HP (Hewlett-Packard) 50g manual

HP  g gr aphing calc ulator user ’s guide H Ed it i on 1 HP part number F2 2 2 9AA-900 06[...]

Notice REG ISTER Y OUR PRODUCT A T: w ww .regis ter .hp .com THI S MANUAL AND ANY EX AMPLES CONT AINED H EREIN ARE PRO VIDED “ AS IS” AND A RE SUBJECT T O CHANGE WITHOUT NOTICE. HEWLET T-P A CKAR D COMP ANY MAKES N O W ARR ANTY OF ANY KIND WIT H REGA RD T O THI S MANU AL , I NCL UDIN G, BUT NOT LIMITED T O, THE IMPLIE D W ARR ANTIE S OF MERCHAN[...]

Pref ace Y ou have in y our hands a compact sy mbolic and numer ical comput er that will fac ilitate calc ulation and mathematical anal ysis o f pr oblems in a var iety of disc iplines, fr om elementary mathematic s to adv anced engineering and s c ience subjec ts. Although r ef err ed to as a calc u lator , because of its compact for mat r esembli[...]

F or sy mbolic oper ations the calc ulator includes a po we rful C omputer A lgebr aic S ystem (CA S) that lets you select diff er ent modes o f oper ation , e.g . , comple x numbers v s. r eal numbers , or ex act (sy mbolic) vs . appro ximat e (numer ical) mode . The displa y can be adjus ted to pr ov ide te xtbook-type e xpres sions , which ca n [...]

Pa g e TO C - 1 T able of contents Chapter 1 - Getting started ,1-1 Basic Operations ,1-1 Batteries ,1-1 Turning the calculator on and 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 date[...]

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 M ENU ,2-67 Selected CHOOSE boxes ,2-69 Chapter 3 - Calculation with real numbers ,3-1 Checking calculators settings ,3-1 Checking calculator mode ,3-2 Real number calculations ,3-2 Changing sign of a number, variable, or expression ,3-3 The inverse function ,3-3 Addition, subtract[...]

Pa g e TO C - 4 Physical constants in the calcula tor ,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 function ,5-23 The FCOEF function ,5-24 The FROOTS functi[...]

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 sub-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 polynomia[...]

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 coordinate sy stem ,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 vectors, and lists ,9-18 Function OBJ  ,9-19 Function  LIST ,9-20 Function DROP ,9-20 Transforming a row vector into a col[...]

Pa g e TO C - 1 0 Function VANDERMONDE ,10-13 Function HILBERT ,10-14 A program to build a matrix out of a number 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 Fun[...]

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 options in the c alculator ,12-1 Plotting an expression of the form y = f(x) ,12-2 Some useful PLOT operations for F[...]

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 REPL[...]

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 Derivatives ,1 3-3 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 serie s ,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 f[...]

Pa g e TO C - 1 6 Checking solutions in the calc ulator ,16-2 Slope field visualization 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 ,16-10 Laplace transform and inverses in the calculator ,16-[...]

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 RKF ,1 6-67 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 keyboard ,2 0-5 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 multiple[...]

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 - 22 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 T O C - 2 3 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 a[...]

Pa g e TO C - 24 Storing objects on an SD card ,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 L[...]

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 Get ting started This c hapter pr ov ides basi c inf ormatio n about the operati on of your calc ulator . It is designed to familiar iz e y ou w ith the basic oper ations and settings befo re y ou perfor m a calc ulation. Basic Operations The f ollow ing secti ons are desi gned to get y ou acquaint ed with the har dw are of y[...]

Pa g e 1 - 2 b . Insert a new CR20 3 2 lithium battery . Make sur e its positi ve (+) si d e is fac ing up. c. Replace the plate and push it to the or iginal place . After installing the batter ies , pre ss [ON] to turn the pow er on. Wa rn i n g : When the lo w bat tery icon is dis play ed, y ou need to r eplace the batteri es as soon as possible [...]

Pa g e 1 - 3 At the top of the displa y you w ill hav e two line s of inf ormati on that descr i be the settings of the calc ulator . T he first line sho ws the c har acter s: R D XYZ HEX R = 'X' F or details on the meaning of thes e s ymbo ls see Chapter 2 . The s econd line show s the char acter s: { HOME } indi cating that the HOME dir[...]

Pa g e 1 - 4 E ach gr oup of 6 entr ies is called a Menu page . The c urr ent menu , know n as the T OOL menu (see be lo w) , has ei ght entri es arr anged in two page s. The ne xt page , containing the next tw o entries o f the menu is av ailable by pr essing the L (NeXT menu) k e y . This k ey is the thir d k ey f r om the left in the third r ow [...]

Pa g e 1 - 5 This CHOO SE box is labeled B ASE MENU and pr ov ides a list of n u mber ed fun cti ons, from 1 . H EX x to 6. B  R. This dis play w ill constitute the f irst page of this CHOOSE bo x menu sho wing si x menu functi ons. Y ou can na v igate thr ough the menu b y using the up and do wn arr ow k ey s, —˜ , located in the upper ri gh[...]

Pa g e 1 - 6 If y ou no w pres s ‚ã , instead of the CHOO SE box that y ou sa w earli er , the displa y will no w show si x soft men u labels as the f irst page o f the S T A CK menu: T o nav igate thr ough the functions of this me nu , pres s the L ke y to m o ve to the ne xt page, o r „« (assoc iated w ith the L k e y ) t o m o v e t o t h [...]

Pa g e 1 - 7 The T OOL menu Th e soft men u ke ys f or the men u cur ren tly dis pla yed , kno wn as the T OOL menu , are a ssoc iated with oper ations r elated to manipulation of v ariable s (see pages for m ore info rmat ion o n variable s) : @EDIT A EDIT the contents of a v aria ble (see Chapter 2 and Appendi x L for mor e informati on on editin[...]

Pa g e 1 - 8 9 k ey the T IME choo se bo x is acti vated . This oper ation can also be r epre sented as ‚Ó . Th e TI ME cho os e b ox i s s hown in th e figu re b el ow: As indicated a bov e, the T IME menu pr ov ides f our differ ent options, number ed 1 thr ough 4. Of inter est to us as this poin t is option 3 . Set time , date.. . U s ing the[...]

Pa g e 1 - 9 Let’s c hange the m inute f ield to 2 5 , b y pressing: 25 !!@@OK#@ . T he seconds fi eld is now hi ghlighted . Suppose that y ou want to c hange the seconds fi eld to 4 5, use: 45 !!@@OK#@ The time f ormat 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 sett ing you ca n either press the W [...]

Pa g e 1 - 1 0 Setting the date After setting the time f ormat option , the SET T IME AND D A TE input f orm w ill look as fo llo ws: T o set the date , f irst s et the date f ormat . The def ault for mat is M/D/Y (month/ day/y ear). T o modif y this f ormat, pr ess the do wn arr o w k ey . T his w ill highli ght the date f ormat as show n below : [...]

P age 1-11 Introducing the calc ulator ’s k eyboar d The f igur e below sho ws a diagr am of the calculator ’s k ey board w ith the numbering of its r ow s 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 k eys eac h, and ro ws 4 th[...]

P age 1-12 shift ke y , k ey ( 9 ,1) , and the ALPHA k ey , ke y (7 ,1) , can be combined with some of the other k ey s to acti vate the alternati ve f unctions sho wn in the k ey board . F or ex ample , the P key , key (4,4) , has the f ollo wing si x functi ons assoc iated wit h it : P Main functi on, to acti vate the S YMBolic menu „´ Left-sh[...]

Pa g e 1 - 1 3 Pr ess the !!@@OK#@ soft men u k ey t o r etur n to normal display . Ex amples of selecting diffe ren t calc ulator modes ar e show n next . Oper ating Mode The calc ulator offer s two oper ating modes: the Algebr aic mode , and the Re vers e P olish Notatio n ( RPN ) mode . The def ault mode is the Algebr aic mode (as indicat ed in [...]

Pa g e 1 - 1 4 T o enter this e xpres sion in the calc ulator w e will f irst us e the equati on wr iter , ‚O . P lease identify the f ollo wing k ey s in the k ey board , besides the numer ic k ey pad k e ys: !@.#*+-/R Q¸Ü‚Oš™˜—` The eq uation w riter is a dis play mode in w hich y ou can build mathematical e xpre ssi ons using e xplic[...]

Pa g e 1 - 1 5 Change the oper ating mode to RPN by f irst pr essing the H butto n. S elect th e RPN operating mode b y either using the k ey , 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 ke y to complete the oper ation . The displa y , f or the RPN mode looks as follo w s: Notice that the displa y show s sev [...]

Pa g e 1 - 1 6 3.` Ent er 3 in lev el 1 5.` Ent er 5 in lev el 1, 3 mov es to y 3.` Ent er 3 in lev el 1, 5 mov es to lev el 2 , 3 to lev el 3 3.* P lace 3 and multiply , 9 appears in le vel 1 Y 1/(3 × 3), last value in lev . 1; 5 in lev el 2; 3 in le vel 3 - 5 - 1/(3 × 3) , occ upi es le vel 1 no w; 3 in lev el 2 * 3 × (5 - 1/(3 × 3)), occu pi[...]

Pa g e 1 - 1 7 Notice ho w the expr ession is placed in stac k lev el 1 after pressing ` . Pr essing the EV AL k ey at this po int will e valuate the numer ical value of that e xpr essi on Note: In RPN mode, pr essing ENTER when ther e is no command line w ill ex ecute the DUP fu nction w hich cop ies the contents of stac k lev el 1 of the stac k o[...]

Pa g e 1 - 1 8 mor e about r eals, see Cha pter 2 . T o illu str ate this and other number f ormats try the fo llo w ing ex erc ises: Θ Standard f ormat : This mode is the mos t used mode as it sho ws number s in the most famili ar notation . Pr ess the ! !@@OK# @ soft menu k ey , with the Number f ormat set to Std , to r eturn to the calc ulator [...]

Pa g e 1 - 1 9 Notice that the Number F ormat mode is set t o Fix follo wed b y a z er o ( 0 ). This n umber indicate s the number of dec imals to be sho wn after t he dec imal point in the calc ulator’s displa y . Press the !!@ @OK#@ sof t menu ke y to r eturn to the calc ulator display . The number no w is sho wn as: This se t ting w ill f or c[...]

Pa g e 1 - 2 0 Press the !!@@OK#@ soft menu ke y to complete the selection: Pr ess the !!@@OK#@ s oft menu k ey r eturn to the calc ulator displa y . The n umber now is sho wn as: Notice ho w the number is r ounded, not tr uncated . Thu s, the number 12 3 .45 6 7 8 9012 3 45 6 , fo r this setting, is dis played a s 12 3 .45 7 , and not as 12 3 .4 5[...]

Pa g e 1 - 2 1 same fashi on that we c hanged the Fixe d number of dec imals in the exa mp l e a b ove ) . Pr ess the !!@@OK#@ soft menu ke y retur n to the calc ulator display . The n umber now is sho wn as: This r esult , 1.2 3E2 , is the calculator ’s versi on of po wer s-of-ten notatio n, i. e. , 1.2 3 5 x 10 2 . In this, so -called , sc ient[...]

Pa g e 1 - 22 Pr ess the !!@@OK#@ s oft menu k ey r eturn to the calc ulator displa y . The number now is sho wn as: Becaus e this number has thr ee fi gur es in the intege r part, it is sho wn w ith four si gnificati ve f igur es and a zer o pow er of ten , while using the Engineer ing for mat. F or e xample , the number 0.00 2 56 , will be sho wn[...]

Pa g e 1 - 23 Θ Pr ess the !!@@OK#@ s oft menu k ey r eturn to the calc ulator displa y . The number 12 3 .45 6 7 8 9012 , enter ed earlie r , no w is sho wn as: Angle Me asure T ri gonometric functi ons, for e xample , requir e arguments r epre senting plane angles . The calc ulator pro vi des three diff erent A ngle Measure mode s fo r wor ki n [...]

Pa g e 1 - 24 ke y . If using the latter appr oach, u se up and do wn arr ow k ey s, — ˜ , to selec t the pref err ed mode , and pr ess the !!@@OK#@ soft m enu key to complete the ope rati on. F or ex ample , in the follo wing s cr een, the R adians mode is select ed: Coordinate S y stem The c oordi na te system se lect ion a ffect s t he way ve[...]

Pa g e 1 - 25 fr om the positi ve z ax is to the r adial distance ρ . The R ectangular and Spher ical coordinate s ys tems are r elated by the follo w ing quantities: T o change the coor dinate s ys tem in yo ur calculat or , follo w these step s: Θ Pr ess the H bu t ton . Next , use the do wn ar r o w k ey , ˜ , thr ee times . Select the Angle [...]

Pa g e 1 - 26 _La st St ack : K eep s the contents o f the last stac k entry for use w ith the functi ons UNDO and ANS (s ee Chapter 2). The _Beep option can be us eful t o adv ise the user abou t err ors. Y ou ma y want to deselec t this option if using y our calc ulator in a cla ssr oom or library . The _K ey Clic k option can be usef ul as an au[...]

Pa g e 1 - 27 Selecting Display modes The calc ulator display can be c ustomi z ed to your pr efer ence by selec ting different disp lay mod es. T o see t he opt ional disp lay setti ngs use t he follow ing: Θ F irst , pr ess the H button to ac tivat e the CAL CULA T OR MODE S input f orm . Within the CAL CULA T OR MODE S input for m, pre ss the @[...]

Pa g e 1 - 28 Pr essing the @CH OOS soft men u k ey w ill pr ov ide a list of a vailable s yst em fo nts, as sho wn belo w: The opti ons availa ble ar e three standar d Sys t e m Fo n t s (siz es 8, 7 , and 6 ) and a Br ow se .. option . The latter w ill let yo u br ow se the calcul ator memory f or additional f onts that y ou may ha ve c reated (s[...]

Pa g e 1 - 2 9 displa y the DISPLA Y MOD E S input for m. Pr ess the do wn arr ow k ey , ˜ , twi ce, to get to the St ack line . This line sho ws tw o propertie s that can be modifi ed. When these pr operties ar e select ed (chec ked) the fo llo wi ng effec ts are acti vated: _Small Changes f ont si ze to small . This max imi zed the amoun t of in[...]

Pa g e 1 - 3 0 times , to get t o the EQW (E quation W riter ) line. T his line shows tw o propertie s that can be modifi ed. When these pr operties ar e select ed (chec k ed) the fo llow ing eff ects ar e activ ated: _Small Changes f ont si z e to small w hile using the equati on editor _Small S tac k Disp Sho ws small font in the s tack f or text[...]

Pa g e 1 - 3 1 ri ght arr ow k ey ( ™ ) t o select the under line in fr ont of the options _Clock or _Analog . T oggle the @  @CHK@@ soft menu k e y until the desir ed setting is achie ved . If the _Clock opti on is selected , the time of the da y and date w ill be sho wn in the upper r ight corner of the display . If the _Analog opti on is al[...]

Pa g e 2 - 1 Chapter 2 Intr oducing the calc ulator In this chapter w e present a n umber of basic operati ons of the calculator including the u se of the E quation W riter and the manipulatio n of data obj e c ts in the calc ulator . S tudy the e xamples in this c hapter to ge t a good gras p of the capabi lities o f the calc ulator f or futur e a[...]

Pa g e 2 - 2 the CAS , it might be a good idea to sw itch dir ectl y into appr ox imate mode. Re fer t o Appendi x C for mor e det ails. Mi xing integers and reals together or mi staking an integer for a real is a common occ urre nce. Th e calc ulator w ill detect su ch mi xing o f objects and as k y ou if y ou want to s wit ch t o appr ox imate mo[...]

Pa g e 2 - 3 Binary integers , obje cts of t ype 10 , are used i n some computer science applications . Graphics objec ts , ob jects o f t ype 11, s tor e graphi cs produced b y the calculator . T agged objec ts , obj ects of ty pe 12 , ar e used in the ou tput of man y progr ams to identify r esults . F or ex ample, in the t agged object: Mean: 2 [...]

Pa g e 2 - 4 The r esulting e xpres sion is: 5.*(1.+1./7.5)/( √ 3.-2.^3). Press ` to get the e xpres sion in the display as f ollow s: Notice that , if your CA S is set to EXACT (s ee Appendix C) and y ou enter y our e xpr essi on using integer number s for in teger v alues, the r esult is a sy mbolic quantity , e . g ., 5*„Ü1+1/7.5™/ „ÜR[...]

Pa g e 2 - 5 T o e valuate the e xpr essi on w e can use the EV AL f u ncti on, as f ollo ws: μ„î` As in the pre vi ous e xample , you w ill be ask ed to appr ov e changing the CAS setti ng to Appro x . Once this is done , you w ill get the same r esult as befo r e. An alter na ti ve w ay to e valuate the e xpres sion en ter ed earlier betw een[...]

Pa g e 2 - 6 This latter r esult is pur ely numer ical , so that the t w o re sults in the stack , although r epre senting the same e xpres sion, seem diff erent . T o ver ify that they ar e not, w e subtr act the t w o values and ev alua te this diff er ence using f unction E V AL: - Subtr act le v el 1 fr om lev el 2 μ Evalua te using function E[...]

Pa g e 2 - 7 The editing c ursor is sho wn as a blinking le ft arr ow o ver the f irst c harac ter in the line to be edited. Since the editing in this case consists of r emov ing some char acter s and replac ing them with other s, w e w ill use the r igh t and left arr o w keys, š™ , to mo ve the c ursor to the a ppropr iate place f or editing, [...]

Pa g e 2 - 8 W e set the calc ulator operating mode t o Algebr aic, the CA S to Exact , and the displa y to T extbook . T o ent er this algebr aic e xpre ssion w e use the f ollo wing keyst ro kes : ³2*~l*R„Ü1+~„x/~r™/ „ Ü ~r+~„y™+2*~l/~„b Press ` to get the follo w ing resul t: Enter ing this expr essi on when the calculator is s [...]

Pa g e 2 - 9 Θ Pr ess the r ight arr ow k ey , ™ , until the c ursor is to the ri ght of the x Θ Ty p e Q2 to enter the pow er 2 fo r the x Θ Pr ess the r ight arr ow k ey , ™ , until the c ursor is to the ri ght of the y Θ Pr ess the delet e ke y , ƒ , once to er ase the char acters y. Θ Ty p e ~„x to enter an x Θ Pr ess the r ight ar[...]

Pa g e 2 - 1 0 Θ Pr essi ng ` once more to re turn to normal display . T o see the entir e expr essi on in the scr een, w e can c hange the option _Small Stack Disp in the DISP LA Y MODE S input for m (see Chapter 1). After effec ting this c hange , the display w ill look as follo ws: Using the Equation W riter (EQW) to create e xpressions The equ[...]

Pa g e 2 - 1 1 The si x soft menu k ey s for the E quation W rit er acti vate the fo llow ing functi ons: @EDIT : lets the u ser edit an entry in the line editor (see e xample s abo ve) @CURS : highli ghts expr ession and adds a graphi cs cur sor to it @BIG : if selec ted (selecti on sho wn by the c harac ter in the label) the font us ed in the wr [...]

Pa g e 2 - 1 2 The r esult is the e xpr essi on The cur sor is sho wn a s a left-fac ing ke y . T he curs or indicat es the c urr ent edition location . T yp ing a char acter , f unctio n name, or oper ation w ill enter the corr esponding char acter or c h ar acters in the c ursor location . F or ex ample, f or the cu rsor in the location indi cate[...]

Pa g e 2 - 1 3 Suppos e that no w y ou want t o add the fr action 1/3 to this entir e expr ession , i .e., y ou wan t to ent er the expr ession: F irst , w e need to highli ght the entir e firs t ter m by using either the r ight ar ro w ( ™ ) or the upper arr o w ( — ) k ey s, r epeatedly , until the entire e xpre ssion is highli ghted , i.e .,[...]

Pa g e 2 - 1 4 Show i ng the expr ession in smaller-si ze T o show the e xpres sion in a smaller -siz e fo nt (w hic h could be usef ul if the e xpre ssi on is long and con volut ed) , simply pr ess the @BIG soft men u k ey . F or this case, the scr e en looks as follo ws: T o r ecov er the larger -font displa y , pre ss the @BIG s oft menu k ey on[...]

Pa g e 2 - 1 5 If y ou wan t a floating-point (n umerical) e valuation , use the  NUM fun ction (i .e., …ï ). T he r esult is as follo ws: Use the function UNDO ( …¯ ) o n c e mo re to re c ove r t he o ri gi n a l ex pre ss io n : Ev aluating a sub-expr ession Suppos e that yo u want t o ev aluate only the e xpres sion in par entheses in [...]

Pa g e 2 - 1 6 A sy mbolic ev aluation once more . Suppose that , at this point , we w ant to ev aluate the left -hand side fr action onl y . Pre ss the upper ar r o w ke y ( — ) three times to s elect that fr action , r esulting in: Then , press the @EVAL soft menu ke y to ob tain: Let ’s tr y a numer ical ev aluation of this t erm at this poi[...]

Pa g e 2 - 1 7 Editing arithmetic e x pr essions W e will sho w some of the editing featur es in the Equati on W riter as an e xer cis e. W e start by e ntering the f ollow ing expr essi on used in the pr ev ious e xe rc ises: And w i ll us e the editing featur es of the E quation E ditor to transf orm it into the follo w ing expr essio n: In the p[...]

Pa g e 2 - 1 8 Pr ess the do wn ar ro w ke y ( ˜ ) to trigger the c lear editing c ursor . The scr een now looks lik e this: By using the left arr ow k ey ( š ) y ou ca n mo ve the c urso r in the gener al left dir ecti on, bu t stopping at eac h indiv idual component of the e xpres sion . F or e xample , suppose that w e will f irst w ill transf[...]

Pa g e 2 - 1 9 Ne xt, w e’ll con vert the 2 in front of the par enth eses in the denominator into a 2/3 by using: šƒƒ2/3 At this point the e xpr essi on looks as fo llow s: The f inal step is to r emov e the 1/3 in the ri ght -hand side o f the expr essi on. Thi s is accomplished by us ing: —————™ƒƒƒƒƒ The f inal ver sion w il[...]

Pa g e 2 - 2 0 Use the follo wing k ey str okes: 2 / R3 ™™ * ~‚n+ „¸ ~‚m ™™ * ‚¹ ~„x + 2 * ~‚m * ~‚c ~„y ——— / ~‚t Q1/3 This r esults in the output: In this ex ample we us ed se ve ral lo we r- case English letter s, e . g ., x ( ~„x ), sev eral Gr eek letters, e .g., λ ( ~‚n ) , and e ven a co mbination of G[...]

Pa g e 2 - 2 1 Editing algebraic e xpressions The editing of algebr aic equations f ollow s the same rules as the editing of algebrai c equations. Name ly : Θ Use the ar r ow k ey s ( š™—˜ ) to highli ght expr essions Θ Use the do wn arr o w ke y ( ˜ ) , re peatedly , to tri gger the clear editing c ursor . In this mode , use the left or r[...]

Pa g e 2 - 2 2 2. θ 3. Δ y 4. μ 5. 2 6. x 7. μ in the expone ntial func tion 8. λ 9. 3 i n t h e √ 3 te rm 10. the 2 in the 2/ √ 3 fr action At an y point we can c hange the clear editing cur sor into the insertio n cur sor by pr essing the delet e k ey ( ƒ ). Let’s use thes e two c ursors (the c lear editing cu rsor and the inserti on [...]

Pa g e 2- 23 Ev aluating a sub-expr ession Since w e alread y have the sub-e xpre ssion highli ghted , let’s pre ss the @EVAL so ft menu ke y to ev aluate this sub-expr ession . The re sult is: Some algebr aic expr essions cannot be simplif ied any more . T r y the follo wing keyst ro kes : —D . Y ou will noti ce that nothing happens, other tha[...]

Pa g e 2- 24 3 in the fi rst ter m of the numerator . Then , pres s the ri ght a rr o w k ey , ™ , to nav igate through the e xp r ession . Simplifying an e xpression Pr ess the @BIG soft menu k ey 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 soft menu k e y , to s ee if it is possible to simplify[...]

Pa g e 2- 25 Press ‚¯ to reco ver the or iginal expr ession . Next , enter the follo wing keyst ro kes : ˜˜˜™™™™™™™———‚™ to sele ct the last two ter ms in the e xpre ssion , i.e ., pr ess the @F ACTO soft menu k ey , to get Press ‚¯ to reco ver the ori ginal expr ession . Now , let’s select the entir e e xpre ssi [...]

Pa g e 2 - 2 6 Ne xt, s elect the command DERVX (the de ri vati ve w ith r espec t to the var iable X, the c urr ent CAS independent v ariable) b y using: ~d˜˜˜ . Command DER VX will no w be sele ct ed: Pr ess the @@ OK@@ s oft men u k ey to get: Ne xt, pr ess the L k ey to r ecov er the or iginal E quation W riter men u , and pr ess the @E VAL@[...]

Pa g e 2- 27 Detailed explanati on on the u se of the help fac ilit y fo r the CA S is pr esented in Chapter 1. T o retur n to the E quation W r iter , pre ss the @EXIT so ft menu k ey . Pre ss the ` ke y to ex it the Eq uation W r iter . Using the editing func tions BEGIN, END , COP Y , CUT and P ASTE T o fac ilitate editing , whether w ith the E [...]

Pa g e 2 - 2 8 Ne xt, w e’ll cop y the fr actio n 2/ √ 3 from t he l ef tm ost fa ctor in the exp ression, and place it in the numerator o f the ar gument for the LN functi on. T ry the follo w ing k ey str ok es: ˜˜šš———‚¨˜˜ ‚™ššš‚¬ The r esulting sc r een is as f ollo ws: The f unctions BEGIN and END ar e no t necessa[...]

Pa g e 2 - 2 9 W e can no w cop y this expr essio n and place it in the denominator o f the LN argume nt, as f ollow s: ‚¨™™ … (2 7 times ) … ™ ƒƒ … (9 times) … ƒ ‚¬ The li ne editor no w looks like this: Pr essi ng ` sho ws the e xpres sion in the Eq uation W r iter (in small-font fo rmat , pr ess the @B IG soft menu key) : [...]

Pa g e 2 - 3 0 T o see the corr esponding e xpres sion in the line editor , pr ess ‚— and the A soft menu k ey , to show : This e xpres sion sho ws the gener al for m of a summation typed dir ectly in the stack or line editor : Σ ( inde x = starting_value , ending_value , summation e xpres sion ) Press ` to r eturn to the E quation W r iter . [...]

Pa g e 2 - 3 1 and the var iable of diff erentiati on. T o f i ll thes e input locatio ns, us e the follo wing keyst ro kes : ~„t™~‚a*~„tQ2 ™™+~‚b*~„t+~‚d The r esu lting scr een is the follo wing: T o see the corr esponding e xpres sion in the line editor , pr ess ‚— and the A soft menu k ey , to show : This indi cates that t[...]

Pa g e 2- 32 Definite integr als W e wi ll use the E quation W r iter to ente r the follo wing def inite integr al: . Pr ess ‚O to acti vate the E quation W r iter . T hen pr ess ‚ Á to enter the integr al sign. Notice that the si gn, w hen entered into the E quation W rit er scr een, pr ov ides input locations f or the limits of integr ation [...]

Pa g e 2- 33 Double integr als are als o possible . F or ex ample, whi ch ev aluates to 3 6. P artial e valuati on is possible , fo r ex ample: This integr al ev aluates to 3 6. Organizing data in the calculator Y ou can organi z e data in y our calculat or by st oring v aria bles in a direc tor y tr ee . T o underst and the calc ulator’s memo ry[...]

Pa g e 2 - 3 4 @CHDIR : Change to selected dir ectory @CANCL : Cancel a ction @@OK@@ : Appro ve a selec tion F or ex ample, to c hange directory to the CA SD IR, pr ess the do wn-arro w ke y , ˜ , and pr ess @ CHDIR . T his action c loses the Fi l e M a n a g e r w indow and r eturns us to nor mal calculator displa y . Y ou w ill notice that the s[...]

Pa g e 2 - 3 5 T o mov e between the differ ent soft men u commands, y ou can use not only the NEXT ke y ( L ), but also the PREV k ey ( „« ). The u ser is in vited to try these f uncti ons on his or her o wn . The ir applicati ons ar e strai ghtforw ard . The HOME dir ector y The HO ME direc tory , as point ed out earli er , is the base dir ect[...]

Pa g e 2- 3 6 This time the CA SD IR is highlight ed in the scr een. T o see the contents of the dir ectory pr ess the @@ OK@@ sof t me nu ke y or ` , to get the f ollow ing scr een: The s cr een sho w s a table des cr ibing the var iables cont ained in the CA SDIR dir ectory . These ar e var iables pr e -def i ned in the calc ulator memory that es[...]

Pa g e 2 - 37 Pr essing the L k ey show s one more var iable stor ed in this directory: • T o see the contents o f the var iable EPS , for e xam p le , use ‚ @EPS@ . T his sho ws the va lue of EP S to be .00 00000001 • T o see the value of a n umeri cal vari able , we need t o pres s only the soft menu k ey f or the v ari able . F or ex ample[...]

Pa g e 2- 3 8 lock the alpha betic k ey board tempor aril y and enter a f ull name bef or e unloc king it again. T he follo w ing combination s of k ey str okes will lock the alphabetic k e yboar d: ~~ locks the alphabeti c ke yboar d in upper case . When lock ed in this fas hion, press in g th e „ befo r e a letter k ey pr oduces a low er case l[...]

Pa g e 2- 39 Creating subdir ectories Subdir ector ies can be cr eated by using the FILES en vir onment or by using the co mm a nd CR D IR. Th e t wo ap proa che s fo r cre at i ng su b- di rect orie s a re pr esent ed next . Using the FILES menu Regar dless of the mode of oper ation of the calc ulator (Algebrai c or RPN) , we can cr eate a direc t[...]

Pa g e 2 - 4 0 The Object input f ield, the f irst input f ield in the fo rm , is highlight ed by def ault. This input f ield can hold the contents of a new v ariable that is be ing cr eated. Since w e hav e no contents f or the new sub-dir ectory at this point, we simpl y skip this input fi eld by pr essing the do wn-arr o w ke y , ˜ , once . The[...]

Pa g e 2 - 4 1 T o mo ve into the MAN S direct ory , pr ess the co rr esponding so ft menu k ey ( A in this case), and ` if in algebraic mode . T he direc tor y tr ee will be show n in the second line of the displa y as {HOME M NS} . Ho we ver , ther e will be no labels as soc iated w ith the soft me nu k ey s, as sho wn belo w , beca use ther e ar[...]

Pa g e 2- 42 Use the do wn ar ro w ke y ( ˜ ) to select the option 2. M E M O RY … , or j ust press 2 . Then, pr ess @@OK@@ . This w ill pr oduce the fo llow ing pull-dow n menu: Use the do wn arr ow k ey ( ˜ ) t o select the 5 . DIRE CT OR Y opti on, or j ust press 5 . Then, pr ess @@OK@@ . T his will pr oduce the follo wing pull-do wn menu: U[...]

Pa g e 2- 4 3 Pr ess the @@ OK@ soft menu ke y to activ ate the command, to cr eate the sub- dir ectory: Mov ing among subdirectories T o mov e dow n the dir ector y tr ee, y ou need to press the s oft menu ke y corr esponding to the sub-dir ectory you w ant to mo ve to . The list o f var iable s in a sub-dir ectory can be pr oduced by pr essing th[...]

Pa g e 2 - 4 4 The ‘S2’ str ing in this f orm is the name o f the sub-direct ory that is being de leted . The s oft menu k ey s pro vi d e the f ollow ing options: @YES@ Pr oceed with deleting the sub-dir ectory (or var iable) @ALL@ Pr oceed w ith deleting all sub-dir ector ie s (or var iables) !ABORT Do not delete sub-dir ectory (or var iable)[...]

Pa g e 2 - 4 5 Use the do wn ar ro w ke y ( ˜ ) to select the option 2. M E M O RY … T h e n , press @@OK@@ . This w ill produce the f ollo w ing pull-do wn menu: Use the do wn ar r o w ke y ( ˜ ) to select the 5 . DIRE CT OR Y option . The n, press @@OK@@ . This w ill produce the f ollo w ing pull-do wn menu: Use the do wn ar ro w k e y ( ˜ )[...]

Pa g e 2 - 4 6 Press @@OK@@ , to get: Then , press ) @ @S3@@ to enter ‘S3’ as the ar gument to PGDIR. Press ` to delete the sub-dir ectory: Command PGDIR in RPN mode T o use the P GD IR in RPN mode y ou need to ha ve the name o f the direc tor y , between q uotes , alread y availa ble in the stac k befor e accessing the command. F or ex ample: [...]

Pa g e 2- 47 Using the PURGE command fr om the TOOL menu The T OOL men u is av ailable by pr essing the I ke y ( Algebr aic and RPN modes sho wn): The P URGE command is av ailable by pr essing the @PURGE s oft menu k e y . In the follo w ing e xample s w e want t o delete su b-dir ectory S1 : • Algebr aic mode: Enter @PURGE J ) @@S1@@ ` • RPN m[...]

Pa g e 2- 4 8 Using the FILES menu W e wi ll 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 to this sub-dir ectory , use the fo llo wing: „¡ and sel ect the INTR O sub-direc tor y as sho wn in this scr e en: Press @@OK@@ t o enter the dir ectory . Y ou w ill get a fil es l[...]

Pa g e 2- 49 T o enter var iable A (see table abov e ), we fir st enter its contents , namely , the number 12 .5, and then its name , A, as follo ws: 12.5 @@OK@@ ~a @@OK@@ . Resulting in the f ollow ing scr een: Press @@OK@@ once more to cr eate the vari able. T he new var iable is show n in the follo w ing var iable listing: The lis ting indicate [...]

Pa g e 2- 5 0 Using the ST O  command A simpler wa y to cr eate a var iable is by u sing the S T O command (i.e ., the K k ey). W e pr ov ide e xamples in both the A lgebrai c and RPN modes, b y cr eating the r emaining of the v ari ables suggested abo ve , namely : • Algebr aic mo de Use the f ollo wing k ey str okes to s tor e the value o f [...]

Pa g e 2 - 5 1 z1: 3+5*„¥ K~„z1` (if needed , accept change t o Comple x mode) p1: ‚å‚é~„r³„ì* ~„rQ2™™™ K~„p1` . The s cr een, at this po int, w ill look as follo ws: Y ou will see si x of the se ven v ariable s listed at the bottom of the sc reen: p1, z1, R, Q, A12 , α . • RPN mode Use the f ollo w ing k ey str okes t[...]

Pa g e 2 - 52 z1: ³3+5*„¥ ³~„z1 K (if needed, accept c hange to Comple x mode) p1: ‚å‚é~„r³„ì* ~„rQ2™™™ ³ ~„p1™` K . The s cr een, at this po int, w ill look as follo ws: Y ou will s ee si x of the se ven v ar iables listed at the bottom o f the sc reen: p1, z1, R, Q, A12 , α . Chec king var iabl es contents As an e[...]

Pag e 2- 53 Pr essing the s oft menu k ey cor r esponding t o p1 w ill pr ov ide an err or messa ge (try L @@@p1@@ ` ): Note: By pre ss i ng @@@p1@@ ` we ar e try ing to acti vate (run) the p1 progr am. Ho we ver , this progr am expects a nume ri cal input . T r y the fo llow ing ex erc ise: $ @@@p1@ „Ü5` . The r esult is: The pr ogra m has the [...]

Pa g e 2 - 5 4 At this point , the scr een looks like 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 progr am in RPN mode, y ou only need to enter the input (5) and pr ess the corr esponding soft menu k ey . (In algebrai c mode , yo u need to place parenth eses to [...]

Pag e 2- 55 Notice that this time the contents o f pr ogr am p1 are liste d in t he scr een. T o see the r emaining var iables in this dir ectory , pr ess L : Listing the content s of all var iables in the screen Use the k ey str oke combinati on ‚˜ to list the contents of all var iables in the sc r een. F or ex ample: Press $ to re turn to nor [...]

Pa g e 2- 56 follo wed b y the var iable ’s soft menu k ey . For e xample , in RPN, if we w ant to change the contents of v ariable z1 to ‘ a+b ⋅ i ’, u s e : ³~„a+~„b*„¥` This w ill place the algebrai c expr ession ‘ a+b ⋅ i ’ in le ve l 1: i n t h e st a ck. To e n t er this result into v aria ble z1 , use: J„ @@@z1@ @ T o[...]

Pa g e 2 - 5 7 Use th e up ar r o w ke y — to select the sub-dir ectory MANS and pres s @@OK@@ . If you no w press „§ , the scr een will sho w the contents of sub-directory MANS (notice that v ariable A is show n in this list, as e xp ect ed): Press $ @INTRO@ ` ( Algebrai c mode), or $ @IN TRO@ (RPN mode) to r eturn to the INTRO dir ectory . P[...]

Pa g e 2- 58 Ne xt, u se the delete k ey thr ee times, to r emo ve the las t three lines in the dis play : ƒ ƒ ƒ . At this poin t , the stac k is r eady to e xec ute the command ANS(1)  z1 . Pr ess ` to e xec ute this command. Then , us e ‚ @@z1@ , to ver ify the contents of the v ariable . Using the stack in RPN mod e T o demonstr ate the [...]

Pa g e 2 - 59 Copy ing two or more v ariables using the stack in RPN mod e The f ollow ing is an ex erc ise to demonstr ate ho w to copy two or mor e var iables using the stac k when the calc ulator is in RPN mode. W e as sume, again, that w e are w ithin sub-dir ectory {HOME MANS INTRO} and that w e want t o copy the var iable s R and Q int o sub-[...]

Pa g e 2- 6 0 The s cr een no w show s the ne w order ing of the var iables: RPN mode In RPN mode, the list o f r e -orde red v ariables is listed in the st ack bef ore apply ing the command ORDER. Su ppose that w e start fr om the same situation as abov e, but in RPN mode , i. e., Th e reo rd ere d l i st i s cre a t ed by us i n g : „ä ) @INTR[...]

Pa g e 2 - 6 1 Notice that v ariable A12 is no longer ther e. If y ou no w press 㤠, the sc r een w ill sho w the contents of sub-dir ectory MANS, inc luding vari able A12 : Deleting var iables V ari ables can be deleted using functi on PUR GE . T his fu nction can be acc essed dir ectl y b y using the T OOLS menu ( I ), or by usin g the FILE S[...]

Pa g e 2 - 62 vari ab le p1 . Pr ess I @PURGE@ J @@p1@@ ` . The scr een will now sho w vari ab le p1 re m ove d : Y ou can use the P URGE command to era se mor e than one var iable b y plac ing their names in a lis t in the argument o f PUR GE. F or ex ample, if no w we w anted to purge v ariable s R and Q , simultaneou s l y , w e can try the foll[...]

Pa g e 2 - 6 3 the HIS T ke y: UNDO r esults fr om the ke ystr oke s equence ‚¯ , w hile CMD r esults fr om the k ey str oke se quence „® . T o illustr ate the us e of UNDO , try the follo w ing ex er c ise in algebr aic (AL G) mode: 5*4/3` . T he UNDO command ( ‚¯ ) will simply er ase the re sult. T he same ex erc ise in RPN mode, w ill f[...]

Pa g e 2 - 6 4 As you can s ee, the number s 3, 2 , and 5, us ed in the fi rst calc ulation abov e, ar e listed in the se lecti on bo x, as w ell as the algebr a i c ‘SIN(5x2)’ , but not the SIN f u ncti on entered pr ev ious to the algebr aic. Flags A flag is a Boo lean value , that can be se t or clear ed (true or fals e), that spec ifies a g[...]

Pa g e 2- 6 5 Ex ampl e of flag setting: general solutions v s. principal value F or ex ample, the def ault v a lue f or s yst em flag 01 is Gener al soluti ons . What this means is that, if an equati on has multiple soluti ons, all the s olutions w ill be r eturned b y the calculator , most lik ely in a list . By pr essing the @  @CHK@@ soft me[...]

Pa g e 2- 6 6 ` (keep ing a second copy in the RPN stack) ³~ „t` Use the follo wing k ey strok e sequence to enter the QU AD command: ‚N~q (use the up and do wn arr ow k ey s, —˜ , to s elect command QU AD) , pr ess @@OK@@ . The sc reen sho ws the princ ipal soluti on: No w , c hange the setting of flag 01 to Gener al soluti ons : H @FLAGS@[...]

Pa g e 2 - 67 CHOOSE bo x es vs. So f t MENU In some of the e xer cises pr esented in this chapter w e ha ve seen men u lists of commands displa yed in the sc reen . Thes e menu lists ar e re fer red to as CHOOSE bo x es . F or ex ample, to us e the ORDE R command to r eorde r var iables in a dir ecto ry , w e use , in algebr aic mode: „°˜ Sho [...]

Pa g e 2- 6 8 The s cr e en sh ow s flag 117 not s et ( CHOO SE box es ), as sho wn here: Pr ess the @  @CHK@@ soft menu k ey to set flag 117 t o soft MENU . The sc reen w ill r efl ect that c hange: Press @@OK@@ twice to r eturn to normal calculator displa y . No w , w e’ll try to find the ORDER command using similar k e y str okes to those u[...]

Pa g e 2- 69 Note: most o f the e xam p les in this us er guide assume that the c urre nt setting of flag 117 is its de fault s etting (that is, not set). If yo u hav e set the flag but w ant to str ictly f ollow the e xam ples in this guide , you should c lear the flag be for e contin uing. Selected CHOOSE box es Some men us w ill only pr oduce CH[...]

Pa g e 2- 70 • The CMD S (CoMmanD S) menu , acti vat ed w ithin the Equati on W riter , i. e. , ‚O L @CMDS[...]

Pa g e 3 - 1 Chapter 3 Calculation with real numbers This c hapter demonstr ates the use of the calc ulator for oper ations and func tions r elated to r eal numbers . Oper ations along the se lines ar e usef ul for mos t common calc ulations in the ph ysi cal sc iences and engineer ing. T he user should be acquainted w ith the ke yboar d to identif[...]

Pa g e 3 - 2 2 . Coordinate sy stem specifi cat ion (XYZ , R ∠ Z, R ∠∠ ). T he s y mb ol ∠ stands f or an angular coor dinate . XYZ: Cartesi an or rectangular (x ,y ,z) R ∠ Z: cylindr ical P olar co or dinates (r , θ ,z) R ∠∠ : Spher ical coordinat es ( ρ,θ,φ ) 3 . Number bas e spec ificati on (HEX, DEC, OCT , BIN) HEX: he xadec i[...]

Pa g e 3 - 3 Real n u mber calc ulations w ill be demonstr ated in both the Algebr aic ( AL G) and Re ver se P olish Notation (RPN) modes . Changing sign of a number , variable, or e xpression Use the ke y . In AL G mode , yo u can pr ess be for e entering the number , e.g., 2.5` . R esult = - 2 . 5. In RPN mode , yo u need to enter at least p[...]

Pa g e 3 - 4 Alter nativ ely , in RPN mode , y ou can separate the oper ands with a space ( # ) befo re pr essing the oper ator ke y . Example s: 3.7#5.2 + 6.3#8.5 - 4.2#2.5 * 2.3#4.5 / Using parentheses P arentheses can be used to gr oup operations , as well as to enc lose arguments of func tions . The par entheses ar e available thr ough the ke y[...]

Pa g e 3 - 5 Squares and squar e roots The s quar e function , SQ, is a vailable thr ough the ke ystr ok e combination: „º . When calc ulating in the stack in AL G mode , enter the func tion befo r e the argument , e.g ., „º2.3` In RPN mode, ent er the number f irst , then the functi on, e .g., 2.3„º The sq uar e root f unction, √ , is[...]

Pa g e 3 - 6 Using po wers o f 10 in entering data P owe rs of te n, i.e. , n u mb e rs of th e for m - 4 .5 ´ 10 -2 , etc., ar e entered b y using the V k ey . For e xample , in AL G mode: 4.5V2` Or , in RPN mode: 4.5V2` Natural logar ithms and exponential function Natur al logarithms (i .e ., logarithms of base e = 2. 7 1 828 1 8282 ) are ca[...]

Pa g e 3 - 7 the inv erse tr igonometri c functi ons re pr esent angles, the ans w er fr om these func tions w ill be give n in the select ed angular measur e (DEG , RAD, GRD). Some e xamples ar e show n next: 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 des cribed abo ve , namely ,[...]

Pa g e 3 - 8 combinati on „´ . With the default s et ting of CHOO SE box es fo r syst em flag 117 (see Chapter 2), the MTH menu is show n as the follo wing menu list: As the y are a gr eat number of mathematic f unctions a vailable in the calc ulator , the MTH menu is so rted by the ty pe of obj ect the fu nctio ns apply on . F or ex ample , opt[...]

Pa g e 3 - 9 Hy perbolic func tions and their inverses Selecting Option 4. HYP ERBOLIC.. , in the MTH menu , and pres sing @@OK@@ , pr oduces the h yperboli c function men u: The h yperbolic f unctions ar e: Hy perbolic sine , SINH, and its inv erse , ASINH or sinh -1 Hy perbolic cosine , CO SH, and its inv erse , AC OSH or cosh -1 Hy perbolic t an[...]

Pa g e 3 - 1 0 The r esult is: The ope rati ons show n abov e assume that you ar e using the defa ult setting for s ys tem flag 117 ( CHOO SE box es ) . If yo u hav e changed the se tting of this flag (see Chapter 2) to SO FT m e nu , the MTH menu w ill sho w as labels o f the soft menu k ey s, as fo llo ws (le f t-hand side in AL G mode, r ight ?[...]

Pa g e 3 - 1 1 F or ex ample, to calc ulate tanh( 2 . 5), in the AL G mode, w hen using SOF T m en us over CHOO S E bo xe s , f ollow this pr ocedure: „´ Select MTH menu ) @@HYP@ Selec t the HYPERBOLIC.. men u @@TANH@ Selec t the TA N H func tion 2.5` Ev aluate tanh(2 .5) In RPN mode, the s ame value is calc ulated using: 2.5` Enter ar gument in[...]

Pa g e 3 - 1 2 Option 19 . MA TH.. retur ns the user to the MTH menu . The r emaining functi ons ar e gr ouped into si x differ ent grou ps descr ibed belo w . If s ys tem flag 117 is set to SO FT m e nu s , the REAL fu nctio ns menu w ill look like this (AL G mode u sed, the s ame soft menu k ey s will be a vailable in RPN mode): The v ery last op[...]

Pa g e 3 - 1 3 The 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 , while ar gument x is located in the f irst le vel o f the stac k. T his means, y ou should enter x fir st, and then , y , j ust as in AL G mode. T hus, the calc ulation of %T(15, 4 5) , in RPN mode , and w ith s yste m [...]

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 should be us ed as y MOD x , and not as MOD(y,x) . Th us, the oper ation of M OD is similar to that of + , - , * , / . As an e xer cise , ver ify that 15 MOD 4 = 15 mod 4 = r esidual o f 15/4 = 3 Absolute value , sign, ma ntissa, e xponent,[...]

Pa g e 3 - 1 5 G AMMA: The Gamma func tion Γ ( α ) P SI: N -th deri vati ve of the digamma f unctio n P si: Digamma f unction , deri vati ve of the ln(Gamma) The Gamma f unction is def ined by . This f unction has applicati ons in applied mathemati cs f or sc ience and engineering , as well as in pr obabil ity and statisti cs. The PSI fu nct ion [...]

Pa g e 3 - 1 6 Example s of these s pec ial func tions ar e show n her e using both the AL G and RPN modes. As an e xe r c ise , verify that G AMMA(2 . 3) = 1.166 711…, PSI(1 .5, 3) = 1 .40909 . ., and Psi ( 1 .5 ) = 3. 6 48997 39 .. E-2 . The se calc ulations ar e sho wn in the follo w ing sc r een shot: Calculator constants The f ollo wing ar e[...]

Pa g e 3 - 1 7 Selecting an y of these en tri es will place the v alue select ed, w hether a sy mbol (e .g., e , i , π , MINR , or MAXR ) or a v alue ( 2 .7 1.., (0,1) , 3 . 14.., 1E-4 99 , 9. 9 9. . E 4 9 9 ) in the stac k. P lease notice that e is a vailable fr om the ke yboar d as exp ( 1) , i.e ., „¸1` , in AL G mode , or 1` „¸ , in RPN [...]

Pa g e 3 - 1 8 The u ser w ill recogni z e mos t of these units (s ome , e.g ., dy ne , are not u sed v ery often no wada ys) fr om his or her ph ysics c lasses: N = ne wtons, dyn = dyne s, gf = gr ams – for ce (to distinguish fr om gram-mas s, or plainly gr am, a unit of mass), kip = kilo -poundal (1000 pounds) , lbf = pound-f or ce (to distingu[...]

Pa g e 3 - 1 9 A vailable units The f ollow ing is a list of the units av ailable in the UNI TS men u . T he unit s ymbo l is show n first f ollow ed by the unit name in parenth eses: LENG TH m (meter), cm (centimeter), mm (millimeter), yd (y ar d) , ft (feet), in (inch) , Mpc (Mega parsec), p c (pars ec) , ly r (light -y ear) , au (astr onomical u[...]

Pa g e 3 - 2 0 SPEED m/s (meter per se cond), cm/s (centimeter per second), ft/s (feet per second), kph (kilometer per hour ) , mph (mile per hour), knot (nautical mile s per hour), c (speed of li ght) , ga (accelerati on of gr av it y ) MA SS k g (kilogram), g (gram), Lb (av oirdu pois pound) , oz (ounce), slug (slug) , lbt (T r oy pound) , ton (s[...]

Pa g e 3 - 2 1 ANGLE (planar and solid angle mea sur ements) o (se xage simal degree), r (radi an) , gr ad (gr ade) , arcmin (minut e of ar c) , arc s (second of ar c) , sr (ster adian) LIGHT (Illumination measur ements) fc (footcan dle) , flam (f ootlambert) , lx (lu x), ph (phot) , sb (stilb), lm (lumem), cd (candela), lam (lamber t) RADI A T IO [...]

Pa g e 3 - 22 Conv er ting to base units T o conv ert an y of these units to the def ault units in the SI s yst em, u se the functi on UB ASE . F or e xample , to find out what is the v alue of 1 pois e ( unit of v iscosi ty) in the SI units , use the follo wing: In AL G mode , sy ste m flag 117 set t o CHOOSE bo xes : ‚Û Select the UNIT S menu [...]

Pa g e 3 - 23 ` Con vert the units In RPN mode , s y stem flag 117 s et to SO FT m e nu s : 1 Enter 1 (no under line) ‚Û Select the UNIT S menu „« @ ) VISC Select the VIS COS ITY option @@@P@@ Select the unit P (po ise) ‚Û Select the UNIT S menu ) @TOOLS Select the T OOLS menu @UBASE Select the UB ASE functi on Attac hing units to numbers [...]

Pa g e 3 - 24 Notice that the under scor e is entered a utomati cally when the RPN mode is acti ve . The r esult is the fo llow ing scr een: As indicated earl ier , if sy stem flag 117 is s et to SO FT m e n us , then the UNIT S menu w ill show up as labels f or the soft menu k eys . This set up is ve r y conv enient fo r extensi ve oper ations wit[...]

Pa g e 3 - 25 Yy o t t a + 2 4 dd e c i - 1 Z z etta + 21 c c enti - 2 E ex a +18 m milli -3 P peta +15 μ mic ro -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 att o -18 h,H he cto +2 z z epto - 21 D(*) dek a +1 y yoct o - 2 4 _______________ ____________________ ________________ (*) In the S I[...]

Pa g e 3 - 26 whi ch sho ws as 6 5_(m ⋅ yd). T o conv ert to units of the SI s ys tem , use f unctio n UB ASE: T o calculat e a div ision, s ay , 3 2 50 mi / 5 0 h, enter it a s (3 2 50_mi)/(5 0_h) ` : whi ch transf ormed to S I units, w ith func tion UB ASE , pr oduces: Addition and su btrac tion can be perfor med, in AL G mode , without u sing [...]

Pa g e 3 - 27 Stac k calculations 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 ` / The se oper ations pr oduce the follo wing outpu t: Also , tr y the f ollow ing operations: 5_m ` 32 0 0 _ m m ` + 12_mm ` 1_cm^2 `* 2_s ` / The se las t two oper ations pr od[...]

Pa g e 3 - 2 8 UF ACT(x ,y) : fac tors a un it y fr om unit object x  UNIT(x ,y): combines value of x w ith units of y The UB ASE f unction w as discu ssed in detail in an earli er secti on in this cha pter . T o access an y of these f unctions f ollow the e xamples pr ov ided earlier f or UB ASE . Notice that , while func tion UV AL r e quir es[...]

Pa g e 3 - 2 9 Ex amples of  UNI T  UNIT( 25,1_m) `  UNIT(11. 3,1_mph) ` Ph ysical constants in the calculator F ollow ing along the treatment of units , we dis cu ss the use of ph ysical const ants that are a vailable in the calc ulator’s memory . These ph ysi cal constan ts ar e contained in a co nstants libr ar y acti vated w ith the [...]

Pa g e 3 - 3 0 The s oft menu k ey s corre sponding to this CONS T ANT S LIBR AR Y sc r een include the fo llo wing f unctions: SI whe n select ed, cons tants values ar e show n in SI units ENGL w hen selec ted, cons tants value s ar e sho wn in English units ( *) UNIT when se lected , constants ar e show n with units attac hed (*) V AL UE w hen se[...]

Pa g e 3 - 3 1 T o see the v alues of the constants in the English (or Imper ial) s ys tem , pre ss the @ENGL optio n: If we de-select the UNIT S opti on (pre ss @UNITS ) only the values ar e shown (English units se lected in this case): T o cop y the value of Vm to the st ack , select the v ariable name , and pre ss ! , then, pr ess @QUIT@ . Fo r [...]

Pa g e 3 - 32 Special phy sical func tions Menu 117 , tri ggered b y using MENU(117) in AL G mode, or 117 ` MENU in RPN mode , produce s the fol low ing menu (labels lis ted in the displa y by u sing ‚˜ ): The fun ct ion s i ncl ud e: ZF A CT OR: gas compr essibility Z fac tor function F AN NI NG : Fann in g frict ion fact or fo r fl uid flow DA[...]

Pa g e 3 - 3 3 ZF ACT O R(x T , y P ) , wher e x T is the reduced temper ature , i . e ., the ratio o f actual temper ature to p seudo -cri tical temper ature , and y P is the r educed pr essur e, i .e., the r atio of the actual pr essur e to the pseudo -cr itical pr essur e . The v alue of x T must be between 1. 05 and 3 .0, while the value of y P[...]

Pa g e 3 - 3 4 Function TINC F unction TI NC(T 0 , Δ T) calculat es T 0 +D T . The ope rati on of this functi on is similar to that of f uncti on TDEL T A in the sens e that it retur ns a re sult in the units of T 0 . Otherwise , it retur ns a simple addition of value s, e .g., Defining and using functions User s can def ine their o wn functi ons [...]

Pa g e 3 - 3 5 Pr ess the J k ey , and you w ill notice that ther e is a ne w vari able in y our so ft menu k ey ( @@@H@@ ). T o see the contents of this var iable pr ess ‚ @@@H@@ . T h e sc r een wi ll s how now: Thu s, the var iable H contains a pr ogram def ined by : <<  x ‘LN(x+1) + EXP(x)’ >> This is a simple pr ogram in t[...]

Pa g e 3 - 3 6 The cont ents of the v ari able K are: <<  α β ‘ α+β ’ >>. Functions defined b y more than one expr ession In this secti on we disc uss the tr eatment of f unctions that ar e def ined b y two or mor e expr essio ns. An e xample o f such f unctions w ould be The fun ctio n I FT E ( IF-Th en -E lse ) d escri bes [...]

Pa g e 3 - 37 Combined IFTE functions T o pr ogram a mor e complicated f u ncti on such as y ou can combine se ver al leve ls of the IFTE func tion, i .e., ‘ g(x) = IFTE(x<- 2 , - x, IFTE(x<0 , x+1, IFTE(x<2 , x-1, x^2)))’ , Define this f unction b y an y of the means pr esented abo ve , and chec k that g(-3) = 3, g(-1) = 0, g(1) = 0, [...]

Pa g e 4 - 1 Chapter 4 Calculations with complex numbers This c hapter sho ws e xam ples of calc ulations and applicati on of func tions to comple x numbers . Definitions A complex number z is a n umber w r itten as z = x + iy , where x and y ar e real numbers , and i is the imaginary unit defined b y i 2 = - 1. T he complex number x+iy has a r eal[...]

Pa g e 4 - 2 Press @@OK@@ , t w ice , to r eturn to the stack . Entering comple x numbers Comple x numbers in the calc ulator can be enter ed in either of the tw o Car tesian representations, nam ely , x+iy , or (x,y) . The r esults in the calc ulator w ill be show n in the or der ed-p air for mat, i .e., (x ,y) . F or e xample , with the calc ulat[...]

Pa g e 4 - 3 Notice that the last entry sho ws a complex n umber in the for m x+iy . This is so because the n u mber w as enter ed between single quot es, w hich r eprese nts an algebrai c expr essi on. T o ev aluate this number use the EV AL ke y( μ ). Once the algebrai c expr ession is e valuated , y ou reco ver the comple x number (3. 5 ,1. 2) [...]

Pa g e 4 - 4 On the other hand , if the coordinate s yste m is set t o cy lindrical coor dinates (use CYLIN), enter ing a complex number (x ,y) , wher e x and y are r eal numbers, w ill pr oduce a polar repr esentati on. F or e xample , in cy lindrical coor dinates, en ter the number (3 .,2 .). T he fi gur e belo w show s the RPN stack , bef ore an[...]

Pa g e 4 - 5 Changing sign of a complex number Changing the sign o f a complex n umber can be accomplished b y using the k e y , e.g ., -(5-3i) = -5 + 3i Entering the unit imaginary number T o enter the unit imaginary number type : „¥ Notice that the n umber i is enter ed as the order ed pair (0,1) if the CA S is set to APP RO X mode . In EX A[...]

Pa g e 4 - 6 CMP LX menu through the MTH menu Assuming that s yst em flag 117 is se t to CHOOSE bo xes (see Chapter 2), the CMPLX sub-men u within the MTH men u is acc essed by using: „´9 @@OK@@ . The follo wing sequence of scr een shot s illustrates these steps : The f irst menu (options 1 thr ough 6) show s the fo llo wing f unctions: RE(z) : [...]

Pa g e 4 - 7 This f irst sc reen sho ws f unctions RE , IM, and C  R. Notice that the la st f unction r eturns a list {3 . 5.} r epre senting the r eal and imaginar y components of the comple x number : The f ollow ing scr een show s functi ons R  C, ABS , and ARG . Notice that the ABS f unction gets tr anslated to |3.+5 .·i|, the notation o[...]

Pa g e 4 - 8 The r esulting menu inc lude some of the f unctions alr eady intr oduced in the pr ev ious s ecti on , namely , ARG , ABS , CONJ , IM, NE G, RE , and S IGN. It also include s func tion i whi ch serve s the s ame purpo se as the ke ys trok e combination „¥ , i .e., to enter the unit imaginary number i in an expr ession . The k ey boa[...]

Pa g e 4 - 9 Functions from the MTH menu The h yper bolic functi ons and their inv erses , as well as the Gamma, P SI, and P si functi ons (special f unctions) w er e introduced and appli ed to r eal numbers in Chapter 3 . Thes e functi ons can also be applied to comple x numbers by follo w ing the procedur es pre sented in Chapter 3 . Some e xampl[...]

Pa g e 4 - 1 0 F unction DROI TE is found in the command catalog ( ‚N ). Using EV AL(ANS(1)) simplif ies the r esult to:[...]

Pa g e 5 - 1 Chapter 5 Algebraic and arithmetic operations An algebr aic obj ect , or simply , algebrai c, is an y number , vari able name or algebrai c expr essi on that can be oper ated upon , manipulated , and combined accor ding to the rules o f algebr a. Ex amples of algebr aic ob jec ts ar e the fo llow ing: • A number : 12 .3, 15 .2_m, ‘[...]

Pa g e 5 - 2 (e xponential , logarithmic , trigonometry , hy perboli c, etc.), as y ou w ould any r eal or comple x number . T o demonstr ate basic oper ations w ith algebr aic obj ects , let’s cr eate a coup le of objects, say ‘ π *R^2’ and ‘ g*t^2/4’ , and stor e them in var iables A1 and A2 (See Chapter 2 to learn ho w to cr eate vari[...]

Pa g e 5 - 3 ‚¹ @@A1@@ „¸ @@ A2@@ The s ame r esults ar e obtained in RPN mode if using the follo 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 The AL G ( Algebr aic) menu is a vaila ble b y using the k ey str ok e sequence ‚×[...]

Pa g e 5 - 4 W e notice that , at the bottom of the sc reen , the line See: E XP AND F A CT OR suggests links to other help fac ility entr ies , the f unctions E XP AND and F A CT OR. T o mov e direc tly to tho se entr ies, pr ess the soft men u ke y @SEE1! for E XP AND , and @SEE2! f or F A CT OR. Pr essing @SEE1! , for e xample , show s the follo[...]

Pa g e 5 - 5 F A CT OR: LNCOLLE CT : LIN: P A R TFR A C: S OL VE: SUB S T : TEXP AND: Note : Re call that, to u se these , or any other f unctions in the RPN mode, y ou mus t enter the ar gument fi rst , and then the func tion . Fo r ex ample , the e xample f or TEXP AND , in RPN mode will be se t up as: ³„¸+~x+~y` At this point , select f unct[...]

Pa g e 5 - 6 Other forms of substitution in algebraic e xpressions F unctions SUB ST , show n ab o ve , is us ed to substitute a v ariable in an e x pr ession . A second f orm of sub stitution can be accomplished b y using the ‚¦ (assoc iated w ith the I k e y) . F or e xample , in AL G mode , the fol low ing entry w ill subs titute the v alue x[...]

Pa g e 5 - 7 A differ ent approac h to subs titution consists in def ining the substitution e xpre ssi ons in calc ulator v ari ables and placing the name o f the var iables in the ori ginal expr ession . F or ex ample, in AL G mode , stor e the follo wing v aria bles: Then , enter the expr ession A+B: The las t expr essi on entered is aut omatical[...]

Pa g e 5 - 8 LNCOLLE CT , and TEXP AND are also contained in the AL G menu pr esented earli er . Functi ons LNP1 and EXP M wer e intr oduced in menu HYPERB OLIC, under the MTH menu (S ee Chapte r 2) . The onl y remai ning fu nct ion is EX PLN. Its desc ripti on is show n in the left-hand side , the ex ample fr om the help fac ility is sho wn to the[...]

Pa g e 5 - 9 Functions in the ARITHME TIC menu The ARI THMET IC menu contains a number o f sub-menu s fo r spec ific appli c ati ons in number theo ry (integers , poly nomials , et c.), as w ell as a n umber of f unctions that appl y to gener al arithme tic ope rati ons. The AR ITHME TIC menu is tr igger ed through the k ey str ok e combinati on ?[...]

Pa g e 5 - 1 0 L GCD (Greatest C ommon Denominator): PROPFRA C (proper f rac tion) SIM P2: The f unctions assoc iated w ith the ARI THMETIC submenu s: INTE GER, POL YNOMIAL , MODUL O , and P ERMUT A TION, ar e the follo wing: INT EG ER men u EUL E R N um b er of in te g ers < n, c o - pri m e wi th n IABCUV Sol v es au + b v = c, w ith a,b ,c = [...]

Pa g e 5 - 1 1 F ACT OR F actori z es an integer number or a poly nomial FCOEF Gener ates fr action gi ven r oots and multipli city FROO T S Returns r oots and multiplic ity giv en a fr action GCD Gr eatest common div isor of 2 numbers or pol ynomials HERMITE n -th degree Her mite poly nomial HORNER Hor ner e valuatio n of a poly nomial LAGRANGE La[...]

Pa g e 5 - 1 2 Applications of the ARI THMET IC m enu This sec tion is inte nded to pr esent some of the back ground necessary for applicati on of the ARITHMET IC menu f unctions. Def initions ar e pres ented next r egarding the su bjec ts of poly nomials , poly nomial fr actions and modular arithme tic . The e xamples pr esented belo w ar e pr ese[...]

Pa g e 5 - 1 3 multiply ing j times k in modulus n arithmeti c is, in essence , the integer r emainder of j ⋅ k / n in infinit e arithmeti c , if j ⋅ k>n . F or ex ample, in modulu s 12 arithme tic we ha ve 7 ⋅ 3 = 21 = 12 + 9 , (or , 7 ⋅ 3/12 = 21/12 = 1 + 9/12 , i .e., the integer r eminder of 21/12 is 9). W e can no w wr ite 7 ⋅ 3 ?[...]

Pa g e 5 - 1 4 Notice that , whene ver a r esult in the ri ght-hand side of the “ c ongr uence ” sy mbol pr oduces a r esult that is larger than the modulo (in this case , n = 6) , you can alw ay s subtr act a multiple of the modulo fr om that result and simplify it to a number smaller than the modulo. T hus, the r esults in the firs t case 8 ([...]

Pa g e 5 - 1 5 [SPC ] entry , and the n pr ess the cor re sponding modular ar ithmetic f uncti on. F or e xam ple , using a modulus o f 12 , try 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 (mo d 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 con vert an y number into a r ing number by using the func tion EXP ANDM OD . For e xample , EXP AN DMO D(1 2 5) ≡ 5 (mod 12) EXP AN DMOD (17 ) ≡ 5 (mod 12) EXP ANDMOD(6) ≡ 6 (mod 12) The modular inverse of a number Let a number k belong to a f inite arithmetic r ing of modulus n , then the mod[...]

Pa g e 5 - 1 7 P ol ynomials P ol ynomials ar e algebraic e xpres sions consisting of one or mor e terms containing dec reasing po wer s of a giv en var iable . F or ex ample, ‘X^3+2*X^2 - 3*X+2’ is a third-o rder poly nomial in X, while ‘S IN(X)^2 - 2’ is a second-or d er poly nomial in SIN(X). A listing of pol ynomi al-r elated f uncti on[...]

Pa g e 5 - 1 8 numbers (f unction ICHINREM). The input consists o f tw o vec tors [e xpressi on_1, modulo_1] and [e xpres sion_2 , modulo_2] . The o utput is a v ector cont aining [e xpre ssion_3, modulo_3] , wher e modulo_3 is related to the product (modulo_1) ⋅ (modulo_2) . Example: CHINREM([X+1, X^2 -1],[X+1,X^2]) = [X+1,-(X^4 -X^2)] Statement[...]

Pa g e 5 - 1 9 An alter nate def inition of the Hermite pol yn omials is wher e d n /dx n = n- th der i vati ve w i th r espect to x . This is the definiti on used in the calculat or . Example s: The Hermit e poly nomials of or ders 3 and 5 ar e giv en by: HERMITE( 3) = ‘8*X^3-12*X’ , And HERMITE(5) = ‘3 2*x^5-160*X^3+120*X’ . The HORNER fu[...]

Pa g e 5 - 2 0 F or ex ample, f or n = 2 , we w ill wr ite: Check this r esult w ith your calc ulator: LAGRANGE([[ x1,x2],[y1,y2] ]) = ‘((y1-y2)*X+(y2*x1-y1*x2))/(x1- x2)’ . Other e xam ples: L A GR ANGE([[1, 2 , 3][2 , 8, 15]]) = ‘( X^2+9*X-6)/2’ LAGRANGE([[0. 5,1.5,2 . 5,3 . 5,4. 5][12 .2 ,13 . 5,19 .2 ,2 7 .3, 3 2 . 5]]) = ‘ -( . 1 3 7[...]

Pa g e 5 - 2 1 The P COEF function Gi ven an arr ay con taining the r oots of a poly nomial , the functi on PC OEF gener a tes an ar ra y containing the coeff ic ients of the cor r esponding pol ynomial . The coe ffi cients cor respond t o decr easing order o f the independent vari able. F or ex ample: PCOEF([- 2 ,–1, 0,1,1,2]) = [1. –1. –5. [...]

Pa g e 5 - 22 The EP SX0 func tion and the CAS vari able EPS The va riab le ε (epsilon) is typi cally used in mathemati cal te xtbooks to repr esent a ve ry small number . The calc ulator’s CA S cr eates a v ariable EP S, w ith defa ult value 0. 000000000 1 = 10 -10 , when y ou use the EPSX0 f unction. Y ou can change this v alue , once cr eated[...]

Pa g e 5 - 23 Frac ti on s F racti ons can be expanded and fact or ed by using func tions EXP AND and F ACT OR , fr om the AL G menu (‚×). F or e xam ple: 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+Y*X)/(X[...]

Pa g e 5 - 24 If y ou hav e the Complex mode ac ti ve , the re sult will be: ‘2*X+(1/2/(X+i)+1/2/(X- 2 )+5/(X-5)+1/2/X+1/2/(X -i))’ The FCOEF function The f unction FC OEF is used to obtain a r ational fr action, gi ven the roots and poles of the fr action . The in put for the func tion is a v ector listing the r oots follo wed b y their m ulti[...]

Pa g e 5 - 25 mode selected , then the re sults wo uld 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 i th poly nomials and fractions By s et ting the CA S modes to Step/s tep the calculat or will sho w simplifi cations of fr actions or oper ations w ith poly nomials in a step-b [...]

Pa g e 5 - 26 The CONVER T M enu and algebraic operations The C ONVERT menu is acti vated b y using „Ú ke y (the 6 key ) . T hi s menu summar iz es all con ver sion menus in the calc ulator . The list o f these menu s is sho wn ne xt: The f unctions a vailable in eac h of the sub-menu s ar e show n next . UNIT S convert menu (Option 1) This men [...]

Pa g e 5 - 27 B ASE conv er t menu (Option 2) This men u is the same as the UNI T S menu obtained b y using ‚ã . The applicati ons of this menu ar e disc uss ed in detail in Chapter 19 . TRIGONOMETRIC convert menu (Option 3) This men u is the same as the TRIG men u obtained b y using ‚Ñ . The appli c ati ons of this menu ar e disc uss ed in d[...]

Pa g e 5 - 2 8 Fu n ct i o n  NUM has the same eff ect as the ke ystr oke comb ination ‚ï (assoc iated w ith the ` key ) . Fu n ct ion  N UM c onver t s a symb ol ic resu l t i nto its floating-poin t v alue . Func tion  Q conv erts a floating-po int value into a fr action. F unction  Q π conv erts a floating-po int value into a fra[...]

Pa g e 5 - 2 9 LIN LNCOLLE CT PO WEREXP AND SIMP LIFY[...]

Pa g e 6 - 1 Chapter 6 Solution to single equations In this chapte r we f eature those f unctions that the calc u lator pr ov ides for s olv ing single equations of the for m f(X) = 0. Assoc iated with the 7 k ey ther e are two men us of eq uation-sol v ing functi ons, the S ymbolic S OL V er ( „Î ), and the NUMer ical SoL V er ( ‚Ï ) . F oll[...]

Pa g e 6 - 2 Using the RPN mode, the soluti on is accomplished by enter ing the equation in the stac k, f ollo wed by the v ari able , befor e enter ing func tion I S OL. R ight bef ore the ex ecuti on of ISOL , the R PN st ack should look as in the fi gure to the left . After appl y ing IS OL, the r esult is sho wn in the f igure to the ri ght: Th[...]

Pa g e 6 - 3 The sc reen shot sho wn abo ve dis plays tw o solutions . In the fir st one , β 4 -5 β =12 5, SOL VE pr oduces no so lutions { }. In the s econd one , β 4 - 5 β = 6, S OL VE pr oduces four s olutions , show n in the last output line . The v ery last solution is not visible becau se the r esult occ upies mor e charac ters than the w[...]

Pa g e 6 - 4 In the fir st case S OL VEVX could not find a solu tion . In the second case , S OL VEVX found a single so lution , X = 2 . The foll owing screen s sh ow th e R PN sta ck for solvin g t he t wo examp les shown abov e (befor e and after applicati on of SOL VEVX) : The equati on used as ar gument for functi on SOL VEVX must be r educible[...]

Pa g e 6 - 5 The S ymbolic So lv er functions pre sented abo ve pr oduce solutions to r ational equations (mainl y , poly nomial equations). If the equation to be so lv ed for has all numer ical coeffi ci ents, a numer ical solu tion is pos sible thr ough the use o f the Numer ical So lv er featur es of the calc ulator . Numerical sol ver menu The [...]

Pa g e 6 - 6 P ol ynomial Equations Using the Solv e p ol y… option in the calc ulator’s SO L V E en vir onment you can: (1) f ind the solutions to a pol ynomial equati on; (2) obtain the coeff ic ien ts of the pol yno mial ha ving a n umber of gi ven r oots; (3) obtain an algebr aic e xpressi on for the poly nomial as a functi on of X. Finding[...]

Pa g e 6 - 7 All the so lutions ar e complex n umbers: (0.43 2 ,-0. 38 9), (0.43 2 , 0.3 89), (-0.7 66 , 0.6 3 2) , (-0.7 66 , -0.6 3 2). Generating polynomial coe fficients giv en the polynomial's r oots Suppos e y ou want t o generate the pol ynomi al whose r oots are the nu mbers [1, 5, - 2 , 4]. T o us e the calculator for thi s purpos e, [...]

Pa g e 6 - 8 Press ˜ to tri g ger the line editor to see all the coe ffi cients . Generating an algebraic expr ession for the poly nomial Y ou can use the calc ulator to gener ate an algebrai c expr ession f or a poly nomial giv en the coeffi c ients or the r oots of the pol yno mial . The r esulting e xpre ssi on w ill be giv en in terms o f the [...]

Pa g e 6 - 9 T o e xpand the produ cts, y ou can use the EXP AND command. T he r esulting e xpr essi on is: ' X^4+-3*X^3+ -3*X^2+11*X-6' . A differ ent approac h to obtaining an expr essi on for the poly nomial is to gener ate the coeffi c ients firs t , then gener ate the algebrai c ex pre ssi on wi th the coeff ic ients highli ghted. F [...]

Pa g e 6 - 1 0 Ex ample 1 – Calc ulating pay ment on a loan If $2 milli on ar e borr ow ed at an annual inter est r ate of 6 .5% to be r epaid in 6 0 monthly pa yments , what should be the monthly pa yment? F or the debt to be totall y repaid in 6 0 months, the fu tur e value s of the loan should be z ero . So , f or the purpos e of using the f i[...]

Pa g e 6 - 1 1 pay ments. Suppose that w e use 2 4 per iods in the first line of the amorti zati on scr e en, i .e., 24 @@OK@@ . Then , pr ess @@AMOR@@ . Y ou will get the f ollo w ing res u l t : This s cr een is interpr eted as indicating that after 2 4 months o f pay i ng bac k the debt , the borr ow er has paid up US $ 7 2 3,211.4 3 into the pr[...]

Pa g e 6 - 1 2 ˜ Skip P MT , since w e will be s olv ing fo r it 0 @@OK@@ Enter FV = 0, the option End is hi ghlighted @@CHOOS ! — @@OK@@ Change pay ment option t o Begin — š @@SOLVE! H ighlight P MT and so lv e for it The s cr een now sho ws the v alue of P MT as –38 , 9 21.4 7 , i .e ., the borr ow er must pay the lender U S $ 38 , 9 21.4[...]

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 of var iable FV ` Exec ute P URGE command The follo w ing two s cr een shots sho w the P URGE co mmand for purging all the var iables in the dir ectory , and the r esult after e xec uting the command. In RPN mode, [...]

Pa g e 6 - 1 4 ³„¸~„x™-S„ì *~„x/3™‚Å 0™ K~e~q` Press J to see the ne wly c reated E Q var iable: Then , enter the SOL VE env ironment and select Sol ve equation… , by u s ing: ‚Ï @@OK@@ . The corr esponding sc r een wi ll be sho wn as: The equati on we sto red in var iable E Q is alr eady loaded in the Eq f ield in the S OL[...]

Pa g e 6 - 1 5 This , ho we ver , is not the only possible s olution f or this equati on. T o obtain a negativ e solutio n, f or e xampl e, ent er a negati ve number in the X: field be for e solv ing the equation. T ry 3 @@@OK@@ ˜ @SOLVE@ . The s olution is no w X: - 3.045. Solution procedur e for Equation Solve ... The n u mer ical sol ver f or [...]

Pa g e 6 - 1 6 The equati on is her e e xx is the unit strain in the x -directi on, σ xx , σ yy , and σ zz , are the normal str esses on the partic le in the dir ection s of the x -, y-, and z -axes , E is Y oung’s modulus or modulus of elastic ity of the materi al, n is the P oiss on rati o of the mater ial , α is the thermal e xpansion coef[...]

Pa g e 6 - 1 7 With the ex: field hi ghlighted , pres s @SOLVE@ to solv e f or ex : The s oluti on can be seen fr om within the S OL VE E QUA TION in put for m by pr essing @EDIT whi l e t he ex: fi eld is highlighted . The r esulting value is 2.47 0 833333333 E- 3. P r es s @@@OK@@ to ex it the EDIT featur e . Suppos e that y ou no w , w ant to de[...]

Pa g e 6 - 1 8 Spec ifi c energ y in an open channel is def ined as the energ y per unit wei ght measur ed with r espect to the c hannel bottom. L et E = spec ific ene rg y , y = chann el depth, V = f low v eloc it y , g = accelerati on of gr av ity , then w e wr ite The f lo w veloc ity , in turn , is giv en b y V = Q/A, wher e Q = water disc harg[...]

Pa g e 6 - 1 9 Θ Solv e for y . The r esult is 0.14 9 8 36 .., i.e ., y = 0.14 98 3 6 . Θ It is kno wn, how ev er , that ther e are ac tually two s oluti ons av ailable f or y in the spec ifi c energ y equation. T he soluti on we j ust found corr esponds to a numer ical soluti on with an initial v alue of 0 (the de faul t va lu e for y , i .e., w[...]

Pa g e 6 - 2 0 In the ne xt e xample w e will u se the D ARCY f unction f or finding fr icti on fac tors in pipelines . Thus , we def ine the functi on in the fo llow ing fr ame. Special function for pipe flo w: DARC Y ( ε /D ,Re) The Dar cy- W eisbac h equation is used to calc ulate the ener g y loss (per unit wei gh t ) , h f , in a pipe flo w t[...]

Pa g e 6 - 2 1 Ex ample 3 – Flow in a pipe Y ou may want t o cr eate a separate sub-dir ectory (PIPE S) to try this ex ample . The main eq uation go vernin g flo w in a pipe is, of cour se, the Dar cy- W eisbac h equation . Thu s, type in the fo llow ing equation into E Q: Also , enter the follo w ing var iables (f , A, V , Re): In this case w e [...]

Pa g e 6 - 2 2 The comb ined equation has pr imitiv e v a r iables: h f , Q, L, g, D, ε , and Nu . Laun ch t he nume rical solver ( ‚Ï @@OK@ @ ) to see the primiti ve v ari ables listed in the S OL VE EQU A T ION input f orm: Suppo se that w e use the v alues hf = 2 m, ε = 0. 00001 m , Q = 0.0 5 m 3 /s, Nu = 0. 000001 m 2 /s, L = 20 m , and g [...]

Pa g e 6 - 23 Ex ample 4 – Universal gr av itation Ne wton ’s law of uni versal gr av itation indi cates that the magnitude of the attrac ti ve fo r ce betw een tw o bodies of mass es m 1 and m 2 separated b y a distance r is gi ven b y the equation Here , G is the uni versal gra vitati onal constant , who se value can be obtained thr ough the [...]

Pa g e 6 - 24 Sol ve for F , and pres s to r eturn to nor ma l calc ulator display . The solu tion is F : 6. 6 7 2 5 9E-15_N , or F = 6.6 7 2 5 9 × 10 -15 N. Different wa ys to enter equations into EQ In all the ex amples sho wn abo ve we ha ve enter ed the equation to be sol ved dir ectl y into v ari able EQ be for e acti vating the n umeri cal s[...]

Pa g e 6 - 2 5 T y pe an equati on, sa y X^2 - 125 = 0, dir ectly on the s tack , and pres s @@@OK@@@ . At this point the equati on is r eady for so lution . Alter nati vel y , y ou can activ ate the equation w riter after pr essing @E DIT to enter y our equation. Pr ess ` to retur n to the numeri cal solver scr een. Another wa y to enter an equati[...]

Pa g e 6 - 2 6 The S OL VE soft menu The SOL VE sof t menu allows acc ess to som e of th e num erical solver funct ions thr ough the soft men u ke ys . T o access this menu us e in RPN mode: 7 4 MENU, or in AL G mode: MENU(7 4) . Alternati vely , y ou can use ‚ (hold) 7 to acti vate the S OL VE soft men u . The sub-menu s pr ov ided b y the SOL V[...]

Pa g e 6 - 27 Example 1 - Sol ving the equati on t 2 -5t = - 4 F or ex ample, if y ou stor e the equation ‘t^2 -5*t=- 4’ into E Q, and pr ess @) SOLVR , it w ill acti vate the f ollo wing menu: This r esult indicates that y ou can solv e for a value o f t for the equati on listed at the top of the display . If you try , for ex ample , „ [ t ][...]

Pa g e 6 - 2 8 Y ou can also solv e mor e than one equation by sol ving one equation at a time , and repeating the pr ocess until a soluti on is found . F or ex ample , if y ou enter the follo w ing list of equati ons into var i able EQ: { ‘ a*X+b*Y = c’ , ‘k*X*Y=s ’}, the k ey str oke seq uence @) ROOT @ ) SOLVR , w ithin the S OL VE so f [...]

Pa g e 6 - 2 9 Using units with the SOL VR sub-menu The se are s ome rules o n the use o f units w ith the SO L VR sub-menu: Θ Enter ing a guess w ith units for a gi ven v ari able , will intr oduce the use of those units in the s olution . Θ If a ne w guess is gi ven w ithout units, the units pr ev iousl y sa ved f or that partic ular v ar iable[...]

Pa g e 6 - 3 0 This f unction 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 , give n a ve c to r of it s ro ot s [r 1 , r 2 , …, r n ]. F or ex ample, a v ector who se r o ots ar e giv en by [-1, 2 , 2 , 1, 0], will pr oduc e the follo wing coeff ic ients[...]

Pa g e 6 - 3 1 Press J to e x it the S OL VR env ir onment . F i nd y our wa y bac k to the TVM sub- menu w ithin the S OL VE sub-me nu to try the other functio ns available . Function TVM ROO T This function requires as argument t he na me of one of the var iables in t he T VM pr oblem. T he functi on r eturns the s olutio n for that var iable , g[...]

Pa g e 7- 1 Chapter 7 Solv ing multiple equations Many pr oblems of sc ience and engineer ing req uir e the simultaneous so lutions of mor e than one equation . The calculator pr ov ides se ve ral pr ocedure s for solv ing multiple equations as pr esented belo w . P lease notice that no discussi on of solv ing sy stems of linear equation s is pr es[...]

Pa g e 7- 2 Use co mmand S OL VE at this poin t (fr om the S. SL V men u: „Î ) After a bout 40 seconds , may be more , you get as r esult a list: { ‘t = (x- x0)/(COS( θ 0)*v0)’ ‘ y 0 = (2*C OS( θ 0)^2*v0^2*y+(g*x^2(2*x0*g+2*SIN( θ 0))*C OS( θ 0)*v0^2)*x+ (x0^2*g+2*SIN( θ 0)*CO S( θ 0)*v0^2*x0)))/(2*CO S( θ 0)^2*v0^2)’]} Press μ t[...]

Pa g e 7- 3 the conten ts of T1 and T2 to the stac k and adding and subtr acting them. Her e is how t o do it with the equati on writ er : Enter and st ore ter m T1: Enter and stor e term T2 : Notice that w e are using the RPN mode in this ex ample, ho we ver , the pr ocedur e in the AL G mode should be v ery similar . Cr eate the equation f or σ [...]

Pa g e 7- 4 Notice that the r esult includes a v ector [ ] contained w ithin a list { }. T o remo ve the list s ymbol, u se μ . F inally , to decompose the vec tor , use functi on OB J  . The r esult is: The se two e xamples constitut e sy stems of linear equatio ns that can be handled equall y we ll w ith functi on LINS OL VE (s ee Chapter 11)[...]

Pa g e 7- 5 Ex ampl e 1 - Ex ampl e fr om the help facilit y As w ith all functi on entries in the help f acility , ther e is an ex ample at tac hed to the MSL V entry as show n abov e. Notice that f unction M SL V r equire s thr ee argume nts: 1. A vec tor containing the equati ons, i .e., ‘[S IN(X)+Y ,X+S IN(Y)=1]’ 2 . A v ector containing th[...]

Pa g e 7- 6 disc harge (m 3 /s o r ft 3 /s) , A is the cr oss-sec tional ar ea (m 2 or ft 2 ), C u is a coeff ic ient that depends on the s yst em of units (C u = 1. 0 for the S I, C u = 1.4 86 fo r the English sy stem of units), n is the Manning’s coe ffi cie nt , a measure o f the channel surface r oughness (e .g., for conc rete , n = 0.012), P[...]

Pa g e 7- 7 μ @@@EQ1@@ μ @@@EQ2@ @ . T he equations ar e listed in the s tack as follo ws (small font opti on selected): W e can see that these equati ons are indeed gi ven in ter ms of the pr imitiv e var iables b, m , y , g , S o , n, Cu , Q, and H o . In order to sol ve f or y and Q w e need to give v a lues to the other vari ables. Suppose w [...]

Pa g e 7- 8 Ne xt, w e’ll ente r var iable EQS: LL @ @EQS@ , follo wed b y vector [y ,Q]: ‚í„Ô~„y‚í~q™ and b y th e init ial guesses ‚í„Ô5‚í 10 . Bef ore pr essing ` , the sc reen w ill look like this: Press ` to sol ve the s yst em of equations . Y ou may , if your angular measur e is not set to r adians, get the follo wing[...]

Pa g e 7- 9 The r esult is a list of thr ee v ectors. The f irst v ector in the list will be the equati ons sol ved . The second v e ctor is the list of unkno wns . The thir d vecto r repr esents the soluti on. T o be able to see the se v ector s, pr ess the do wn-arr ow k ey ˜ to acti vate the line editor . The sol ution w ill be show n as follo [...]

Pa g e 7- 1 0 The co sine la w indicate s 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 ⋅ cos γ . In orde r to solv e any tr iangle , yo u need to know at leas t thr ee of the fol lo w ing si x v ari ables: a, b, c, α, β, γ . Then , you can use the equati [...]

Pa g e 7- 1 1 ‘SIN( α )/a = S IN( 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*COS( γ )’ ‘b^2 = a^2+c^2 - 2*a*c*CO S( β )’ ‘ a^2 = b^2+c^2 - 2*b*c*COS( 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 eate[...]

Pa g e 7- 1 2 Press J , if needed , to get y our var iables me nu . Y our men u should sho w the vari ab le s @LVARI! !@ TITLE @@EQ@@ . Preparing to run t he ME S The ne xt step is to acti vate the ME S and try one sample solution . Befor e we do that, ho we ver , we w a nt to se t the angular units to DEGr ees, if the y a r e not alr eady s et to [...]

Pa g e 7- 1 3 Let ’s tr y a simple s oluti on of Case I, using a = 5, b = 3, c = 5 . Us e the follo 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 sol ve f or the angles u se: „ [ α ] [...]

Pa g e 7- 1 4 Pr essi ng „ @@ALL@@ will so lv e for all the v ari ables, tempor aril y show ing the intermediate re sults. Press ‚ @@ALL@@ to see t he solu tions: When done , pres s $ to retur n to the MES en vir onment. Pr ess J to ex i t the ME S env ir onment and r eturn to the normal calc ulator display . Org anizing the variabl es in the s[...]

Pa g e 7- 1 5 Progr amming t he MES triangle solution using User RP L T o fac ilitate acti vating the ME S for f utur e so lutions , we w ill cr eate a pr ogr am that w ill load the MES w ith a single ke ystr ok e . The pr ogram should look lik e this: << DEG MINI T T ITLE L V ARI MI TM MS OL VR >>, and can be t y ped in b y using: ‚?[...]

Pa g e 7- 1 6 Use a = 3, b = 4 , c = 6. T he solution pr ocedure us ed her e consists of sol ving fo r all var iables at once , and then recalling the soluti ons to the stack: J @TRISO T o clear up data and r e -start ME S 3 [ a ] 4 [ b ] 6 [ c ] T o ent er data L T o mov e to the next v ariable s menu. „ @ALL! So lve f or all the unkno wns . ‚[...]

Pa g e 7- 1 7 Adding an I NFO but ton to your directory An inf ormati on button can be us eful f or your dir ectory to help y ou remember t he oper ation o f the functi ons in the direc tory . In this dir ectory , all we need to r emember is to pr ess @ TRISO to get a tr iangle solution s tarted. Y ou may w ant to type in the fo llo w ing pr ogr am[...]

Pa g e 7- 1 8 An e xplanation of the v ari ables follo ws : SOL V EP = a progr am that tri g gers the m u ltiple equati on sol ver f or the partic ular set of equations s tor ed in var iable PEQ ; NAME = a var iable st oring the name o f the multiple equati on solv er , namel y , "ve l. & acc . p olar coor d." ; LIST = a list of the v[...]

Pa g e 7- 1 9 Notice that after y ou enter a partic ular value , the calc ulator displa ys the var iable and its value in the upper left co rner of the dis play . W e hav e now enter ed the kno wn v aria bles . T o calc ulate the unkno wns w e can proceed in tw o ways: a). Solv e for indi vidual v ariable s, f or ex a mple , „ [ vr ] giv es vr: 0[...]

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

Pa g e 8 - 1 Chapter 8 Operations w ith lists L ists ar e a type of calc ulator’s ob ject that can be u seful f or data pr ocessing and in pr ogramming . This Chapt er pr esents e xamples of oper ations w ith lists. Definitions A list , within the conte xt of the calculator , is a ser ies o f objec ts enclosed between br aces and separated b y sp[...]

Pa g e 8 - 2 The f igur e belo w show s the RPN stack be fo r e pre ssing the K key : Composing and decomposing lists Compo sing and decomposing lis ts mak es sense in RPN mode onl y . Under suc h oper ating mode , decomposing a list is achi ev ed by u sing functi on OBJ  . With this functi on, a list in the RPN stac k is decomposed into its ele[...]

Pa g e 8 - 3 In RPN mode, the f ollow ing scr een show s the three lists and the ir names read y to be stor ed. T o stor e the lists in this case y ou need to pres s K three time s. Changing sign The si gn -change k ey ( ) , w hen applied t o a list of number s, will c hange the sign o f all elements in the list . Fo r exam ple: Addition, subtr a[...]

Pa g e 8 - 4 Subtr action , multiplication, and di vision o f lists of numbers o f the same length pr oduce a list of the same length w ith term-by-ter m oper ations. Ex amples: The di visi on L4/L3 will pr oduce an infinity entry becaus e one of the eleme nts in L3 is z er o: If the lists in vo lv ed in the oper ation hav e diffe rent lengths , an[...]

Pa g e 8 - 5 ABS 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 unctions of inter est fr om the MTH menu include , fr om the HYPERBOLIC men u: SINH , AS INH, CO SH, A COS H, T ANH, A T ANH, and fr om the REAL menu: %, %CH, %T , MIN, MAX, MOD , S IGN, MANT ,[...]

Pa g e 8 - 6 T ANH , A T ANH SIGN , MANT , XPON IP , FP FL OOR, CEIL D  R, R  D Ex ampl es of functions that use two arguments The s cr een shots below sho w applications o f the functi on % to lis t arguments . F unction % r equires two ar g uments. T he first tw o ex amples sho w c ases in w hich only one of the tw o ar guments is a list . [...]

Pa g e 8 - 7 %({10,20, 30},{1,2 ,3} ) = {%(10,1),%(20,2),%(30, 3)} This de sc ripti on of func tion % for lis t ar guments sh o ws the gener al pattern of ev aluation of an y functi on w ith two ar guments when one or both ar guments are lists . Example s of appli cations of f unctio n RND ar e show n next: Lists o f comple x numbers The f ollo wi [...]

Pa g e 8 - 8 The f ollow ing ex ample sho ws appli cations o f the functi ons RE(Real part) , IM(imaginary par t), AB S(magnitude), and ARG(argument) of comple x numbers . The r esults are lists of r eal numbers: Lists o f algebraic objects The f ollow ing are e xamples o f lists of algebr aic obj ects w i th the func tion SIN appl ied t o them: Th[...]

Pa g e 8 - 9 This me nu cont ains the fo llo w ing func tions: Δ LIS T : Calculate inc rement among consec utiv e elements in list Σ LIS T : Calc ulate summation o f elemen ts in the list Π LIS T : Calculate pr oduct of elements in the list S ORT : Sorts elements in inc reasing or der REVLIS T : Re ver ses orde r of list ADD : Oper ator for ter [...]

Pa g e 8 - 1 0 M anipulating elements of a list The P RG (pr ogramming) men u includes a LI ST su b-menu w ith a number o f func tions to mani pulate ele ments of a li st . With s ys tem f lag 117 set to CHOOSE bo x es: Item 1. ELEMENT S.. con tains the fol low ing func tions that can be us ed for the manipulation o f elements in lists: List si ze [...]

Pa g e 8 - 1 1 F unctions GET I and P UTI , als o av ailable in sub-menu PR G/ ELEMENT S/, can also be used to extr act and place elements in a list . Thes e two f unctions , ho we ver , ar e usef ul mainly in pr ogramming . F unction GE TI us es the same argume nts as GET and r eturns the list , the element locati on plus one , and the element at [...]

Pa g e 8 - 1 2 SEQ is u seful to pr oduce a list of v alues gi ven a partic ular expr essi on and is desc r ibed in more de tail her e . The SE Q functi on tak es as arguments an e xpressi on in terms of an inde x, the name of the index , and starting, ending , and inc rement v alues for the inde x, and re turns a list consis ting of the ev aluatio[...]

Pa g e 8 - 1 3 In both cases , you can either ty pe out the M AP command (as in the e xamples abo ve) or s elect the command from the CA T menu . The f ollow ing call to func tion MAP us es a pr ogram instead o f a functi on as second argument: Defining functions t hat use lists In Chapter 3 w e intr oduced the use of the DEFINE functi on ( „à )[...]

Pa g e 8 - 1 4 to r eplace the plus sign (+) w ith ADD: Ne xt, w e stor e the edited expr ession in to v ari able @@@G@@@ : Ev alua ting G(L1,L2) no w produces the f ollow ing result: As an alternati ve , yo u can define the f unction w ith ADD rather than the plus sign (+), fr om the start, i .e ., use DEFINE(' G(X,Y)=(X DD 3)* Y') : Y o[...]

Pa g e 8 - 1 5 Applications of lists This sec tion show s a couple of applications o f lists to the calc ulation of statisti cs of a samp le. B y a sample we un derstand a list of valu es, say , { s 1 , s 2 , …, s n }. Suppos e that the sampl e of inter est is the list {1, 5, 3, 1, 2, 1, 3, 4, 2, 1} and that we st or e it into a var iable called [...]

Pa g e 8 - 1 6 3 . Di vide the r esult abov e by n = 10: 4. Apply the INV() f unction to the late st re sult: Thu s, the harmonic mean of lis t S is s h = 1.63 48… Geometric mean of a list The geometr ic mean of a sample is def ined as T o find the geometr ic mean of the list stor ed in S, we can u se the follo wing pr ocedur e: 1. Apply f unctio[...]

Pa g e 8 - 1 7 Thu s, the geometri c mean of list S is s g = 1.00 3 2 0 3… W eighted aver age Suppos e that the data in list S , defined a bo ve , namely : 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 define the w eight list as W = {w 1 ,w 2 ,…,w n }, we notice that the k -t h element in [...]

Pa g e 8 - 1 8 3. Us e fu n c t io n Σ LIS T , once mor e, to calc ulate the denominator o f s w : 4. Use the e xpre ssion AN S(2)/ANS(1) to cal culat e the w eigh ted av er age: Thu s, the wei ghted av er age of list S w i th w eights in list W is s w = 2 .2 . Statistics of grouped data Gr ouped data is ty pi call y giv en by a t able sho wing th[...]

Pa g e 8 - 1 9 The c lass mark dat a can be stor ed in var iable S , whi le the fr equency coun t can be stored in v ariable W , as f ollow s: Giv en the list of class marks S = {s 1 , s 2 , …, s n }, and the list of fr equency coun ts W = {w 1 , w 2 , …, w n }, the w ei ghted a v erage o f the data in S with w eights W r epre sents the mean va[...]

Pa g e 8 - 2 0 T o calc ulate this last r esult , we can us e the fo llow ing: The s tandar d dev iation o f the gr ouped data is the squar 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 ectors This Cha pter pr ov ides e xamples o f enter ing and oper ating with v ectors , both mathematical ve ctors o f many e lements, as w ell as ph ysi cal vectors of 2 and 3 components . Definitions F rom a mathematical po int of v ie w , a vec tor is an arr ay of 2 or mor e elements arr anged into a r ow or a column . Th[...]

Pa g e 9 - 2 wher e θ is the angle between the two v ectors . The cr oss pr oduct pr oduces a vec tor A × B whos e magnitude is | A × B | = | A || B |sin( θ ), and its dir ection is gi ven b y the so -called right-hand rule (consult a te xtbook on Math, Ph ysi cs, or Mechani cs to see this oper ation illustr ated gra phically). In terms of Ca r[...]

Pa g e 9 - 3 Stor ing vectors into v ariables V ectors can be stor ed into var iables . The sc reen shots belo w show the vec tors u 2 = [1, 2] , u 3 = [-3, 2, -2] , v 2 = [3,- 1] , v 3 = [1, -5, 2] stored into var iabl es @ @@u2@@ , @@@u3@@ , @@@v2@@ , and @@@v3@@ , r especti vel y . F irst , in AL G mode: Then , in RPN mode (bef ore pr essing K ,[...]

Pa g e 9 - 4 The ← WID k ey is used to decr ease the wi dth of the columns in the spr eadsheet . Pr ess this k ey a couple of time s to see the column w idth decr ease in y our Matri x W riter . The @ W I D → k e y is used to incr ease the w idth of the columns in the spr eadsheet . Pr ess this k ey a couple of time s to see the column w idth i[...]

Pa g e 9 - 5 The @+ROW@ k ey w ill add a ro w full of z er os at the location o f the selected cell of the s pr eadsheet . The @-ROW ke y will dele te the ro w corr esponding to the selec ted cell of the spr eadsheet. The @+COL@ k ey w ill add a column full of z er os at the location of the select ed cell of the spr eadsheet . The @-COL@ ke y will [...]

Pa g e 9 - 6 Building a vector with  ARR Y The fun ct ion → ARR Y , av ailable in the function catalog ( ‚N‚é , u se —˜ to locate the f unction), can also be used to build a ve ctor or arr ay in the f ollo wing wa y . In AL G mode , enter  ARR Y( vector elements, nu mber of elements ), e.g., In RPN mode: (1) Enter the n elements of [...]

Pa g e 9 - 7 In RPN mode, the f unction [ → ARR Y] tak es the obje cts fr om stac k lev els n+1, n, n-1 , …, dow n to stack le vels 3 and 2 , and conv erts them into a vec tor of n elements . The ob ject or iginally at s tack le vel n+1 becomes the f irst element , the objec t ori gina ll y at lev el n becomes the second element, and so on . Id[...]

Pa g e 9 - 8 Highli ghting the entire e xpr essio n and using the @ EVAL@ so ft menu k e y , w e get the res u l t : -15 . T o r eplace an element in an arr ay use f unction P UT (y ou can find it in the func tion catalog ‚N , or in the P RG/LI S T/ELEMENTS su b-menu – the later wa s intr oduced in Chapter 8). In AL G mode , you need to use f u[...]

Pa g e 9 - 9 Simple operations w it h vectors T o illustr ate oper atio ns wi th vec tors w e will u se the ve ctor s A, u2 , u3, v2 , and v3, stor ed in an earli er ex er cise . Changing sign T o change the si gn of a v ector u se the k ey , e .g., Addition, subtr ac tion Addition and subtr action o f vec tors r equir e that the two v ector oper[...]

Pa g e 9 - 1 0 Absolute value function The ab solute v alue functi on (ABS), when appli ed to a vec tor , produce s the magnitude of the vec tor . F or a vec tor A = [ A 1 ,A 2 ,…,A n ], the magnitude is def ined as . In the AL G mode, ent er the functi on name follo wed b y the vector ar gument . F or ex ample: BS([1,-2,6]) , BS( ) , BS(u3) , w [...]

Pa g e 9 - 1 1 Dot pr oduc t F unction DO T is used to calc ulate the dot produc t of two vect ors o f the same length. So me ex amples of applicati on of functi on DO T , using the v ectors A, u2 , u3, v2 , and v3, stor ed earlier , ar e sho wn ne xt in AL G mode. A ttempts to calc ulate the dot pr oduct of tw o v ectors of diff erent length pr od[...]

Pa g e 9 - 1 2 In the RPN mode , application o f func tion V  w ill list the components of a vec tor in the stac k, e .g., V  (A ) will pr oduce the fo llo w ing output in the RPN stack (vector A is listed i n stack lev el 6:) . Building a two -dimensional vector Fu n ct i o n  V2 is used in the RPN mode to bu ild a vector w ith the values[...]

Pa g e 9 - 1 3 When the r ectangular , or Cartesian , coordinat e s ys tem is selec ted , the top line of the displa y will sho w an XY Z fi eld, and an y 2 -D or 3-D vector e nter ed in the calculator is r eproduced as the (x ,y ,z) components of the vec tor . Thus, to ente r the vec tor A = 3 i +2 j -5 k , we u se [3,2 ,-5], and the v e c tor is [...]

Pa g e 9 - 1 4 The f igur e belo w show s the tr ansfor mation of the v e ct or fr om spheri cal to Cartesi an coor dinates , with x = ρ sin( φ ) co s( θ ), y = ρ sin ( φ ) cos ( θ ), z = ρ cos( φ ). F or this case , x = 3.20 4, y = 1.4 9 4 , and z = 3. 5 36 . If the CYLINdr ical s yst em is selected , the top line of the display w ill sho [...]

Pa g e 9 - 1 5 equi valent (r , θ ,z) w ith r = ρ sin φ , θ = θ , z = ρ cos φ . F or ex ample , the follo w ing fi gure sho ws the v ector enter ed in spheri cal coordinat es, and tr ansformed to polar coor dinates . F or this case, ρ = 5, θ = 2 5 o , and φ = 4 5 o , w hile the transf ormation sho ws that r = 3. 5 6 3, and z = 3. 5 36 . ([...]

Pa g e 9 - 1 6 Suppose that y ou want t o find the angle between v ectors A = 3 i -5 j +6 k , B = 2 i + j -3 k , y ou could try the f ollow ing operati on (angular measur e set to degree s) in AL G mode: 1 - Enter vect ors [3,-5, 6], press ` , [2 ,1,-3], pre ss ` . 2 - DO T(ANS(1),ANS(2)) calc ulates the dot product 3 - ABS( ANS(3))*ABS(( ANS(2)) c[...]

Pa g e 9 - 1 7 Thus, M = (10 i +2 6 j +2 5 k ) m ⋅ N. W e kno w that the magnitude of M is such that | M | = | r || F |sin( θ ), where θ is the angle betw een r and F . W e can f ind this angle as, θ = si n -1 (| M | /| r || F |) by the f ollo wing oper ations: 1 – ABS( ANS(1))/(AB S(ANS( 2))*ABS( ANS(3)) calc ulates sin( θ ) 2 – A SIN( A[...]

Pa g e 9 - 1 8 Ne xt, w e calculate v e ct or P 0 P = r as ANS(1) – ANS( 2), i.e ., F inally , we tak e the dot produc t of ANS(1) and AN S( 4) and make it equal to z ero to complete the operatio n N • r =0: W e can no w use f unctio n EXP AND (in the AL G menu) to expand this ex pre ss io n : Thu s, the equation of the plane thr ough point P 0[...]

Pa g e 9 - 1 9 In this secti on w e will sho wing y ou wa ys to transf orm: a column vec tor into a r o w vect or , a ro w vecto r into a column v ector , a list into a vec tor , and a vec tor (or matr ix) into a list . W e fir st demonstr ate these tr ansfor mations using the RPN mode. In this mode , w e wi ll use func tions OB J  ,  LIS T ,[...]

Pa g e 9 - 2 0 If w e no w apply f uncti on OB J  once more , the list in st ack le vel 1:, {3 .}, w ill be decomposed as f ollows: Function  LIS T This f uncti on is used to c reate a list gi ven the elements o f the list and the list length or si ze . In RPN mode , the list si ze , sa y , n , should be placed in st ack le vel 1:. The ele me[...]

Pa g e 9 - 2 1 3 - Use f u ncti on  ARR Y to build the column vec tor The se thr ee steps can be put toge ther into a U serRP L progr am, e nter ed as follo ws (in RPN mode , still): ‚å„° @) TYPE! @ OBJ  @ 1 + !  ARRY@ `³~~rxc` K A ne w var iabl e , @@RX C@@ , w ill be av ailable in the soft menu labels after pr essing J : Press ‚[...]

Pa g e 9 - 22 2 - Use f u ncti on OBJ  to deco mpose the list i n stack level 1: 3 - Pr ess the delet e k ey ƒ (als o kno wn as f unction DROP) t o eliminate the number in stac k lev el 1: 4 - Use f u ncti on  LIST to cr eate a list 5 - Use f u ncti on  ARR Y to c r eate the r ow v ector The se f i ve s teps can be put toge ther into a Us[...]

Pa g e 9 - 23 Thi s va riab le, @@CXR@@ , can no w be used to dir ectl y transf orm a column v ector to a r ow v ector . In RPN mode, enter the column v ector , a nd then pr ess @@CXR@ @ . T ry , for e xample: [[1],[2], [3]] ` @@CXR@@ . After ha ving def ined var iable @@CXR@@ , we can use it in AL G mode to transf orm a r ow v ector into a column [...]

Pa g e 9 - 24 A ne w var iabl e , @@LX V@@ , w ill be av ailable in the soft menu labels after pr essing J : Press ‚ @@LXV@@ to see the pr ogram co ntained in the var iable LXV : << OBJ  1  LIST  RRY >> Thi s vari ab le, @@LXV@@ , can no w be used to dir ectly tr ansfor m a list into a vec tor . In RPN mode , enter the list ,[...]

Pa g e 1 0 - 1 Chapter 10 ! Creating and manipulating matr ices This c hapter sho ws a number of e xamples aimed at cr eating matri ces in the calc ulator and demonstrating manipulati on of matri x elements. Definitions A matri x is simpl y a rec tangular arr ay of ob ject s (e.g ., numbers , algebr aics) hav ing a number of r ow s and columns. A m[...]

Pa g e 1 0 - 2 Entering matr ices in the stac k In this secti on w e pre sent tw o differ ent methods to enter matr ices in the calc ula tor s tack: (1) using the Matr ix W r iter , and (2) ty ping the matri x direc tly in to th e s ta ck. Using the M atri x Wr iter As with th e case of vectors, di sc ussed in Chapter 9 , matrices ca n be entered i[...]

Pa g e 1 0 - 3 If y ou hav e selected the te xtbook display opti on (using H @) DISP! and c hecking off  Textbook ), the matr ix w ill look lik e the one sho wn abo ve . Otherwise , the displa y w ill sho w: The dis play in RPN mode w ill look very similar to these . T y ping in t he matri x directly into the stack The s ame r esult as abo ve ca[...]

Pa g e 1 0 - 4 or in the MA TRICE S/CREA TE menu a vaila ble thr ough „Ø : The MTH/MA TRIX/MAKE sub menu (let’s call it the MAKE menu) contains the fo llo w ing func tio ns: while the MA TRICE S/CREA TE sub-menu (let’s call it the CREA TE menu) has the fo llo w ing func tio ns:[...]

Pa g e 1 0 - 5 As yo u can see f rom e xploring these men us (MAKE and CREA TE) , they both hav e the same functi ons GET , GE TI , PUT , P U T I, S UB, REPL , RDM, R ANM, HILBERT , V ANDERMOND E , IDN, CON, → DIA G , and DIA G → . The CREA TE menu inc ludes the C OL UMN and RO W sub-menus , that are also av ailable under the MTH/MA TRIX menu .[...]

Pa g e 1 0 - 6 Functions GET and P UT F unctions GET , GET I, PUT , and PUT I, oper ate w ith matri ces in a similar manner as w ith lists or vec tors , i.e ., you need to pr ov ide the locati on of the element that y ou want to GE T or PUT . How ev er , w hile in lists and vector s only one inde x is r equired to identify an element , in matr ices[...]

Pa g e 1 0 - 7 Notice that the s cr een is prepar ed for a su bseq uent appli cation o f GET I or GET , by inc reasing the column index o f the original r efer ence by 1, (i .e., fr om {2 ,2} to {2 , 3}) , whil e sho wing the ex trac ted value , namely A(2 ,2) = 1.9 , in st ack le vel 1. No w , suppo se that y ou want to insert the value 2 in eleme[...]

Pa g e 1 0 - 8 If the ar gument is a real matr ix , TRN simply pr oduces the tr anspose of the r eal matri x. T ry , f or ex ample , TRN(A ) , and compare it w ith TRAN(A). In RPN mode, the tr ansconjugat e of matri x A is calculated by using @@@A@@@ TRN . Function CON The f unction tak es as ar gument a list of tw o elements, corr esponding to the[...]

Pa g e 1 0 - 9 In RPN mode this is accomplished by u sing {4,3} ` 1.5 ` CON . Function IDN F unction IDN (IDeNtit y matri x) cr eates an identity matri x giv en its si ze . Recall that an identity matr i x has to be a squar e matri x, ther efor e, onl y one value is r equir ed to des cr ibe it completely . F or ex ample, t o cr eate a 4 × 4 iden[...]

Pa g e 1 0 - 1 0 vec tor ’s dimension , in the latter the number of r ow s and columns of the matri x. The f ollow ing ex amples illus tr ate the use o f functi on RDM: Re-dim ensioning a vector into a matri x The f ollow ing ex ample show s how to r e -dimension a vec tor of 6 ele ments into a matri x of 2 r ow s and 3 columns in AL G mode: In R[...]

Pa g e 1 0 - 1 1 If using RPN mode , we as sume that the matr ix is in the st ack and u se {6} ` RDM . Function RANM F unction RANM (R ANdom Matr ix) w ill gener ate a matri x with 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 ex ample, in AL G mode , t w o diff er ent 2[...]

Pa g e 1 0 - 1 2 In RPN mode , assuming that the ori ginal 2 × 3 matr ix is alr eady in the stack , use {1,2} ` {2,3} ` SUB . Function REP L F unction REPL r eplaces or inserts a sub-matr ix int o a larger one . The input for this func tion is the matr i x wher e the r eplacement w ill tak e place, the location wher e the replacement begins , and [...]

Pa g e 1 0 - 1 3 In RPN mode, w ith the 3 × 3 matri x in the stack , we simpl y have to acti vate fun ctio n  DI G to obtain the same r esult as abov e. Function DIA G → Fu n ct i o n D I AG → takes a v ector and a list o f matr ix dimensions {r o ws , columns}, and cr eates a diago nal matri x wi th the main diagonal r eplaced with the pr [...]

Pa g e 1 0 - 1 4 F or ex ample, the f ollo wing command in AL G mode f or the list {1,2 , 3, 4}: In RPN mode, enter {1, 2,3,4} ` V NDERMON DE . Function HILBERT F unction HILBERT c reates the Hilbert matr i x corr esponding to a dimension n . By def inition , the n × n Hilbe rt matri x is H n = [h jk ] n × n , so that The H ilber t matri x has ap[...]

Pa g e 1 0 - 1 5 enter ed in the display as y ou perform tho se ke ystr ok es . F irst , we pres ent the steps ne cessar y to produce program CRMC. Lists r epresent columns of the matri x The p rogra m @CRMC allo ws y ou to put together a p × n matri x (i .e., p r ow s, n columns) out of n lists of p elements each . T o cr eate the progr am enter [...]

Pa g e 1 0 - 1 6 ~„n # n „´ @) MATRX! @ ) COL! @COL!  COL  ` Pr ogram is dis play ed in lev 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 progr am use J ‚ @CRMC . T he progr am listing is the fo llo w ing: « DUP → n « 1 SWAP FOR j OB J →→ RRY IF j n < THEN j 1 + ROLL END NEXT IF n 1 > TH[...]

Pa g e 1 0 - 1 7 Lists r epresent ro ws of the matrix The pr ev ious pr ogram can be easil y modified to c reate a matr ix w hen the input lists w ill become the r ow s of the r esulting matri x. The onl y change to be perfor med is to change C OL → for ROW → in the progr am listing. T o perfor m this change u se: ‚ @CRMC L ist progr am CRMC [...]

Pa g e 1 0 - 1 8 Both appr oaches w ill show the same f unctions: When s ystem f lag 117 is set to S OFT menus , the COL menu is acces sible thr ough „´ !) MATRX ) !)@@COL@ , or thr ough „Ø !) @CREAT@ ! ) @@ COL@ . Both appr oaches w ill sho w the same set of f unctions: The operation of these functions is presented be lo w . Function → COL[...]

Pa g e 1 0 - 1 9 In this re sult, the f irst column occ upies the highe st stac k lev el after decompositi on, and st ack le vel 1 is occ upied b y the number of co lumns of the ori ginal matri x. T he matri x does not survi ve decompositi on, i .e., it is no longer av ailable in the stack . Function COL → Fu n ct i o n C O L → has the opposite[...]

Pa g e 1 0 - 2 0 In RPN mode, ent er the matr i x fir st , then the v ector , and the column number , bef or e apply ing func tion COL+. T he fi gure belo w show s the RPN stack be fo re and after apply ing functi on COL+. Function COL- F unction COL - tak es as ar gument a matri x and an integer number r epr esenting the positi on of a column in t[...]

Pa g e 1 0 - 2 1 In RPN mode, f unction CS WP lets you s wap the columns of a matr ix listed in stac k lev el 3, who se indices ar e listed in stac k lev els 1 and 2 . F or ex ample , the fo llow ing fi gure sho ws the RPN st ack bef ore 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 yo u can see , the col[...]

Pa g e 1 0 - 2 2 When s yst em flag 117 is set to S OFT menus , the RO W menu is acces sible thr ough „´ !) MATRX ! )@@ROW@ , or thr ough „Ø !) @CREAT@ ! ) @@ROW@ . Both appr oaches w ill sho w the same set of f unctions: The operation of these functions is presented be lo w . Function → ROW Fu n ct i o n → RO W tak es as argument a matr [...]

Pa g e 1 0 - 23 matri x does not survi ve decompo sition , i.e ., it is no longer av ailable in the stack. Function RO W → Fu n ct i o n R OW → has the opposite e ffect o f the functio n → RO W , i .e ., giv en n vec tor s of the same le ngth, and the number n , func tion R OW  builds a matri x by plac ing the input v ectors as r o ws o f [...]

Pa g e 1 0 - 24 Function RO W- F unction RO W - tak es as argument a matr ix and an in teger number r epre senting the position o f a r ow in the matri x. T he functi on returns the or iginal matr ix , minus a r o w , as w ell as the e xtrac ted ro w sho wn as a v ector . Here is an e xam ple in the AL G mode using the matr ix st or ed in A: In RPN[...]

Pa g e 1 0 - 2 5 As y ou can see , the ro ws that or iginally occ upi ed positions 2 and 3 ha ve been s wapped . Function RCI F unction R CI stands f or multipl y ing R ow I by a C ons tant v alue and r eplace the r esulting r ow at the same location . The follo wi ng ex ample, w ritten in AL G mode , tak es the matri x stor ed in A, and multiplies[...]

Pa g e 1 0 - 26 In RPN mode, ent er the matr ix f irst , follo wed by the const ant value , then by the r o w to be multiplied b y the constant value , and finall y enter the ro w that will be r eplaced. T h e f ollo wing f igure sho ws the RPN stac k befor e and after apply ing func tion R CIJ under the same conditi ons as in the AL G ex ample sho[...]

P age 11-1 Chapter 11 M atr ix Operations and L in ear Algebr a In Chapter 10 w e introduced the concept of a matr ix and pr esent ed a number of func tions f or enter ing, c r eating, o r manipulating matri ces. In this Chapt er w e pr esent e xamples o f matr ix oper ations and applicatio ns to pr oblems of linear algebra . Operations with matr i[...]

P age 11-2 Addition and subtr action Consi der a pair of matr ices A = [a ij ] m × n and B = [b ij ] m × n . A ddition and subtr action of thes e t w o matri ces is only pos sible if the y have the s ame number of r ow s and columns. The r esulting matr i x, C = A ± B = [c ij ] m × n has elem ents c ij = a ij ± b ij . Some e xample s in AL G m[...]

P age 11-3 By comb ining add ition and subtr action w ith multiplicatio n by a scalar w e can fo rm linear combinati ons of matr ices o f the same dimensions , e.g ., In a linear combinati on of matr ices, w e can multiply a matr i x by an imaginary number to obtain a matr ix o f complex n umbers, e .g., Matrix -vector multipli cation Matri x -vec [...]

P age 11-4 Matrix multiplication Matri x multiplicati on is defined b y C m × n = A m × p ⋅ B p × n , w h er e A = [a ij ] m × p , B = [b ij ] p × n , and C = [c ij ] m × n . Notice that matr ix multipli cation is onl y possible if the number of columns in the f irst oper and is equal to the number o f r o ws of the second oper and. T he ge[...]

P age 11-5 (another r ow vect or). Fo r the calculator to identify a ro w vector , you mus t use double br acke ts to enter it: T erm -b y-term multiplication T erm-b y-term multiplication o f two matri ces of the same dimensions is possible thr ough the us e of func tion HAD AMARD . The r esult is, o f cours e , another matr i x of the same dime n[...]

P age 11-6 In algebr aic mode , the k eys trok es are: [enter or s elect the matri x] Q [enter the po wer] ` . In RPN mode , the ke ys tr ok es ar e: [enter or select the matr ix] † [enter the po we r] Q` . Matri ces can be r aised to negativ e po we rs . In this case , the result is equi valent to 1/[matr i x]^ABS(po we r). The identity matrix I[...]

P age 11-7 T o ver ify the pr operties of the in verse matr ix , consider the follo wing multiplications: Characteri zing a matrix (T h e matr ix NORM menu) The matr ix NORM (NORMALI ZE) menu is accesse d thr ough the k ey str oke sequenc e „´ (sy stem flag 117 s et to CHOO SE box es): This me nu cont ains the fo llo w ing func tions: Thes e fun[...]

P age 11-8 Function ABS F unction ABS calc ulates what is kno wn as the F robeniu s norm of a matr ix . For a matri x A = [a ij ] m × n , the F robeniu s norm of the matr ix is def ined as If the matri x under consider ation in a ro w vec tor or a column vector , then the F robeniu s norm , || A || F , is simply the v ector ’s magnitude . F unct[...]

P age 11-9 Functions RNRM and CNRM F unction RNRM r eturns the Ro w NoRM of a matr i x , while f unction CNRM r eturns the C olumn NoRM of a matri x. Ex amples, Singular value decomposition T o underst and the oper ation of F uncti on SNRM, w e need to introduce the concept of matri x decompositi on. Ba sicall y , matr i x decompositi on inv olv es[...]

P age 11-10 Function SRAD F unction SRAD determine s the Spectr al R ADius o f a matri x, def ined as the large st of the ab solute v alues of its eigen values . F or ex ample, Function COND F unction COND deter mines the condition number of a matr i x: Definition of eigenvalues and eigen vec tors of a matri x The e igenv alues of a sq uare matr i [...]

P age 11-11 T ry the follo wing e xer cis e fo r matri x condition nu mber on matr i x A3 3. T he condition number is C O ND( A3 3 ) , r o w norm, and column norm for A3 3 are sho wn to the left . The cor r esponding numbers f or the inv erse matr ix , INV(A3 3) , ar e show n to th e r ight: Since RNRM(A3 3) > CNRM(A3 3) , then w e tak e ||A3 3|[...]

P age 11-12 F or ex ample, try finding the r ank for the matr ix: Y ou will f i nd that the r ank is 2 . That is because the second r ow [2 , 4, 6] is equal to the f irst r ow [1,2 , 3] multiplied b y 2 , thu s, ro w two is linear ly dependent o f r o w 1 and the max imum number of linearl y independent r o ws is 2 . Y ou can chec k that the maximu[...]

P age 11-13 The determinant of a matri x The de ter minant of a 2x2 and o r a 3x3 matri x ar e r e pr esented b y the same arr angement of elements o f the matr ices, but enc losed betw een ve rtical lines , i. e. , A 2 × 2 deter minant is calcul ated b y multiply ing the elements in its di agonal and adding those pr oducts accompanied b y the pos[...]

P age 11-14 Function TRACE F unction TRA CE calculates the tr ace of squar e matri x, def ined as the sum of the elements in its main diagonal , or . Example s: F or squar e matrice s of hi gher or der determinants can be calc ulated by using smaller or der determinant called cof actors . The gener al idea is to "expand" a determinant o f[...]

P age 11-15 Function TRAN F unction TRAN re turns the tr anspose o f a r eal or the conj ugate transpo se of a comple x matri x. TRAN is equi valent t o TRN. The oper ation of func tion TRN wa s pr esent ed in Chapter 10. Additional matri x operations (T h e matri x OPER menu) The matr ix OP ER (OPERA TION S) is availa ble through the k e y str oke[...]

P age 11-16 MAD and RSD ar e related t o the soluti on of s yste ms of linear equati ons and wil l be pr esent ed in a subsequen t sec tion in this Cha pter . In this sectio n we ’ll disc uss only f unctions AXL and AXM. Function AXL F unction AXL conv erts an arra y (ma tr ix) into a list , and vi ce ver sa: Note : the latte r oper ation is simi[...]

P age 11-17 The im plementation of functi on L CXM f or this case r equires y ou to enter : 2`3`‚ @@P1@@ LCXM ` The f ollow ing fi gure sho ws the RPN st a c k befo r e and after apply ing func tion LC X M : In AL G mode , this ex ample can be obtained b y using: The pr ogram P1 mu st still ha ve been c reated and stor ed in RPN mode. Solution of[...]

P age 11-18 , , Using the numerical solv er for linear s ystems Ther e are man y way s to solv e a sy stem of linear equations w ith the calculator . One possib ility is through the numer ical sol v er ‚Ï . Fr om the numer ical sol ver s cr een, sho wn belo w (left) , select the opti on 4. So lv e lin sy s.., and pr ess @@@OK@@@ . The f ollo win[...]

P age 11-19 This s yst em has the same number of equations as of unknow ns, and will be r efer red to as a squar e sy stem. In gener al, there sho uld be a unique soluti on to the s ystem . The soluti on will be the po int of intersec tion o f the thr ee planes in the coor dinate sy stem (x 1 , x 2 , x 3 ) r epre sented b y the three eq uations. T [...]

P age 11-20 T o chec k that the solution is cor r ect , ent er the matri x A and multiply times this soluti on vector (e xample in algebr aic mode) : Under-deter mined sy stem The s ys tem of linear eq uations 2x 1 + 3x 2 –5x 3 = -10, x 1 – 3x 2 + 8x 3 = 85, can be wr itten as the matri x equation A ⋅ x = b , if This s yst em has mor e unkno [...]

P age 11-21 T o see the details of the so lution v ector , if needed , press the @E DIT! button . T his w ill acti vate the Matr ix W r iter . Within this en vir onment, u se the r ight- and left- arr ow k e ys t o mov e about the vec tor: Thu s, the solution is x = [15 .3 7 3, 2 .46 2 6, 9 .6 2 6 8] . T o re turn to the numer ical solv er env i r [...]

P age 11-22 Let’s stor e the latest result in a v ari able X, and the matr i x into var iable A, as fo llow s: Press K~x` to stor e the solution vect or 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 vari able A No w , let’s ver if y the so lution b y using: @@@A@@@ * @@@ X@@@ ` , whi[...]

P age 11-2 3 can be wr itten as the matri x equation A ⋅ x = b , if This s ystem has mo r e equations than unkno wns (an ov er-determined s yst em) . The s ys tem does not hav e a single solution . E ach of the linear equati ons in the sy stem presented abo v e r e pr esen ts a straig ht line in a t w o -dimensional Cartesian coor dinate s ys tem[...]

P age 11-2 4 Press ` to retur n to the numer ical solv er env ironment . T o check that the soluti on is correc t, try the follo wing: • Press —— , to highlight the A: field . • Press L @CALC@ ` , to cop y matri x A onto the stack . • Press @@@OK@@@ to r eturn to the n umeri cal sol ver e nv ironment . • Press ˜ ˜ @CALC@ ` , to cop y [...]

P age 11-2 5 • If A is a squar e matri x and A is non-singular (i .e., it’s in ver se matr ix e xis t , or its determinant is non- z ero), LSQ r eturns the ex act soluti on to the linear s y stem . • If A has less than f ull ro w rank (under determined s yst em of equatio ns) , LS Q retur ns the soluti on with the minimum E uclidean le ngth o[...]

P age 11-2 6 Under-deter mined sy stem Consider the s yst em 2x 1 + 3x 2 –5x 3 = -10, x 1 – 3x 2 + 8x 3 = 85, wi th The s oluti on using LS Q is sho wn ne xt: Over-determin ed s ystem Consider the s yst em x 1 + 3x 2 = 15, 2x 1 – 5x 2 = 5, -x 1 + x 2 = 2 2 , wi th The s oluti 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 soluti ons w ith the ones calculated w ith the numer ical solver . Solution with the in verse matri x The s olution t o the sy stem A ⋅ x = b , w here A is a squar e matri x is x = A -1 ⋅ b . This r esults fr om multiply ing the firs t equation b y A -1 , i.e ., A -1 ⋅ A ⋅ x = A -1 ⋅ b . B y def inition [...]

Pa g e 1 1 - 2 8 The pr ocedure f or the case of “ di viding ” b by A is illustr ated belo w for the cas e 2x 1 + 3x 2 –5x 3 = 13, x 1 – 3x 2 + 8x 3 = -13, 2x 1 – 2x 2 + 4x 3 = -6 , The pr ocedure is show n in the follo wing s cr een shots: The s ame soluti on as found abo ve w ith the inv erse matr i x. Solv ing multiple set of equations[...]

P age 11-29 [[14,9,-2] ,[2,-5,2],[5, 19,12]] ` [[1,2,3],[3, -2,1],[4,2,- 1]] `/ The r esult of this oper ation is: Gaussian and Gauss-Jordan elimination Gaussi an elimination is a pr ocedure b y whi ch the squar e matri x of coeff ic ients belonging to a sy stem of n linear equati ons in n unkno wns is r e duced to an upper - tri angular matri x ( [...]

P age 11-30 T o start the pr ocess of f orwar d elimination , we di vi de the firs t equation (E1) b y 2 , and st or e it in E1, and sho w the three eq uatio ns again to pr oduce: Next , we r eplac e the second equati on E2 by (equation 2 – 3 × equation 1, i .e ., E1-3 × E2) , and the thir d by (equati on 3 – 4 × equation 1), to get: Ne xt, [...]

P age 11-31 an expr ession = 0. T hus, the las t set of equati ons is interpr eted to be the follo w ing equiv alent set of equatio n s: X +2Y+3Z = 7 , Y+ Z = 3, - 7Z = -14. The pr ocess of backw ard subs titution in Gaussi an elimination consis ts in finding the value s of the unknow ns, starting fr om the last equation and w orking upw a r d s. T[...]

P age 11-3 2 T o obtain a solution to the s yst em matr ix equati on using Gaussian eliminati on, we f i rs t c re a t e w h a t i s k n ow n a s t h e augmented matri x corr esponding to A , i .e., The matr ix A aug is the same as the or iginal matr ix A w ith a new r ow , corr esponding to the elements o f the vec tor b , added (i.e ., augmented)[...]

P age 11-3 3 Multiply r ow 2 by –1/8: 8Y2 @RCI! Multiply r ow 2 by 6 add it to r ow 3, r eplacing it: 6#2#3 @RCIJ! If y ou we r e perfor ming these oper ations by hand , you w ould wr ite the fo llow ing: The symb ol ≅ (“ is eq uiv alent to ”) indicate s that what f ollo ws is equi valent to the pr ev ious matr ix w ith some r ow (or colum[...]

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

Pa g e 1 1 - 3 5 While perfo rming pi voting in a matr ix elimination pr ocedure , yo u can impr ov e the numer ical solutio n e ven more b y selecting as the pi vot the ele ment wi th the large st absolute v alue in the column and r ow o f inter est . This oper ation ma y r equir e e xc hanging not only r ow s, but also columns, in s ome pi voting[...]

Pa g e 1 1 - 3 6 No w we ar e read y to start the Gauss-Jor dan elimination w ith full pi vo ting. W e will need to k eep track of the per mutation matri x by hand, s o take y our notebook and w rite the P ma t rix s how n a b ove. F irst, w e check the piv ot a 11 . W e notice that the element w ith the larges t absolu te value in the f irst r ow [...]

P age 11-3 7 Hav ing f illed up w ith zer os the elements o f column 1 belo w the pi vot , now w e pr oceed to chec k the piv ot at position (2 ,2) . W e find that the number 3 in position ( 2 ,3) w ill be a bet ter pi vot , thus, w e ex change columns 2 and 3 by using: 2#3 ‚N @@@OK@@ Checking the p iv ot at position ( 2 ,2) , we no w find that t[...]

P age 11-38 2 Y #3#1 @RCIJ F inally , w e eliminate the –1/16 fr om position (1,2) b y using: 16 Y # 2#1 @RCIJ W e now ha ve an identity matri x in the por tion o f the augmented matr ix corr esponding to the or iginal coeff ici ent matri x A, thu s w e can proceed to obtain the soluti on while accounting f or the ro w and column ex changes code[...]

P age 11-3 9 Then , for this partic ular ex ample , in RPN mode , use: [2,-1,41] ` [[1,2,3],[2,0,3],[8 ,16,-1]] `/ The calc ulator sho ws an a ugmented matr ix consis ting of the coeff ic ients matr ix A and the identity matr ix I , w hile, at the same time , show ing the next pr ocedure to calculate: L2 = L2 - 2 ⋅ L1 stands fo r “ r eplace ro [...]

P age 11-40 T o see the int ermedi ate steps in calc ulating and inv erse , jus t ente r the matri x A fr om abov e, and pr ess Y , while k eeping the step-b y-step option acti ve in the calc ulator’s CA S. Use the f ollow ing: [[ 1,2,3],[3,- 2,1],[4,2,-1 ]] `Y After going thr ough the differ ent steps , the solu tion r eturned is: What the calc [...]

P age 11-41 The r esult ( A -1 ) n × n = C n × n / det ( A n × n ) , is a gener al result that appli es to any non -singular matr i x A . A general f orm for the ele ments of C can be written based on the Gaus s-Jor dan algorithm . Based on the equation A -1 = C /det( A ), sk etched abo ve , the inve rse matri x, A -1 , is not def ined if det ( [...]

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 solution: [X=-1, Y=2,Z = -3]. F unction LINS OL VE w orks w ith s ymboli c expr essions . F unctions REF , rre f, and RREF , w ork wi th the augment ed matri x in a Gaussi an elimination a ppr oach . Functions REF , rr ef, RREF The u pper tr iangular f orm to [...]

P age 11-43 The di agonal matr ix that r esults f r om a Gaus s -Jor dan elimination is called a r o w-reduced ec helon for m. F unction RREF ( R ow-R educed Ec helon F orm) The r esult of this f unction call is to pr oduce the r o w-r educed echelon f orm so that the matri x of coeff ici ents is r educed to an identity matri x. The e xtra column i[...]

P age 11-44 The r esult is the augmented matr i x corr esponding to the sy stem of equations: X+Y = 0 X- Y =2 Residual err ors in linear sy stem solutions (Function RSD) F unction R SD calculate s the Re SiDuals or error s in the so lution of the matri x equation A ⋅ x = b , repr esenting a s ystem o f n linear equations in n unkno wns. W e can t[...]

P age 11-45 Eigenv alues and eig env ec tors Gi ven a sq uare matr ix A , w e can w rite the ei gen value equation A ⋅ x = λ⋅ x , wher e the values of λ that satisfy the equation ar e know n as the eigen values of matri x A . F or each v alue of λ , w e can find , from the s ame equ ation , values of x that satisfy the ei genv alue equati on[...]

Pa g e 1 1 - 4 6 Using the var iable λ to r epres ent e igen values , this char acter istic pol ynomial is t o be interpr eted as λ 3 -2 λ 2 -2 2 λ +21=0. Function EG VL F unction E GVL ( E iGenV aL ues) pr oduces the ei gen value s of a sq uar e matri x. F or e xam ple , the eigen values o f the matri x sho wn belo w are calc ulated in AL G mo[...]

P age 11-4 7 of a matri x, w hile the cor r esponding ei genv alues are the compone nts of a vec tor . F or ex ample, in AL G mode , the e igen vector s and ei genv alues of the matr i x listed belo w are f ound by a pply ing func tion E G V: The r esult sho ws the e igen values as the columns of the matr ix in the r esult list . T o see the ei gen[...]

P age 11-48 • A list w ith the eigen vect ors cor res ponding to eac h eigen value of matr ix A (stac k level 2) • A v ector w ith the eige nv ector s of matr i x A (stac k le ve l 4) F or ex ample, try this ex erc ise in RPN mode: [[4,1,-2],[1 ,2,-1],[-2,-1 ,0]] JORD N The ou tput is the fo llo w ing: 4: ‘X^3+-6*x^2+2*X+8’ 3: ‘X^3+-6*x^2[...]

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 eige nvalue equati on A ⋅ x = λ⋅ x . As an e xample , in RPN mode , tr y: [[4,1,-2] [1, 2,-1][-2,-1,0 ]] M D The r esult is: 4: -8. 3: [[ 0.13 –0.2 5 –0.3 8][-0.25 0. 50 –0.2 5][-0.38 –0.2 5 –0.88]] 2: {[[1 0 0][0 1 0][0 0 1][...]

P age 11-50 Function L U F unction L U tak es as input a s quar e matri x A , and r eturns a low er - tri angular matri x L , an upper tr iangular matri x U , and a perm utation matri x P , in stack lev els 3, 2 , and 1, respec ti ve ly . The r esults L , U , and P , satisfy the equation P ⋅ A = L ⋅ U . When you call the L U func tion , the cal[...]

P age 11-51 decomposition , while the v ector s r epresents the main di agonal of the matr ix S used earli er . F or ex ample, in RPN mode: [[5,4 ,-1],[2,-3,5 ],[7,2,8]] S VD 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 –0.5 4][-0. 6 0 0.6 7 0.44]] 1: [ 12 .15 6.88 1.[...]

Pa g e 1 1 - 52 Function QR In RPN, f unction QR pr oduces the Q R fact oriz at ion of a matrix 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 permut ation matri x, in s tack le ve ls 3, 2 , and 1. T he matri ces A , P , Q and R ar e rel a te d by A ⋅ P = Q ⋅ R . For e xample , [[ 1,-2[...]

Pa g e 1 1 - 5 3 This menu inc ludes functi ons AXQ, CHOLE SKY , G A US S, QXA, and S YL VE STER . Function AX Q In RPN mode , function AXQ pr oduces the quadr atic f orm cor responding t o a matri x A n × n in stac k leve l 2 using the n var iables in a v ector placed in st ack lev el 1. F unction r eturns the quadr atic f orm in stac k lev el 1 [...]

P age 11-54 suc h that x = P ⋅ y , by 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 unction S YL VE STER t ak es as ar gument a s ymmetr ic squar e matr i x A and re turns a vec tor cont aining the diagonal ter ms of a diagonal matr ix D , and a matr[...]

Pa g e 1 1 - 5 5 Infor mation on the func tions list ed in this menu is pr esented belo w by using the calc ulator’s o w n help fac ility . The f igure s show the help f acility 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 aphi c s In this chapt er we intr oduce some of the gr aphics capab ilities o f the calc ulator . W e wi ll pre sent graphi cs of functi ons in Cartesian coor dinates and polar coor dinates , parametr ic plots , gr aphics of co nics , ba r plots, s cat ter plots, and a var iety of thr ee -dimensi onal gr aphs. Graphs op[...]

Pa g e 1 2 - 2 The se gr aph options ar e desc ri bed bri efl y next . Fu n ct i o n : f or equations of the f orm y = f(x) in plane Cartesi an coordinates P olar : for equations o f the f ro m r = f( θ ) in polar coordinate s in the plane Pa r a m e t r i c : for plotting equati ons of the for m x = x(t) , y = y(t) in the plane Diff E q : f or pl[...]

Pa g e 1 2 - 3 Θ Enter the P L O T env ir onment b y pres sing „ñ (pre ss them simultaneou sly if in RPN mode). Pr ess @ADD to get y ou into the equati on wr iter . Y ou will be pr ompted to fill the r ight-hand side of an equation Y1(x) =  . T ype the functi on to be plotted so that the E quation W r iter sho ws the f ollow ing: Θ Press ` [...]

Pa g e 1 2 - 4 Θ Enter the P L OT WINDO W env ironme nt by ent er ing „ò (pr ess them simultaneously if in RPN mode). Use a r ange of –4 to 4 for H- VIEW , t hen press @AUT O to generate the V - VIEW automatically . The PL O T WINDO W sc r een looks as f ollow s: Θ Pl ot t he g rap h : @ERASE @D RAW (w ait till the calculator f inishes the g[...]

Pa g e 1 2 - 5 Some useful PL O T operations fo r FUNCTION plots In orde r to disc uss these P L O T options , w e'll modif y the func tion to f or ce it to hav e some real r oots (Since the curr ent curve is totall y contained abov e the x axis , it has no real r oots.) Pr ess ‚ @@@Y1@@ to list the contents of the f unction Y1 on the stac k[...]

Pa g e 1 2 - 6 ROO T : 1.66 3 5... T he calculator indicated , befor e show ing the root , that it wa s found thr ough SIGN REVER SAL . Press L to r ecover the menu . Θ Pr es sing @ISECT w ill gi ve y ou the intersecti on of the c urve w i th the x -axis, whi ch is esse ntiall y the roo t . Place the c ursor e xac tly at the r oot and press @ISECT[...]

Pa g e 1 2 - 7 Θ Enter the PL O T env i r onment by pres sing, simultaneousl y if in RPN mode, „ñ . Notice that the highlighted f ield in the PL O T en v ir onment now contains the deri vati ve of Y1(X) . Pr ess L @@@OK@@@ to r eturn to return to nor mal calculator dis play . Θ Press ‚ @@EQ@@ to check the conte nts of EQ. Y ou will noti ce t[...]

Pa g e 1 2 - 8 T o r eturn t o nor mal calculato r func tion , pres s @) PICT @CANCL . Graphics of tr anscendental func tions In this secti on we us e some of the gr aphics f eatures of the calc ula tor t o sho w the typi cal beha vior of the natur al log, e x ponenti al, tr igonometr ic and h yperboli c functi ons. Y ou will not see mor e graphs i[...]

Pa g e 1 2 - 9 10 by us i ng 1 @@@OK@@ 10 @@@OK@@@ . Ne xt , pr ess the soft k ey labeled @AUTO to let the calc ulator determine the cor r esponding v er ti cal range . After a co uple of seconds this r ange w ill be shown in the P L O T WINDO W-FUNCT ION w indo w . At this point w e are r eady to pr oduce the graph o f ln(X) . Pr ess @ERASE @DRAW[...]

Pa g e 1 2 - 1 0 Graph of the e x ponential function F irst , loa d the f unction e xp(X) , by pr essing , simultaneousl y if in RPN mode , the left-shif t k ey „ and the ñ ( V ) k ey to acce ss the PL O T-FUNCTI ON windo w . Pr ess @@DEL@@ to remo ve the f unction LN( X) , if y ou didn’t dele te Y1 as suggested in the pr ev ious no te . Pr es[...]

Pa g e 1 2 - 1 1 The P P AR v ariable Press J to reco ver y our vari ables menu , if needed. In your v ariables me nu y ou should ha ve a v ar iable labe led PP AR . Pr ess ‚ @PPAR to get the contents of this var iable in the stack . Pres s the dow n -arr o w key , , to launch the stack editor , and use the up- and do wn -ar r o w ke ys to vi ew [...]

Pa g e 1 2 - 1 2 As indicated earl ier , the ln(x) and exp(x) f unctions ar e inv erse of eac h other , i .e., ln(e xp(x)) = x, and e xp(ln(x)) = x. This can be v erif ied in the calculato r b y typing and e valuating the f ollow ing expr essi ons in the Eq uation W r iter: LN(EXP(X)) and EXP(LN(X)). The y should both ev aluate to X. When a func ti[...]

Pa g e 1 2 - 1 3 Summary of FUNCTION plot oper ation In this secti on w e pre sent inf ormati on regar d ing the PL O T SETUP , PL O T - FUNCTION , and P L O T WINDOW sc reens accessible thr ough the left-shif t k ey combined w ith the soft-menu k ey s A thr ough D . Based on the gr aphing e xam ples pr esented abo ve , the procedur e to follo w to[...]

Pa g e 1 2 - 1 4 Θ Use @CANCL to cancel an y changes to the P L O T SE TUP windo w and retur n to nor mal calc ulator displa y . Θ Press @@@OK@@@ to save c h anges to the op tions in the PL O T S E TUP window and r eturn t o normal calc ulator display . „ñ , simultaneously if in RPN mode: Access to the PL O T w i ndo w (in this case it w ill b[...]

Pa g e 1 2 - 1 5 Θ Enter lo w er and upper limits fo r hor iz ont al v ie w (H-V ie w) , and pr ess @AUTO , whi le the cur sor is in one of the V - Vie w fi elds, to gener ate the verti cal vie w (V- Vie w) range automatically . Or , Θ Enter lo wer and u pper limits fo r vertical v iew (V -V iew), and pr ess @AUTO , whi le the cur sor is in one o[...]

Pa g e 1 2 - 1 6 „ó , simultaneou sly if in RPN mode: Plots the gr aph based on the setting s stor ed in var iable P P AR and the cur rent func tions def ined in the PL O T – FUNCTION s cr een. If a gr aph, diff eren t fr om the one y ou ar e plotting, alr eady ex ists in the graphi c display s cr een, the ne w plot w ill be superimpos ed on t[...]

Pa g e 1 2 - 1 7 Generating a table of v alues for a function The co mbinati ons „õ ( E ) and „ö ( F ), pressed simultaneousl y if in RPN mode , let’s the us er pr oduce a table of values o f functi ons. F or ex ample, w e will pr oduce a table of the f unction Y(X) = X/(X+10), in the r ange -5 < X < 5 fo llow ing these instru ctio ns[...]

Pa g e 1 2 - 1 8 the corr esponding values o f f(x) , listed as Y1 b y default . Y ou can us e the up and do wn ar ro w k ey s to mo ve abou t in the table . Y ou w ill notice that w e did not hav e to indicate an ending value f or the independent var iable x. T hus, the table contin ues bey ond the maximum v alue for x sugges ted earl y , namel y [...]

Pa g e 1 2 - 1 9 W e wi ll try to plot the f unction f( θ ) = 2(1-sin( θ )), as follo ws: Θ F irst , mak e sure that y our calculator ’s angle measure is s et to r adians. Θ Press „ô , simultaneousl y if in RPN mode, to access to the PL O T SETUP w indo w . Θ Chang e TYPE to Polar , b y pres sing @CHOOS ˜ @@@OK@@@ . Θ Press ˜ and type:[...]

Pa g e 1 2 - 2 0 Θ Press L @CAN CL to retu rn to th e PL O T W I ND OW scre en. Press L @@@OK@@@ to r eturn t o normal calc ulator display . In this ex erc ise w e ent er ed the equation to be plotted dir e ctl y in the PL O T SETUP w indo w . W e can also enter equations f or plotting using the PL O T w indow , i.e ., simultaneous ly if in RPN mo[...]

Pa g e 1 2 - 2 1 The calc ulator has the ability of plotting one or more coni c curv es b y selecting Con ic as the functi on TYPE in the PL O T en vir onment . Make sur e to delete the var iables PP AR and E Q befor e continuing. F or e xample , let's sto re the list of equations { ‘(X-1)^2+(Y - 2)^2=3’ , ‘X^2/4+Y^2/3=1’ } into the v [...]

Pa g e 1 2 - 2 2 Θ T o see labels: @EDIT L @) LABEL @MENU Θ T o r ecov er the menu: LL @) PICT Θ T o estimate the coordinate s of the po int of inter section , pres s the @ ( X, Y ) @ menu k ey and mo ve the c ursor as c lose as pos sible to those points using the arr ow k ey s. The coor dinates of the c ursor ar e show n in the display . F or e[...]

Pa g e 1 2- 23 whi ch inv olve constant values x 0 , y 0 , v 0 , and θ 0 , we need to stor e the values of those par ameters in v ari ables . T o de ve lop this ex ample, c reate a sub-dir ectory called ‘PR OJM’ for P RO J ectile Moti on, and w ithin that sub-direc tory stor e the follo w ing var iables: X0 = 0, Y0 = 10, V0 = 10 , θ 0 = 30, a[...]

Pa g e 1 2 - 24 Θ Press @AUTO . This w i ll gener ate automatic values of the H-V iew and V - Vi ew r anges based on the v alues of the independent v ariable t and the def initions o f X(t) and Y(t) us ed. The r esult w ill be: Θ Press @ERASE @DRAW to dr aw the par ametri c plot . Θ Press @EDIT L @LABEL @MENU to see the gr aph w ith labels. T he[...]

Pa g e 1 2 - 2 5 parameter s. The other v ariables contain the v alues of constants us ed in the def initions of X(t) and Y(t). Y ou can store diff erent v alues in the var iables and pr oduce ne w parametr ic plots of the pr ojectile equati ons used in this ex ample. If y ou want to er ase the c urr ent pic ture contents bef ore pr oducing a ne w [...]

Pa g e 1 2 - 26 P lot ting the solution to simpl e differ ential equations The plot o f a simple differ ential equati on can be obtained by selecting Diff Eq in the TYPE fi eld of the PL O T SETUP en vir onment as follo ws: su ppose that w e want t o plot x(t) fr om the differ ential equati on dx/dt = e xp(- t 2 ), w it h i ni ti al conditi ons: x [...]

Pa g e 1 2- 27 Θ Press L to reco ver the menu . P r ess L @) PICT to reco ver the original graphi cs menu. Θ When w e observed the gr aph being plotted, y ou'll notice that the gr aph is not v er y smooth . That is becau se the plotter is u sing a time step th at is too lar ge . T o r ef ine the gr aph and make it smoothe r , use a step of 0[...]

Pa g e 1 2 - 28 T ruth plots T ruth plots ar e used to pr oduce two -dimensi onal plots of r egions that satisfy a certain mathematical co ndition that can be eithe r true or fals e. F or ex ample , suppos e that y ou want to plot the r egion f or X^2/3 6 + Y^2/9 < 1, pr oceed as fo llow s: Θ Press „ô , sim ultaneously if in RPN mode , to ac[...]

Pa g e 1 2 - 2 9 Θ Press „ô , sim ultaneously if in RPN mode , to access t o the PL O T SETUP wi nd ow . 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 to be plotted . Θ Press @ERASE @DRAW to dr aw the truth plot . Again , yo u have to be patient while the calc ulator produce s t[...]

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 unction S T O Σ (a vailable in the functi on c atalog , ‚N ) . Pr ess V AR to reco ve r your v ariable s menu . A soft menu ke y labeled Σ D A T should be a vailable in the s tac k. T he figur e below sho ws the stor age of this matri x[...]

Pa g e 1 2 - 3 1 accommodate the max imum value in column 1 of Σ D A T . Bar plots ar e use ful when plotting categori cal (i .e., non -numer ical) data. Suppos e that y ou want t o plot the data in column 2 of the Σ DA T m a t rix : Θ Press „ô , sim ultaneously if in RPN mode , to access t o the PL O T SETUP wi nd ow . Θ Press ˜˜ to highl[...]

Pa g e 1 2- 32 Θ Press @ERASE @DR AW to dr aw the bar plot . Pre ss @EDIT L @LABEL @ MENU to see the plot unenc umber ed by the menu and w i th ide ntifying la bels (the c ursor w i ll be in the middle of the plot , how ev er ): Θ Press LL @) PICT to lea ve the EDI T en vir onment. Θ Press @CANCL to retur n to the PL O T WINDOW en vir onment . T[...]

Pa g e 1 2- 3 3 Slope fields Slope fi elds ar e us ed to vi suali z e the solutions to a diffe r ential equation of the fo rm y’ = f(x ,y) . Basi cally , w hat is pres ented in the plot ar e segmen ts tangenti al to the soluti on curve s, since y’ = d y/dx, e valuated at an y point (x,y), repr esents the slope of the tangent line at point (x ,y[...]

Pa g e 1 2 - 3 4 of y(x ,y) = constant , for the soluti on of y’ = f(x,y). Thus , slope fi elds ar e usef ul tools f or v isualizing parti cul arl y diffic ult equations to sol ve . T ry also a slope field plot f or the functi on y’ = f(x ,y) = - (y/x) 2 , by using: Θ Press „ô , sim ultaneously if in RPN mode , to access t o the PL O T SETU[...]

Pa g e 1 2 - 35 Θ Press @ERASE @DRAW to dr aw the thr ee -dimensional surface . The r esult is a w i r efr ame pi ctur e of the surface w ith the r efer ence coor dinate s y stem sho wn at the lo wer le ft corner of the sc reen . B y using the arr o w k ey s ( š™— ˜ ) you can c hange the orient ation of the surface . The or ientati on of the[...]

Pa g e 1 2 - 36 Θ Press „ô , simult aneousl y if in RPN mode, t o access the P L O T SETUP wi nd ow . Θ Press ˜ and type ‘S IN(X^2+Y^2)’ @@@OK@@@ . Θ Press @ERASE @DRAW to dra w the plot . Θ When done , pres s @EXIT . Θ Press @CANCL to r eturn to PL O T W INDO W . Θ Press $ , or L @@@OK@@@ , to return to nor mal calculator displa y . [...]

Pa g e 1 2 - 37 Θ Press @EDIT L @ LABEL @MENU to s ee the gr aph w ith labels and range s. This partic ular v ersio n of the gr aph is limited to the lo we r part of the display . W e can change the v ie wpoint to see a differ ent versi on of the graph . Θ Press LL @) PICT @CANCL to r eturn to the PL O T WINDO W env ironment . Θ Change the e ye [...]

Pa g e 1 2 - 3 8 T ry also a Wir efr ame plot for the surface z = f(x ,y) = x 2 +y 2 Θ Press „ô , simult aneousl y if in RPN mode, t o access the P L O T SETUP wi nd ow . Θ Press ˜ and type ‘X^2+Y^2’ @@@OK@@@ . Θ Press @ERASE @DRAW t o dr aw the slope f ield plot . Pr ess @EDIT L @) MENU @LAB EL to see the plot unenc umbered b y the menu[...]

Pa g e 1 2 - 3 9 Θ Press @EDIT ! L @LABEL @MENU t o see the gr aph w ith labels and r anges. Θ Press LL @) PICT@CANCL to r eturn to the PL O T WINDOW en vir onment . Θ Press $ , or L @@@OK@@@ , to return to nor mal calculator displa y . T ry also a P s-Contour plot f or the sur face z = f(x ,y) = sin x cos y . Θ Press „ô , simult aneousl y i[...]

Pa g e 1 2 - 4 0 Θ Make sur e that ‘X’ is selected as the Indep: and ‘ Y’ as the Depnd: variab le s. Θ Press L @@@OK@@@ to re turn to nor mal calculator dis play . Θ Press „ò , simultaneou sly if in RPN mode , to access the P L O T WINDO W scr e en . Θ Change the defa ult plot windo w ranges t o r ead: X-L eft:-1, X-Ri ght:1, Y -Near[...]

Pa g e 1 2 - 4 1 Θ Press „ô , sim ultaneou sly i f in RPN mode , to access t o the P L O T SETUP w indow . Θ Cha nge TYPE to Gr idmap. Θ Press ˜ and type ‘SIN(X+i*Y)’ @@@OK@@@ . Θ Make sur e that ‘X’ is selected as the Indep: and ‘ Y’ as the Depnd: variab le s. Θ Press L @@@OK@@@ to re turn to nor mal calculator dis play . Θ P[...]

Pa g e 1 2 - 42 F or ex ample, to pr oduce a Pr- Sur fa ce plot for the surf ace x = x(X,Y) = X sin Y , y = y(X,Y) = x co s Y , z=z(X,Y)=X, u se the f ollo w ing: Θ Press „ô , sim ultaneou sly i f in RPN mode , to access t o the P L O T SETUP w indow . Θ Cha nge TYPE to Pr -Surface . Θ Press ˜ and t y pe ‘{X*S IN(Y), X*CO S(Y), X}’ @@@OK[...]

Pa g e 1 2 - 4 3 Interactiv e draw ing Whene ve r we pr oduce a two-dimensional gr aph, w e find in the gr aphics s cr een a soft menu k e y label ed @) EDIT . Pres sing @) EDIT produce s a menu that include the fo llow ing options (pr ess L to see additi onal func tions): Thr ough the ex amples abov e , y ou have the opportunity to try out functio[...]

Pa g e 1 2 - 4 4 Ne xt, w e illustr ate the use o f the differ ent dra w ing functi ons on the resulting gr aphic s sc reen . The y requir e use of the c ursor and the arr ow k ey s ( š™— ˜ ) to mo ve the cu rsor about the gr aphic s scr een. DO T+ and DO T- When DO T+ is selec ted , pi xels w ill be activ ated wher ev er the cursor mov es le[...]

Pa g e 1 2 - 4 5 should hav e a str aight angle tr aced by a hor iz on tal and a ve rtical segmen ts. The c ursor is still acti ve . T o deacti vate it , without mo ving it at all , pre ss @LINE . The c ursor r eturns to its n ormal shape (a c ro ss) and the LINE f unction is no longer acti ve . TLINE (T oggle LINE) Mo ve the c ursor to the se cond[...]

Pa g e 1 2 - 4 6 DEL This command is us ed to remo ve parts of the gr aph betw een two MARK positions . Mov e the cur sor to a point in the gr aph, and pre ss @MARK . Mov e the cu rsor to a diff er ent point , press @MARK again. T hen, pr ess @@DEL@ . T he section of the gr aph bo xed betw een the two marks w ill be deleted. ERASE The f unction ERA[...]

Pa g e 1 2- 47 X,Y  This command copi es the coordinates o f the cur r ent cur sor position, in us er coor dinates , in the stac k . Zooming in and out in the gr aphics display Whene ve r y ou produce a tw o -dimensional FUNCT ION gr aphic in ter activ ely , the fir st soft-menu k ey , labeled @) ZOOM , lets y ou access functi ons that can be us[...]

Pa g e 1 2 - 4 8 Y ou can alwa ys r eturn to the v ery last z oom windo w by using @ZLAST . BO XZ Z ooming in and out of a gi ven gr aph can be perfor med by u s ing the so ft-menu ke y BO XZ . With BO XZ you selec t the rect angular sector (the “bo x ”) that y ou want to z oom in into. Mov e the cursor to one of the corners of the bo x (using [...]

Pa g e 1 2 - 4 9 cu rsor at the center of the scr een, the w indow gets z oomed so that the x -ax is extends f rom –64. 5 to 6 5 . 5 . ZSQR Z ooms the graph s o that the plotting scale is maintained at 1:1 b y adjus ting the x scale , keeping the y s c ale f ix ed, if the w indow is w ider than taller . This f or ces a pr oportional z ooming. ZTR[...]

Pa g e 1 2- 5 0 S OL VER .. „Î (the 7 key ) C h. 6 TRIGONO METRIC. . ‚Ñ (the 8 key ) Ch. 5 EXP&LN .. „Ð (the 8 key ) Ch. 5 The S YMB/GRAPH menu The GRAP H sub-men u w ithin the S YMB menu inc ludes the follo w ing func tions: DEFINE: same as the ke ystr oke s e quence „à (the 2 key) GROB ADD: paste s two GR OBs fir st ov er the seco[...]

Pa g e 1 2 - 5 1 T AB V AL(X^2 -1,{1, 3}) produ ces a list of {min max} v alues of the f u ncti on in the interval {1, 3}, w hile SIGNT AB(X^2 -1) show s the sign of the fu ncti on in the interval ( - ∞ ,+) , 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 wing t abl[...]

Pa g e 1 2 - 52 of F . The question ma rks indicates u ncertai nty or non -definition. Fo r example , for X<0, LN(X) is not def ined, thu s the X lines sho ws a que stion mark in that interval . Ri ght at z er o (0+0) F is inf inite, f or X = e, F = 1/e. F inc reas es bef ore r eaching this v alue , as indicated by the u p war d arr ow , and dec[...]

P age 13-1 Chapter 13 Calculus Applications In this Chapter w e discu ss applicati ons of the calculator ’s functions to oper ations r elated to Calc ulus, e .g., limits , der iv ativ es, integr als, pow er ser ies, etc. The CAL C (Calculus) m enu Many o f the functi ons pres ented in this Chapter ar e contained in the calc ula tor ’s CAL C men[...]

P age 13-2 Function lim The calc ulator pro vi des functi on lim t o ca l cu l a t e l im i t s o f fu n ct i on s . Th i s fu n c ti o n uses a s input an expr ession r epr esenting a fu nction and the v alue wher e the limit is to be calculated . Functi on lim is av ailable thro ugh the command catalog ( ‚N~„l ) or thr ough option 2 . LIMIT S[...]

P age 13-3 T o calculat e one -sided limits, add +0 or -0 to the v alue to the vari able. A “+0” means limit fr om the ri ght , w hile a “-0” means limit fr om the left . F or ex ample , the limit of as x approac hes 1 fr om the left can be deter mined with the follo w ing k ey str ok es (AL G mode): ‚N~„l˜ $OK$ R!ÜX- 1™@íX@Å1+0` [...]

P age 13-4 in AL G mode . R ecall that in RPN mode the arguments mu st be en ter ed befor e the functi on is applied. The DERI V&INTEG menu The f unctions a vailable in this sub-me nu ar e listed belo w: Out of thes e functi ons D ERIV and DER VX ar e used for der iv ati ve s. T he other functi ons include functi ons r elated to anti-deri vati [...]

P age 13-5 be differ entiated . Thus , to calculate the deri vati ve d(sin(r ) ,r), use , in AL G mod e: ‚¿~„r„ÜS~„r` In RPN mode , this expr ession mu st be enclos ed in quot es befo re ente ring it in to th e sta ck. Th e re su lt in ALG mo d e i s: In the E quati on W r iter , when y ou pre ss ‚¿ , the calc ulator pr o vi des the fo[...]

P age 13-6 T o ev aluate the deri vati ve in the E quation W r iter , pres s the up-arr o w k e y — , fo ur times, to s elect the entir e expr essi on, then , pr ess @EVAL . The der ivati ve w ill be ev aluated in the E quation W riter as: The chain rule The c hain rule for der ivati ves appli es to deri vati ves of composit e functi ons. A gener[...]

P age 13-7 Deri vati ves of equations Y ou can us e the calculat or to calculat e deri vati ves of eq uations , i .e., expr essi ons in whi ch deri vati ves w ill e xist in both side s of the equal sign . Some ex amples ar e sho wn belo w: Notice that in the e xpressi ons wher e the deri vati ve sign ( ∂ ) or f unction DERIV was u sed , the equal[...]

P age 13-8 Analyzing gr aphic s of functions In Chapter 11 w e pres ented some functi ons that ar e a vailable in the gr ap hic s sc r een f or anal yzing gr aphics of f unctions of the for m y = f(x) . The se functi ons include (X,Y ) and TRACE f or determining point s on the gra ph, a s wel l as functi ons in the Z OOM and FCN menu . The f unctio[...]

P age 13-9 Θ Press L @PICT @CANCL $ to re turn to nor mal calculator dis play . Notice that the slope and tangent line that y ou reques ted ar e listed in the stac k. Function DOMAIN F unction DOMAIN , av ailable through the command catalog ( ‚N ), pr o vi des the domain of def inition of a func tion as a list of numbers and spec ificati ons. F [...]

P age 13-10 This r esult indicat es that the r ange of the functi on corr esponding to the domain D = { -1,5 } is R = . Function SIGNT AB F unction SIGNT AB, av ailable through the command cat alog ( ‚N ), pro vides informa tion on th e sign of a function th r ou gh it s domai n . For e xample, for the T AN(X) functi on, SIGNT AB ind icat es that[...]

P age 13-11 Θ Le vel 3: the f uncti on f(VX) Θ T wo lists , the fir st one indicate s the vari ation of the f unction (i .e., whe r e it incr eases or dec reas es) in ter m s of the independent v ari able VX, the second one indicate s the var iation of the f unction in ter ms of the dependent v ariable . Θ A gr aphi c obj ect sho w ing ho w the [...]

P age 13-12 The interpr etation of th e var iation table show n abo ve is as f ollow s: the functi on F(X) incr eases f or X in the interval (- ∞ , -1), reac hing a max imum equal to 3 6 at X = -1. Then, F(X) dec reas es until X = 11/3, reac hing a minimum of –400/2 7 . After that F(X) inc reas es until r eaching + ∞. Also, at X = ±∞ , F(X[...]

P age 13-13 W e find tw o cr itical po ints, one at x = 11/3 and one at x = -1. T o ev aluate the second der iv ativ e at each point use: The last sc reen sho ws that f ”(11/3) = 14, thus , x = 11/3 is a r elativ e minimum. F or x = -1, we ha ve the f ollow ing: This r esult indi cates that f ” (-1) = -14, thu s, x = -1 is a r elativ e max imum[...]

P age 13-14 Anti-deri vativ es and integrals An anti-der iv ative o f a func tion f(x) is a func tion F(x) such that f(x) = dF/dx. F or e xam ple , since d(x 3 ) /dx = 3x 2 , an anti-der iv ativ e of f(x) = 3x 2 is F(x) = x 3 + C, wher e C is a constant. One wa y to represent an anti-deri vati ve is as a indefinite integral , i .e ., , if and only [...]

P age 13-15 abo ve . Their r esult is the so -called discr ete der iv ativ e, i .e., one de fined f or integer number s only . Definite integr als In a def inite integr al of a f unction , the resulting an ti-der i vati ve is ev aluated at the upper and lo wer limit of an interv al (a,b) and the e valuated v a lues subtr acted . S ymbolicall y , wh[...]

P age 13-16 This is the gener al format f or the definite integr al when typed dir ectly into the stack , i .e., ∫ (low er limit , upper limit, int egr and, v ari able of in tegr ation) Pr essi ng ` at this point w ill evaluat e the integral in the stac k: The integr al can be ev aluated also in the E quation W r iter by se lecting the entir e e [...]

P age 13-17 The f ollow ing ex ample sho ws the ev aluation of a def inite integral in the E quation W r iter , step-by-s tep: ʳʳʳʳʳ Notice that the step-b y-step pr ocess pr ov ides infor mation on the inter mediate steps f ollow ed by the CAS to solv e this integral . F irst , CAS ide ntif ies a squar e r oot integral , next , a rational f r[...]

P age 13-18 T echniques of integration Se ver al techni ques of integr ation can be implemented in the calc ulators, as sho wn in the f ollo wing e xamples . Substitution or chang e of v ariables Suppose w e w ant to calculate the integral . If w e use step-by- step calc ulation in the Eq uation W r iter , this is the sequence o f var iable substit[...]

P age 13-19 Integration b y par ts and differentials A differ ential of a f unction y = f(x), is defined as d y = f’(x) dx , wher e f’(x) is the deri vati ve of f(x). Differ entials ar e used to repr esent small inc r ements in the var iables . The differ ential o f a produc t of tw o functions , y = u(x)v(x) , is gi ven b y dy = u(x)d v(x) +du[...]

P age 13-20 Integration b y par tial fr actions F unction P AR TFRAC, presented in Chapte r 5, pr ovi d es the decomposition of a fr action into partial f rac tions. This t echni que is use ful to r educe a complicated fr action into a sum of simple fr actio ns that can then be integrated t erm b y te rm . F or ex ample , to integr ate w e can deco[...]

P age 13-21 Using the calc ulator , we pr oceed as fo llo ws: Alternati vel y , you can e valuate the integra l to inf inity from the start , e.g . , Integration w it h units An integr al can be calculated w ith units incorporat ed into the limits of integr ation , as in the e xample sh o wn belo w that uses AL G mode , with the CAS set to Appr ox [...]

P age 13-22 Some no tes in the us e of units in the limits of integr ations: 1 – The units of the lo wer limit o f integrati on will be the ones used in the f inal r esult , as illustr ated in the two e xamples belo w: 2 - Upper limit units mus t be consiste nt w ith lo we r limit units . Otherwise , the calc ulator simply r eturns the unev aluat[...]

Pa g e 1 3 - 23 T ay lor and M aclaurin’s series A func tion f(x) can be e xpanded into an infinit e ser ies ar ound a point x=x 0 by using a T a ylor ’s ser ies, namel y , , wher e f (n) (x) represen ts the n- th der iv ativ e of f(x) with r esp ect to x , f (0) (x) = f(x). If the value x 0 is z er o , the ser ies is r eferr ed to as a Maclaur[...]

P age 13-2 4 wher e ξ is a number near x = x 0 . Since ξ is typ icall y unknow n, inst ead of an estimate o f the residual , we pr ov ide an estimate of the or der of the re sidual in ref e ren c e t o h , i. e., we s ay t h a t R k (x) has an err or of order h n+1 , or R ≈ O(h k+1 ). If h is a small number , say , h<<1, then h k+1 wi ll [...]

P age 13-2 5 incr ement h. T he list r eturned as the fir st output objec t includes the f ollow ing items: 1 - Bi-dir ectional limit o f the funct ion at po int of e xpansion , i .e., 2 - An equi valent v alue of the functi on near x = a 3 - Expr essi on for the T ay lor poly nomial 4 - Or der of the r esidual or r emainder Becaus e of the r elati[...]

Pa g e 1 4 - 1 Chapter 14 Multi-var iate Calculus Applications Multi-vari ate calculus re fers to functi ons of two or mor e vari ables. In this Chapter w e discu ss the basi c concepts of multi-v ari ate calculu s including partial deri vati ves and multiple int egrals . Multi-var iate functions A functi on of two or mor e vari ables can be define[...]

Pa g e 1 4 - 2 . Similarl y , . W e wi ll use the multi-var iate functi ons def ined earlier to calc ulate partial deri vati ves using the se def initions. Her e are the der iv ative s of f(x,y) w ith r espect to x and y , r especti vely : Notice that the def inition of partial der i vati ve w ith r espect t o x, f or e xample , r equir es that we [...]

Pa g e 1 4 - 3 ther ef or e , w ith DERVX y ou can only calculate der iv ativ es with r espect to X. Some e xamples of f irst-order partial der iv ative s are sho wn ne xt: ʳʳʳʳʳ Hi gh er- ord e r d erivat ives The f ollo wing s e cond-or der deri vati ves can be def ined The la st two e xpre ssions r epres ent cr oss-der iv ativ es, the parti[...]

Pa g e 1 4 - 4 Thir d-, fourth-, and higher or der der i vati ves ar e def ined in a similar manner . T o calc ulate higher o rde r der iv ativ es in the calculator , simply r epeat the deri vati ve func tion as man y times as needed. So me e xample s are sho wn belo w: The chain rule for partial deriv atives Consi der the func tion z = f(x ,y) , s[...]

Pa g e 1 4 - 5 A diffe r ent ver sion of the c hain rule appli es to the case 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 ollow ing form ulas r epre sent the chain r ule for this situatio n: Determining e xtrema in functions of t w o variables In or der fo r the functi on z = f(x,y) to hav e an extr em[...]

Pa g e 1 4 - 6 W e find c riti cal points at (X,Y) = (1, 0) , and (X,Y) = (-1, 0) . T o calculate the disc riminant , we pr oceed to calculate the second der iv ativ es, fXX(X,Y) = ∂ 2 f/ ∂ X 2 , fXY(X,Y) = ∂ 2 f/ ∂ X/ ∂ Y , and fYY(X,Y) = ∂ 2 f/ ∂ Y 2 . The la st r esult indicat es that the disc riminant i s Δ = -12X, thus , for (X,[...]

Pa g e 1 4 - 7 Applicati ons of function HE S S are easi er to visuali z e in the RPN mode . Consi der as an ex ample the function φ (X,Y ,Z) = X 2 + XY + XZ , we ’ll apply fun ctio n H E SS 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 befo re and after appl y ing functi on HES S . Wh[...]

Pa g e 1 4 - 8 The r esulting 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. The discr iminant , f or this criti cal point s2(1, 0) is Δ = ( ∂ 2 f/ ∂ x 2 ) ⋅ ( ∂ 2 f/ ∂ y 2 )- [ ∂ 2 f/ ∂ x ∂ y] 2 = (6 .)(- 2 .) = -12 .0 < 0, indi c ating a sa[...]

Pa g e 1 4 - 9 Jacobian of coordinate tr ansformation Consi der the coordinate tr ansfor mation x = x(u ,v) , y = y(u ,v) . The Jacobi an of this tr ansfor mation is def ined as . When calc ulating an integr al using suc h transf ormation , the e xp r ession to us e is , whe re R’ is the r egion R expr essed in (u ,v) coordinates . Double integra[...]

Pa g e 1 4 - 1 0 wher e the region R’ in polar coor dinates is R ’ = { α < θ < β , f ( θ ) < r < g( θ )}. Double integr als in polar coordinat es can be enter ed in the calculator , making sur e that the Jacobian |J| = r is inc lud ed in the integrand. T h e f ollo w ing is an e xam ple of a double int egr al calculat ed in pol[...]

P age 15-1 Chapter 15 V ector Analy sis Applications In this Chapter w e pres ent a number of functio ns fr om the CAL C menu that apply t o the analy sis of scalar and vec tor fields . The CAL C menu w as pr esent ed in detail in Chapter 13 . In par tic ular , in the DERIV & INTE G menu we identif ied a number of functi ons that have appli cat[...]

P age 15-2 At an y partic ular point , the maximum r ate of change of the functi on occurs in the dir ection o f the gradien t , i .e ., along a unit vec tor u = ∇φ /| ∇φ |. The v alue of that directi onal deri vativ e is equal to the magnitu de of the gr adient at any po int D max φ (x ,y ,z) = ∇φ •∇φ /| ∇φ | = | ∇φ | The eq u[...]

P age 15-3 as the matri x H = [h ij ] = [ ∂φ / ∂ x i ∂ x j ], the gradie nt of the f unction w ith r espec t to the n-vari ables, gr ad f = [ ∂φ / ∂ x 1 , ∂φ / ∂ x 2 , … ∂φ / ∂ x n ], and the list of vari ab le s [ ‘ x 1 ’ ‘ x 2 ’…’x n ’]. Consider as an e xample the func tion φ (X,Y ,Z) = X 2 + XY + XZ , we ?[...]

P age 15-4 not hav e a potential functi on asso c iated with it , sinc e , ∂ f/ ∂ z ≠∂ h/ ∂ x. Th e calcula tor response in th is case is shown below : Div ergence The di ver gence of a vector f unction , F (x ,y ,z ) = f(x ,y ,z) i +g(x ,y ,z) j +h(x ,y ,z) k , is defi ned by t aking a “ dot-product ” of the del oper ator with the fu[...]

P age 15-5 Cur l The c url of a v ector field F (x ,y ,z) = f(x ,y ,z) i +g(x ,y ,z) j +h(x,y ,z) k , is def ined by a “ c r oss-pr oduct” of the del oper ator with the v ector fi eld, i .e. , The c url of v ector fi eld can be calc ulated with f unction C U RL . For e xample , for the func tion F (X,Y ,Z) = [XY ,X 2 +Y 2 +Z 2 ,YZ], the c url i[...]

P age 15-6 As an ex ample, in an ear lier ex ample w e attempted to f ind a potenti al func tion for the vect or fie ld F (x,y ,z) = (x+y) i + (x-y+z) j + xz k , and got an err or message back f r om functi on PO TENT IAL. T o ver if y that this is a r otational f ield (i .e., ∇× F ≠ 0), we use func tion CURL on this fi eld: On the other hand [...]

P age 15-7 pr oduces the v ector potential f unction Φ 2 = [0, ZYX- 2YX, Y -( 2ZX-X)], w hic h is differ ent fr om Φ 1 . The last command in the sc r een shot show s that indeed F = ∇× Φ 2 . Thu s, a vector potenti al function is not uniquel y d eter mined. The compone nts of the giv en vect or fi eld, F (x ,y ,z ) = f(x ,y ,z) i +g(x ,y ,z) [...]

Pa g e 1 6 - 1 Chapter 16 Differential Equations In this Chapter w e pres ent e xample s of sol ving or dinary differ ential equati ons (ODE) using calc ulator functi ons. A differ ential equatio n is an equati on inv olv ing deri vativ es of the independent var iable . In most cases , we seek the dependent func tion that satisf ies the differ enti[...]

Pa g e 1 6 - 2 ( H @) DISP ) is not selec ted. Pr ess ˜ to see the equati on in the Equati on Wr i t e r. An alter native no tation fo r deri vati ves typed dir ectly in the s tack is to use ‘ d1’ for the deri vati ve w ith respect t o the firs t independent vari able, ‘ d2’ f or the deri vati ve w ith res pect to the s econd independent v[...]

Pa g e 1 6 - 3 EV AL(ANS(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 ex ample, yo u could also use: ‘ ∂ t( ∂ t(u(t))))+ ω 0^2*u(t) = 0’ to enter the differ ential equation . Slope field visuali zation of solutions Slope fi eld plots, introduc[...]

Pa g e 1 6 - 4 The se func tions ar e brief ly desc ribed next . The y will be desc ribed in mor e detail in later parts of this Chapte r . DE SOL VE: Differe ntial E quation S OL VEr , pro vi des a soluti on if possible IL AP: Inv erse LAPlace transf orm, L -1 [F(s)] = f(t) LA P: LAPlace tr ansform , L[ f(t)]=F(s) LDEC: so lv es Linear Diff erenti[...]

Pa g e 1 6 - 5 Both of these inputs mu st be giv en in terms of the def ault independent v ari able for the calc ulator’s CAS (ty pi cally ‘X’). The output fr om the functi on is the gener al solution o f the ODE . The functi on LDEC is av ailable thr ough in the CAL C/DIFF menu . The e xample s are sho wn in the RPN mode , how ev er , transl[...]

Pa g e 1 6 - 6 The s olution , show n partially her e in the Equati on W riter , is: Replac ing the combination o f constants accompan ying the e xponential ter ms with sim pler values , the e xpres sion can be simplifi ed to y = K 1 ⋅ e –3x + K 2 ⋅ e 5x + K 3 ⋅ e 2x + ( 4 50 ⋅ x 2 +3 30 ⋅ x+2 41)/13 500. W e re cogni z e the firs t thr[...]

Pa g e 1 6 - 7 2x 1 ’(t) + x 2 ’(t) = 0. In algebr aic fo rm , this is wr it te n as: A ⋅ x ’(t) = 0, wher e . T he s ys tem can be so lv ed by using fu nctio n LDEC w ith argumen ts [0, 0] and matri x A, as sho wn in the f ollo wing sc reen u sing AL G mode: The s oluti on is giv en as a vect or containing the func tions [x 1 (t) , x 2 (t)[...]

Pa g e 1 6 - 8 Example 2 -- Sol ve the second-o rde r ODE: d 2 y/dx 2 + x (dy/dx) = e xp(x) . In the calculator u se: ‘ d1d1y(x)+x*d 1y(x) = EXP(x) ’ ` ‘ y(x) ’ ` DESOLVE The r esult is an e xpr essi on hav ing two impli c it integrations , namely , F or this par tic ular equation , how ev er , w e reali z e that the left -hand side of the [...]

Pa g e 1 6 - 9 P erf orming the int egr ation b y hand, we can onl y get it as far as: because the in tegr al of exp(x)/x is no t av ailable in c losed f orm. Example 3 – Sol v ing an equatio n w ith initial conditi ons. Sol ve d 2 y/dt 2 + 5y = 2 cos(t/2) , w ith initial conditions y(0) = 1.2 , y’(0) = -0. 5. In the calculator , use: [‘ d1d1[...]

Pa g e 1 6 - 1 0 Press J @ODETY to get the str ing “ Linear w/ cst coeff ” for the ODE type in this case . Laplace T ransf orms The L aplace transf orm o f a functi on f(t) pr oduces a functi on F(s) in the image domain that can be utili zed t o find the solu tion of a linear diff erential eq uation inv olv ing f(t) thr ough algebr aic me thods[...]

Pa g e 1 6 - 1 1 Laplace tr ansform and in verses in the calc ulator The calc ulator pr o vi des the f uncti ons LAP and IL AP to calc ulate the L aplace transf orm and the in verse L aplace transf orm, r especti vel y , of a f unction f(VX), wher e VX is the CAS def ault independent var iable, w hich y ou should set to ‘X’ . Thu s, the calcula[...]

Pa g e 1 6 - 1 2 Example 3 – Deter mine the in ver se Laplace tr ansfor m of F(s) = sin(s). Use: ‘SIN(X)’ ` ILAP . The calc ulator tak es a fe w seconds to r eturn the re sult: ‘ILAP(SIN(X))’ , meaning that there is no cl osed-f orm expr essi on f(t), such that f(t) = L -1 {sin(s)}. Example 4 – Determine the in ve rse L aplace tr ansfor[...]

Pa g e 1 6 - 1 3 Θ Differ entiati on theorem f or the n -th deri vati ve . 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 inearity theor em . L{af(t)+bg(t)} = a ⋅ L{f(t)} + b ⋅ L{g(t)}. Θ Differ entiati on theor em for the image f unction[...]

Pa g e 1 6 - 1 4 Θ Shift theore m for a shift to the r ight . Let F(s) = L{f(t)}, then L{f(t-a)}=e –as ⋅ L{f(t)} = e –as ⋅ F(s). Θ Shift theore m f or a shift to the left . Let F(s) = L{f(t)}, and a >0, then Θ Similar ity theor em . Let F(s) = L{f(t)}, and a>0, then L{f(a ⋅ t)} = (1/a) ⋅ F(s/a). Θ Dampi ng theor em . L et F(s)[...]

Pa g e 1 6 - 1 5 Dirac’s delta function and Heaviside’s step function In the analy sis of contr ol s ys tems it is cu stomary to utili ze a ty pe of functi ons that r epre sent certain ph ysi cal occurr ences suc h as the sudden acti vati on of a sw itch (Heav iside’s s tep func tion , H(t)) or a sudden , instantaneou s, peak in an input to t[...]

Pa g e 1 6 - 1 6 Y ou can pro ve that L{H(t)} = 1/s, from whi ch i t fol lows th at L {U o ⋅ H(t)} = U o /s, wher e U o is a constant. Also , L -1 {1/s}=H(t), and L -1 { U o /s}= U o ⋅ H(t). Also , using the shift theor em for a shift to the ri ght , L{f(t -a)}=e –as ⋅ L{f(t)} = e –as ⋅ F (s ) , we c an wri t e L { H( t - k )} = e –ks[...]

Pa g e 1 6 - 1 7 Applications of Laplace tr ansform in the solution of linear ODEs At the beginning of the se ction on L aplace transf orms w e indicated that y ou could us e these tr ansfor ms to conv er t a linear ODE in the time domain into an algebrai c equation in the image domain . The r esulting equati on is then solv ed for a f unction F(s)[...]

Pa g e 1 6 - 1 8 The r esult is ‘H=((X+1)*h0+a)/(X^2+(k +1)*X+k)’ . T o fi nd the soluti on to the ODE , h(t) , w e need to use the inv erse L aplace transf orm, as f ollow s : OB J  ƒ ƒ Isolate s ri ght-hand side of last e xpres sion ILAP μ Obtains the inv erse L aplace tr ansfor m The r esult is . R eplac ing X w ith t in this e xpr ess[...]

Pa g e 1 6 - 1 9 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(0) and y 1 = h ’(0), the transf ormed eq uation is s 2 ⋅ Y(s) – s ⋅ y o – y 1 + 2 ⋅ Y(s) = 3/(s 2 +9). Use the calculator to solve f or Y (s), b y wr iting: ‘X^2*Y -X*y0 -y1+2*Y=3/(X^2+9)’ ` ‘Y’ I SOL The r esult is ‘Y=[...]

Pa g e 1 6 - 2 0 Example 3 – Consider the equati on d 2 y/dt 2 +y = δ (t-3) , wher e δ (t) is Dir ac’s delta functi on. Using Laplace tr ansforms , we 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)}. Wit h ‘ Delta(X-3) ’ ` LA P , the calc ulator produc es EXP(-3*X), i.e ., L{ δ (t -3)} = e –3s . Wi [...]

Pa g e 1 6 - 2 1 Check w hat the solution t o the ODE would be if y ou use the f unction LDEC: ‘Delta(X- 3)’ ` ‘X^2+1’ ` LDEC μ Notes : [1]. An alter nativ e wa y to obtain the in ver se Laplace tr ansform of the e xpr essi on ‘(X*y0+(y1+EXP(-(3*X))))/(X^2+1)’ is b y separating the e xpr essi on into partial f r actions , i.e ., ‘ y0[...]

Pa g e 1 6 - 2 2 The r esult is: ‘SI N(X-3)*Heav iside(X-3) + cC1*SIN(X) + cC0*CO S(X)’ . P lease notice that the var iable X in this expr essi on actuall y re presen ts the var iable t in the ori ginal ODE . Thu s, the translati on of the soluti on in paper may be wr itten as: When compar ing this result w ith the pre vio us r esult for y(t), [...]

Pa g e 1 6 - 23 Use of the func tion H(X) w ith LD E C, L AP , or IL AP , is not allo wed in the calc ulator . Y ou have to u se the main re sults pr ov ided earlier w hen dealing with the Heav iside step f unction , 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 –as [...]

Pa g e 1 6 - 24 wher e H(t) is Heavisi de’s step f u ncti on. Using L aplace transfo rms, w e can writ e: 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 last t erm in this expr 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 , wher e y o = h(0) and [...]

Pa g e 1 6 - 2 5 Example 4 – P lot the solution to Ex ample 3 using the same v alues of y o and y 1 used in the plot of Example 1, abo ve . W e no w plot the functi on 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 graph should look lik e this: Again ,[...]

Pa g e 1 6 - 26 f(t) = U o ⋅ [1-(t-a)/(b-1)] ⋅ [H(t-a) -H(t -b)]. Example s of the plots generated b y these functi ons, fo r Uo = 1, a = 2 , b = 3, c = 4, hor iz ontal r ange = (0,5) , and verti cal range = (-1, 1. 5) , ar e show n in the fig ures b el ow: Four ier series F ourie r ser ies are s eri es inv olv ing sine and cosine fu nctions ty[...]

Pa g e 1 6 - 27 The f ollow ing ex erc ises ar e in AL G mode, w ith CAS mode s et to Ex act . (When y ou pr oduce a gr aph, the CA S mode wi ll be re set to A ppr o x. Mak e sure to s et it back to Ex act after pr oduc ing the gr aph .) Suppose , for e xample , that the functi on f(t) = t 2 +t is periodi c with per iod T = 2 . T o determine the co[...]

Pa g e 1 6 - 28 Function FOURIER An alter nati ve w ay to def ine a F ouri er ser ies is by using comple x numbers as fo llow s: whe re F unction FOURIER pr ov ides the coeff ic ient c n of the comple x -for m of the F our ier ser ies giv en the functi on f(t) and the value of n . The f unction FOURIER r equir es y ou to sto r e the value of the pe[...]

Pa g e 1 6 - 2 9 Next , we mo ve to the CASDIR sub-dir ectory under HOME to change the value of var iable PERIOD, e .g., „ (hold) §`J @) CASDI `2 K @PERIOD ` Retur n to the sub-dir ectory wher e you def ined functions f and g , and calculate the coeff ic ients (A ccept change to C omplex mode w hen requ ested): Thu s, c 0 = 1/3, c 1 = ( π⋅ i+[...]

Pa g e 1 6 - 3 0 The f itting is somew hat acceptable for 0<t<2 , although not as good as in the pr ev ious e xample . A general expr ession for c n The f unction FO URIER can pro vi de a gener al expr ession for the coe ffi cien t c n of the comple x F our ier ser ies e xpansion. F or ex ample , using the same functi on g(t) as befor e, the [...]

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 Fou rier series Hav ing determined the gener al expr ession f or c n , w e can put together a finit e comple x Fo uri er seri es b y using the summati on functi on ( Σ ) in the calc ulator as fo llow s: Θ F irst , define a f unction c(n) r epr[...]

Pa g e 1 6 - 32 Or , in the calc ulator 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))’), wher e T is the period , T = 2 . The f ollo wing sc reen shots sho w the definiti on of func tion F and the stor ing of T = 2 : The fun ct ion @@@F@@@ can be us ed to gener ate the expr ession f or[...]

Pa g e 1 6 - 3 3 Accept change t o Approx mode if reque sted . The re sult is the value –0.40 46 7…. The ac tual value o f the functi on g(0.5) is g(0. 5) = -0.2 5. T he fo llow ing calc ulations sho w ho w well the F our ier se ri es appr o ximat es this value as the number of componen ts in the ser ies, gi ven b y k , inc reas es: F (0. 5, 1,[...]

Pa g e 1 6 - 3 4 peri odic ity in the graph of the ser ies. T his periodi city is eas y to v isuali ze b y expanding the hori z ontal range of the plot to (-0.5, 4) : Four ier series for a triangular w ave Consider the f unction whi ch we assume to be per iodic w ith per iod T = 2 . This f uncti on can be def ined in the calc ula tor , in AL G mode[...]

Pa g e 1 6 - 35 The calc ulator r eturns an int egr al that cannot be evaluat ed numer icall y because it depends on the parame ter n . The coeff ic ient can still be calc ulated by typing its def inition in the calc ulator , i.e ., wher e T = 2 is the perio d. T he value of T can be st or ed using: T y ping the firs t integral a bo ve in the E qua[...]

Pa g e 1 6 - 36 Press `` to copy this r esult to the scr een. T hen , re acti vat e the Eq uation W rit er to calc ulate the second integral de fining the coeff ic ient c n , namel y , Once again, r eplacing e in π = (-1) n , and u sing e 2in π = 1, we get: Press `` to cop y this second re sult to the sc reen . Now , add ANS(1) and ANS( 2) to get[...]

Pa g e 1 6 - 37 This r esult is used to def ine the functi on c(n) as follo ws: DEFINE(‘ c(n) = - (((-1)^n-1)/(n^2* π ^2*(-1)^n)’) i. e. , Next , we def ine function F(X,k,c0) to calculate the F ourie r series (if y ou completed e xample 1, yo u already ha ve this functi on stor ed): DEFINE(‘F(X,k,c0) = c0+ Σ (n=1,k ,c(n)*EXP(2*i* π *n*X/T[...]

Pa g e 1 6 - 3 8 F rom the plot it is very diffi cult to distinguish the or iginal f unction f rom the F ourier ser ies appr ox imation . Using k = 2 , or 5 terms in the seri es, show s not so good a f itting: The F our ier s eri es can be used to gener ate a peri odic tri angular wa ve (or sa w tooth w av e) by changin g the hor iz ontal ax is r a[...]

Pa g e 1 6 - 3 9 In thi s case, the peri od T , is 4. Make sure to c hang e the valu e of var iabl e @@@T@@@ to 4 (use: 4K @@@T@@ ` ) . F unction g(X) can be def ined in the calculator by u s in g DEFINE(‘ g(X) = IFTE((X>1) AND (X<3),1, 0)’) The function plot ted as follo ws (hori zontal r ange : 0 to 4, vertical range: 0 to 1.2 ): Using [...]

Pa g e 1 6 - 4 0 Th e s i mp l i fica t io n o f th e rig ht -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). Then, r etype the e x pr ession f or c(n) as sho wn in the f igure to the left abo ve , to define fu ncti on c(n). The F ourie r ser ies is calculat ed with F(X,k,c0), as in ex amples 1 and 2 abo ve , w [...]

Pa g e 1 6 - 4 1 W e can use this r esult as the firs t input to the f unction LDE C when used to obt ain a soluti on to the s yste m d 2 y/dX 2 + 0.2 5y = SW(X), wher e S W(X) stands for Squar e W av e function o f X. The second inpu t item w ill be the char acter istic equation cor responding t o the homogeneous ODE sho wn abo ve , i.e ., ‘X^2+[...]

Pa g e 1 6 - 42 The s olution is sho wn belo w: Four ier T ransfor ms Befor e presen ting the concept of F our ier tr ansforms , we ’ll discus s the general def i nitio n of an integr al transf orm. In gener al, an int egr al transf orm is a transf ormation that r elates a functi on f(t) to a ne w function F(s) b y an integratio n of the f orm Th[...]

Pa g e 1 6 - 4 3 The am plitudes A n w ill be ref err ed to as the spectr um of the functi on and will be a measure of the magnitude of the component of f(x) with f requency f n = n/T . The basi c or fundamental fr equency in the F ouri er ser ies is f 0 = 1/T , th us, all other fr equenc ies ar e multiples o f this basic fr equenc y , i .e ., f n [...]

Pa g e 1 6 - 4 4 and The continuous spectrum is giv en by The fun ct ion s C ( ω ), S ( ω ), and A( ω ) ar e continuous functions of a var iable ω , whi ch becomes the tr ansfor m vari able for the F ourier tr ansforms de fined below . Example 1 – Determine the coef fi cients C( ω ), S ( ω ) , and the continuous s pectrum A( ω ), for the f[...]

Pa g e 1 6 - 4 5 Define this expr essio n as a f unction by u s ing func tion DEFINE ( „à ) . T hen, plot the continuou s spectr um, in the r ange 0 < ω < 10, as: Definition of F ourier transfor ms Differ ent types of F our ier tr ansforms can be def ined. The f ollow ing are the def i nitio ns of the sine , cosine , and full F ourie r tr[...]

Pa g e 1 6 - 4 6 The continuous spectrum, F( ω ) , is calculated w ith the integral: This r esult can be r ationali z ed b y multiply ing numer a to r and denominator b y the conjugate o f the denominator , namely , 1-i ω . The r esult is now: which is a co mpl ex fun ct ion. The ab solute v alue of the real and imaginary par ts of the func tion [...]

Pa g e 1 6 - 47 Properties of the F ourier transf orm L inearity: If a and b ar e constants , and f and g functi ons, then F{a ⋅ f + b ⋅ g} = a F{f }+ b F{g}. T r ansfor mation of partial deri vati ves . Let u = u(x ,t) . If the F ouri er transf orm transf orms the var iable x , then F{ ∂ u/ ∂ x} = i ω F{u}, F{ ∂ 2 u/ ∂ x 2 } = - ω 2 [...]

Pa g e 1 6 - 4 8 the number o f oper ations using the FF T is reduced b y a factor of 10000/66 4 ≈ 15 . The FFT operates on t he sequenc e {x j } by partitioning it into a n umber of shorte r sequence s. The DFT ’s of the shorter seq uences ar e calculated and later combined t ogether in a highl y effi c ient manner . F or details on the algor [...]

Pa g e 1 6 - 4 9 The f igur e belo w is a box plot o f the data pr oduced. T o obtain the gra ph, f irst copy the arr ay ju st cr eated, then tr ansform it into a column vector b y using: OB J  1 +  ARR Y (F unctions OB J  and  ARR Y are av ailable in the command catalog , ‚N ). Stor e the arr ay int o var iable Σ DA T by us i ng fun[...]

Pa g e 1 6 - 5 0 Example 2 – T o produce the si gnal giv en the spectrum , we modify the progr am GD A T A to inc lude an abso lute v alue, so that it r eads: <<  m a b << ‘2^m’ E V AL  n << ‘(b-a)/(n+1)’ EV AL  Dx << 1 n F OR j ‘ a+(j-1)*Dx ’ EV AL f ABS NE XT n  ARR Y > > >> >> &[...]

Pa g e 1 6 - 5 1 Except f or a large peak at t = 0, the signal is mo stl y noise . A smaller vertical scale (-0. 5 to 0.5) sho ws the si gnal as follo ws: Solution to specific second-order diff erential equations In this secti on w e pre sent and sol ve spec ifi c types of or dinar y differ ential equations w hose solu tions ar e defined in te rms [...]

Pa g e 1 6 - 52 wher e M = n/2 or (n -1)/2 , whi chev er is an integer . Legendr e’s pol ynomials ar e pre -pr ogrammed in the calculat or and can be r ecalled by us ing the func tion LE GEND RE gi ven the or der of the poly nomial , n. The f unction LEGENDRE can be obtained fr om the command catalog ( ‚N ) or thr ough the menu ARITHME TIC/POL [...]

Pa g e 1 6 - 53 wher e ν is not an integer , and the function Gamma Γ ( α ) is def ined in Chapter 3. If ν = n, an int eger , the Bes sel func tio ns of the fi rst kind f or n = integer ar e def ined by Regar dless of whether we use ν (non-intege r) or n (int eger) in the calc ulator , we can def i ne the Bes sel f uncti ons of the f irst kind[...]

Pa g e 1 6 - 5 4 Y ν (x) = [J ν (x) co s νπ – J −ν (x)]/sin νπ , for n on-integer ν , and fo r n integer , w ith n > 0, by wher e γ is the Euler constant , def ined by and h m repr esents the har monic se ries F or the case n = 0, the Bess el f unction o f the seco nd kind is def ined as With these def i nitions, a gener al solution[...]

Pa g e 1 6 - 55 The modif ied Bessel f unctions o f the second kind, K ν (x) = ( π /2) ⋅ [I - ν (x) − I ν (x)]/sin νπ , ar e also solu tions of this ODE . Y ou can implement functi ons repr esenting Bess el’s functi ons in the calc ulator in a similar manner to that used to def ine Bessel’s f unctions of the f irst kind, but k eeping [...]

Pa g e 1 6 - 5 6 Laguerr e’s equation Laguer re ’s equation is the second-o rde r , linear ODE of the for m x ⋅ (d 2 y/dx 2 ) +(1 − x) ⋅ (d y/dx) + n ⋅ y = 0. Laguer re poly nomials, de fined as , ar e soluti ons to L aguerr e’s equati on. Laguerr e’s po ly nomials can also be calc ulated with: The te rm is the m-th coeffi ci ent of[...]

Pa g e 1 6 - 57 L 2 (x) = 1- 2x+ 0. 5x 2 L 3 (x) = 1-3x+1.5x 2 -0 . 16666 … x 3 . W eber ’s equation and Hermite poly nomials W eber’s equati on is defined as d 2 y/dx 2 +(n+1/2 - x 2 /4)y = 0, for n = 0, 1, 2 , … A partic ular solutio n of t his equatio n is gi ven b y the functi on , y(x) = ex p( -x 2 /4)H * (x/ √ 2) , wher e the functi[...]

Pa g e 1 6 - 5 8 F irst , cr eate the e xpressi on defi ning the der iv ativ e and stor e it into var iable E Q. The f igur e to the left sho ws the AL G mode command, w hile the ri ght -hand side fi gure sho ws the RPN s tack be for e pre ssing K . Then , enter the NUMERICAL SOL VER env ir onment and select the diff ere ntial equation sol ver : ?[...]

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

Pa g e 1 6 - 6 0 Θ „ô (simultaneousl y , if in RPN mode) to enter PL O T en vir onment Θ Highl ight t he fi eld in fr ont o f TYPE , using the —˜ ke ys . Then , p r ess @CHOOS , and highlight Diff Eq , u sing the —˜ k ey s. Pre ss @@OK@@ . Θ Change f ield F: to ‘EXP(- t^2)’ Θ Make sur e that the follo wing par ameters ar e set to: [...]

Pa g e 1 6 - 6 1 LL @) PICT T o re c ove r m e nu a n d ret u rn to P IC T en viro n m en t. @ ( X,Y ) @ T o deter mine coordinates o f any po int on the gr aph . Use the š™ ke ys to mov e the cursor ar ound t he plot area. At the bottom of the sc r een yo u w ill see the coor dinates of the c ursor as (X,Y ) , i .e., the calc ulator uses X and [...]

Pa g e 1 6 - 62 time t = 2 , the input for m for the diff erenti al equati on sol ver sho uld look as fo llow s (notice that the Init: v alue for the Soln: is a v ector [0, 6]): Press @SOLVE (wai t) @EDIT to so lv e f or w(t=2). The solution r eads [.16 716… - .6 2 71…], i .e ., x(2) = 0.16 716 , and x'(2) = v( 2) = -0.6 2 71. Pr ess @CANC[...]

Pa g e 1 6 - 63 (Changes initial v alue of t to 0.7 5, and final v alue of t to 1, solv e again for w(1) = [-0.4 6 9 -0.6 0 7]) Repeat f or t = 1.2 5, 1.5 0, 1.7 5, 2 . 00. Pre ss @@OK@@ after v ie wing the last r esult in @EDIT . T o return to nor mal c alc u lator displa y , pr ess $ or L @@OK@@ . Th e differ ent soluti ons will be sho wn in the [...]

Pa g e 1 6 - 6 4 Notice that the opti on V- V ar : is set to 1, indicating that the fi rst element in the vec tor so lution , namely , x’ , is to be plotted against the independent var iable t . Accept c hanges to PL O T SETUP b y pres sing L @@OK@@ . Press „ò (simultaneousl y , if in RPN mode) to enter the P L O T WINDO W env iro nment . Modi[...]

Pa g e 1 6 - 65 Press LL @PICT @CANCL $ to r eturn t o nor mal calc ulator displa y . Numerical solution for stiff first-order ODE Consi der the ODE: dy/dt = -100y+100t+ 101, sub ject t o the initial conditi on y(0) = 1. Ex act solution This equati on can be wri tten as dy/dt + 100 y = 100 t + 101, and sol ved using an integr ating factor , IF(t) =[...]

Pa g e 1 6 - 6 6 Here w e are try ing to obtain the value of y( 2) giv en y(0) = 1. With the Soln: Final fi eld highlighted , pres s @SOLVE . Y ou can chec k that a so lution t ak es about 6 s ec on ds, whi le in t he previous fir st - orde r exa mp le th e s olu tio n wa s a lm ost instantaneous . Press $ to cancel the calc ulation. This is an e x[...]

Pa g e 1 6 - 67 Note: T he option Stiff is also av ailable for gr aphical soluti ons of differ ential equations . Numerical solution to ODEs with the S OL VE/DIFF menu The S OL VE soft menu is acti vated b y using 7 4 MENU in RPN mode. This men u is pre sented in detail in Cha pter 6 . One of the sub-menu s, DIFF , contains functi ons for the numer[...]

Pa g e 1 6 - 6 8 The value o f the solu tion , y fi n al , will be av ailable in var iable @@@y@@@ . This f unctio n is appr opriate f or progr amming since it leav es the differ ential equation spec ificati ons and the tolerance in the stac k read y for a ne w solution . Notice that the soluti on uses the initi al conditions x = 0 at y = 0. If , y[...]

Pa g e 1 6 - 69 contain only the v alue of ε , and the s tep Δ x w ill be taken as a small def ault value . After running func tion @@RKF@@ , the stack w ill show the lines: 2 : {‘ x’ , ‘ y’ , ‘f(x ,y)’ ‘ ∂ f/ ∂ x’ ‘ ∂ f/vy’ } 1: { εΔ x } The v alue of the soluti on, y fina l , w ill be avail able in v ari able @@@y@@@[...]

Pa g e 1 6 - 70 The se r esults indicate that ( Δ x) ne xt = 0.3 404 9… Function RRKS TEP This f unction u ses an input list similar to that o f functi on RRK, as w ell as the toler ance for the sol ution , a possible step Δ x , and a number (L A S T) specify ing the last method u sed in the soluti on (1, if RKF was used , or 2 , if RRK was use[...]

Pa g e 1 6 - 7 1 The se r esults indicate that ( Δ x) ne xt = 0.00 5 5 8… and that the RKF method (CURRENT = 1) should be used. Function RKFERR This f unction r eturns the abs olute er r or estimate f or a gi ven s tep w hen sol v ing a pr oblem as that desc ribed f or func tion RKF . T he input st ack looks as f ollow s: 2: ʳʳ ʳ {‘ x’ , [...]

Pa g e 1 6 - 72 The f ollow ing scr een shots sho w the RPN stack be for e and after applicati on of functi on RSBERR: The se r esults indicate that Δ y = 4.1514… and err or = 2 .7 6 2 ..., for Dx = 0.1. Chec k that , if Dx is reduced t o 0. 01, Δ y = -0. 003 0 7… and e rr or = 0. 0005 4 7 . Note : As yo u ex ecut e the commands in the D IFF [...]

Pa g e 1 7- 1 Chapter 17 Pr obability Applications In this Chapter we pr ov ide ex amples of appli cations of calc ulator’s func tions to pr obabil ity distribu tions . The MTH/P ROB ABILITY .. sub-m enu - part 1 The MTH/P ROB ABI LI TY .. sub-menu is acces s ible thr ough the ke ystr ok e sequence „´ . With s ys tem flag 117 set to CHOOSE bo [...]

Pa g e 1 7- 2 T o simplify notation , use P(n ,r) fo r p er mutations , and C(n,r ) for combinati ons. W e can calculate comb inations , perm utations , and factor ials with f unctions COMB , PERM, and ! fr om the MTH/PROB ABILITY .. sub-men u . T he oper ation of those f unctions is desc ribed next: Θ COMB(n ,r): Combinati ons of n items tak en r[...]

Pa g e 1 7- 3 Random n u mber gene rat ors , in gener al, oper ate b y taking a v alue, called the “ seed” of the gener ator , and perfor ming some mathematical algor ithm on that “ seed” that gener ates a ne w (pseudo)r andom number . If y ou want to gener ate a sequence of n umber and be able to r epeat the same sequence later , yo u can [...]

Pa g e 1 7- 4 fun ctio n (pmf) is r e pr esente d by f (x) = P[X=x], i .e., the pr obability that the ran d om vari ab le X ta kes th e va l ue x. The mas s distr ibution func tion mu st satisfy the conditions that f(x) >0, f or all x , and A cu mulativ e distributi on fu nctio n (cdf) is defined as Next , we w ill define a number of f unctions [...]

Pa g e 1 7- 5 P oisson distribution The probabilit y mass function of the P oisson di stribution is g i ven by . In this expr ession , if the random var iable X r epresen ts the number of occur rences of an ev ent or observati on per unit time, length , area , vo lume , etc., then the par ameter l repr esents the a ver age number of occ u rr ences [...]

Pa g e 1 7- 6 Continuous probability distr ibutions The pr obability distributi on for a continuou s random v ari able , X, is char acter i zed b y a function f(x) kno wn as the pr obability density functi on (pdf) . The pdf ha s the follo wing pr operties: f(x) > 0, f or all x , and Pr obabiliti es ar e ca lc ulated using the cum ulati ve dis t[...]

Pa g e 1 7- 7 , while its cdf is gi ven b y F(x) = 1 - exp(- x/ β ) , for x>0, β >0. The beta distr ibution The pdf f or the gamma distributi on is giv en by As in the case of the gamma distr ibution , the corres pond ing cdf f or the beta distr ibution is also gi ven b y an integr al wi th no clo sed-f orm soluti on. The W eibull distribut[...]

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 define all the se func tions . Ne xt, ente r the values of α and [...]

Pa g e 1 7- 9 Continuous distributions f o r statistical infer ence In this secti on we dis cu ss f our continu ous pr obability distr ibutions that ar e commonl y used f or pr oblems relat ed to statis tical infer ence . These distr ibutions ar e the normal distr ibution , the Student ’s t distr ibutio n, the C hi-squar e ( χ 2 ) distr ibution,[...]

Pa g e 1 7- 1 0 wher e μ is the mean , and σ 2 is the v ariance of the distr ibution . T o calculate the val ue of f( μ , σ 2 ,x) for the normal distr ibution , use f unction NDI S T with the follo w ing arguments: the mean , μ , the v ariance , σ 2 , and, the v alue x , i.e ., NDIS T( μ , σ 2 ,x) . F or ex ample, c heck that for a normal d[...]

Pa g e 1 7- 1 1 wher e Γ ( α ) = ( α -1)! is the GAMMA f unction de fined in Chapt er 3 . The calc ulator pro vi des for v alues of the upper - tail (cumulati ve) distr ibution functi on for the t-distr ibution , functi on UTPT , giv en the paramet er ν and the v alue of t , i .e., UTPT( ν ,t) . The def inition of this functi on is, ther ef or[...]

Pa g e 1 7- 1 2 The calc ulator pro vi des for v alues of the upper - tail (cumulati ve) distr ibution fun ctio n for th e χ 2 -distribu tion u sing [UTPC] gi ve n the value of x and the paramet er ν . The def inition of this f uncti on is, ther efor e, T o use this f u ncti on, w e need the degrees o f fr eedom, ν , and the value o f the chi-sq[...]

Pa g e 1 7- 1 3 The calc ulator pro vi des for v alues of the upper - tail (cumulati ve) distr ibution functi on for the F distr ibution, f unction UTPF , gi ven the paramete rs ν N and ν D, and t he value of F . The definition of this function is, t heref ore , F or ex ample, to calc ulate UTPF(10,5, 2 .5 ) = 0.1618 34… Diffe rent pr obability[...]

Pa g e 1 7- 1 4 Exponential: W eibull: F or the Gamma and Beta distr ibutions the e xpressi ons to sol ve w ill be mor e compli cated due to the pr esence of in tegr als, i . e ., • Gamma, • Beta, A numer ical soluti on w ith the numerical s olv er will n ot be feasible beca use of the integr al sign in vo lv ed in the expr ession . How ev er ,[...]

Pa g e 1 7- 1 5 Ther e are two r oots of this functi on found b y using function @ROOT w ithin the plot env iro nment . Becaus e of the integr al in the equation , the r oot is appr o ximated and w ill not be sho wn in the plot sc reen . Y ou will onl y get the message Constant? Sho wn in the sc reen. Ho we ver , if you pr ess ` at this point , the[...]

Pa g e 1 7- 1 6 Notice that the second par ameter in the UTPN functi on is σ 2, n o t σ 2 , r epre senting the var iance of the distr ibution . Also , the s ymbol ν (the lo wer -case Gr eek letter no) is not a vailable in the calc ulator . Y ou can u se , for e xample , γ (gamma) instead o f ν . T he letter γ is av ailable thought the char ac[...]

Pa g e 1 7- 1 7 Thu s, at this point, y ou will hav e the f our equations av ailable for solution . Y ou needs ju st load one of the equations into the E Q f ield in the numer ical sol ver and proceed w ith solv ing fo r one of the var iables . Examples of the UTP T , UTPC, and UPTF ar e show n below : Notice that in all the e xamples sho wn abo ve[...]

Pa g e 1 7- 1 8 With thes e four equati ons, w henev er you launch the n u mer ical solv er you hav e the fo llo w ing cho ices: Example s of soluti on of equations E QNA, E QT A, E QCA, and EQ F A ar e show n belo w: ʳʳʳʳʳ[...]

P age 18-1 Chapter 18 Statistical Applications In this Chapter w e introdu ce statisti cal applicati ons of the calc ulator including statisti cs of a sample , fr equency dis tributi on of data, simple r egre ssi on, conf idence int ervals , and hy pothesis te sting . Pre-programmed statistical featur es The calc ulator pr ov ides pr e -progr ammed[...]

P age 18-2 Stor e the progr am in a var iable called LX C. After s tor ing this pr ogram in RPN mode yo u can also use it in AL G mode . T o stor e a column vec tor into v ariable Σ DA T use function S T O Σ , av ailable thr ough the catalog ( ‚N ) , e .g., S T O Σ (ANS(1)) in AL G mode . Example 1 – Using the pr ogram LX C, defi ned abo ve [...]

P age 18-3 Example 1 -- F or the data stor ed in the pr ev ious ex ample, the single -v ari able statis tics re sults ar e the f ollo wing: M e a n : 2. 1 3333333333 , S t d D e v: 0 . 96 42 0 79 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 Definitions The d efi ni tio n s used for thes e quantities ar [...]

P age 18-4 Example s of calculati on of these measur es, using lis ts, ar e available in C hapter 8. The medi an is the v alue that splits the data set in the mi ddle w hen the elements ar e placed in incr easing order . If you ha ve an odd number , n, of or der ed elements, the medi an of this sample is the v alue located in position (n+1)/2 . If [...]

P age 18-5 The ran ge of the sample is the differ ence between the max imum and minimum value s of the sample . Since the calculat or , through the pr e -progr ammed statisti cal functi ons pro vides the max imum and minimum v alues of the sample , y ou can easily calculate the r ange. Coefficient of variation The coe ffi cient o f var iation of a [...]

P age 18-6 Definitions T o unders tand the meaning of thes e paramet ers w e pre sent the f ollow ing def initions : Gi ven a set o f n data values: {x 1 , x 2 , … , x n } listed in no partic ular or der , it is ofte n req uired t o gr oup these data int o a ser ies o f clas ses by counting the f requenc y or number of values cor responding to ea[...]

P age 18-7 Θ Gener ate the list of 200 number b y using RDLIS T(200) in AL G mode , or 200 ` @RD LIST@ in RPN mode . Θ Use pr ogram LX C (see abo ve) to conv ert the list thus gener ated into a column vec tor . Θ Stor e the colu mn v ector into Σ DA T, b y us i n g f u n c t io n STO Σ . Θ Obtain single- var iable inf ormation u sing: ‚Ù @[...]

P age 18-8 to calculate f or uniform-si ze classes (or b ins) , and the class mark is j ust the av erage of the c lass boundari es for eac h class . F inally , the c u mulati ve fr equency is obtained by adding to eac h value in the last column , ex cept the fir st, the f requenc y in the next r o w , and replac ing the r esult in the last column o[...]

P age 18-9 « DUP S IZE 1 GET  f req k « {k 1} 0 CON  cfr eq « ‘fr eq(1,1)’ EV AL ‘ cfr eq(1,1)’ S T O 2 k FOR j ‘ cf r eq(j-1,1) +fr eq(j,1)’ EV AL ‘ cfr eq (j,1)’ ST O NEXT cfr eq » » » Sa ve it unde r the name CFREQ. Use this pr ogram t o gener ate the list of cu mulativ e fr equenc ies (pr ess @CFREQ wi th the column [...]

P age 18-10 Θ Press @CANCEL to r eturn t o the pre vi ous sc reen . Change the V-v iew and Bar Wi dth once mor e, n o w to r ead V - Vi ew: 0 3 0, Bar Width: 10. T he new histogr am, based on the same dat a set , now looks lik e this: A plot of fr equency count , f i , v s. c lass marks, xM i , is kno wn as a fr equency poly gon. A plot of the c u[...]

P age 18-11 Θ F irst, ente r 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 function S T O Σ . Θ T o access the pr ogr am 3. Fit data.. , use the f ollow ing k ey str okes: ‚Ù˜˜ @@@OK@@@ The input f orm w ill show the c urr ent Σ D A T , already loaded. If needed , change y our set up s c[...]

P age 18-12 Wher e s x , s y ar e the standar d dev iations of x and y , res pectiv ely , i.e . The va lu es s xy and r xy ar e the "Cov ariance" and "Cor relati on," r especti vel y , obtained by u sing the "F it data" featur e of the calc ulator . Lineari zed relationships Many c urvilinear r elatio nships "str [...]

P age 18-13 The ge neral f orm of the r egressi on equation is η = A + B ξ . Best data fitting The calc ulator can determine w hich one of its linear or lineari z ed relati onship offer s the best fitting f or a set of (x ,y) data points. W e w ill illustrate the u se of this featur e wit h an e xample . Suppose y ou wan t to f ind whi ch one of [...]

P age 18-14 X-Col, Y -Col: these opti ons apply only w hen you ha ve mor e than two columns in the matr ix Σ D A T . By def ault, the x co lumn is column 1, and the y column is co lumn 2 . _ Σ X _ Σ Y… : summary statisti cs that yo u can choo se as re sults of this pr ogram b y chec king the appropr iate f ield using [  CHK] when that f iel[...]

P age 18-15 B. I f n ⋅ p is an integer , say k , calculat e the mean of the k - th and (k -1) th or der ed observati ons. This algor ithm can be implemented in the fo llo w ing pr ogr am typed in RPN mode (See C hapter 21 for pr ogr amming informati on) : « S ORT DUP S IZE  p X n « n p *  k « IF k CEIL k FL OOR - NO T THEN X k GET X k 1 [...]

P age 18-16 The D A T A sub-menu The D A T A sub-menu contains fu nctions u sed to manipulate the s tatisti cs matr ix Σ DA TA: The oper ation of thes e func tions is as f ollo ws: Σ + : add ro w in lev el 1 to bottom of Σ DA T A m a trix. Σ - : r emov es last r ow in Σ D A T A matri x and places it in le vel of 1 of the st ack . The modifi ed[...]

P age 18-17 Σ P A R: sho ws statisti ca l par ameters. RE SET : r eset parameter s to default v alues INFO: sho ws s tatist ical par ameter s The MODL sub-menu within Σ PA R This sub-me nu cont ains func tio ns that let yo u change the data-fitting model t o LINFIT , L OGFIT , EXPFIT , PWRFIT o r BE STFI T by pr essing the appr opri ate button . [...]

P age 18-18 The f unctions inc luded ar e: B A RP L: pr oduces a bar plot with dat a in Xcol column of the Σ D ATA m a t r i x . HIS TP: produce s histogr am of the data in Xcol column in the Σ DA T A m a t rix, using the def ault width cor res ponding to 13 bins unle ss the bin si z e is modifi ed using functi on BINS in the 1V AR sub-menu (see [...]

P age 18-19 Σ X^2 : pr o vi des the sum of squar es of values in Xcol co lumn. Σ Y^2 : pro vi des the sum of squar es of values in Ycol column . Σ X*Y : pr ov ides the sum of x ⋅ y , i.e ., the pr od uc ts of data in columns Xcol and Ycol. N Σ : pr o vi des the number of col umns in the Σ DA T A m a trix. Ex ampl e of S T A T menu oper ation[...]

P age 18-20 @) STAT @ ) £PAR @ RESET resets s tatistical paramet ers L @ ) STAT @PLOT @ SCATR pr oduces scatt erpl ot @STATL dr aws data f it as a strai g ht line @CANCL r eturns to main displa y Θ Deter mine the fitting equati on and some of its statis tics: @) STAT @ ) FIT@ @ £LINE pr oduces '1.5+2*X' @@@LR@@@ produce s Intercept: 1.[...]

P age 18-21 Θ F it data using columns 1 (x) and 3 (y) using a logarithmi c fitting: L @ ) STAT @ ) £PAR 3 @YCOL selec t Ycol = 3, and @) MODL @LOGFI select Mod el = Logfit L @ ) STAT @PLOT @SCATR produce s catter gram o f y vs . x @STATL sho w line for log f itting Obv iousl y , the log-f it is not a good choi ce. @CANCL r eturns to normal dis pl[...]

P age 18-22 L @ ) STAT @PLOT @SCATR produce s catter gram o f y vs . 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 back use: J . Confidence inter vals Statis tical infer ence is the proces s of making conclusi ons about a population based on info rmation f rom sample dat a. In or [...]

P age 18-2 3 Θ P o int estimati on: when a single value of the par ameter θ is pro vided . Θ Conf idence interval: a n umerical interv al that contains the par ameter θ at a giv en leve l of pr obability . Θ Es timator : rule or method of estimati on of the par ameter θ . Θ Es timate: value that the estimator y ields in a partic ular applica[...]

P age 18-2 4 Θ The par ameter α is kno wn as the si gnificance le vel . T ypi cal values of α ar e 0.01, 0. 05, 0.1, corr esponding to conf idence lev els of 0.99 , 0.9 5, and 0.90, r espectiv ely . Confidence intervals for the population mean when the population var iance is know n Let ⎯ X be the mean of a random sample of siz e n, dra wn fr [...]

P age 18-2 5 Small samples and large samples The beha vi or of the Student’s t distr ibution is such that f or n>30, the distr ibution is indistinguishable fr om the standar d normal distributi on. Th us, for sample s larger than 3 0 elements when the population v ariance is unkno wn , y ou can use the same conf idence interval as when the pop[...]

P age 18-2 6 Es timators for the mean and s tandar d dev iation o f the diff er ence and sum of the statisti cs S 1 and S 2 ar e giv en by : In t hese expressions, ⎯ X 1 and ⎯ X 2 ar e the values of the statisti cs S 1 and S 2 fr o m samples tak en fr om the t w o populations, and σ S1 2 and σ S2 2 ar e the var iances of the populations o f t[...]

P age 18-2 7 In this case , the centered conf idence intervals fo r the sum and difference o f the mean value s 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 degr ees of fr eedom in the Student’s t distr ibution . In the last tw o options we spec ify that the population var iances, although un[...]

P age 18- 28 These options are to be interpr eted as follow s : 1. Z -INT : 1 μ .: Single sample confi dence in terval for the populati on mean, μ , w ith know n population var iance , or for lar ge samples with unkno wn populatio n var iance . 2. Z - I N T : μ1−μ2 .: Conf idence interval f or the differe n ce of the populati on means, μ 1 -[...]

P age 18-29 Press @HELP to obtain a scr een explaining the meaning of the conf idence interval in terms o f random number s generated b y a calculator . T o scr oll d o wn the r esulting sc r een use the do wn-arr ow k ey ˜ . Pr ess @@@OK@@@ when done w ith the help sc ree n. T his w ill retur n you to the sc reen sho wn abo ve . T o calculate the[...]

P age 18-30 Example 2 -- Data f r om two s amples (sample s 1 and 2) indicate that ⎯ x 1 = 5 7 .8 and ⎯ x 2 = 60. 0. The sample si z es ar e n 1 = 45 and n 2 = 7 5 . If it is kno wn that the populations ’ standar d dev iations ar e σ 1 = 3.2 , and σ 2 = 4. 5, determine the 9 0% confi dence interval f or the differ ence of the population mea[...]

P age 18-31 When done , pres s @@@OK@@@ . The r esults, as t ext and gr aph, are sho wn be lo w: Example 4 -- Determine a 90% conf idence interval for the diff er ence between two pr oportions if sample 1 sho ws 20 succe sses out of 120 tr ials, and sample 2 shows 15 s uccesses out of 1 00 trial s. Press ‚Ù— @@@OK@@@ to access the confidence i[...]

P age 18-3 2 Example 5 – Determine a 9 5% confi dence interval f or the mean of the population if a sample of 5 0 elements has a mean of 15 .5 and a standar d dev iatio n of 5 . The population ’s standar d dev iation is unkno wn . Press ‚Ù— @@@OK@@@ to access the confidence inte rval featur e in the calc ulator . Press —— @@@OK@@@ to s[...]

P age 18-3 3 The se re sults assume that the v alues s 1 and s 2 ar e the population s tandard dev iations . If these v alues actually r eprese nt the samples ’ standar d d e viati ons, y ou should enter the same v alues as befor e, but w ith the option _pooled selected . T he re sults no w become: Confidence intervals for the var iance T o dev e[...]

P age 18-34 The conf idence interv al for the populati on var iance σ 2 is t herefor 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 varia bl e, wit h ν = n -1 degr ees of f reedom , e x ceeds with pr obabiliti es α /2 and 1- α /2 , res pe[...]

P age 18-35 Hy pot hesis testing A h ypo thesis is a declar ation made about a population (f or instance , w ith r espect t o its mean) . Acceptance o f the h ypothesis is bas ed on a statis tical te st on a sample tak en fr om the population . The consequent acti on and decision- making ar e called h ypo thesis testing . The pr ocess of h ypothesi[...]

Pa g e 1 8 - 3 6 Err ors in h ypothesis testing In hy pothesis testing w e use the ter ms err ors of T ype I and T ype II to def ine the cases in w hich a true h ypothesis is r ejec ted or a false h ypothe sis is accepted (not rejected) , respect i vely . Let T = value of test statistic, R = r eject ion r egion, A = acceptance r egion , thus , R ?[...]

P age 18-3 7 The va lu e of β , i .e ., the pr obability of making an err or of T ype II, depends on α , the sample si z e n, and on the true v alue of the paramete r tested . Thus , the val ue of β is determined after the h ypothesis te sting is perfor med. It is c usto mar y to dr aw gr aphs show ing β , or the po wer of the te st (1- β ), a[...]

P age 18-38 The c riter ia to us e for h ypothesis t esting is: Θ Rej ec t H o if P - v alue < α Θ Do not r ej ect H o if P -value > α . The P -value fo r a two-sided test can be calculat ed using the pr obability func tio ns in the calc ulator as f ollo ws: Θ If using z , P -value = 2 ⋅ UTPN(0,1,|z o |) Θ If using t , P - value = 2 ?[...]

P age 18-3 9 Next , we us e the P -value assoc iated with eithe r z ο or t ο , and compare it to α to dec ide whether or no t to r ej ect the nul l hy pothesis. T he P -value f or a two-sided test is def ined as either P -value = P(z > |z o |), or , P -value = P(t > |t o |) . The c riter ia to us e for h ypothesis t esting is: Θ Rej ec t [...]

P age 18-40 val ue s ⎯ x 1 and ⎯ x 2 , and standar d dev iations s 1 and s 2 . If the populations standar d dev iati ons cor re sponding to the samples, σ 1 and σ 2 , ar e kno wn, or if n 1 > 30 and n 2 > 30 (large sa mples) , th e test stati stic to be used is If n 1 < 30 or n 2 < 3 0 (at least one small sample), use the fo llo w[...]

P age 18-41 The c riter ia to us e for h ypothesis t esting is: Θ Rej ec t H o if P - v alue < α Θ Do not r ej ect H o if P -value > α . P aired sample tests When w e deal with two s a mple s of si ze n w ith paired data po ints, inst ead of test ing the null h ypothesis , H o : μ 1 - μ 2 = δ , using the mean v alues and standar d dev [...]

P age 18-4 2 wher e Φ (z) is the c umulativ e distributi on func tion (CDF ) of the standar d normal distr ibution (see Cha pter 17). Re ject the null hy pothesis, H 0 , if z 0 >z α /2 , or if z 0 < - z α /2 . In other w ords , the r ejecti on regi on is R = { |z 0 | > z α /2 }, w hile the acceptance r egion is A = {|z 0 | < z α /2[...]

P age 18-43 T w o - tail ed test If using a two- tailed test w e will f ind the v alu e of z α /2 , f r 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 func tion (CDF ) of the standar d normal distr ibution . Re ject the null hy pothesis, H 0 , if z 0 >z α /2 , or[...]

P age 18-44 1. Z - T es t: 1 μ .: Single s ample h ypothesis tes ting for the populati on mean, μ , w ith kno wn populati on var iance , or for lar ge samples w ith unknow n populatio n var iance . 2. Z - Te s t : μ1−μ2 .: H ypothesis t esting f or the diffe rence of the populati on means, μ 1 - μ 2 , with e ither know n population var ianc[...]

P age 18-45 Then , we r ejec t 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 99 64, cor r esponding t o cr itical ⎯ x r ange of {14 7 .2 15 2 .8}. This inf ormati on can be observed gr aphically b y pres sing the so ft -men[...]

P age 18-46 W e re ject the null h ypothe sis, H 0 : μ 0 = 15 0, against the alternati ve h ypothesis , H 1 : μ > 15 0. The te st t value is t 0 = 5.6 5 68 5 4, w ith a P - v a lue = 0. 0000003 9 35 2 5. T he c ri tical v alue of t is t α = 1.6 7 65 51, corr esponding to a crit ica l ⎯ x = 15 2 . 3 71. Press @GRAPH to see the results gr aph[...]

P age 18-4 7 Th us, w e accept (mor e accurat el y , we do not r ejec t) the h ypothe sis: H 0 : μ 1 −μ 2 = 0 , or H 0 : μ 1 =μ 2 , against the alternati ve h ypothesis H 1 : μ 1 −μ 2 < 0 , or H 1 : μ 1 =μ 2 . The test t value is t 0 = -1.3 4177 6 , with a P -value = 0. 09130 9 61, and cr itical t is –t α = -1.6 5 9 7 8 2 . The gr[...]

P age 18-48 The t est c r iter ia are the s ame as in h ypothesis te sting of means, name ly , Θ Rej ec t H o if P - v alue < α Θ Do not r ej ect H o if P -value > α . P lease notice that this pr ocedure is v alid only if the populati on fr om whic h the sample wa s tak en is a Nor mal population . Example 1 -- Consi der the case in w hic[...]

P age 18-4 9 The f ollow ing table sho ws ho w to select the n umer ator and denominator f or F o depending on the alternati ve h ypothe sis cho sen: _______________ ____________________ _____________________ ____________ Alternat i ve T est De gr ees h ypothesis statisti c of fr eedom _______________ ____________________ _____________________ ____[...]

P age 18-50 Ther efor e, the F test statistics is F o = s M 2 /s m 2 =0.3 6/ 0.2 5=1. 44 The P -v alue is P -value = P(F>F o ) = P(F>1.44) = UTPF( ν N , ν D ,F o ) = UTPF(20, 3 0,1.44) = 0.17 88… Since 0.17 88… > 0. 05, i .e., P -value > α , theref ore , we cannot r ej ect the null h ypothesis that H o : σ 1 2 = σ 2 2 . Additio[...]

P age 18-51 W e get the, s o -called, nor mal equations: This is a s ys tem o f linear equati ons w ith a and b as the unkno wns , whi ch can be sol ved u sing the linear equation featur es of the calculator . Ther e is, ho we ver , no need to bother w ith these calculati ons because you can use the 3. Fit Data … option in the ‚Ù men u as pr e[...]

Pa g e 1 8 - 52 F rom w hic h it fo llow s that the standar d dev iations o f x and y , and the cov ariance of x ,y are giv en, r especti ve ly , by , , and Also , the sample corr elation coeff ici ent is In ter ms of ⎯ x, ⎯ y, S xx , S yy , and S xy , the s oluti on to the nor m al equati ons is: , Prediction error The r egr essi on curve o f [...]

Pa g e 1 8 - 5 3 Θ Conf idence limits for r egres sion coeff ic 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 . Procedur e for inference statistics f or linear regression using the calculator 1) Enter (x,y) as columns of data in the statis tical matr ix Σ D AT. 2) Produce a s catterplot for the a ppr opri ate columns of Σ D A T , and us e appr opri ate H- and V -VIEW[...]

Pa g e 1 8 - 5 5 1: Covariance: 2.025 The se r esults are int erpr eted as a = -0.8 6 , b = 3 .2 4 , r xy = 0.9 89 7 20 22 9 7 4 9 , and s xy = 2 . 02 5 . The corr elation coeff ic ient is clo se enough to 1.0 t o conf irm the linear tr end obse rved in the gr aph . Fro m t h e Single-var… option of the ‚Ù men u we f ind: ⎯ x = 3, s x = 0.7 [...]

P age 18-5 6 Example 2 -- Su ppose that the y-data used in Ex ample 1 repr esent the elongation (in h undr edths of an inc h) of a me tal w ire w hen sub jec ted to a f orce x (in tens of pounds) . T he phy sical phenomenon is suc h that we e xpect the inter cept, A, to be z er o. T o check if that should be the case , we test the n ull h ypothes i[...]

P age 18-5 7 Multiple lin ear fitting Consi der a data set of the for m Suppos e that w e searc h for a data f itting of the fo rm 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 c oeffic ients b = [b 0 b 1 b 2 b 3 … b n ], by putting together the matri[...]

P age 18-5 8 With the calc ulator , in RPN mode , yo u can pr oceed as fo llo ws: F irst , w ithin your HO ME direc tory , c r eate a sub-dir ectory to be called MPFIT (Multiple linear and P o ly nomial data FI Tting) , and enter the MPFI T sub- dir ectory . Within the sub-dir ectory , t ype this pr ogram: «  X y « X TRAN X * INV X TRAN * y * [...]

P age 18-5 9 Compar e these f itted values w ith the ori ginal data as sho wn in the ta ble belo w: P ol ynomial fitting Consider the x -y data set {(x 1 ,y 1 ), ( x 2 ,y 2 ), …, (x n ,y n )}. Suppose that w e w ant to fit a po ly nomial or order p to this data s et . In other wor ds, w e seek a f it ting of the f orm y = b 0 + b 1 ⋅ x + b 2 ?[...]

P age 18-60 If p > n-1 , then add columns n+1, …, p-1, p+1 , to V n to f orm matr ix X . In step 3 f r om this lis t , w e hav e to be aw are that column i ( i = n+1, n+2 , …, p+1 ) is the vec tor [x 1 i x 2 i … x n i ]. If we w er e to us e a list of data value s for x rathe r than a vec tor , i.e ., x = { x 1 x 2 … x n }, we can easil [...]

P age 18-61 « Open pr ogram  x y p Enter lis ts x and y , and p (lev els 3,2 ,1) « Open subpr ogram 1 x SI ZE  n Determine si z e of x list « Open subpr ogram 2 x V ANDERMONDE Place x in stac k, obtain V n IF ‘ p<n -1’ THEN This IF implements step 3 in algor ithm n P lace n in stac k p 2 + Calculate p+1 FOR j Start loop j = n -1, n-[...]

P age 18-6 2 Becaus e w e w ill be using the same x -y data for f itting poly nomials of diff er ent or ders , it is adv isable to sav e the lists of data v alues x and y into var iables xx and yy , r especti vel y . This w ay , we w ill not hav e to type the m all ov er again in each a pplicati on of the pr ogram P OL Y . Thu s, proceed as f ollow[...]

P age 18-63 Θ The cor relation coe ffi cient , r . T h is value is constr ained to the r ange –1 < r < 1. Th e cl ose r r is to +1 or –1, the better the data fitting . Θ The sum o f squar ed erro rs, S SE . This is the quantity that is to be minimi zed b y least-squar e approac h. Θ A plot of r esiduals . This is a plot of the err or c[...]

P age 18-64 x V ANDERMONDE P lace x in stac k, obtain V n IF ‘ p<n -1’ THEN This IF is step 3 in algo rithm n P lace n in stac k p 2 + Calc ulate p+1 FOR j Start loop, j = n-1 to p+1, step = -1 j COL − D R OP Remo ve column , drop f rom stac k -1 S TEP Close F OR-S TEP loop ELSE IF ‘ p>n -1’ THEN n 1 + C alculate n+1 p 1 + Calc ulat[...]

P age 18-65 “SSE”  T A G T ag r esult as S SE » Close sub-pr ogram 4 » Clos e sub-progr am 3 » Clo se sub-pr ogram 2 » Clo se sub-pr ogram 1 » Clos e main progr am Sa ve this pr ogram unde r the name PO L YR, to em phasi z e calc ulation of the correlation c oeffic ient r . Using the POL YR pr ogram f or values of p betw een 2 and 6 pro[...]

P age 19-1 Chapter 19 Numbers in Different Bases In this Chapter w e pre sent e xamples o f calculati ons of number in base s other than the dec ima l basis . Definitions Th 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) di gits, namely 0-9 , to wr ite out any[...]

P age 19-2 With s yst em flag 117 set to S OFT menus , the B ASE men u show s the follo wing: With this f ormat , it is ev ident that the L OGIC, BIT , and B Y TE entr ies w ithin the B ASE menu a r e themselves sub-menus. These me nus are discussed later in this Chapter . Functions HEX, DEC, OCT , and B IN Numbers in non-decimal sy stems ar e wr i[...]

P age 19-3 As the deci mal (DEC) s ystem has 10 di gits (0,1,2 , 3,4 ,5, 6, 7 , 8, 9) , the hex adecimal (HEX) s yst em 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 binary (BIN) sy stem has only 2 digits (0,1). Conv ersion bet ween number s ystems Whatev er the n[...]

P age 19-4 The onl y effect o f selecting the DEC imal sy stem is that dec imal number s, w hen started w ith the sy mbol #, are w ritten with the suff ix d . W ordsi ze The w ordsi z e is the number of bits in a binary obj ect . By defa ult , the wor dsiz e is 64 bites . F unction RCW S (ReC all W ordSi z e) shows the c urr ent wor d si z e. F unc[...]

P age 19-5 The L OGIC m enu The L OGIC menu , av ailable through the B ASE ( ‚ã ) pr ov ides the f ollo wing fun ctio ns : The f unctions AND , OR, X OR (ex clusi ve OR), and NO T ar e logical f uncti ons. The in put to these f unctions ar e two v alues or e xpre ssi ons (one in the case of NO T) that can be e xpresse d as b inary logical result[...]

P age 19-6 AND (BIN) OR (BIN) XO R (BIN) NO T (HEX) The BI T menu The BI T menu , available thr ough the BA SE ( ‚ã ) pro vide s the follo wing fun ctio ns : F unctions RL, SL , ASR , SR, RR , contained in the BI T menu , are u sed to manipulate bits in a b inar y integer . The de fi nition of these f unctions ar e show n belo w: RL: R otate Lef[...]

P age 19-7 The B Y TE menu The B YTE menu , a vaila ble thr ough the B A SE ( ‚ã ) pr ov ides the f ollo wing fun ctio ns : F unctions RLB, SLB , SRB, RRB, co ntained in the BIT menu , ar e used to manipulate bits in a b inar y integer . The de fi nition of these f unctions ar e show n belo w: RLB: Rotate L ef t one byte , e.g ., # 1100b  #11[...]

Pa g e 2 0 - 1 Chapter 20 Customi zing menus and k ey board Thr ough the use of the man y calculator menu s yo u hav e become familiar w ith the oper ation of men us f or a var iety of a pplicatio ns. Also , you are f amiliar w ith the man y functi ons availa ble by u sing the ke ys in the k ey board , whether thr ough their main f unction , or by [...]

Pa g e 2 0 - 2 Menu numbers (RCLMENU and MENU func tions) E ach pr e -defined men u has a number attached to it . F or e xample , suppose that y ou acti vate the MTH menu ( „´ ). Then , using the functi on catalog ( ‚N ) find f u nc tion R CLMENU and acti vate it . In AL G mode simple pres s ` after RCLMENU() sho ws u p in the sc reen . The r [...]

Pa g e 2 0 - 3 T o acti vate an y of those f unctions y ou simply need to enter the function argume nt (a number ) , and then pr ess the corr esponding soft menu k ey . In AL G mode , the list to be en ter ed as argument of func tion TMENU or MENU is mor e complicated: {{“ e xp” , ”EXP( “},{“ln ” , ”LN(“},{“Gamma” , ”G AMMA( ?[...]

Pa g e 2 0 - 4 Y ou can tr y using this list w ith TMENU or MENU in RPN mode to ver if y that y ou get the same menu as obt ained earli er in AL G mode. Menu spec ification and CST v ariable F rom the tw o ex erc ises sho wn abo ve w e notice that the most gener al menu spec ificati on list include a n umber of sub-lists equal to the number of item[...]

Pa g e 2 0 - 5 Customizing the k e yboard E ach k ey in the k ey board can be iden tifi ed by two n umbers r e pr esenting their r o w and column. F or ex ample , the V AR k ey ( J ) is located in r ow 3 of column 1, and w ill be r eferr ed to as k ey 31. Now , since eac h ke y has up to t en functi ons assoc iated w i th it , each f uncti on is sp[...]

Pa g e 2 0 - 6 The f unctions av ailable are: AS N: Assigns an objec t to a ke y specif ied by XY .Z S T OK E Y S: Stores user -defined key list RCL KEYS: Ret ur ns cu rrent us er-d efin ed ke y l ist DELKEY S: Un -assi gns one or mor e ke ys in the c urr ent user -d ef ined ke y list, the ar guments are e ither 0, to un-assign all use r -def ined [...]

Pa g e 2 0 - 7 Operating user-defined ke ys T o operate this us er -defined k ey , enter „Ì befo re pre ssing the C key . Notice that after pr essing „Ì the sc reen sho ws the spec ificati on 1USR in the second displa y line. Pr essing f or „Ì C for this e xample , y ou should r ecove r the PL O T menu as foll o ws: If y ou hav e more than[...]

Pa g e 2 0 - 8 T o un -assign all user-def ined k eys use: AL G mode: DELKEYS (0) RPN mode: 0 DELKEYS Chec k that the user -k e y def initions w ere r emov ed by using f u ncti on RCLKEY S.[...]

P age 21-1 Chapter 21 Pr ogramming in User RP L language Use r RPL language is the pr ogramming language mo st commonl y used to pr ogram the calc ulator . The progr am components can be put together in the line editor by inc luding them bet w een progr a m containers « » in the appr opriat e orde r . Becau se ther e is mor e exper ience among ca[...]

P age 21-2 „´ @LIST @ADD@ ADD Calc ulate (1+x 2 ), / / then di v ide ['] ~„x™ 'x' „° @) @MEM@@ @ ) @DIR@@ @ PURGE PURGE Purg e variab l e x ` Pr ogram in le vel 1 _______________ ________ __________ _________________ ____ T o sa ve the pr ogram u se: ['] ~„gK Press J to reco ver y our var iable menu , and ev aluate g[...]

P age 21-3 use a local v ari able within the pr ogram that is only de fined f or that progr am and w ill not be availa ble fo r use after pr ogram e xec ution. T he pre vi ous pr ogram could be modif ied to r ead: « → x « x SINH 1 x SQ ADD / »» The ar ro w sy mbol ( → ) is obtained b y combining the r ight-shift k ey ‚ w ith the 0 key , i[...]

P age 21-4 Global V ariable Scope An y vari able that you def ine in the HOME direc tory or any o ther dir ectory or sub-dir ectory will be consider ed a global var iable fr om the point of v iew o f pr ogram de velopment . How ev er , the scope of such v ariable , i .e ., the location in the dir ectory tr ee wher e the var iable is accessible , w [...]

P age 21-5 Local V ariable Scope Local v ariable s are ac tiv e only w ithin a progr am or sub-pr ogr am. The ref ore , their s cope is limited to t he pr ogram or sub-pr ogram w her e the y’r e defined . An e xam ple of a local var iable is the inde x in a FOR loop (desc ribed later in this chapter ) , for e xample « → n x « 1 n FOR j x NEXT[...]

P age 21-6 S T ART : S T ART -NEXT- STEP cons truc t for br anching FOR: FOR-NE XT- STEP constr uct for loops DO: DO-UNTIL -END constr uct f or loops WHILE: WHILE -REPEA T-END c onstru ct for loop s TE S T : Compar ison operator s, logical oper ators, f lag testing f unctions TYPE: F unctions f or conv er ting obje ct types , splitting ob jects , e[...]

P age 21-7 Functions listed b y sub-menu The f ollow ing is a listing of the func tions w ithin the PRG sub-me nus list ed by sub- menu . ST A CK MEM/DIR BRCH/IF BRCH/WHILE TYP E DUP PUR GE IF WHILE OBJ  SW A P R C L T H E N R E PE AT  ARR Y DRO P S T O ELSE END  LIS T O V ER P A TH END  ST R RO T CRDIR TEST  TAG UNRO T P GD IR BRCH/[...]

P age 21-8 LIST/ELEM GROB CHARS MODES/FLAG MODES/MISC GET  GR OB SUB SF BEEP GET I BLANK REPL CF CLK PUT G O R POS FS? S Y M PUTI G XOR SIZ E F C? S T K SI ZE SUB NUM F S?C ARG PO S REPL CHR F S?C CMD HEAD  LC D O BJ  FC?C INFO TA I L LC D  ST R STO F SIZE H EA D RCLF IN LIS T/PR OC ANIMA TE T AIL RE SET INFORM DOLIS T SREPL NOV AL D[...]

P age 21-9 Shortc uts in the PR G menu Many o f the functi ons listed abo ve f or the PRG menu ar e readil y av ailable thr ough other means: Θ Compar ison operators ( ≠ , ≤ , <, ≥ , >) ar e av ailable in the k eyboar d. Θ Many f unctions and s ettings in the MODE S sub-menu can be acti vated by u s ing the input f unctions pr ov ided[...]

P age 21-10 „ @ ) @IF@@ „ @CASE@ „ @ ) @IF@@ „ @CASE@ „ @ ) START „ @) @FOR @ „ @ ) START „ @) @FOR @ „ @ ) @@DO@@ „ @WHIL E Notice that the inse rt prompt (  ) is a vailable after the k ey w or d for eac h constr uct so y ou can start typing at the r ight location. K e ystr oke sequence f or commonly used commands The f ollo[...]

P age 21-11 @) STACK DUP „° @) STACK @@ DUP@@ SW A P „° @) STACK @SWAP@ DRO P „° @) STACK @DROP@ @) @MEM@@ @ ) @DIR@@ PUR GE „° @) @ME M@@ @ ) @DIR@ @ @PURGE ORDER „° @) @MEM@@ @ ) @DIR@@ @ ORDER @) @BRCH@ @ )@IF@@ IF „° @) @BRCH@ @ ) @I F@@ @@@IF@@@ THEN „° @) @BRCH@ @ ) @IF@@ @THEN@ ELSE „° @) @BRCH@ @ ) @IF@ @ @ELSE@ END [...]

P age 21-12 @) @BRCH@ @ ) WHILE@ WHILE „° @) @BRCH@ @ ) WHILE@ @WHILE REPE A T „° ) @BRCH@ @ ) WHILE@ @REPEA END „° ) @BRCH@ @ ) WHILE@ @@ END@ @ ) TEST@ == „° @ ) TEST@ @@ @ ≠ @@@ AND „° @ ) TEST@ L @@AND@ OR „° @ ) TEST@ L @@@ OR@@ XOR „° @ ) TEST@ L @@XOR@ NO T „° @ ) TEST@ L @@NOT@ SA M E „° @ ) TEST@ L @SAME SF „[...]

P age 21-13 @) LIST@ @ ) PRO C@ REVLI S T „° @) LIST@ @ ) PROC@ @REVLI@ SO RT „° @) LIST@ @ ) PROC@ L @SORT@ SEQ „° @) LI ST@ @ ) PROC@ L @@SEQ@@ @) MODES @ ) ANGLE@ DE G „°L @) MODES @ ) ANGLE @ @@DEG@ @ RAD „°L @) MODES @ ) ANGLE@ @@ RAD@@ @) MODES @ ) MENU@ CST „°L @) MO DES @ ) MENU@ @@CST@@ MENU „°L @) MODES @ ) MENU@ @@ME[...]

P age 21-14 fun ction s from the MT H m enu. Specific ally , you ca n use fun ctio ns for li st oper ations such as S ORT , Σ LI S T , etc. , av ailable thr ough the MTH/LIS T menu . As additional pr ogramming e xer cise s, and to try the ke ystr ok e sequences lis ted abo ve , we pr esent her ein thr e e pr og rams f o r c r eating or manipulatin[...]

P age 21-15 Ex amples of sequential progr amming In gener al, a pr ogram is an y sequence o f calculato r instructi ons enclosed between the pr ogram container s and ». Subprogr ams can be inc luded as part of a progr am. The e xamples pr esented pr ev iousl y in this guide (e.g ., in Chapte rs 3 and 8) 6 can be cla ssif ied basi cally into tw o t[...]

P age 21-16 wher e C u is a constant that depends on the sy stem of units used [C u = 1.0 f or units of the International S ys tem (S.I .) , and C u = 1.48 6 for units o f the English S yste m (E .S .)], n is the Ma nning ’s re sistance coeff ic ient, w hich depends on the type of c hannel lining and other f actor s, y 0 is the flo w depth, and S[...]

P age 21-17 Y ou can also separate the in p ut data w ith spaces in a single stac k line rathe r than using ` . Progr ams that simulate a sequence of stac k operations In this case , the terms to be inv olv ed in the sequence of oper ations are assumed to be pr esent in the stac k. The pr ogram is ty ped in by f irst opening the pr ogram cont ainer[...]

P age 21-18 As yo u can see , y is used fir st, then w e use b , g, and Q, in that or der . Ther efor e, f or the purpose of this calc ulation we need to enter the v ariables in the inv erse or der , i .e., (do not ty pe the fo llo w ing): Q ` g ` b ` y ` F or the spec ifi c values under consider ation w e use: 23 ` 32. 2 ` 3 ` 2 ` The pr ogram its[...]

P age 21-19 Sav e the progr am into a v ari able called hv: ³~„h~„v K A ne w var iable @@@hv @@@ should be a vailable in y our soft k ey men u . (Pre ss J to see y our var iable list .) The pr ogram le f t in the s tack can be e valuated b y using functi on EV AL. T he re sult should be 0.2 2817 4…, as befor e. Also , the progr am is av aila[...]

P age 21-20 it is alw ay s pos s ible to r ecall the pr ogr am def inition into the s tack ( ‚ @@@q@@@ ) to see the or der in whic h the v ariabl es mus t be enter ed, namel y , → Cu n y0 S0 . Ho we ver , for the case o f the pr ogram @@hv@@ , its de finiti on « * SQ * 2 * S W AP SQ S W AP / » does not pr ov ide a clue of the or der in whi ch[...]

P age 21-21 whi ch indicates the positi on of the differ ent stack in put lev els in the form ula. B y compar ing this r esult with the or iginal f ormula that w e progr ammed, i .e., w e find that w e must enter y in st ack lev el 1 (S1) , b in stac k lev el 2 (S2), g in stac k leve l 3 (S3), and Q in stack le ve l 4 (S4) . Prompt w ith an input s[...]

P age 21-22 The r esult is a stac k prom pting the user for the value o f a and placing the c ursor ri ght in fr ont of the pr ompt :a: Enter a value fo r a, sa y 35, then pr ess ` . The r esult is the input str ing :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 of code to calc ulate the func tion , f(a)[...]

P age 21-2 3 @SST ↓ @ R esult: empty stac k, e xec uting → a @SST ↓ @ R esult: empty stack , entering subpr ogram « @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’ , leavi ng main pr ogram» F ur ther pr essing the @SST ↓ @ s oft menu k ey pr oduces no more ou [...]

P age 21-2 4 Fi xing the program The onl y possible explanati on fo r the failur e of the pr ogram to pr oduce a numer ical re sult seems to be the lac k of the command  NUM after the algebrai c expr ession ‘2*a^2+3’ . Let ’s edit the progr am by adding the missing EV AL f u ncti on. T he pr ogram , after editing, should r ead as follo ws:[...]

P age 21-2 5 Input string pr ogram for two input v alues The in put str ing pr ogram for tw o input values, sa y a and b, looks as f ollo ws: « “ Enter a and b: “ { “  :a:  :b: “ {2 0} V } INPUT OBJ → » This pr ogram can be easil y cr eated by modify ing the contents of INP T a. Sto r e this pr ogr am into v ari able INPT2 . Appli[...]

P age 21-2 6 ` . The r esult is 4 98 8 7 . 06_J /m^3 . The units of J/m^3 are equi valent to P ascals (P a) , the pref err ed pressur e unit in the S.I . s ys tem . In put stri ng progra m for thre e i npu t valu es The in put str ing progr am for thr ee input v alues, sa y a ,b, and c , looks as fo llow s: « “ Enter a, b and c: “ { “  :a[...]

P age 21-2 7 Enter v alues of V = 0. 01_m^3, T = 300_K , and n = 0.8_mol. Be fo r e pre ssing ` , the stac k will look lik e this: Press ` to get the result 19 9 54 8.2 4_J/m^3, or 199 54 8.2 4_P a = 199 . 5 5 kP a . Input through input f orms F unction INFORM ( „°L @) @@IN@ @ @INFOR@ .) c an be used t o cr eate detailed input fo rms for a pr og[...]

Pa g e 2 1 - 2 8 The lis ts in items 4 and 5 can be em pty lists. A lso , if no value is t o be selected f or these options y ou can use the NO V AL command ( „°L @) @@IN@@ @NOVAL@ ). After f unction INF ORM is a cti vated y ou will get as a r esult either a z er o, in case the @CANCEL opti on is enter ed , or a list w ith the values ente r ed i[...]

P age 21-29 3 . F ield f ormat info rmation: { } (an empty list , thu s, def ault values us ed) 4. List of reset values: { 120 1 .0001} 5 . L ist of initial values: { 110 1.5 .00001} Save th e prog ram in to vari ab le IN FP 1 . P ress @INFP1 t o run the pr ogram . The input fo rm, w ith initial values loaded , is as follo ws: T o see the eff ect o[...]

P age 21-30 Th us , we demons tr ated the us e of f unction INF ORM. T o see how t o use thes e input v alues in a calculati on modif y the pr ogram as fo llo ws: « “ CHEZY’S EQN” { { “C:” “Che zy’s coefficient” 0} { “R:” “Hydraulic radius” 0 } { “S:” “Channel bed slope” 0} } { } { 120 1 .0001} { 110 1.5 .00001 } I [...]

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 Acti vati on of the CHOO SE function w ill r eturn e ither a ze r o , if a @CANCEL ac tion is used , or , if a ch oice is made , the cho ice selected (e .g., v) and the numbe r 1, i . e ., in the RPN stack: Example 1 – Manning’s equati on f or calc ulating the veloc it y in an open cha n nel flow in clu de s a co ef ficient, C u , [...]

P age 21-3 3 commands “Operation cancelled” MSGBOX w ill sho w a message bo x indicating that the oper ation wa s cancelled. Identif y ing output in progr ams The simple st wa y to identify numer ical progr am output is to “tag” the pr ogram r esults . A tag is simply a str ing at tac hed to a number , or to an y object . The str i ng w ill[...]

P age 21-34 Ex ampl es of tagged output Example 1 – tagging output fr om function FUNC a Let ’s modify the function FUNCa , defined ear lier , to pr oduce a tagged output . Use ‚ @FUNCa to recall the cont ents of FUNCa to the st ack . The or iginal functi on progr am reads « “ Enter a: “ { “  :a: “ {2 0} V } INPUT OBJ →→ a « [...]

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 »» (Recall that the f uncti on S W AP is availa ble by u sing „° @) STACK @SWAP@ ). Stor e the progr am back into FUNCa b y using „ @FUNCa . Next , run the pr ogram b y pres sing @FUNCa . Enter a v alue of 2 wh[...]

Pa g e 2 1 - 3 6 Example 3 – tagging input and output f rom func tion p(V ,T) In this ex ample we modify the pr ogram @@@p@@@ so that the o utput tagged inpu t value s and tagged r esult . Use ‚ @@@p@@@ to r ecall the contents of the pr ogram to the stac k: « “ Enter V, T, and n: “ { “  :V :  :T:  :n: “ {2 0} V } INPUT OBJ →[...]

P age 21-3 7 Stor e the progr am back into var iable p by using „ @@@p@@@ . Ne xt, r u n the pr ogram b y pres sing @@@p@@@ . Ent er v alues of V = 0. 01_m^3, T = 3 00_K, and n = 0.8_mol, w hen prom pted . Befor e pre ssing ` for input , the stack w ill look like this: After e xec uti on of the pr ogram , the stac k w ill look like this: Using a [...]

P age 21-38 The r esult is the f ollo wing message bo x: Press @@@OK@@@ to cancel the mes sage bo x. Y ou could use a mes sage bo x for o utput fr om a progr am by us ing a tagged output , conv erted to a str ing, as the output st ring f or MS GBOX . T o con ve rt any tagged r esult , or any algebr aic or non- tagged v alue , to a string , use the [...]

P age 21-3 9 Press @@@OK@@@ to can cel message box output. T he stack will now look lik e this: Including input and output in a m essage bo x W e could modify the p r ogram so that not onl y the output , but also the input , is included in a mes sage bo x. F or the case of pr ogram @@@p@@@ , the modifi ed pr ogram w ill look lik e: « “ Enter V, [...]

P age 21-40 Y ou will noti ce that after typing the k ey strok e sequence ‚ë a ne w line is gener ated in the stack . The las t modificati on that needs to be included is to type in the plus si gn three times after the call to the f unction at the v ery end of the sub-pr ogram . T o see the pr ogr am oper ating: Θ St or e the progr am back int [...]

P age 21-41 Incorporating units w ithin a program As yo u have bee n able to obse rve fr om all the ex amples fo r the diffe r ent vers ion s of prog ram @@@p@@@ pr esented in this cha pter , attaching units to inpu t value s may be a tedi ous proce ss. Y ou could hav e the progr am itself attach those units to the in put and output v alues. W e w [...]

P age 21-4 2 2. ‘ 1_m^3 ’ : T he S.I. units cor respo nding to V are then placed in stac k lev el 1, the tagged input f or V is mo ved to stack lev el 2 . 3 . * : By multiply ing the contents of s tack le vels 1 and 2 , we gen er a te a nu mber wi th units (e .g ., 0. 01_m^3), but the tag is lo st . 4. T ‘ 1_K ’ * : Calc ulating value of T [...]

P age 21-43 Press @@@OK@@@ to cancel mes sage box ou tput . Messag e bo x output without units Let ’s modify the progr am @@@p@@@ once mor e to eliminate the u se of units thr oughout it . T he unit-less progr am will look lik e this: « “ Enter V,T,n [S.I.]: “ { “  :V:  :T:  :n: “ {2 0} V } INPUT OBJ →→ V T n « V DTAG T DTA[...]

P age 21-44 oper ators ar e used to mak e a statement r egarding the r elativ e position of tw o or mor e real number s. Depending on the actual numbers used , such a st atement can be true (r epres ented b y the numer ical value o f 1. in the calc ulator), or false (r epr esented b y the numeri cal value of 0. in the calc u lator ) . The r elation[...]

P age 21-45 Logical oper ators Logi cal oper ators ar e logical partic les that are u sed to jo in or modify simple logical stat ements. T he logical operat ors a vailable in the calc ulator can be easily acc essed thr ough the ke ystr ok e sequence: „° @ ) TEST@ L . The a vailable logi cal oper ator s ar e: AND , OR , XOR (e xc lusiv e or), NO [...]

Pa g e 2 1 - 4 6 The calc ulator include s also the logi cal oper ator S AME . This is a non-standar d logical oper ator used t o determi ne if two ob jects ar e identical . If they ar e identi cal, a v alue of 1 (true) is r eturned, if not , a value of 0 (f alse) is r eturned. F or ex ample, the f ollow ing ex erc ise , in RPN mode , r eturns a v [...]

P age 21-4 7 Branching with IF In this secti on w e pre sents ex amples using the constr ucts IF…THEN…END and IF…THEN…ELSE…END . The IF…THEN…END construct The IF…THEN…END is the simple st of the IF pr ogram constr ucts . T he general for mat of this construc t is: IF logical_statement THEN program_statements END . The ope rati on [...]

P age 21-48 With the c ursor  in fr ont of the IF stat ement pr ompting the user f or the logical statement that w ill acti vate the IF constr uct w h en the pr ogram is e xecu ted. Example : T ype in the follo w ing progr am: « → x « IF ‘ x<3 ’ THEN ‘ x^2 ‘ EVAL END ” Done ” MSGBOX » » and sa ve it under the name ‘f1 ’ [...]

P age 21-4 9 Example : T ype in the follo wing pr ogram: « → x « IF ‘ x<3 ’ THEN ‘ x^2 ‘ ELSE ‘ 1-x ’ END EVAL ” Done ” MSGBOX » » and sa ve it under the name ‘f2 ’ . Pres s J and ver ify that vari able @@@f2@@@ is indeed av ailable in your v aria ble menu . V erify the f ollow ing results: 0 @@@f2@@@ Result: 0 1.2 @@@f[...]

P age 21-50 IF x<3 THEN x 2 ELSE 1-x END While this simple constr uct w orks f ine when y our functi on has only tw o branc hes, y ou may need to nes t IF…THEN…ELSE…END constru cts to deal with func tion w ith three or mor e branc hes . F or ex ample, co nsider the functi on Her e is a possible w ay t o ev aluate this functi on using IF…[...]

P age 21-51 A comple x IF construct lik e this is called a set of neste d IF … THEN … ELSE … END constr ucts . A possible w ay to e valuate f3(x), based on the nested IF constr uct show n abov e, is to wr ite the pr og r am: « → x « IF ‘ x<3 ‘ THEN ‘ x^2 ‘ ELSE IF ‘ x<5 ‘ THEN ‘ 1-x ‘ ELSE IF ‘ x<3* π ‘ THEN [...]

Pa g e 2 1 - 52 pr ogram_s tatements , and pa sses pr ogr am flo w to the statement follo wing the END statement . The CA SE , THEN, and END stat ements ar e available f or selecti ve typ ing by using „° @) @BRCH@ @ ) CASE@ . If y ou are in the BR CH menu, i .e., ( „° @) @ BRCH@ ) y ou can use the f ollo wing shortcuts t o type in yo ur CASE [...]

Pa g e 2 1 - 5 3 5. 6 @@f3c@ Resu l t : -0.6312 66… (i .e., sin(x), with 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 pr oduces ex actly the same r esults as f3. The onl y diffe rence in the pr ogr ams is the branc hing constructs u sed . For the cas e of fun cti[...]

P age 21-54 Commands in volv ed in the ST ART constr uct are a vailable thr ough: „° @) @BRCH@ @ ) START @START Within the BRCH men u ( „° @) @BRCH@ ) the follo wing k ey str ok es are a vaila ble to gener ate S T AR T construc ts (the s ymbol indi ca tes c ursor positi on) : Θ „ @START : S tarts the S T AR T…NEXT construc t: S T AR T ?[...]

Pa g e 2 1 - 5 5 1. This pr ogram needs an in teger numbe r as inpu t . Th us , bef or e e xec ution , that number (n) is in stac k lev el 1. T he progr am is then e xec uted . 2 . A z er o is enter ed , mov ing n to stac k lev el 2 . 3 . T he command DU P , whi ch can be typed in as ~~dup~ , cop ies the contents of st ack le ve l 1, mov es all the[...]

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 ( « - start subpr ogram) @SST ↓ @ SL1 = 0., (st art value of loop inde x) @SST ↓ @ SL1 = 2 .(n), SL2 = 0. (end value of loop inde x) @S[...]

P age 21-5 7 @SST ↓ @ SL1 = 1. (S + k 2 ) [S tore s value of SL2 = 2 , into SL1 = ‘k ’] @SST ↓ @ SL1 = ‘S’ , SL2 = 1. (S + k 2 ) @SST ↓ @ Empty stac k [Stor es value of SL2 = 1, into SL1 = ‘S’] @SST ↓ @ Empty stac k (NEXT – end of loop) --- loop e xec ution nu mber 3 f or k = 2 @SST ↓ @ SL1 = 2 . (k) @SST ↓ @ SL1 = 4. (S Q[...]

P age 21-5 8 3 @@@S1@@ Res u lt : S:14 4 @@@S1@@ Res ul t: S:30 5 @@@S1@@ Res u lt : S:55 8 @@@S1@@ Res ul t: S:204 10 @@@S1@@ Res ul t: S:385 20 @@@S1@@ Resu l t: S:2870 30 @@@S1 @@ Resu lt : S:9455 100 @@@S1@@ Res u l t: S:338350 The ST AR T…STEP construct The ge neral f orm of this statemen t is: start_value end_value START program_statements [...]

P age 21-5 9 J 1 # 1.5 # 0. 5 ` Enter p ar ameters 1 1. 5 0.5 [ ‘ ] @GLIST ` Ent er the pr ogram name in lev el 1 „°LL @) @RUN@ @ @DBG@ St art the debugger . Use @SST ↓ @ to s tep into the pr ogram and see the detailed oper ation of eac h command . The F OR construct As in the case of the S T AR T command, the F O R command has tw o var iati[...]

P age 21-60 T o av oid an infinit e loop , make sur e that start_value < end_value . Ex am ple – calculate the summati on S using a FOR…NEXT construct The f ollow ing progr am calculat es 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 progr am in a var iable [...]

P age 21-61 Example – gene rat e a list of numbers u sing 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 iable @GLI S2 . Θ Chec k out that the pr ogram call 0. 5 ` 2. 5 ` 0. 5 ` @GLI S2 pr oduces the list {0. 5 1. 1.5 2 . 2 .5}. Θ[...]

P age 21-6 2 The f ollow ing progr am calculat es 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 progr am in a var iable @@S3@@ . V er if y the f ollo w ing ex erc ises: J 3 @@@S3@@ Res u lt : S:14 4 @@@S3@@ Res ul t: S:30 5[...]

Pa g e 2 1 - 6 3 The WHILE const ruct The ge ner al stru ctur e of this command is: WHILE logical_statement REPEAT program_statements END The WHILE st atement w ill r epeat the program_statements whi l e logical_statement is true (n on z er o ). If not , pr ogram contr ol is passed to the stateme nt right afte r END . T he program_statements must i[...]

P age 21-64 and stor e it in var iable @GLI S4 . Θ Chec k out that the pr ogram call 0. 5 ` 2. 5 ` 0. 5 ` @GLI S4 pr oduces the list {0. 5 1. 1.5 2 . 2 .5}. Θ T o see step-b y-step operation u se the pr ogram DBUG for a short list, f or e xample: J 1 # 1.5 # 0. 5 ` Enter p ar ameters 1 1. 5 0.5 [‘] @GLIS4 ` En ter the pr ogram name in lev el 1 [...]

P age 21-65 If y ou enter “ TR Y A GAIN” ` @DOERR , p r od uces t he follow in g message: TR Y AGA I N F inally , 0` @DOERR , produces t he messag e: Interrupted ERRN This f unction r eturns a number r epres enting the most r ecent err or . For e xample , if y ou try 0Y$ @ERRN , y ou get the number #30 5h. T h is is the binary integer r epr ese[...]

P age 21-66 The se ar e the components of the IFERR … THEN … END construc t or o f the IFERR … THEN … ELSE … END construc t. Both logical cons truc ts ar e used f or tra pping er ror s dur ing pr ogram e xec ution . W ithin the @) ERROR su b-menu , enter ing „ @) IFERR , or ‚ @) IFERR , w ill place the IFERR structur e components in t[...]

P age 21-6 7 User RP L progr amming in alg ebr aic mode While all the pr ograms pr esent ed earlier ar e pr oduced and run in RPN mode, y ou can alw ay s type a pr ogram in U ser RP L when in algebr aic mode by us ing functi on RPL>. T his functi on is availa ble thr ough the command catalog . As an e xam ple , tr y cr eating the follo w ing pr [...]

P age 21-68 Wher eas, using RP L, ther e is no proble m when loading this pr ogram in algebrai c mode:[...]

Pa g e 2 2- 1 Chapter 2 2 Pr ograms f or graphics manipulation This c hapter includes a n u mber o f ex amples show ing how to u se the calc ulator’s func tions f or manipulating graphi cs inte rac tiv ely or thr ough the us e of pr ograms . As in Chapt er 21 w e recommend u sing RPN mode and setting s yst em flag 117 to S OFT menu labels. « » [...]

Pa g e 2 2- 2 T o user -def ine a k ey y ou need to add to this list a command or pr ogram fo llow ed by a r efer ence to the k ey (see det ails in Chapter 20). T ype the lis t { S << 81.01 MEN U >> 13.0 } in the stac k and use f unction S T OKEY S ( „°L @) MODES @ ) @K EYS@ @@STOK @ ) to user-define k ey C as the access to the PL O [...]

Pa g e 2 2- 3 LABEL (10) The f unction LABEL is used to label the axe s in a plot including the v ari able names and minimum and max imum values of the axes . The var iable names ar e selected f rom info rmation con tained in the var iable PP AR. AU TO ( 1 1 ) The functi on A UT O (A UT Oscale) calculat es a display r ange for the y-ax is or for bo[...]

Pa g e 2 2- 4 EQ (3 ) The v ariable name E Q is res erved b y the calc ulator to stor e the c urren t equatio n in plots or soluti on to equations (see c hapter …). The soft menu k ey la beled E Q in this menu can be used a s it wo uld be if you ha ve y our var iable menu av ailable, e .g., if y ou pres s [ E Q ] it will lis t the curr ent conten[...]

Pa g e 2 2- 5 The f ollow ing diagr am illustr ates the functi ons available in the P P AR menu . The letters attac hed to each f unction in the di agram ar e used for r efe r ence purpo ses in the desc ription o f the functi ons show n below . INFO (n) and PP AR (m) If y ou pr ess @INFO , or enter ‚ @PPAR , while in this menu , you w ill get a l[...]

Pa g e 2 2- 6 INDEP (a) The command INDEP spec ifie s the independent var iable and its plotting r a nge . The se spec ificati ons are st or ed as the third par ameter in the var iable P P AR. T he def ault value is 'X'. T he values that can be as signed to the independen t var iable spec ificati on are: Θ A var iable name , e .g., &apos[...]

Pa g e 2 2- 7 CENTR (g) The command CENTR tak es as ar gument an order ed pair (x,y) or a v alue x, and adjus ts the f irst tw o elements in the var iable P P AR, i .e ., (x min , y min ) and (x max , y max ), so that the center of the plot is (x,y) or (x , 0) , res pecti vel y . S CALE (h) The S CALE command determines the plotting scale r epres e[...]

Pa g e 2 2- 8 A list of tw o binary integers {#n #m}: sets the ti ck annotations in the x - and y- axes t o #n and #m pix els, r espectiv ely . AXE S (k) The in put value f or the axes command consis ts of either an order ed pair (x,y) or a list {(x ,y) atick "x -axis la bel" "y-axis la bel"}. The par ameter atick st ands for th[...]

Pa g e 2 2- 9 The PTYP E menu within 3D (IV) The P TYPE menu under 3D contains the f ollow ing functi ons: The se fu nctions cor res pond to the gr aphics options Slope field , Wir efr ame, Y - Slice , Ps-C ontour , Gridmap and Pr -Sur face pr esented ear lier in this c hapter . Pr essing one o f these s oft menu k ey s, whil e typing a pr ogram , [...]

Pag e 22- 1 0 XV OL (N) , YV OL (O) , and ZVOL (P) The se func tions tak e as input a minimum and maxi mum value and ar e used to spec ify the extent of the par allelepiped wher e the graph w ill be generated (the vi ew ing parallelepiped). Thes e v alues ar e stor ed in the var iable VP AR . The defa ult values f or the ranges XV OL, YV OL, and ZV[...]

Pag e 22- 1 1 The ST A T menu within PL OT The S T A T menu pro vi des access to plots r elated to statis tical anal ysis. W ithin this menu w e find the fo llow ing menus: The di agr am belo w show s the branc hing of the S T A T menu wi thin PL O T . The numbers and letter s accompany ing each func tion or men u are us ed f or r efe r ence in the[...]

Pag e 22- 1 2 The PTYP E m enu within ST A T (I) The P TYPE menu pr ov ides the follo w ing func tions: Thes e ke ys corr espond to the plot t y pes Bar (A ) , Histogr am (B) , and Scatter(C ) , pr esented ear lier . Pres sing one of these so ft menu ke ys , while typ ing a pr ogram , will pl ace the corr esponding f uncti on call in the pr ogram .[...]

Pag e 22- 1 3 XC OL (H) The command X COL is used to in dicate w hich o f the columns of Σ DA T , if mor e than one , w ill be the x - column or independent var iable column. YC O L ( I ) The command Y COL is used to indicat e whi ch o f the columns of Σ DA T, i f m o re than one , w ill be the y- column or dependent v ari able column. MODL (J) T[...]

Pag e 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 retur n to the PL O T menu . Generating plots with pr ograms Depending on whe ther w e are deal ing w ith a two-dimensional gr aph defined by a fun ctio n, by d at a from Σ DA T ,[...]

Pag e 22- 1 5 Thr ee -dimensional graphics The thr ee -dimensional gr aphics a vaila ble , namely , options Slopefi eld, Wir efr ame , Y -Slice , P s -Co ntour , G r idmap and Pr -Surface , use the VP AR var ia ble w ith the follo wing f ormat: { x left , x right , y near , y far , z low , z high , x min , x max , y min , y max , x eye , y eye , z [...]

Pag e 22- 1 6 @) PPAR Sho w plot paramet ers ~„r` @INDEP Def i ne ‘ r’ as the indep . var iable ~„s` @DEPND Def ine ‘ s ’ as the dependent va ri able 1 # 10 @XRNG De fine (- 1, 10) as the x -r ange 1 # 5 @YRNG L Def ine (-1, 5) as t he y-r ange { (0, 0) {.4 .2} “Rs ” “Sr”} ` Axes de finiti on list @AXES Define axes center , ti[...]

Pag e 22- 1 7 @) PPAR Sho w plot paramet ers { θ 0 6. 2 9} ` @INDEP De fine ‘ θ ’ as the indep. V ari ab le ~y` @DEPND Def ine ‘Y’ as the dependent v ariab le 3 # 3 @XRNG Def ine (-3,3) as the x -range 0. 5 # 2.5 @YRNG L Def ine (-0.5,2 .5) as the y-r ange { (0, 0) {.5 . 5} “ x ” “ y”} ` Axes def i nition lis t @AXES Define axes[...]

Pag e 22- 1 8 « St art pr ogram {PPAR EQ} PURGE P urge c urr ent P P AR and E Q ‘ √ r’ STEQ St ore ‘ √ r’ into E Q ‘r’ INDEP S et independent v ari able to ‘ r’ ‘s’ DEPND S et dependent v ariable t o ‘ s ’ FUNCTION Select FUNCT ION as the plot type { (0.,0.) {.4 .2} “Rs” “Sr” } AXES Set axe s infor mation –1. [...]

Pag e 22- 1 9 Example 3 – A polar plot . Enter the follo wing pr ogram: «S t a r t p r o g r a m RAD {PPAR EQ} PURGE Change to r adians , purge v ars. ‘1+SIN( θ )’ STEQ St ore ‘ f( θ )’ into EQ { θ 0. 6.29} INDEP Set indep . var iable to ‘ θ ’ , w ith r ange ‘Y’ DEPND S et dependent v ariable t o ‘ Y’ POLAR Select P OL AR[...]

Pag e 22- 2 0 PICT This so ft ke y re fer s to a v ari able called PICT that stor es the c urr ent contents of the gr aphic s w indow . This v ariable name , ho w ev er , cannot be placed w ithin quotes, and ca n only store graph ics obje cts. In tha t sen se, PICT i s li k e n o oth er calc ulator va ri ables. PDI M The f unction P DIM tak es as i[...]

Pag e 22- 2 1 BO X This command t ake s as input 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 aws the bo x who se diagonals ar e r epre sented b y the t w o pairs of coor dinates in the input. ARC This command is us ed to dr aw an arc . ARC tak es as input the f ollow ing obj e[...]

Pag e 22- 22 Θ PI X? Checks if pi xel at locati on (x,y) or {#n , #m} is on. Θ PI XOFF turns o ff pi xel at location (x ,y) or {#n, #m}. Θ PI XON turns on p ix el at location (x ,y) or {#n, #m}. PVIEW This command tak es as input the coor dinates of a point as use r coor dinates (x,y) or p ix els {#n, #m}, and places the conte nts of PICT w ith [...]

Pa g e 22- 23 (5 0., 50.) 12 . –180. 180. ARC Dr aw a c irc le center (5 0,50), r= 12 . 1 8 FOR j Dr aw 8 lines w ithin the ci rc le (50., 5 0.) D UP L ines ar e centered as (5 0,5 0) ‘12*COS( 45*(j-1))’  NUM Calculate x , other end at 50 + x ‘12*SIN( 45*(j-1))’  NUM Calculat es y , other end at 5 0 + y R  C C on ve rt x y to (x [...]

Pa g e 22 - 24 It is suggested that y ou cr eate a separate sub-dir ectory to stor e the progr ams. Y ou could call the sub-direc tory RIVER , since we ar e dealing with ir regular open channel c r oss-s ectio ns, typ ical of ri ver s. T o see the pr ogram XSE CT in action , use the f ollow ing data sets . Enter them as matri ces of two columns, th[...]

Pa g e 22 - 2 5 P ixel coor dinates The f igur e belo w show s the graphi c coordinat es fo r the t yp ical (minimum) scr e en of 131 × 64 pix els. P ix els coordinates ar e measured fr om the top left corner of the screen {# 0 h # 0h}, w hich corresponds to user-defined c oordinates Data set 1 Da ta set 2 xy x y 0.4 6. 3 0.7 4.8 1. 0 4.9 1. 0 3 .[...]

Pag e 22- 26 (x min , y max ) . The max imum coor dinates in terms of p i xels cor respond to the lo wer r ight corner of the sc reen {# 8 2h #3Fh}, whic h in user-coor dinates is the point (x max , y min ) . The coordinates of the two other cor ners both in pi xel as well as in user-defined coordinates ar e show n in t he fi gure . Animating graph[...]

Pa g e 22- 27 Animating a collection of graphics The calc ulator pr ov ides the func tion ANIMA TE t o animate a n umber of gr aphic s that hav e been placed in the stac k. Y ou can generat e a gra ph in the gr aphic s sc r een by u sing the commands in the PL O T and PICT menu s. T o place the gener a ted gr aph in the stack , use P I CT RCL . Whe[...]

Pag e 22- 28 ANIMA TE is available by u sing „°L @) GROB L @ANIMA ) . The animati on will be r e -started. Pr ess $ to stop the animati on once more . Notice that the number 11 w ill still be list ed in stac k leve l 1. Pr ess ƒ to dr op it fr om the stack. Suppose that y ou want t o keep the f igures that compo se this animation in a var iable[...]

Pa g e 2 2- 2 9 Example 2 - Animating the plotting of differ ent po wer f unctions Suppose that y ou want t o animate the plotting of the functi ons f(x) = x n , n = 0, 1, 2 , 3, 4, in the same set of ax es. Y ou could use the follo wing pr ogram: «B e g i n p r o g r a m RAD Set angle units to radians 131 R  B 64 R  B PD IM Set PIC T scree [...]

Pag e 22- 3 0 pr oduced in the calculator ’s scr een. T here for e, w hen an image is converted into a GR OB, it becomes a s equence of binary digits ( b inary dig its = bits ), i . e . , 0’s and 1’s . T o illus trate GR OBs and con ve rsi on of image s to GR OBS consider the fo llo w ing ex er c ise . When w e pr oduce a graph in the calc ul[...]

Pag e 22- 3 1 1` „°L @) GROB @  GROB . Y ou w ill now ha ve in le vel 1 the GR OB desc r ibed as: As a gra phic obj ect this equation can no w be placed in the graphi cs display . T o re cov er the graphic s display pr ess š . Then , mov e the c u rs or to an empty sector in the gr aph, and pr ess @) EDIT LL @REPL . The equation ‘X^2 -5’[...]

Pa g e 2 2- 32 BLANK The f unction BLANK, w ith arguments #n and #m , cr eates a blank gra phics objec t of w idth and height spec ifie d by the v alues #n and #m, r especti vely . This is similar to the functi on PDIM in the GR APH menu . GOR The fun ctio n GO R ( Graph ics O R) takes as in put gr ob 2 (a target GROB) , a set of coor dinates , and[...]

Pa g e 2 2- 3 3 An ex ample of a progr am using GROB The f ollow ing progr am produ ces the gr aph of the sine f unctio n inc luding a fr ame – dra wn w ith the func tion B OX – and a GROB to label the gr aph. Her e is the listing of the pr ogram: «B e g i n p r o g r a m RAD Set angle units to r adians 131 R  B 64 R  B PD IM Set PIC T s[...]

Pa g e 2 2- 3 4 sho ws the state of s tres ses w hen the element is r otated b y an angle φ . In this case, the normal st r esses are σ ’ xx and σ ’ yy , while the shear str esses are τ ’ xy and τ ’ yx . The relationshi p between the origina l state of str esses ( σ xx , σ yy , τ xy , τ yx ) and the state o f stres s w hen the ax e[...]

Pa g e 22- 35 The stress condit ion for which the she ar stress, τ ’ xy , is ze ro , indicated by segment D’E’ , produces the so-called pr inc i pal str esses , σ P xx (at point D’) and σ P yy (at point E’). T o obtai n the princ ipal stre sses y ou need to r otate the coor dinate s y stem x ’-y’ by an angle φ n , counte rc lockw [...]

Pag e 22- 3 6 separ ate vari ables in the calculator . These sub-pr ograms ar e then linked b y a main pr ogram , that we w ill call MOHRCIRCL . W e will fir st cr eate a sub- dir ectory called MOHRC w ithin the HOME dir ectory , and mo ve into that dir ectory to type the pr ograms . The ne xt step is to c reat e the main pr ogram and sub-pr ograms[...]

Pa g e 2 2- 37 At this point the pr ogram MOHR CI RC L starts calling the sub-progr ams to pr oduce the fi gure . Be patient. T he resulting Mohr’s c ir cle will loo k as in the pic ture to the left . Becaus e this v ie w of P ICT is invo ked thr ough the function PVIEW , we cannot get any othe r infor mation out of the plot besides the f igur e [...]

Pa g e 22 - 3 8 infor mation tell us is that some where 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, press $ . T h en type the list cor re sponding to the value s { σ x σ y τ xy}, f or this case, it w ill be { 25 75 50 } [ENTER] Then , press @ CC&r . T he last r esul[...]

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

Pag e 22- 4 0 T o find the v alues of the str esse s corr esponding to a ro tation of 3 5 o in the angle of th e stressed pa rticle, we use: $š Clear scr e en, show PICT in g r aphics scr e en @TRACE @ ( x,y ) @ . T o move c ursor o ver the c irc le sho w ing φ and (x ,y) Ne xt, pr ess ™ until y ou read φ = 3 5. T he corr esponding coordinat e[...]

Pag e 22- 4 1 Since pr ogr a m IND A T is us ed also f or pr ogr am @PRNST (P RiNc ipal ST r esses) , running that partic ular progr a m w ill now us e an input fo rm , f or e xample , The r esult , after pres sing @@@OK@@@ , is the fo llow ing:[...]

Pa g e 2 3 - 1 Chapter 2 3 Character strings Char acter strings ar e calculator ob jects enc losed betw een double quotes . The y ar e tr eated as te xt by the calculat or . For e xample , the string “S INE FUNCTION” , can be transf ormed int o a GROB (Gr aphics Ob ject), to label a gr aph, o r can be used as output in a pr ogr am. S ets of c h[...]

Pa g e 2 3 - 2 String concatenation Str ings can be concatenated (j oined together ) by using the plus sign +, f or exa mp le : Concatenating s tring s is a prac tical w ay to cr eate output in pr ogr ams. F or e xample , concatenating "Y OU ARE " A G E + " YEAR OLD" cr eates the str ing "Y OU ARE 2 5 YEAR OLD", wher e[...]

Pa g e 2 3 - 3 The ope rati on of NUM, CHR, OB J  , and  S TR was pr esent ed earlie r in this Chapter . W e ha ve als o seen the func tions SUB and REP L in relati on to gra phics earli er in this chapter . F unctions S UB, REP L, P OS, SI ZE , HEAD , and T AIL have similar eff ects as in lists , namely: SI ZE: number of a sub-str ing in a s[...]

Pa g e 2 3 - 4 scr een the ke ystr oke sequence to get suc h char acter (  . f or this case) and the numer ical code corr esponding to the char acter (10 in this cas e) . Char acters that ar e not def ined appear as a dark squar e in the charac ter s list (  ) and sho w ( None ) at the bottom of the displa y , ev en though a numer ical code e[...]

Pa g e 24 - 1 Chapter 2 4 Calculator objec ts and flags Numbers , lists, v ectors, matr ices, algebr aics, etc ., are calc ulator objects . The y ar e classif ied accor ding to its nature into 30 diff erent ty pes, w hic h are desc r ibed belo w . Fl ags ar e var iables that can be u sed to con tr ol the calculat or properties. Flags wer e introduc[...]

Pa g e 24 - 2 Number T ype Ex ample _______________ ____________________ _____________________ ____________ 21 Extended Real Number Long Real 2 2 Extended Comple x Number Long Complex 2 3 Link ed Arr ay Lin ked rray 2 4 Char acter Obj ect Character 25 Co d e O b j e ct Code 2 6 Libr ar y Data Library Data 2 7 External Ob ject External 28 I n t e g [...]

Pa g e 24 - 3 Calculator flags A flag is a v ariable that can e ither be set or unse t . The statu s of a flag affec ts the behav ior of the calc ulator , if the flag is a s ys tem flag , or of a pr ogr am, if it is a user f lag. T hey ar e descr ibed in mor e detail next . S ystem flags S yste m flags can be accesse d by using H @) FLAGS! . Pr ess[...]

Pa g e 24 - 4 The f unctions contained w ithin the FL A G menu are the f ollow ing: The oper ation of thes e func tions is as f ollo ws: SF Set a flag CF Clear a f lag F S? Retur ns 1 if flag is set, 0 if not set FC? Returns 1 if flag is clear (not set), 0 if flag is set F S?C T ests flag as F S does, then c lea rs it FC?C T ests flag as FC does, t[...]

Pa g e 25 - 1 Chapter 25 Date and T ime Functions In this Chapter w e demonstr ate some of the func tions and calc ulations using times and dates . The T I ME menu The T IME menu , av ailable thro ugh the ke ystr ok e sequence ‚Ó (the 9 k ey) pr o vi des the follo wing f uncti ons, w hich ar e desc ribed ne xt: Setting an alarm Option 2 . Set al[...]

Pa g e 25 - 2 Bro wsing alarms Option 1. Br o ws e alarms... in the TIME me nu lets yo u r ev iew your c urr ent alarms . F or ex ample, after ente ring the alarm u sed in the e xample abo ve , this option w ill show the fo llo w ing scr een: This sc reen pr ov ides f our soft menu ke y labels: EDIT : F or editing the selected alarm , pro viding an[...]

Pa g e 25 - 3 The appli cation of these f u ncti ons is demonstrated belo w . D A TE: P laces c urr ent date in the stack  D A TE: Set sy stem date to specif ied value TIME: Place s cu rr ent time in 2 4 -hr HH.MMS S for mat  TIME: Set sy stem time to spec ified v alue in 2 4 -hr HH.MM. SS f ormat TICK S: Pr ov ides s yst em time as binary in[...]

Pa g e 25 - 4 Calculating with tim es The fun ct ion s  HMS , HMS  , HMS+, and HM S - are us ed to manipulate value s in the HH.MM SS f ormat . This is the same f ormat us ed to calc ulate with angle measur es in degree s, minu tes , and seconds. T hus , these oper ations ar e usef u l not onl y fo r time calculati ons, but also for angular c[...]

Pa g e 2 6 - 1 Chapter 2 6 Managing memor y In Chapter 2 w e intr oduced the basic concepts of , and oper ations f or , cr eating and managing var iables and dir ector ies . In this Chapter w e disc uss the management of the calc ulator’s memory , inc luding the par tition o f memory and techni ques for backing u p data. Mem ory St r uct ur e The[...]

Pa g e 2 6 - 2 P ort 1 (ERAM ) can contain up to 12 8 KB of data. P or t 1, together w ith P ort 0 and the HOME direc tory , constitut e the calculator ’s R AM (R andom Access Memory) segment of calc ulator’s memory . T he RAM memor y segment r equires contin uous elec tri c pow er supply f r om the calculat or batter ies to operate . T o av oi[...]

Pa g e 2 6 - 3 Chec king objec ts in memory T o see the obj ects stor ed in memory you can u se the FILE S functi on ( „¡ ). Th e scre e n be l ow sh ows th e H OM E d i rec to r y wi th five d ire c to ri es, n a m ely , TRIANG , MA TRX , MPFIT , GRP HS, and CA SD IR. Additional dir ector ies can be vi ew ed by mo ving the c u r sor do wnw ards[...]

Pa g e 2 6 - 4 Bac k up objec ts Back up obj ects ar e used to copy dat a fr om your home dir ectory into a memory port. T he purpose o f bac king up objects in me mory port is to pr eserve the contents of the ob jects f or futur e usage . Ba c k up obj ects hav e the follo wing cha ra cte rist ic s: Θ Back up obj ects can onl y ex ist in port mem[...]

Pa g e 2 6 - 5 Bac king up and restor ing HOME Y ou can back up the contents of the c urr ent HOME dir ectory in a single back up obje ct . This ob jec t w ill contain all v ari ables , k ey as signments , and alarms c urr ently def ined in the HOME direc tory . Y ou can also r estor e the conte nts of y our HOME dir ectory fr om a back up ob ject [...]

Pa g e 2 6 - 6 Stor ing, deleting, and rest oring bac k up objec ts T o cr eate a back up obj ect us e one of the f ollow ing appr oache s: Θ Use 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 bac kup obj ect will ha ve the same name as the o ri ginal object . Θ Use the S T O command to co[...]

Pa g e 2 6 - 7 Using data in backup objects Although y ou cannot directl y modif y the contents of back up objec ts, y ou can use thos e contents in calculat or oper ations. F or ex ample, y ou can run pr ograms stor ed as back up objec ts or use dat a fr om back up objects t o run pr ograms . T o run back up-obj ect pr ograms or use data f rom bac[...]

Pa g e 2 6 - 8 T o re move an SD car d, turn o f f the HP 5 0g, pr ess gentl y on the expo sed edge of the car d and push in . The car d should spring out of t he slot a small distance , allo w ing it now to be easil y r emov ed fr om the calculator . For m atting an SD card Most SD car ds will alr eady be for mat ted , but they ma y be formatted w[...]

Pa g e 2 6 - 9 Accessing objects on an SD card Acces sing an obj ect fr om the SD car d is similar to w hen an object is located in ports 0, 1, or 2 . Ho we ver , P ort 3 wi ll not appear in the menu when us ing the LIB func tion ( ‚á ). T he SD file s can only be managed using the F iler , or F ile M anager ( „¡ ). When starting the F iler ,[...]

Pa g e 2 6 - 1 0 Note that if the name of the object y ou intend to stor e on an SD card is longer than ei ght char acters , it will appear in 8. 3 DOS f ormat in port 3 in the Filer once it is stor ed on the card . Recalling an object from an SD car d T o recall an obj e ct f r om the SD card onto the sc reen, u se func tion RCL , as fo llow s: Θ[...]

Pa g e 2 6 - 1 1 Note that in the case of obj ects with long f iles names , you can s pecify the full name of the ob ject , or its truncate d 8. 3 name , when ev aluating an objec t on an SD car d. P urging an object from the SD card T o pur ge an obj ect fr om the SD card onto the s creen , use f unction P URGE , as fo llow s: Θ In algebr aic mod[...]

Pa g e 2 6 - 1 2 This w ill stor e the objec t pr ev iou sly on the stac k onto the SD card into the dir ectory named PR OGS into an objec t named PR OG1. Note: If PR OGS does not ex ist, the dir ectory will be au tomaticall y cr eated. Y ou can specify an y number of nes ted subdirect ori es. F or ex ample, to r efer to an objec t in a thir d-leve[...]

Pa g e 2 6 - 1 3 Libr ar y numbers If y ou use the LIB men u ( ‚á ) and pr ess the soft menu k ey cor r esponding to port 0, 1 or 2 , you w ill see library numbers lis ted in the soft menu k ey labe ls. E ach libr ar y has a thr ee or four -digit number ass oc iated with it . (F or e xam p le , the two libr aries that mak e up the Eq uation L ib[...]

Pa g e 2 6 - 1 4 w ill indicate w hen this battery needs r eplacement. T he diagram belo w sho ws the location of the bac kup battery in the top compartment at the back of the calc ulator .[...]

Pa g e 27- 1 Chapter 2 7 T he Equation Libr ar y The E quation L ibrary is a collection o f equations and commands that enable y ou to sol ve simple s c ience and engin eer ing pr oblems. T he library consists o f mor e than 300 equations gr ouped into 15 techni cal subj ects containing mor e than 100 pr oblem titles . Eac h p r oblem title contain[...]

Pa g e 27- 2 7 . For eac h know n vari able, type its v alue and press the cor respo nding menu k ey . If a vari able is not sho wn , press L to display furth er v ariables. 8. Optional: suppl y a guess fo r an unknow n var iable . This can speed up the soluti on pr ocess or help to f o c us on one of se ver al solutions . Enter a guess ju st as y [...]

Pa g e 27- 3 Using the menu ke ys The ac tions of the unshifted and shifted var iable menu k ey s for both s olv ers ar e identi cal. Noti ce that the Multiple E quation S olver u ses tw o for m s of men u labels: black and whit e . The L k ey dis plays additi onal menu labels, if r equir ed. In additi on, each s olv er has spec ial menu k ey s, w [...]

Pa g e 27- 4 Bro wsing in the Equation L ibrary When y ou select a sub ject and title in the E quation L ibrary , you spec ify a set of one or mor e equati ons. Y ou can get the f ollow ing infor mation about the equation s et from the E quatio n Libr ary catalogs:  The equations thems elves and the number of equati ons.  The v aria bles us e[...]

Pa g e 27- 5 Vie wing var iables and selec ting units After y ou select a subj ect and title , y ou can vi e w the catalog of names , desc r iptions , and units for the v ari ables in the equation s et b y pre ssing #VARS# . The t able belo w summari ze s the oper ations av ailable to y ou in the V ar iable catalogs . Oper ation s in V ariable cata[...]

Pa g e 27- 6  Pr ess to stor e the pictur e in PI C T , the gra phics memory . T hen y ou can use © PIC T ( or © PICTURE) t o vi ew the pic ture again after y ou hav e quit the Equati on Libr ar y .  Pr ess a menu k ey or to v iew other equation inf ormati on. Using the M ultiple-Equation Solver The E quation L ibrary starts the Multiple-Eq[...]

Pa g e 27- 7 The men u labels for the v ariable k ey s are w hite at fir st, but c hange during the soluti on proces s as des cr ibed below . Becaus e a soluti on inv olv es man y equations and man y var iable s, the Multiple- E quation S olver mu st keep tr ack of var iables that ar e user -def ined and not def ined—those it can ’t change and [...]

Pa g e 27- 8 Meani ngs of Menu Labe ls Defining a set of equations When y ou design a s et of eq uations , you sh ould do it w ith an understanding o f ho w the Multiple -Equati on Solv er uses the equati ons to sol ve pr oblems. The Mul tiple -E quation S olv er uses the same pr ocess y ou’d us e to solv e f or an unkno wn v ariable (a ssuming t[...]

Pa g e 27- 9 F or ex ample, the f ollo wing thr ee equations def ine initial v elocity and acceler a ti on based on tw o observed dis tances and times. T he firs t two equations alone ar e mathematicall y suffi c ient f or solv ing the problem , but each equation con tains tw o unkno wn v aria bles. A dding the third equati on allo ws a succes sful[...]

Pa g e 27- 1 0 6. P ress !MSOLV! to launc h the sol ver w ith the new se t of equati ons. T o chang e 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 (as the y are u sed when the Multiple -E quation Sol ver is launc hed) . 2 . Ent er a te xt str ing containing the new titl e onto the stac k.[...]

Pa g e 27- 1 1  Constant? T he initial value of a v ariable ma y be leading the r oot - finder in the w rong dir ection . Supply a guess in the oppo site dir e cti on fr om a cr itical v alue. (If negati ve value s are v alid, try one . ) Chec king solutions The va riab les h avin g a š mark in their menu la bels ar e re lated for the mo st r e[...]

Pa g e 27- 1 2  Not r elated . A var iable ma y not be inv olv ed in the solution (no mark in the label), so it is not compatible w ith the var iables that w ere in volv ed.  W r ong dir ection . The initial v alue of a v ariable ma y be leading the r oot - finder in the w rong dir ection . Supply a guess in the oppo site dir e cti on fr om a[...]

Pa g e A - 1 Appendix A Using input forms This e xample o f setting time and date illu str ates the use o f input f orms in the calc ulator . S ome general r ules: Θ Use the arr ow k ey s ( š™˜— ) to mov e from one field to the ne xt in the input f orm. Θ Use an y the @CHOOS soft m enu k ey to see the opt ions av ailable for an y gi ven f i[...]

Pa g e A - 2 In this particular ca se w e can giv e v alues to all but one of the var iables, sa y , n = 10, I%YR = 8. 5, PV = 10000, FV = 1000, and s ol ve f or var iable P MT (the meaning of thes e var iables w ill be pre sented later ) . T r y the f ollow ing: 10 @@OK@@ Enter n = 10 8. 5 @@OK@ @ Enter I%Y R = 8. 5 10000 @@ OK@@ Enter PV = 10000 [...]

Pa g e A - 3 !CALC Pr ess to access the stac k for calc ulations !TYPES Press to determine the type of object in highlighte d fi eld !CANCL Cancel o peration @@OK@@ Accep t en tr y If y ou pre ss !RESET y ou w ill be ask ed to se lect between the tw o options: If y ou select R eset value onl y the highlighted v alue w ill be rese t to the defa ult [...]

Pa g e A - 4 (In RPN mode , we w ould hav e used 113 6.2 2 ` 2 `/ ). Press @@ OK@@ to ent er this ne w value . The input f orm w ill now look lik e this: Press !TYPES to s ee the type of dat a in the P MT fi eld (the highligh ted fi eld) . As a r esult , y ou get the follo wing spec ifi cation: This indi cates that the v alue in the P MT field mu s[...]

Pa g e B - 1 Appendix B T he calc ulator ’s ke y board The f igur e belo w show s a diagram o f the calc ulator ’s ke yboar d w ith the number ing of its ro ws and columns . The f igure sho ws 10 r ow s of ke ys combined w ith 3, 5, or 6 columns. Ro w 1 has 6 k ey s, ro ws 2 and 3 hav e 3 ke ys eac h, and r o ws 4 thr ough 10 hav e 5 ke ys eac [...]

Pa g e B - 2 fi ve f uncti ons. T he main ke y f uncti ons ar e sho wn in the fi gure belo w . T o oper ate this main k ey func tions simpl y press the cor responding k ey . W e will r efer to the k ey s b y the r ow and column w here the y are located in the sk etc h abo ve , thus , ke y (10,1) is the ON key . Main key functions in the calc ulator[...]

Pa g e B - 3 Main ke y functions Key s A through F ke ys ar e assoc iated w ith the soft menu opti ons that appear at the bottom of the calculat or’s displa y . Th us, these k e ys w ill acti vate a var iety of func tions that change acco rding t o the acti ve menu .  The a rrow keys, —˜š™ , are u sed to mov e one char acter at a time in[...]

P age B-4  The left-s hi ft k ey „ and the ri ght -shift ke y … are combined w ith other ke ys to ac ti vate menu s, enter char acters , or calc ulate functi ons as descr ibed else wher e.  The numerical k ey s ( 0 to 9 ) are used to ent er the digits of the decimal number sy stem.  Ther e is a deci mal po int k e y (.) and a space ke [...]

P age B-5 the other three f unctions is ass oci ated with the left-shif t „ ( MTH ) , right-shift … ( CA T ) , and ~ ( P ) k ey s. Diagr am s sho w ing the function or c haracter r esulting fr om combining the calculator k ey s with the left-shift „ , ri g ht-shift … , ALPHA ~ , ALPHA-left - shift ~„ , and ALPHA-r ight-shif t ~… , are p[...]

Pa g e B - 6  The CM D func tion sho ws the most r ecent commands, the PRG fun ctio n acti vates the pr ogramming men us, the MTR W functi on acti vates the Matri x Wr i t e r, Left-shift „ func tions of the calc ulator ’s ke yboard  The CM D func tion sho ws the most r ecent commands.  The PRG functi on acti vates the pr ogramming men[...]

Pa g e B - 7  The e x ke y calcul ates the e xponential func tion of x .  The x 2 ke y calculat es the squar e of x (this is ref erred to as the SQ fun ctio n) .  The A SIN , A COS , and A T AN func tions calc ulate the ar csine, ar ccosine , and ar ctangent f unctions, r especti vel y .  The 10 x functio n calculates the anti-logar ith[...]

Pa g e B - 8 Righ t-s hif t … func tions of the calculator ’s ke yboard Right-shift functions The sk etch abo ve sho ws the functi ons, char acters, or menus ass o c iated w i th the differ ent calculator k ey s when the r ight-shift ke y … is activ ated.  The fun ct ion s BEGIN , END, C OP Y , CUT and PA S T E are u sed for editing purpo [...]

Pa g e B - 9  The CA T function is u sed to acti vate the command catalog.  The CLEAR functi on clear s the sc reen .  The LN functi on calc ulates the natur al logarithm .  The func tion calc ulates the x – th root o f y .  The Σ function is us ed to enter summati ons (or the upper case Gr eek letter sigma).  The ∂ func tion[...]

Pa g e B - 1 0 is used mainl y to enter the upper -case letter s of the English alphabet ( A thr ough Z ) . T he numbers , mathematical s ymbols ( - , + ), decimal po int ( . ) , and the space ( SPC ) ar e the same as the main functions o f these k ey s. The ~ fun ction pr oduces a n aster isk ( * ) whe n combined wi th the times k e y , i.e ., ~* [...]

Pa g e B - 1 1 Notice that the ~„ combinatio n is used ma inly to enter the lo wer -c ase letters of the English alphabet ( A thr ough Z ). The number s, mathematical sym bo l s ( - , +, × ), dec imal point ( . ) , and t he space ( SP C ) are the same a s the main functi ons of these ke ys . The ENTER and CONT k ey s also w ork as their main fun[...]

Pa g e B - 1 2 Alpha-right-shift c har ac ters The f ollow ing sketc h show s the c har acter s assoc iated w ith the differ ent calc ulator k ey s when the ALP HA ~ is combined with the r ight -shift k ey … . Alpha ~… functions of the calc ulator’s ke yboard Notice that the ~… combination is used mainl y to enter a number of spec ial char [...]

Pa g e B - 1 3 ~… combination inc lude Greek letters ( α, β, Δ, δ, ε, ρ, μ, λ, σ, θ, τ , ω , and Π ), other c har acter s gener ated by the ~… co mbinati on ar e |, ‘ , ^, =, <, >, /, “ , , __, ~, !, ?, <<>>, and @.[...]

Pa g e C - 1 Appendix C CAS settings CA S stands f or C omputer A lgeb r a ic S yste m. T his is the mathemati cal core o f the calc ulator wher e the sy mbolic mathematical oper ations and func tions ar e pr ogrammed . The CA S offe rs a number of settings can be adj usted accor ding to the type of oper ation of inter est . T o see the optional CA[...]

Pa g e C - 2 Θ T o r ecov er the ori ginal menu in the CAL CULA T OR MODES inpu t bo x, pr ess the L k e y . Of inter est at this point is the changing o f the CAS se ttings. This is accompli shed by pr essing the @@ CAS@@ soft menu k e y . The def ault value s of the CA S setting ar e sho wn belo w: Θ T o nav igate thr ough the man y options in [...]

Pa g e C - 3 A var iable called VX ex ists in the calc ulator’s {HO ME CASDIR} directory that take s, by def ault , the value of ‘X’ . This is the name of the pr efe rr ed independent v ari able fo r algebr aic and calculu s applicati ons. F or that reason , most e xamples in this C hapter us e X as the unknow n var iable . If y ou use other [...]

Pa g e C - 4 The s ame e xample , corr esponding to the RPN oper ating mode , is show n next: Appr o ximate v s. Ex act CAS mode When t he _ Appro x is selected, s ymbolic oper ations (e.g ., def inite integrals, squar e roots , etc .) , will be calc ulated numeri cally . When the _Appr ox is unselect ed (Ex a ct mode is ac tiv e) , s ymboli c oper[...]

Pa g e C - 5 The k ey str ok es necessary for ent er ing these v alues in Algebrai c mode are the follo wing: …¹2` R5` The s ame calculati ons can be pr oduced in RPN mod e . Stack le vels 3: and 4: sho w the case of Ex act CAS setting (i .e ., the _Numeri c CAS optio n is unselect ed) , while s tack lev els 1: and 2: sho w the case in w hic h t[...]

Pa g e C - 6 It is r ecommended that y ou se lect EXA CT mode as default CA S mode, and change t o APP ROX mode if r equested b y the calcul ator in the perfor mance of an oper ation . F or additional infor mation on real and integer n umbers , as we ll as other ca lcul ato r’s obje cts, refer to Cha pte r 2 . Comple x vs . Real CAS mode A comple[...]

Pa g e C - 7 If y ou pre ss the OK soft menu ke y () , then the _Comple x op ti on is for ced, and the r esult is the f ollo wing: The k ey str okes us ed abov e ar e the fo llo w ing: R„Ü5„Q2+ 8„Q2` When ask ed to change to C OMP LEX mode , use: F . If you dec ide not to accept the change to C OMPLE X mode , yo u get the follo wing er r or [...]

Pa g e C - 8 F or ex ample, ha v ing selec ted the St ep/step optio n, the f ollow ing scr eens show the step-b y-step di visi on of two pol ynomials , namely , (X 3 -5X 2 +3X- 2)/(X- 2). This is accomplished by u sing functi on DIV2 a s sho wn belo w . Pr ess ` to s ho w the fir st step: The s cr een infor m us that the calc ulator is operating a [...]

Pa g e C - 9 . Increasing-po wer CA S mode When t he _Incr po w CA S option is se lected , polynomi als will be list ed so that the ter ms will ha ve incr easing po wer s of the independent var iable . If the _Incr po w CAS option is not s elected (defa ult value) then pol yn omials w ill be listed so that the terms w ill hav e decr easing po we rs[...]

Pa g e C - 1 0 Rigor ous CAS set ting When t he _Rigor ous CAS option is se lected , the algebrai c expr essi on |X|, i.e ., the absolute v alue, is not simplif ied to X . If the _R igor ous CA S option is not select ed, the algebr aic e xpressi on |X| is simplified t o X . The CAS can s ol ve a lar ger var iety of pr oblems if the r igorou s mode [...]

Pa g e C - 1 1 Notice that , in this instance, s oft menu k ey s E and F are the onl y one wi th ass oci ated commands , namely : !!CANCL E CANCeL the help fac ility !!@@OK#@ F OK to acti vate help fac ility fo r the selected comman d If y ou pr ess the !!CAN CL E k ey , the HELP fac ilit y is skipped , and the calculator r eturns t o normal displa[...]

Pa g e C - 1 2 Notice that ther e are si x commands assoc iated w ith the soft menu k ey s in this case (y ou can chec k that there ar e only si x command s because pr essing the L pr oduces no additional menu items). The soft menu k ey commands ar e the fo llo w ing: @EXIT A EXIT the help fac ilit y @ECHO B Copy the e xample command to the s tack [...]

Pa g e C - 1 3 T o nav igate quic kly to a partic ular command in the help f ac ility list w ithout hav ing to use the arr o w k e ys all the time , we can us e a shortcu t consisting of typing the f irst letter in the command’s name . Suppose that w e want to find infor mation on the co mmand IBP (Integr ation B y P arts) , once the help f ac il[...]

Pa g e C - 1 4 In no ev ent unless r equired b y appli cable law w ill any cop yr ight holder be liable to y ou for damage s, inc luding any gene ral , speci al, inc idental or conseq uential damage s arising ou t of the use or ina bility to use the CA S Softwar e (including but not limited to lo ss of data or data being r ender ed inaccur ate or l[...]

Pa g e D - 1 Appendix D Additional character set While y ou can use an y of the upper -case and low er -case English letter fr om the ke yboar d, ther e are 2 5 5 char acters usable in the calc ulator . Including spec ial cha ract ers li ke θ , λ , etc ., that that can be used in algebr aic expr essi ons. T o access thes e char acters w e use the[...]

Pa g e D - 2 functi ons ass oc iated w ith the soft menu k ey s, f4, f5, and f6. T h ese f unctions ar e: @MODIF : O pens a gra phics scr een wher e the user can modify highlighted char acter . Use this option car efull y , since it will alt er the modifi ed charac ter up to the ne xt r eset o f the calc ulator . (Imagine the eff ect of c hanging t[...]

Pa g e D - 3 Gr ee k lett ers α (alpha) ~‚a β (beta) ~‚b δ (delta) ~‚d ε (epsilon) ~‚e θ (theta) ~‚t λ (lambda) ~‚n μ (mu) ~‚m ρ (rh o) ~‚f σ (sigma) ~‚s τ (tau) ~‚u ω (omega) ~‚v Δ (upper -case delta) ~‚c Π (upper -case pi) ~‚p Other c har acters ~( t i l d e ) ~‚1 !( f a c t o r i a l ) ~‚2 ? (questi on [...]

Pa g e E - 1 Appendix E The Selec tion T ree in th e Equation W riter The e xpre ssion tr ee is a diagr am sho wing h o w the E quation W r iter interpr ets an ex pre 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 hie rar ch y of oper ation . The rules ar e as fo llo ws: 1. Operati ons [...]

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 of the hier arc hy-of-oper ation rules in this selecti on. F irst the y (Step A1) . Then , y- 3 (S tep A2 , par enth eses ). Then , (y-3)x (Step A3, multiplicati on) . T h en (y-3)x+5, (Ste p A4, additi on) . Then , ((y-3)x+5)(x 2 +4) (Step A5, multipli cati[...]

Pa g e E - 3 Step B1 Step B2 Step B3 Step B4 = Step A5 Step B5 = S tep A6 W e can also follo w the ev aluation o f the expr essi on starting fr om the 4 in the argume nt of the SIN func tion in the denominator . Pr ess the dow n arr ow k e y ˜ , continuously , until the clear , editing cur sor is tr igger ed around the y , once mor e . T hen, pre [...]

Pa g e E - 4 Step C3 Step C 4 Ste p C5 = Step B5 = S tep A6 The expr ession tree for the expression presented above is show n next: The s teps in the e valuation of the thr ee terms ( A1 through A6 , B1 thro u gh B5, and C1 thr ough C5) ar e sho wn ne xt to the c irc le containing number s, var iables , or oper ators .[...]

Pa g e F - 1 Appendix F T he Applications (APP S) menu The A pplicati ons (AP PS) men u is av ailable through the G key ( fi rst key i n second r o w fr om the ke yboard’s top). The G ke y sho ws the f ollo w ing applications: The diff erent appli cations ar e desc ribed ne xt. P lot func tions.. Selecting opti on 1. Plot f u ncti ons.. in the AP[...]

Pa g e F - 2 I/O functions.. Selecting opti on 2 . I/O f uncti ons .. in the APP S menu w ill produce the f ollow ing menu list o f input/output func tions The se appli cations ar e descr ibed next: Send to C alculat or Se nd data to another calc ulator (or to a PC w ith an infr ared port) Get fr om Calculator Recei ve data fr om another calculato [...]

Pa g e F - 3 The C onstants Libr ar y is disc ussed in detail in C hapter 3 . Numeric solv er .. Selecting opti on 3. C onstants lib .. in the APP S menu pr oduces the numer ical solver menu : This oper ation is equi valent to the k ey strok e sequence ‚Ï . The nu meri cal sol ver men u is present ed in detail in Chapt ers 6 and 7 . Time & d[...]

Pa g e F - 4 Equation wr iter .. Selecting opti on 6.E quation w riter .. in the AP P S menu opens the equati on writ er: This oper ation is eq ui val ent to the k ey str oke s equence ‚O . The eq uation wr iter is intr oduced in detail in Chapte r 2 . Examples that u se the equatio n wr iter are a vailable thr oughout this guide. F ile manager .[...]

Pa g e F - 5 M atr ix W riter .. Selecting opti on 8.Matri x W riter .. in the APP S menu launc hes the matri x wr iter: This oper ation is eq ui val ent to the k ey str oke s equence „² .T he Matri x W rit er is pre sent ed in detail in Chapter 10. T e xt editor .. Selecting opti on 9 .T ext editor .. in the APP S men u launches the line te xt [...]

Pa g e F - 6 This oper ation is eq ui val ent to the k ey str oke s equence „´ . T he MTH menu is intr oduced in Chapte r 3 (real n umb er s) . Other f uncti ons fr om the MTH menu ar e pr esented in Chapters 4 (comple x numbers) , 8 (lists) , 9 (vec tors) , 10 (matri x cr eation), 11 (matri x operatio n), 16 (f ast F our ier tr ansfor ms), 17 ([...]

Pa g e F - 7 Note that flag –117 should be set if y ou are go ing to use the E quatio n Libr ary . Note too that the E quation L ibrary will onl y appear on the APP S menu if the two E quation L ibrary files ar e stor ed on the calculator . The E quation L ibrary is explained in det ail in chapter 2 7 .[...]

P age G-1 Appendix G Useful shortc uts Pr esented her ein ar e a number of k eyboar d shortcuts commonl y used in the calc ulator : Θ Adjust di splay co ntrast: $ (hold) + , or $ (hold) - Θ T oggle between RPN and AL G modes: H @@@OK@@ or H` . Θ Set/c lear sy stem flag 9 5 (AL G v s. RPN oper ating mode) H @) FLAGS —„—„—„ — @@CHK[...]

P age G-2 Θ Set/clear s yst em flag 117 (CHOO S E bo xes vs . SOFT menu s): H @) FLAGS —„ — ˜ @@CHK@ Θ In AL G mode , SF(-117) selects S OFT menus CF(-117) se lects CHOO SE BOXE S . Θ In RPN mode, 117 ` SF selec ts SOFT men us 117 ` CF selects S OFT menus Θ Change angular measur e: o T o degr ees: ~~deg` o T o r adian: ~~rad` Θ Spec i[...]

P age G-3 Θ S ystem-lev el op er ation (H old $ , r elease it after enter ing second or thir d k e y): o $ (hold) AF : “Cold” r estart - all memory er ased o $ (hold) B : C ancels ke ys tr ok e o $ (hold) C : “W arm ” r estart - memor y pr eserved o $ (hold) D : S tar ts inter acti ve self-test o $ (hold) E : S tar ts continuou s self -tes[...]

P age H-1 Appendix H T he CAS help facilit y The CA S help fac ility is available thr ough the ke ystr oke seq uence I L @HELP ` . The fo llo w ing scr een shots sho w the fir st menu page i n the listing of th e CAS help fac ili ty . The commands ar e listed in alphabeti cal or der . Using the v ertical arr ow k ey s —˜ one can na vigat e thr o[...]

P age H-2 Θ Y ou can t ype two or more letters of the com mand of interest, b y locking the alphabeti c ke yboar d. T his will t ake y ou to the command of int eres t , or to its neighbor hood. Afterwar d s, y ou need to unlock the alpha k ey board , and use the ve r tical ar r o w ke ys —˜ to locate the command , if needed. Pr ess @@OK@@ to lo[...]

Pa g e I - 1 Appendix I Command catalog list This is a list o f all commands in the command catalog ( ‚N ) . Those commands that belong to the CA S (C omputer A lgebrai c Sy stem) ar e listed also in Appendi x H. CAS help fac ility entrie s are av ailable for a gi ven command if the soft menu k ey @HELP sho ws up w hen yo u highligh t that partic[...]

Pa g e J - 1 Appendix J T he MA THS menu The MA THS menu , accessible thr ough the command MA THS (a vailable in the catalog N ), contains the follo w ing sub-menu s: The CMP LX sub-menu The CMP L X sub-men u contains functi ons per tinent to oper ations with comple x numbers: The se fu nctions ar e descr ibed in Chapter 4. The CONS T ANT S sub-men[...]

Pa g e J - 2 The HYP ERBOLI C sub-menu The HYP ERBOLIC sub-menu co ntains the h yperboli c functi ons and their in ver ses . The se func tions ar e descr ibed in Chapter 3 . The INTEGER sub-menu The INTE GER sub-menu pr ov ides functi ons f or manipulating integer numbers and some poly nomials. T hese functi ons are pr esent ed in Chapter 5: The MO[...]

Pa g e J - 3 The P OL YNOMIAL sub-menu The P OL YNO MIAL sub-menu inc ludes func tions for gener ating and manipulating poly nomials . The se func tions ar e pres ented in Chapt er 5: The TE ST S sub-m enu The TE S TS su b-menu inc ludes r elational oper ators (e .g., ==, <, etc .) , logical oper ators (e .g., AND , OR, etc .) , the IFTE functio[...]

Pa g e K- 1 Appendix K Th e M A I N m en u The MAIN men u is av ailable in the command catalog . This men u include the fo llow ing sub-menu s: The CA SCF G command This is the f irst entry in the MAIN menu . This command conf igure s the CAS . F or CAS conf igur ation inf ormatio n see Appendi x C. The AL GB sub-menu The AL GB sub-men u includes t[...]

Pa g e K- 2 The DIFF sub-menu The DIFF sub-menu contains the fo llo w ing funct ions: The se func tions ar e also av ailable thr ough the CAL C/DIFF sub-menu (start with „Ö ). T hese f uncti ons ar e desc r ibed in Chapter s 13, 14 , and 15, ex cept fo r functi on TRUNC, whi ch is desc ribed next u sing its CAS help f acility entry: The MA THS s[...]

Pa g e K- 3 The se fu nctions ar e also av ailable in the TRIG menu ( ‚Ñ ) . Description of these f unctions is incl uded in Chapter 5 . The S OL VER sub-menu The S OL VER men u includes the follo w ing functi ons: The se fu nctions ar e av ailable in the CAL C/S OL VE menu (st art with „Ö ). The f unctions ar e descr ibed in Chapter s 6 , 11[...]

Pa g e K- 4 The su b-menus INTE GER, MODULAR, and POL YNOMIAL ar e pr esented in detail in Appendi x J. The E XP &LN sub-menu The EXP &L N menu contains the f ollow ing functi ons: This men u is also acces sible thr ough the k eyboar d by using „Ð . T he functi ons in this menu are pr esented in Chapter 5 . The MA TR sub-menu The MA TR m[...]

Pa g e K- 5 The se f unctio ns are av ailable through the C ONVERT/REWR ITE me nu (start w ith „Ú ) . T h e f unctio ns ar e pres ented in Chapt er 5, e xcept f or func tions XNUM and XQ, w hich ar e descr ibed next u sing the corr esponding entr ies in the CAS help f acility ( IL @HELP ): XNUM X Q[...]

Pa g e L- 1 Appendix L L ine editor commands When y ou trigger the line editor b y using „˜ in the RPN stack or in AL G mode , the follo w ing soft menu f u ncti ons ar e pro vided (pr ess L to see the r emaining func tions): The f unctions ar e brief ly desc ribed as fo llo ws:  SKIP: Skips char acter s to beginning of w ord . SKIP  : Ski[...]

Pa g e L- 2 The it ems sho w in this scr een are self-e xplanatory . F or ex ample , X and Y positions mean the positi on on a line (X) and the line number (Y). Stk Size means the number of obj ects in the AL G mode history or in the RPN stac k. Mem(KB) means the amount of fr ee memory . Clip Siz e is the number of char acter s in the clipboar d. S[...]

Pa g e L- 3 The SE ARCH sub-menu The f unctions 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 vi ded w ith this command is show n next: Rep la c e : Use this command t o fi nd and replace a str ing. T he input f orm pr o vi ded for this command is: F ind next .. : Finds the ne[...]

Pa g e L- 4 The GO T O sub-menu The f unctions in the GO T O sub-men u are the f ollow ing: Goto L ine: to mo ve to a spec ifie d line. T he input fo rm pr ov ided w ith this command is: Goto P osition : mo ve to a spec ified positi on in the command line. The input for m pro vided f or this command is: Labe ls : mo ve t o a spec ified label in the[...]

Pa g e L- 5[...]

Pa g e M - 1 Appendix M T able of Built-In Equations The E quation L ibrary consists o f 15 subj ects corr esponding to the s ections in the table belo w) and more than 100 titles. T he numbers in par e ntheses below indicate the n u mber of eq uations in the set and the number of v ariables in the set . Ther e are 315 equations in total us ing 3 9[...]

Pa g e M - 2 3: Fluids (29 , 2 9 ) 1: Pr essur e at D epth (1, 4) 3: F low w ith Lo sses (10, 17) 2 : Bernoulli E quation (10, 15) 4: Flo w in F ull P ipes (8 , 19) 4 : Forces and Energy ( 3 1 , 36) 1: L inear Mechanics (8 , 11) 5 : ID Elastic Collisi ons (2 , 5 ) 2 : Angular Mec hanics (12 , 15) 6: Dr ag F orce (1, 5 ) 3: Centripe tal Fo rce ( 4, [...]

Pa g e M - 3 9: Op tics ( 1 1 , 1 4) 1: La w of Ref racti on (1, 4) 4: Spher ical Ref lection (3, 5 ) 2 : Criti cal Angle (1, 3) 5: Spheri cal Refr action (1, 5) 3: Bre wst er’s L aw (2 , 4) 6: Thin L ens (3, 7) 1 0: Oscillations (1 7 , 1 7) 1: Mass–S pring S ys tem (1, 4) 4: T orsional P endulum (3, 7) 2 : Simple P endulum (3, 4) 5: Simp le Ha[...]

Pa g e N - 1 Appendix 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 ALRM[...]

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 norm 11- 7 Column vectors 9-18 COL- 10-20 COMB 17- 2 Combinations 17-1 Command catalog list I-1 Complex CAS mode C-6 Complex Fourier series 16-26 C[...]

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 pr ograms 22-22 DRAX 22-4 DROITE 4-9 DROP 9-20 DTAG 23-1 [...]

Pa g e N - 6 ERRN 21-65 Error trapping in programming 21-64 Errors in hypothesis testing 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 Ext[...]

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 variabl e 21-2 Glo[...]

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 Histograms 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 differenti al 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 menu 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 P[...]

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 RE[...]

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 5-29 Simplif[...]

Pa g e N - 1 7 Stiff differential equations 16-67 Stiff ODE 16-66 Stiff ODEs numerical solution 16-67 STOALARM 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, 5[...]

Pa g e N - 1 8 TINC 3-34 TITLE 7-14 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 Transpose 10 -1 Triangle solution 7-9 Triangular wave Fourier series 16-34 TRIG menu 5-8 Trigonometric function[...]

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 Limited W arranty HP 50g gr aphing calculator ; W arr anty period: 12 months 1. HP war rants to y ou, the end-us er cu stomer , that HP har dw a r e, access ori es and supplies w ill be fr ee fr om defects in mat er ials and w orkmanship after the dat e of pur chase , for the per iod s pecif ied abo ve . If HP r e cei ves noti ce of su[...]

Pa g e LW- 2 W ARRANTY S T A TEMENT ARE Y OUR SOLE AND E X CL USI VE REMED IE S. EX CEPT A S INDICA TED ABO VE , IN NO EVENT WILL HP OR I T S S UPPLIER S BE LIABLE FOR L O S S OF D A T A OR FOR DIRE CT , SP E CIAL, INCIDENT AL, C ONSE QUENTIAL (INCL UDING L OS T PR OFIT OR D A T A ) , OR O THER D AMA GE , WHETHER B ASED IN C ONTRAC T , T OR T , OR [...]

Pa g e LW- 3 Swi t ze r la n d +41-1-43 9 5 35 8 (Ger man) + 4 1 -2 2- 8 27878 0 ( F r e n c h ) +3 9-0 2 - 7 5 419 7 8 2 (Itali an) T urk ey +4 20 -5 - 414 2 25 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 fri ca +2 7 -11- 2 3 7 6 200 Lu xembour g +3 2 - 2 -712 6 219 Other Eu r opean countr ies +4 20 -5- 414 22 5[...]

Pa g e LW- 4 Regulatory infor mation Fe deral Communications Commission Notic e This eq uipment has bee n test ed and found t o compl y with the limits f or a Class B digital de vice , pursuant t o P art 15 of the FCC R ules. T hese limits are de signed to pr ov ide r e asona ble pr otecti on against harmf ul interfer ence in a resi dential install[...]

Pa g e LW- 5 This de vi ce complies with P ar t 15 of the FCC Rules . Operati on is s ub ject to the follo wing two conditi ons: (1) this dev ice may not cause harmf ul interfer ence, and (2) this dev ice must accept an y interf er ence rece iv ed, incl uding interfer ence that may cau se undesir ed operation . F or ques tions r ega r ding y our pr[...]

Pa g e LW- 6 This compliance is indi c ated b y the fo llow ing conformity marking placed on the pr oduc t: Japanese No tice 䈖 䈱ⵝ⟎䈲䇮 ᖱႎಣℂⵝ⟎╬㔚ᵄ㓚ኂ⥄ਥⷙ೙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 ?[...]