac ters , it will a ppear in 8. 3 DOS f or mat in por t 3 in the F iler once it is stor ed on the ca r d. Recalling an object from an SD car d T o r ecall an ob ject f r om the SD card onto the sc r een, u se functi on RCL , as fo llo[...]

Pa g e 26 - 1 1 Note that in the case of objects w ith long file s names , yo u can spec ify the f ull name of the objec t , or its truncat ed 8. 3 name , when ev aluating an obj ect on an SD car d. P urging an object from the SD card T o pur ge an ob ject f r om the SD car d onto the s cr een , us e functi on P URGE , as fo llo w s: Θ In algebr a[...]

Pa g e 26 - 1 2 T his will s tor e the obj ect pr ev iousl y on the stac k onto the SD card int o the dir ect or y named P ROG S into an obj ect named P ROG1. Not e: If PR OGS doe s not e xis t, the dir ectory will be au tomaticall y cr eated. Y ou can spec ify an y number of nested subdir ector ies . F or ex ample , to re fer t o an obj ect in a t[...]

Pa g e 26 - 1 3 Libr ary numbers If y ou us e the LIB menu ( ‚á ) and pr ess the so ft menu k e y corr es ponding to port 0, 1 or 2 , yo u wi ll see libr ar y n umbers list ed in the soft menu k e y labels . E ac h library has a thr ee or f our -digit n umber assoc iated w ith it . (F or e x ample , the two libr ar ies that mak e up the Eq uatio[...]

Pa g e 26 - 1 4 w ill indicat e when this battery needs r eplacement . The diagr am belo w sho ws the location o f the back up bat t er y in the top compartment at the back o f the calc ulat or .[...]

Pa g e 27- 1 Chapter 2 7 T he Equation Libr ar y T he E quation L ibrary is a collection o f equations and commands that enable y ou to so lv e simple s c ience and e ngineer ing pr oblems. T he libr ary consists o f mor e than 300 equatio ns gr ouped int o 15 techni cal subj ects con taining mor e than 100 pr oblem titles . E ach pr oblem title co[...]

Pa g e 27- 2 7 . F or eac h know n var iable , type its value and pr es s the corr espo nding menu k e y . If a v ari able is not show n , pre ss L to disp la y furt h er variables. 8. Optional: su pply a gues s f or an unkno wn v ar iable . This can speed up the soluti on pr ocess or help to f oc us on one of s ev er al soluti ons. Enter a gue ss [...]

Pa g e 27- 3 Using the m enu k ey s T he actions o f the unshifted and shifted var iable menu k ey s f or both sol ver s ar e identi cal. No tice that the Multiple Eq uation S olv er us es two f orms o f menu labels: blac k and w hite . The L k e y displa y s additional menu la bels, if r equir ed . In addition , each s olv er ha s spec ial me nu k[...]

Pa g e 27- 4 Br o wsing in the Equation L ibrary When y ou se lect a sub ject and title in the E quation L ibrary , y ou spec ify a set of one or mor e equati ons. Y ou can get the f ollo w ing infor mation a bout the equati on set fr om the E quatio n Li brary catalogs:  The equati ons themsel ves and the number o f equa ti ons .  T he var i[...]

Pa g e 27- 5 Vie wing v ariables and sel ecting units After y ou select a sub jec t and title , y ou can vi e w the catalog of names , desc r iptions , and units for the v ari ables in the equati on set b y pre ssing #VARS# . T he table belo w summari z es the oper ations av a i lable to y ou in the V ari able catalogs . Oper atio ns i n V ariable [...]

Pa g e 27- 6  Pr ess to stor e the pic tur e in PIC T , the gra phics memory . T hen y ou can use © PIC T ( or © P ICTURE) to v ie w the pi ctur e again af t er y ou hav e quit the E q uation L ibr ar y .  Pr ess a menu k ey or to v ie w other equati on infor mation . Using the M ultiple -Equation Sol ver T he E quation L ibrary starts the [...]

Pa g e 27- 7 T he menu labels f or the var iable k ey s ar e w hite at fir st , but c hange during the solu tion pr ocess as des cr ibed belo w . Becau se a solu tion in v olv es man y equations and man y v ar ia bles, the Multiple - E quati on Sol ver mu st k eep tr ack o f var ia bles that are u ser -def ined and not def ined—those it can ’t [...]

Pa g e 27- 8 Mea nings of Menu Labe ls Defining a set o f equations When y ou design a s et of eq uations , y ou should do it w ith an under standing o f ho w the Multiple -E quation Sol ver use s the equations to sol ve pr oblems. T he Multiple -E quati on Sol v er uses the sa me pr ocess y ou ’d use t o sol ve f or an unkno wn v ar ia ble (assu[...]

Pa g e 27- 9 F or ex ample , the f ollo w ing thr ee equati ons defi ne initial v eloc ity a nd acceler atio n based on tw o observed dis tances and times . T he fir st tw o equations alone ar e mathematicall y suff ic ient f or solv ing the problem , but eac h equati on contains tw o unkno w n var ia bles. Adding the thir d equation allo ws a succ[...]

Pa g e 27- 1 0 6. P ress ! MSOLV! to launc h the sol ver w ith the ne w set of equati ons . T o c hange the title and menu for a set of equations 1. Mak e sur e that the set o f equati ons is the c urr ent set (a s the y are u sed w hen the Multiple -E quati on Sol ve r is launc hed) . 2 . En ter a te xt str ing cont aining the new titl e onto the [...]

Pa g e 27- 1 1  Constant? The initi al value of a v ar iable ma y be leading the r oot - f inder in the wr ong direc tion . Suppl y a guess in the oppo site dir ectio n fr om a cr iti cal value . (If negati ve v alues ar e vali d , tr y one . ) Chec king solutions Th e va riab le s h avin g a š mark in their men u labels ar e r elated fo r the [...]

Pa g e 27- 1 2  Not r elated . A var iable ma y not be in v olv ed in the s oluti on (no mark in the label) , s o it is not com patible wi th the var ia bles that w er e inv ol ved .  W r ong dir ecti on . The initial v alue of a var iable ma y be leading the roo t - f inder in the wr ong direc tion . Suppl y a guess in the oppo site dir ecti[...]

Pa g e A - 1 Appendi x A Using input forms T his ex ample o f setting time and date illu str ates the use of input f orms in the calc ulator . Some gener al rules: Θ Use the ar ro w ke ys ( š™˜— ) to mov e fr om on e f ield to th e ne xt in the input f or m. Θ Use an y the @C HOOS sof t menu k ey to see the options av ailab le for an y gi v[...]

Pa g e A - 2 In this par ti c ular case w e can giv e v alues to all but one of the var iables, s ay , n = 10, I%YR = 8. 5, PV = 10000, FV = 1000, and sol ve fo r va ri able P MT (the meaning of thes e var iables w ill be pr esent ed later ) . T r y the f ollo w ing: 10 @@OK@@ Enter n = 10 8. 5 @@ OK@@ Ente r I%YR = 8. 5 10000 @@ OK@@ Ent er PV = 1[...]

Pa g e A - 3 !CALC Pr es s to access the stac k f or calculati ons !TYPES Pr ess to determin e the t y pe of object in highlighte d f ield !CANCL Cancel operation @@OK@ @ Ac cep t en tr y If y ou pr ess !RESET y ou w ill be ask ed to se lect between the tw o options: If y ou select R es et value onl y the highli ghted v alue w ill be r eset t o the[...]

Pa g e A - 4 (In RPN mode , w e would ha v e used 113 6 .2 2 ` 2 `/ ). Press @ @OK@@ to en ter this ne w v alue. T he inpu t for m w ill no w look lik e this: Press !TYPES t o see the type o f data in the P MT fi eld (the highligh ted f ield). As a r esult , y ou get the f ollo w ing spec if icati on: T his indicates that the v alue in the P MT f i[...]

Pa g e B - 1 Appendi x B T he calc ulator ’s k e y board T he fi gur e belo w sho w s a diagr am o f the calc ulato r’s k e y board w ith the number ing of its r o ws and columns . T he fi gure sho ws 10 r ow s of k e y s combined w ith 3, 5, or 6 columns. R o w 1 has 6 k ey s, r o ws 2 and 3 ha ve 3 k ey s eac h , and r o w s 4 thro ugh 10 hav[...]

Pa g e B - 2 f i ve f uncti ons. T he main k e y func tions ar e sho wn in the f igur e belo w . T o oper ate this main k e y func tions simpl y pr ess the cor r esponding k e y . W e w ill r ef er to the ke y s by the r o w and column wher e the y are located in the sk etc h abo v e , th us , k e y (10,1) is the ON key . Mai n k ey functio ns in t[...]

Pa g e B - 3 M ain k e y functions Ke ys A thr ough F k ey s ar e assoc iated w ith the soft men u options that appear at the bottom of the calc ulator’s dis play . T hus , thes e k ey s will ac tiv a t e a v ari ety of func tions that c hange acco rding t o the acti v e menu .  Th e a rrow keys, —˜š™ , ar e used to mo ve one c har act e[...]

P age B-4  The l eft- shi ft k e y „ and the r ight-shift key … ar e combi ned with other k ey s to acti vat e menus, en ter char acters , or calc ulate functi ons as desc r ibed else wher e.  The n umeri cal k ey s ( 0 to 9 ) ar e used to enter the digits of the dec imal number s ys tem.  Ther e is a d ec imal po in t k e y (.) and a [...]

P age B-5 the other thr ee functi ons is a ssoc iated w ith the le f t-shift „ ( MTH ) , r ight-shift … ( CA T ) , and ~ ( P ) ke y s. Diagr ams show ing the f uncti on or char acter r esulting fr om com b ining the calculat or k ey s w ith the lef t-shift „ , r ight-shift … , ALPH A ~ , ALPHA-left - shift ~„ , and ALP HA -r ight-shif t ~[...]

Pa g e B - 6  Th e CMD fu nction sho ws the most r e cent commands , the PRG fun ctio n acti v ates the pr ogramming men us , the MTRW f uncti on acti vat es the Matri x Wr i t e r, Left-shift „ func tions of th e calculator ’s k e yboard  Th e CMD fu nction sho w s the most r ecent commands.  Th e PRG functi on acti v ates the pr ogr [...]

Pa g e B - 7  Th e e x k e y calc ulates the e xponenti al func tion o f x .  Th e x 2 k e y calc ulates the sq uar e of x (this is re fer red to as the SQ fu nct ion) .  T he AS IN , A CO S, and A T AN fu ncti ons calc ulate the ar csine , ar ccosine , and ar c tangent f uncti ons, r especti vel y .  Th e 10 x func tion calc ulates the[...]

Pa g e B - 8 Rig ht-s hif t … func tions of the calculator ’s k ey board Right-shift functions The sk etch abo v e show s the functi ons , char acter s, or men us ass oci ated w ith the diffe r ent calculator k ey s w hen the r igh t -shift k e y … is acti vated .  Th e fu nct ion s BE GIN, END , COP Y , CUT and PA S T E ar e used f or edi[...]

Pa g e B - 9  Th e CA T functi on is used to ac tiv ate the command c atalog .  Th e CLE AR functi on c lears the sc r een .  Th e LN func tion calc ulates the natur al logarithm .  T he functi on calc ulates the x – th r oot of y .  Th e Σ f uncti on is used to ent er summations (or the upper case Gree k letter sigma).  Th e ?[...]

Pa g e B - 1 0 is used mainl y to e nter the upper -case letter s of the English alpha bet ( A thr ough Z ) . T he numbers , mathematical s ymbols ( - , + ), dec imal poi nt ( . ) , and the s pace ( SP C ) ar e the same as the main functi ons of these k ey s. T he ~ fu nc tion pr oduc es an aster isk ( * ) whe n combined w ith the time s ke y , i .[...]

Pa g e B - 1 1 Notice that the ~„ combinati on is used mainl y to enter the lo wer -c ase letters of the English alphabet ( A thr ough Z ) . T he numbe rs , mathematical sym bo l s ( - , +, × ) , dec imal p o int ( . ) , and the spac e ( SP C ) are the s ame as the main func tions of these k ey s. The ENTER and CONT k e y s also w ork as their m[...]

Pa g e B - 1 2 Alpha-right-shift c har ac ters T he follo wing sk etch sho ws the c harac ter s assoc iated w ith the diffe r ent calc ulat or k e y s w hen the ALPH A ~ is combined w ith the ri ght -shift k e y … . Alpha ~… functions of the calculator’s ke y board Notice that the ~… combinati on is used mainly to enter a n umber of spec ia[...]

Pa g e B - 1 3 ~… combination inc lude Gr eek let ter s ( α, β, Δ, δ, ε, ρ, μ, λ, σ, θ, τ , ω , and Π ) , other c har acte rs gener ated b y the ~… combinati on ar e |, ‘ , ^, =, <, >, /, “ , , __, ~, !, ?, <<>>, and @.[...]

Pa g e C - 1 Appendi x C CAS settings CA S stands f or C omputer A lge br aic S y stem . T his is the mathematical cor e of the calc ulator w her e the sy mbolic mathematical oper atio ns and functi ons ar e pr ogr ammed. T he CA S offe rs a number of settings can be adj ust ed according to the type of oper ation of inter est . T o see the optional[...]

Pa g e C - 2 Θ T o r eco ver the or iginal menu in the CAL CUL A T OR MODE S input bo x , pr ess the L k e y . Of inter est at this point is the c hanging of the CA S settings . T his is accomplished by pr essing the @ @ CAS@@ s oft menu k e y . The def ault v alues of the CA S setting ar e sho w n belo w: Θ T o na vi gate thr ough the man y opti[...]

Pa g e C - 3 A v ari able called VX ex ists in the calc ulator ’s {HOME CA SDI R} dir ect or y that tak es, b y def ault , the v alue of ‘X’ . T his is the name of the pr efer r ed independent v ar iable f or algebr aic and calc ulus a pplicati ons. F or that re ason , most e xamples in this C hapter u se X as the unkno wn v ar iable . If y o[...]

Pa g e C - 4 T he same e x ample , corr es ponding to the RPN oper ating mode, is sho wn ne xt: Appr o x imate v s. Ex ac t CA S mode When the _ A ppr ox is s elected , sy mbolic oper ati ons (e.g ., def inite integrals , squar e roots , etc .) , w ill be calc ulated numer i cally . When the _A ppr o x is unselec ted (Ex act mode is acti v e) , s y[...]

Pa g e C - 5 T he k ey str ok es nece ssary for ent er ing these v alues in Algebr ai c mode ar e the fo llow ing: …¹2` R5` T he same calc ulations can be pr oduced in RPN mode . Stac k lev els 3: and 4: sho w the case of Ex act CAS se tting (i .e ., the _Numeri c CAS opti on is unselec ted) , w hile stac k lev els 1: and 2: show the cas e in wh[...]

Pa g e C - 6 It is r ecommended that y ou se lect EXA CT mode as def ault CA S mode , and c hange to APP R O X mode if r equest ed b y the calc ulator in the perf ormance of an oper ation . F or add iti onal inf ormati on on r eal and integer numbers , as w ell as other c alcul at or’s obje cts, r efe r to Cha pte r 2 . Comple x vs . R eal CAS mo[...]

Pa g e C - 7 If y ou pr ess the OK so ft menu ke y (), then the _Comple x optio n is for ced, and the r esult is the f ollo wing: T he k ey str ok es us ed abo ve ar e the follo w ing: R„Ü5„Q2+ 8„Q2` When ask ed to change to C OMP LEX mode , u se: F . If y ou deci de not to accept the change t o COMP LEX mode , y ou get the f ollo wing er r [...]

Pa g e C - 8 F or ex ample , hav ing selec ted the S tep/step opti on, the f ollo wing s cr eens sho w the step-b y-step di v ision of tw o poly nomials , namel y , (X 3 -5X 2 +3X - 2)/(X - 2) . T his is accomplished b y using f uncti on DIV2 a s sho w n belo w . Pr ess ` to show the f irst s tep: T he scr een infor m us that the calc ulator is ope[...]

Pa g e C - 9 . Increasing-po w er CAS mode When the _Incr po w CA S option is selec ted , poly nomi als wi ll be listed so that the ter ms w ill hav e incr easing po we rs of the independent v ar iable . If the _Inc r po w CAS opti on is not select ed (defa ult v alue) then pol ynomi als w ill be list ed so that the ter ms wi ll hav e dec r easing [...]

Pa g e C - 1 0 Rigor ous CAS setting When the _Ri gorous CA S option is se lected , the algebrai c e xpr essi on |X|, i .e., the absolute v alue , is not simplified to X . If the _R igor ous CA S option is not selec ted , the algebrai c e xpr essi on |X| is simplif ied to X . T he CA S can sol v e a lar ger v ar iety of pr oblems if the r igor ous [...]

Pa g e C - 1 1 Notice that , in this ins tance , soft menu k ey s E and F are the o nly o ne w ith as soc iated commands , namel y: !!CANCL E CANCeL the help f ac ilit y !!@@OK#@ F OK to ac ti vate help fac ilit y f or the selected comman d If y ou pr ess the !! CANCL E k e y , the HELP f aci lit y is skipped, and the calc ulator r eturns t o norma[...]

Pa g e C - 1 2 Notice that the re ar e six co mmands assoc iated w ith the s oft menu k e y s in this case (y ou can chec k that ther e are onl y si x commands because pr essing the L pr oduces no additi onal menu items). T he so ft menu ke y commands are the f ollo w ing: @EXIT A EX IT the help fac ilit y @ECHO B Cop y the ex ample command to the [...]

Pa g e C - 1 3 T o nav igate qui ckl y to a partic ular command in the help fac ility list w ithout ha ving to u se the arr o w k e ys all the time , we can us e a shortcu t consisting of typing the f irs t letter in the command’s name . Suppose that w e w ant to find inf ormati on on the co mmand IBP (Integr ation B y P ar ts), once the help f a[...]

Pa g e C - 1 4 In no e vent unle ss r equir ed b y applicable la w w ill an y copy r ight holde r be liable t o yo u for damage s, inc luding an y general , speci al , inc ident al or cons equential damage s ar ising out of the us e or inability to us e the CA S Softwar e (including but not limit ed to loss o f data or data being r ender ed inacc u[...]

Pa g e D - 1 Appendi x D Additional c har acter set While y ou can us e an y of the u pper -case and lo w er -case English letter f r om the k e yboar d, ther e are 2 5 5 char acter s usable in the calc ulator . Including spec ial ch arac ter s l ike θ , λ , et c., that that can be used in algebr ai c expr essions . T o access the se char acters [...]

Pa g e D - 2 func tions assoc iated w ith the soft menu k e y s, f4 , f5, and f6. The se func tions ar e: @MODIF : Opens a graphi cs sc r een whe r e the user can modify highlight ed c harac ter . Use this opti on car ef ull y , since it w ill alter the modif ied c har acter u p to the ne xt r ese t of the calc ulator . (Imagine the e ffec t of c h[...]

Pa g e D - 3 Gr ee k lett er s α (alpha) ~‚a β (beta) ~‚b δ (delta) ~‚d ε (epsilon) ~‚e θ (theta) ~‚t λ (lambda) ~‚n μ (mu) ~‚m ρ (r ho) ~‚f σ (sigma) ~‚s τ (tau) ~‚u ω (omega) ~‚v Δ (upper -case delta) ~‚c Π (upper -case pi) ~‚p Ot her char ac ters ~( t i l d e ) ~‚1 !( f a c t o r i a l ) ~‚2 ? (questi o[...]

Pa g e E - 1 Appendi x E T h e Selec tion T ree in the Equation W riter T he expr essi on tr ee is a diagr am sho w ing ho w the E quati on W r iter inte rpr ets an ex p r e ss io n. The fo rm of th e exp re ss io n t re e i s de t erm i ne d by a n u mb er o f r ul es kno wn as the hi er ar ch y of oper ation . T he rules ar e as follo ws: 1. Oper[...]

Pa g e E - 2 Step A1 Ste p A2 Step A3 Ste p A4 Step A5 Ste p A6 W e notice the appli cation o f the hier ar ch y-of-oper ation r ules in this selecti on. F i r st the y (Step A1) . T hen, y-3 (S tep A2 , par enth eses) . Then , (y-3)x (Step A3, multiplicati on) . T hen (y-3)x+5, (Step A4 , additi on) . T hen , ((y-3)x+5)(x 2 +4) (St ep A5, multipli[...]

Pa g e E - 3 Step B1 S te p B2 Step B3 St ep B4 = Step A5 St ep B5 = Step A6 W e can also fol lo w the ev aluation o f the expr essi on starting fr om the 4 in the ar gument of the S IN func tion in the denominat or . Pr ess the do wn ar r o w k e y ˜ , continuousl y , until the clear , editing cu rsor is tr igger ed around the y , once mor e . T [...]

Pa g e E - 4 Step C3 Step C 4 St ep C5 = St ep B5 = Step A6 The expr ession t r ee f or t he expr ession p r esente d abov e is s ho wn next: T he steps in the e v aluation of the thr ee terms ( A1 thr ough A6 , B1 thro ugh B5, and C1 thr ough C5) ar e sho w n ne xt to the c ir c le containing numbers , v ari able s, or oper ators .[...]

Pa g e F - 1 Appendi x F T he Applications (APP S) menu T he Applicati ons ( APP S) menu is av ailable thr ough the G key ( fi rst key i n second r o w fr om the k e yboar d’s top) . T he G k ey sh o ws the f ollo w ing applicati ons: T he differ ent appli cations ar e desc ribed ne xt . P lot func tions.. Selec ting option 1. P lot f u nc tions [...]

Pa g e F - 2 I/O func tions .. Selecting opti on 2 . I/O f uncti ons .. in the APP S menu w i ll pr oduce the f ollo w ing menu lis t of input/ou tput func tions T hese appli cations ar e desc r ibed next: Send to C alc ulator Send data to another calc ulator (or to a P C with an infr ared port) Get fr om C alculator R ecei ve data f r om another c[...]

Pa g e F - 3 T he Const ants Libr ar y is disc us sed in detail in C hapter 3 . Numeric sol ver .. Selec ting option 3 . Constan ts lib .. in the APP S menu pr oduces the nume ri cal solver me nu: This oper ation is equi valent to the k e y str ok e sequence ‚Ï . T he numer ical sol v er menu is pr esent ed in detail in Chapt ers 6 and 7 . Time [...]

Pa g e F - 4 Equation wr iter .. Selec ting option 6 .E quation w r iter .. in the APP S menu opens the equation wri ter: T his oper ation is eq ui val ent to the k ey str ok e seq uence ‚O . The equati on w rit er is intr oduced in det ail in Chapter 2 . Examples that u se the equatio n w rite r are a v ailable thr oughout this guide . F ile man[...]

Pa g e F - 5 M atr ix W riter .. Selec ting option 8.Matr i x W r iter .. in the APP S menu launc hes the matr i x wr iter : T his oper ation is eq ui val ent to the k ey str ok e seq uence „² .The Matr i x W r iter is pr esen ted in detail in Chapter 10. T e xt editor .. Selec ting option 9 .T e xt editor .. in the APP S menu launc hes the line[...]

Pa g e F - 6 T his oper ation is eq ui val ent to the k ey str ok e seq uence „´ . T he MTH menu is intr oduced in Chapt er 3 (r eal numbers). Other func tions f r om the MTH menu ar e pr esented in Chapters 4 (comple x numbers), 8 (lists) , 9 (vec tors) , 10 (matr i x cr eation) , 11 (matr ix oper ation), 16 (f as t F our ier tr ansfor ms) , 17[...]

Pa g e F - 7 Note that flag –117 should be se t if you ar e going to us e the E quatio n L ibrary . Note too that the E quation L ibr ary w ill only appear on the AP P S menu if the two E quation L ibrary files ar e stor ed on the calculator . T he E quation L ibrary is e xplained in de tail in chapt er 2 7 .[...]

P age G-1 Appendi x G Useful shortc uts Pr esented her ein ar e a number o f k e yboar d shor tc uts commonl y used in the calc ulat or : Θ Adjust d isplay c ontrast: $ (hold) + , or $ (hold) - Θ T oggle between RPN and AL G modes: H @@@OK@@ or H` . Θ Set/c lear s ys tem flag 9 5 (AL G vs. RPN oper ating mode) H @) FLAGS —„—„—„ —[...]

P age G-2 Θ Set/c lear sy stem flag 117 (CHOO SE bo xe s vs . S OFT menus): H @) FLAGS —„ —˜ @@CHK@ Θ In AL G mode , SF(-117) selects S O FT menus CF(-117) se lects CHOO SE BO XE S . Θ In RPN mode , 117 ` SF se lects S OFT me nus 117 ` CF selec ts SOF T menus Θ Change angular mea sur e: o T o degr ees: ~~deg` o T o r adian: ~~ra d` Θ [...]

P age G-3 Θ S ystem-lev el o per ation (H old $ , r elease it after enter ing second or thir d k e y) : o $ (ho ld) AF : “C old” r estart - all memory er ased o $ (ho ld) B : Cancels k ey str ok e o $ (ho ld) C : “W arm ” re start - memor y pr eserved o $ (ho ld) D : Starts inter acti ve se lf- test o $ (ho ld) E : Starts continuou s self-[...]

P age H-1 Appendi x H T he CAS help facilit y T he CAS help f ac ilit y is a vaila ble thro ugh the k ey str ok e sequence I L @HELP ` . The f ollo w ing sc r een shots sh o w the fir st menu page in the listing of the CAS help fac i lity . T he commands ar e listed in alphabeti cal or der . Using the v er ti cal arr o w k e ys —˜ one can na v i[...]

P age H-2 Θ Y ou c an type t w o or mor e let ters of the c ommand of inter est , by locking the alphabeti c k e y boar d. T his w ill tak e yo u to the command of int er est , or to its nei ghborhood. A fterwar d s, y ou need to unloc k the alpha k e yboar d, and u se the v ertical arr ow k ey s —˜ to locate the command , if needed. Pr ess @@O[...]

Pa g e I - 1 Appendi x I Command catalog list T his is a l ist of all commands in the command catalog ( ‚N ) . Those commands that belong t o the CA S (C omput er Algebr aic S y stem) ar e lis ted also in Appendi x H. CAS help f ac ilit y en tri es ar e a vailabl e for a gi v en command if the so ft menu k ey @HELP sho ws up w hen yo u highli ght[...]

Pa g e J - 1 Appendi x J T he MA THS me nu T he MA THS menu , accessible thr ough the command MA THS (a v ailable in the catalog N ), contains the fo llo w ing sub-menu s: T he CMPLX sub-menu T he CM P L X su b-menu contains fu nctions pertinent to oper ations w ith complex numbers: T hese f uncti ons are des cr ibed in Chapter 4. T he CONST A NT S[...]

Pa g e J - 2 T he HYPERBOLIC sub-menu T he HYPERB OLIC sub-menu co ntains the h y perboli c func tio ns and their in v ers es . T hese f unctions ar e descr ibed in Chapter 3 . T he I NTE GER sub-menu T he INTEGER su b-menu pr o v ides f uncti ons for manipulating integer number s and some pol ynomi als. T hese f unctions ar e pre sented in Cha pte[...]

Pa g e J - 3 T he POL YNOM IAL sub-menu T he POL YNOMIAL sub-men u includes f uncti ons for ge ner ating and manipulating pol yno mials . The se func tions ar e pr es ented in Chapte r 5: T he TES T S sub-menu T he TE S TS su b-menu inc ludes r elati onal oper ator s (e .g ., ==, <, etc .) , logical oper ators (e .g., AND , OR, et c.), the IFTE [...]

Pa g e K - 1 Appendi x K Th e MA I N m en u T he MAIN menu is av ailable in the command catalog . This menu inc lude the fo llo w ing sub-menu s: T he CASCF G command T his is the f irs t entr y in the MAIN menu . T his command conf igur es the CA S . F or CA S conf igur ation inf orm atio n see A ppendi x C. T he AL GB sub-menu T he AL GB sub-menu[...]

Pa g e K - 2 T he DIFF sub-m enu T he DI FF sub-me nu contains the f ollo w ing f unctio ns: T hese f unctions ar e also av ailable thr ough the CAL C/DI FF sub-menu (s tart wi th „Ö ) . T hese f uncti ons ar e desc r ibed in Chapte rs 13, 14, and 15, e x cept fo r func tion TR UNC, w hic h is desc r ibed next us ing its CAS help f ac ilit y en [...]

Pa g e K - 3 T hese f uncti ons are als o av ailable in the TRIG menu ( ‚Ñ ) . Description of these f uncti ons is incl uded in C hapter 5 . T he SOL VER sub-m enu T he S OL VER menu include s the fo llo w ing func tions: T hese f uncti ons are a v ailable in the CAL C/S OL VE menu (st art with „Ö ). T he functi ons ar e des cr ibed in Cha pt[...]

Pa g e K - 4 T he sub-menus INTE GER , MODUL AR , and P OL YNOMIAL ar e pre sented in detail in Appe ndi x J. The E XP &LN sub-menu T he EXP&LN menu contains the follo w ing functions: T his menu is also acces sible thr ough the k e yboar d by using „Ð . T he functi ons in this menu are pr esented in Chapter 5 . T he MA TR sub-m enu T he[...]

Pa g e K - 5 T hese f uncti ons ar e av ailable thr ough the CONVER T/REWR ITE me nu (start w ith „Ú ) . T he func tions ar e pr esent ed in Chapter 5, ex cept for f uncti ons XNUM and XQ , whi ch ar e desc ribed ne xt using the corr es ponding entr ies in the CA S help fac i lity ( IL @HELP ): XNUM X Q[...]

Pa g e L - 1 Appendi x L L ine editor commands When y ou tr igger the line editor b y u sing „˜ in the RPN stac k or in AL G mode , the follo wing s oft menu f unctions ar e pr ov ided (pr ess L to see the r emaining fu nctions): T he functi ons ar e br ief ly de sc ribed as follo ws:  SKIP: Skip s char acters to beginning o f wor d. SKIP [...]

Pa g e L - 2 T he items sho w in this scr e en are s elf-e xplanator y . F or e x ample , X and Y positi ons mean the po sition on a line (X) and the line number (Y ) . Stk Siz e means the number of ob jects in the AL G mode history or in the RPN stac k. Mem(KB) means the amount o f fr ee memory . Clip Si z e is the number of c har acte rs in the c[...]

Pa g e L - 3 T he SEARCH sub-menu T he functi ons of the SE ARCH sub-me nu ar e: Fi n d : Use this functi on to find a str ing in the command line . The input f orm pr o v ided w ith this command is sho wn next: Rep l ac e : Use this co mmand to f ind and r eplace a s tr ing. T he input f or m pr o v ided for this co mmand is: F ind next .. : F ind[...]

Pa g e L - 4 T he GO T O sub-menu T he functi ons in the GO T O sub-men u are t he follo w ing: Goto L ine: to mo ve to a spec ifi ed line. T he input fo rm pr o v ided w ith this command is: Goto P ositi on : mov e to a spec ifi ed position in the command line . The input fo rm pr o v ided f or this command is: Lab els : mo v e to a spec if ied la[...]

Pa g e L - 5[...]

Pa g e M - 1 Appendi x M T abl e o f Built-In Equations T he E quation Libr ar y consists o f 15 sub jects cor r esponding t o the secti ons in the table belo w) and mor e than 100 titles. T he n umbers in par e ntheses belo w indicat e the number of equati ons in the set and the number of v ari ables in the set . T here ar e 315 equati ons in tota[...]

Pa g e M - 2 3: Fluids ( 2 9 , 29) 1: Pr essur e a t D epth (1, 4) 3: F lo w w ith Lo ss es (10, 17) 2 : Bernoulli E quation (10, 15 ) 4: F lo w in F ull P ipes (8 , 19) 4 : F o r ces an d Energy (3 1 , 3 6) 1: L inear Mechanic s (8, 11) 5: ID Elas tic Collisi ons (2 , 5) 2 : Angular Mec hanics (12 , 15 ) 6: Dr ag F or ce (1, 5 ) 3: Centr ipetal F [...]

Pa g e M - 3 9: Op ti cs ( 1 1 , 1 4) 1: La w of Ref r acti on (1, 4) 4: Spher i cal Ref lecti on (3, 5) 2 : Criti cal Angle (1, 3) 5: Spher i cal Ref r acti on (1, 5) 3: Br ew ster’s L a w (2 , 4) 6: Thin Le ns (3, 7) 1 0: Osc illations ( 1 7 , 1 7) 1: Mass–S pr ing S ys tem (1, 4) 4: T ors ional P endulum (3, 7) 2 : Sim ple P endulum (3, 4) 5[...]

Pa g e N - 1 Appendi x N Inde x A ABCUV 5-10 ABS 3-4, 4-6, 11-8 ACK 25-4 ACKALL 25-4 ACOS 3-6 ADD 8-9, 12-20 Additional character set D-1 ADDTMOD 5-11 Alarm functions 25-4 Alarms 25-2 ALG menu 5-3 Algebraic objects 5-1 ALOG 3 -5 ALPHA characters B-9 ALPHA keyboard lock-unlock G-2 Alpha-left-shift characters B-10 Alpha-right-shift characters B-12 AL[...]

Pa g e N - 2 Bar plots 12-29 BASE menu 19-1 Base units 3-22 Beep 1-25 BEG 6-31 BEGIN 2-27 Bessel’s equation 16-52 Bessel’s functions 16-53 Best data fitting 18-13, 18-62 Best polynomial fitting 18-62 Beta distribution 17-7 BIG 12-18 BIN 3-2 Binary numbers 19-1 Binary system 19-3 Binomial distribution 17-4 BIT menu 19-6 BLANK 22-32 BOL L-4 BOX 1[...]

Pa g e N - 3 Clock display 1-30 CMD 2-62 CMDS 2-25 CMPLX menus 4-5 CNCT 22-13 CNTR 12-48 Coefficient of variation 18-5 COL+ 10-19 COL  10-19 "Cold" calculator restart G-3 COLLECT 5-4 Column no rm 11-7 Column vectors 9-18 COL- 10-2 0 COMB 1 7-2 Combinations 17-1 Command catalog list I-1 Complex CAS mode C-6 Complex Fourier series 16-26 [...]

Pa g e N - 4 Dates calculations 25-4 DBUG 21-35 DDAYS 25-3 Debugging programs 21-22 DEC 19-2 Decimal comma 1-22 Decimal numbers 19-4 decimal point 1-22 Decomposing a vector 9-11 Decomposing lists 8-2 Deep-sleep shutdown G-3 DEFINE 3-36 Definite integrals 13-15 DEFN 12-18 DEG 3-1 Degrees 1-23 DEL 12-46 DEL L L-1 DEL  L-1 DELALARM 25-4 Deleting su[...]

Pa g e N - 5 DISTRIB 5-28 DIV 15-4 DIV2 5-10 DIV2MOD 5-11, 5-14 Divergence 15-4 DIVIS 5-9 DIVMOD 5-11, 5-14 DO construct 21-61 DOERR 21-64 DOLIST 8-11 DOMAIN 13-9 DOSUBS 8-11 DOT 9-11 Dot product 9-11 DOT+ DOT- 12-44 Double integrals 14-8 DRAW 12-20, 22-4 DRAW3DMATRIX 12-52 Drawing functions programs 22-22 DRAX 22-4 DROITE 4-9 DROP 9-20 DTAG 23-1 E[...]

Pa g e N - 6 ERRN 21-65 Error trapping in programming 21-64 Errors in hypothesis testi ng 18-36 Errors in programming 21-64 EULER 5-10 Euler constant 16-54 Euler equation 16-51 Euler formula 4-1 EVAL 2-5 Exact CAS mode C-4 EXEC L-2 EXP 3-6 EXP2POW 5-28 EXPAND 5-4 EXPANDMOD 5-11 EXPLN 5-8, 5-28 EXPM 3-9 Exponential distribution 17-6 Extrema 13-12 Ex[...]

Pa g e N - 7 Function, table of values 12-17, 12-25 Functions, multi-variate 14-1 Fundamental theorem of algebra 6-7 G GAMMA 3-15 Gamma distribution 17-6 GAUSS 11-54 Gaussian elimination 11- 14, 11-29 Gauss-Jordan elimination 11-33, 11-38, 11-40 , 11-43 GCD 5-11, 5-18 GCDMOD 5-11 Geometric mean 8-16, 18-3 GET 10-6 GETI 8-11 Global vari able 21-2 Gl[...]

Pa g e N - 8 HELP 2-26 HERMITE 5-11, 5-18 HESS 15-2 Hessian matrix 15-2 HEX 3-2, 19-2 Hexadecimal numbers 19-7 Higher-order derivatives 13-13 Higher-order partial derivatives 14-3 HILBERT 10-14 Histogram s 12-29 HMS- 25-3 HMS+ 25-3 HMS  25-3 HORNER 5-11, 5-19 H-VIEW 12-19 Hyperbolic functions graphs 12-16 Hypothesis testing 18-35 Hypothesis test[...]

Pa g e N - 9 Integrals step-by-step 13-16 Integration by partial fractions 13-20 Integration by parts 13-19 Integration change of variable 13-19 Integration substitution 13-18 Integration techniques 13-18 Interactive drawing 12-43 Interactive input programming 21-19 Interactive plots with PLOT menu 22-15 Interactive self-test G-3 INTVX 13-14 INV 4-[...]

Pa g e N - 1 0 Left-shift functions B-5 LEGENDRE 5-11, 5-20 Legendre’s equation 16-51 Length units 3-19 LGCD 5-10 lim 13-2 Limits 13-1 LIN 5-5 LINE 12-44 Line editor commands L-1 Line editor properties 1-28 Linear Algebra 11-1 Linear Applications 11-54 Linear differ ential equations 16-4 Linear regression additional notes 18- 50 Linear regression[...]

Pa g e N - 1 1 Mass units 3-20 Math menu.. F-5 MATHS menu G-3, J-1 MATHS/CMPLX m enu J-1 MATHS/CONSTANTS menu J-1 MATHS/HYPERBOLIC menu J-2 MATHS/INTEGER menu J-2 MATHS/MODULAR menu J-2 MATHS/POLYNOMIAL menu J-3 MATHS/TESTS menu J-3 matrices 10-1 Matrix "division" 11-27 Matrix augmented 11-32 Matrix factorization 11-49 Matrix Jordan-cycle[...]

Pa g e N - 1 2 Multiple integrals 14-8 Multiple linear fitting 18-57 Multiple-Equation Solver 27-6 Multi-variate calculus 14-1 MULTMOD 5-11 N NDIST 17-10 NEG 4-6 Nested IF...THEN..ELSE..END 21-49 NEW 2-34 NEXTPRIME 5-10 Non-CAS commands C-13 Non-linear differential equations 16-4 Non-verbose CAS mode C-7 NORM menu 11-7 Normal distribution 17-10 Nor[...]

Pa g e N - 1 3 Partial fractions integration 13-20 Partial pivoting 11-34 PASTE 2-27 PCAR 11-45 PCOEF 5-11, 5-21 PDIM 22-20 Percentiles 18-14 PERIOD 2-37, 16-34 PERM 17-2 Permutation matrix 11-50, 11-51 Permutations 17-1 PEVAL 5-22 PGDIR 2-44 Physical constants 3-29 PICT 12-8 Pivoting 11-34 PIX? 22-22 Pixel coordinates 22-25 Pixel references 19- 7 [...]

Pa g e N - 1 4 17-6 Probability distributions discrete 17-4 Probability distributions for statistical inference 17-9 Probability mass function 17-4 Program branching 21-46 Program loops 21-53 Program-generated plots 22-17 Programming 21-1 Programming choose box 21-31 Programming debugging 21-22 Programming drawing commands 22-19 Programming drawing[...]

Pa g e N - 1 5 RCLMENU 20-1 RCWS 19-4 RDM 10-9 RDZ 17- 3 RE 4-6 Real CAS mode C-6 Real numbers C-6 Real numbers vs. Integer numbers C-5 Real objects 2-1 Real part 4-1 RECT 4-3 REF. RREF, rref 11-43 Relational operators 21-43 REMAINDER 5-11, 5-21 RENAM 2-34 REPL 10-12 Replace L-3 Replace All L-3 Replace Selection L-3 Replace/Find Next L-3 RES 22-6 R[...]

Pa g e N - 1 6 SEARCH menu L-2 Selection tree in Equation Writer E-1 SEND 2-34 SEQ 8-11 Sequential programming 21-15 Series Fourier 16-26 Series Maclaurin 13-23 Series Taylor 13-23 Setting time and date 25-2 SHADE in plots 12-6 Shortcuts G-1 SI 3-30 SIGMA 13-14 SIGMAVX 13-14 SIGN 3-14, 4-6 SIGNTAB 12-50, 13-10 SIMP2 5-10, 5-23 SIMPLIFY 5-2 9 Simpli[...]

Pa g e N - 1 7 Stiff differential equations 16-67 Stiff ODE 16- 66 Stiff ODEs numerical solution 16-67 STOALA RM 25-4 STOKEYS 20-6 STREAM 8-11 String 23-1 String concatenation 23-2 Student t distribution 17-11 STURM 5-11 STURMAB 5-11 STWS 19-4 Style menu L-4 SUB 10-11 Subdirectories creating 2-39 Subdirectories deleting 2-43 SUBST 5-5 SUBTMOD 5-11,[...]

Pa g e N - 1 8 TINC 3-34 TITLE 7-1 4 TLINE 12-45, 22-20 TMENU 20-1 TOOL menu CASCMD 1-7 CLEAR 1-7 EDIT 1-7 HELP 1-7 PURGE 1-7 RCL 1-7 VIEW 1-7 TOOL menu 1-7 Total differential 14-5 TPAR 12-17 TRACE 11-14 TRAN 11-15 Transforms Laplace 16-10 Transpos e 10-1 Triangle solution 7-9 Triangular wave Fourier series 16-34 TRIG menu 5-8 Trigonometric functio[...]

Pa g e N - 1 9 Vector elements 9-7 Vector fields 15-1 Vector fields curl 15-5 Vector fields divergence 15-4 VECTOR menu 9-10 Vector potential 15-6 Vectors 9-1 Verbose CAS mode C-7 Verbose vs. non-verbose CAS mode C-7 VIEW in plots 12-6 Viscosity 3-21 Volume units 3-19 VPAR 12-42, 22-10 VPOTENTIAL 15-6 VTYPE 24-2 V-VIEW 12-19 VX 2-37, 5-19 VZIN 12-4[...]

Pa g e N - 2 0 ! 17-2 % 3-12 %CH 3-12 %T 3-12  ARRY 9-6, 9-20  BEG L-1  COL 10-18  DATE 25-3  DIAG 10-12  END L-1  GROB 22-31  HMS 25-3  LCD 22-32  LIST 9-20  ROW 10-22  STK 3-30  STR 23-1  TAG 21-33, 23-1  TIME 25-3  UNIT 3-28  V2 9-12  V3 9-12 Σ DAT 18-7 Δ DLIST 8-9 Σ PAR 22-13 Π PLIST 8-9[...]

Pa g e LW- 1 L imited W arr ant y HP 5 0g graphing calc ulator ; W arr anty peri od: 12 months 1. HP w arr ants to y ou , the end-us er cu stomer , that HP hard w ar e, access or ies and suppli es w ill be fr ee fr om d e fec ts in mater ials and w orkmanship afte r the date of pur chas e , for the per iod s pecif ied abo v e . If HP r ecei ves not[...]

Pa g e LW- 2 W ARR ANTY S T A TEMENT ARE Y OUR SOLE AND EX CL US IVE REMEDIE S . EX CEPT A S INDICA TED ABO VE , IN NO EVENT WILL HP OR I T S S UP PLIER S BE LIABLE FOR L OS S OF D A T A OR F OR DIRE CT , SPE CIAL, INCIDENT AL , CON SE QUENT IAL (INCL UDING L O S T P ROFI T OR D A T A), OR O THER D AMA GE , WHETHER B ASED IN C ONTR A CT , T ORT , O[...]

Pa g e LW- 3 Swi t ze r l a n d +41-1- 4 3 9 5 3 5 8 (German) + 4 1 -2 2- 8 27878 0 ( F r e n c h ) +3 9-0 2 - 7 5419 7 8 2 (Italian) T urk e y +4 20 -5- 414 2 2 5 2 3 UK +44 - 20 7 - 4 5 80161 Cz ech R epubli c +4 20 -5- 414 2 2 5 2 3 South A f ri ca +2 7 -11- 2 3 7 6 200 L u xembour g + 3 2 - 2 - 712 6 219 Other E ur opean coun tr ies +4 20 -5- 4[...]

Pa g e LW- 4 Regulat or y inf ormation F edera l C o mmunications Commission Notice T his equipment has bee n tes ted and fo und to compl y w ith the limits for a C lass B digital de vi ce , pursuant t o P art 15 of the FCC R ules . T hese limits ar e designed to pr o v ide r easonable pr otection agains t harmf ul interfer ence in a r esidenti al [...]

Pa g e LW- 5 This de v ice complie s with P ar t 15 of the FCC R ules. Oper ation is sub ject to the follo wing tw o c ondi tions: (1) this dev ice may not caus e harmful interf er ence , and (2) this de vi ce must accept an y interfer ence rece iv ed , including interf er ence that may ca use undesir ed oper ation . F or questi ons r egarding y ou[...]

Pa g e LW- 6 This compli ance is indicated b y the follo w ing confor mit y marking placed on the pr oduc t: Japanese Notice 䈖 䈱ⵝ⟎䈲䇮 ᖱႎಣℂⵝ⟎╬㔚ᵄ㓚ኂ⥄ਥ ⷙ೙ද⼏ળ (V CCI) 䈱ၮḰ 䈮 ၮ䈨 䈒 ╙ੑᖱႎᛛⴚⵝ⟎ 䈪 䈜 䇯 䈖 䈱ⵝ⟎䈲䇮 ኅᐸⅣႺ 䈪 ૶↪䈜 䉎 䈖 䈫 䉕 ⋡ ⊛ 䈫 䈚 [...]