HP 40gs manuale d’uso

HP 4 0gs gr aphing calc ulator user's guide Ed i t io n 1 P ar t Number F2 22 5AA -90001 hp40g+.book Page i Friday, December 9, 2005 1:03 AM[...]

Notice REG ISTER Y OUR PRODUCT A T: w ww .register . hp.com THI S MANUAL AND ANY EXAMPLES C ONT AINED HEREI N ARE PRO VI DED "AS I S" AND ARE SUBJECT TO CHANGE WITHOUT NO TICE. HEWLETT-P A CKARD COMP ANY MAKE S NO W AR- RANT Y O F ANY KIND WITH R EGARD T O THI S MANUAL , INCL UDING , BU T N O T LI MIT ED T O, TH E I MP LIE D W A RRAN T IE[...]

iii Contents Preface Manual conventions ................ ................ ................. ............. P-1 Notice ................ ................ ................ ................ ................. P-2 1 Getting started On/off, cancel operatio ns ........... .................... ................ ....... 1-1 The display ...................... .[...]

iv Function aplet interactive a nalysis .... .................... ................ ... 3-9 Plotting a piecewise -defined function ........ .................... .... 3-12 4 Parametric aplet About the Pa rametric aplet .......... ................ ................ .......... 4-1 Getting started with the Parame tric aplet ............. ................[...]

v About the Inference aplet ............................. ................ ........ 11-1 Getting started with the Infe rence aplet ........................... .. 11-1 Importing sample statistics from the S tatistics aplet ......... ..... 11-4 Hypothesis tests ...................... ................ ................. ........... 1 1-8 One-Sample Z-T[...]

vi Symbolic calculations .. ................ ................ ................ ...... 13-20 Finding derivatives .............. ................ ................ ......... 13-21 Program constants and physical constants ......................... .. 13-24 Program constants ........ ................ .................... ............ 13 -25 Physical con[...]

vii Accessing CAS function s .................... ................ ................ 15-12 Equation Writer variable s . ................ ................ ................ 1 5-16 Predefined CAS variables ...... .................... ................ ... 15-16 The keyboard in the Eq uation Writer ........ ................ ...... 15-17 16 Step-by-Step E[...]

viii Aplet naming convention ........ ................ ................ ...... 21-10 Example .................. ................ ................ ................. .. 21-10 Programming commands ................... ................. ............... 21-13 Aplet commands .... ................ ................... ................. .. 21-14 Branch comma[...]

ix Solve aplet variables ........... ................ ................. ........... R -11 Statistics aplet variables ............. ................... ................ .. R-12 MATH menu categ ories . ................. ................ ................ ..... R-13 Math functions ... ................ ................ ................. ........... R-13[...]

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P-1 Preface The HP 40gs is a feature-rich graphing calculator. It is also a powerful mathematics learning tool, with a built-in computer algebra system (CAS). The HP 40gs is designed so that you can use it to explore mathematical functions and their properties. You can get more information on the HP 40gs from Hewlett-Packard’s Calcula tors web si[...]

P-2 Notice This manual and any examples contained herein are provided as-is and are subject to change without notice. Except to the extent prohibit ed by law, Hewlett-Packard Company makes no express or implied warranty of any kind with regard to this manu al and specifically di sclaims the implied warranties and conditions of merchantability and f[...]

Getting started 1-1 1 Get ting star ted On/off, cancel operations To turn on Press to turn on the calculator. To cancel When the calculator is on, the key cancels the current operation. To turn off Press OFF to turn the calculator off. To save power, the calculat or turns itself off after several minutes of inactivity. All stored and displayed info[...]

1-2 Getting started The display To adjust the contrast Simultaneously press and (or ) to increase (or decrease) the contrast. To clear the display • Pres s CANCEL to clear the edit line . • Pres s CLEAR to cle ar the edit line and the display history . Parts of the display Menu key or soft key labels. The la bels for the menu k ey s ’ cur r e[...]

Getting started 1-3 Annunciators . Annunciators are sy mbo ls that appear above the title bar and give you important status information. The keyboard Annunciator Description Shift in effect for next keystroke. To cancel, press again. α Alpha in effect for next keystroke. To cancel, press again. (( • )) Low battery power. Busy. Data is being tran[...]

1-4 Getting started Menu keys • On the calculato r ke yboar d , the top ro w of k ey s ar e called menu k ey s. T heir meanings depend on the conte xt—that’s w hy the y ar e blank. T he menu k e y s are so metimes called “ soft k ey s” . • The bo ttom line of the displa y sho ws the labels f or the menu k ey s ’ cur rent meanings . Ap[...]

Getting started 1-5 Entry/Edit keys The entry and edit keys are: K ey Meaning ( CANCEL ) Cancels the current operation if the calculator is on by pressing . Pressing , then OFF turns the calculator off. Accesses the function printed in blue above a key. Returns to the HOME view, for performing calculations. Accesses the alphabetical characters prin[...]

1-6 Getting started Shifted keystr okes There are two shift keys that you use to access the operations and characters printed above the keys: and . CHARS Displays a menu of all avai lable characters. To type one, use the arrow keys to highlight it, and press . To select multiple characters, select each and press , then press . Ke y Meaning (Continu[...]

Getting started 1-7 HELPWITH The HP 40gs built-in help is available i n HOME only. It provides syntax help for bu ilt-in math functions. Access the HELPWITH command by pressing SYNTAX and then the math key for which you require syntax help. Example Pr ess SYNTAX Note: R emov e the left paren thesis fr om built-in functi ons suc h as sine, co sine, [...]

1-8 Getting started • Pr essing display s the list of Pr ogr am Const ants. Y ou can use the se in pr ograms that you d eve l op. • Pr essing display s a menu of ph ys ical constants fr om the fields of c hemistry , physi cs, and quantum mechani cs. Y o u can use these constan ts in calculati ons . (pSee “Ph y sical constants ” on page 13- [...]

Getting started 1-9 To search a menu • Pres s or to scr oll through the list. If y ou pres s or , y ou’ll go all the w ay to the end or the beginning of the list . Hi ghlight the item y ou want to s elect , then pres s (or ). • If there ar e two columns , the left column shows gener al categor ies and the r ight column sho ws spec ifi c conte[...]

1-10 Getting started Mode settings You use the Modes input form to set the modes for HOME. HINT Although the numeric setting in Modes affects only HOME, the angle setting controls HOME and the current aplet. The angle setting selected in Modes is the angle setting used in both HOME and current aplet. To further configure an aplet, you use the SETUP[...]

Getting started 1-11 Setting a mode This example demonstrates how to change the angle measure from the default mode, radians, to degrees for the current aplet. The procedure is the same for changing number format and decimal mark modes. 1. Pres s MODES t o o p e n t h e H O M E M O D ES i n p u t form. Engineering . Displays result with an exponent[...]

1-12 Getting started The c ursor (hi ghlight) is in the firs t fie ld, Angle Measure . 2 . Pres s to display a lis t of choices. 3. P re s s to sel ect Degrees , and pr ess . The angle measur e changes to degrees. 4. Pres s to return to HOME . HINT Whenever an input form has a list of choices for a field, you can press to cycle through them instead[...]

Getting started 1-13 symbolic views of the aplets in the following table. See “Aplet view configuration” on page 1-18 for further information. In addition to these aplets, wh ich can be used in a variety of applications, the HP 40 gs is supplied with two teaching aplets: Quad Explorer and Trig Explorer. You cannot modify configuration settings [...]

1-14 Getting started charge and transferred to the HP 40gs using the provided Connectivity Kit. Quad Explorer aplet The Quad Explorer aplet is used to investigate the behaviour of as the values of a , h and v change, both by manipulati ng the equation and seeing the change in the graph, and by manipulating the graph and seeing the change in the equ[...]

Getting started 1-15 Trig Explorer aplet The Trig Explorer aplet is used to investigate the behaviour of the graph of as the values of a , b , c and d change, both by manipulating the equation and seeing the change in the graph, or by manipulating the graph and seeing the change in the equation. Press , select Trig Explorer , and then press to disp[...]

1-16 Getting started Aplet library Aplets are stored in the Aplet library. To open an aplet Press to display the Aplet library menu. Select the aplet and press or . From within an aplet, you can return to HOME any time by pressing . Aplet views When you have configured an aplet to define the relation or data that you want to explore, you can displa[...]

Getting started 1-17 Numeric view Press to display the aplet’s Numeric view. In this view, the functions that you have defined are displayed in tabular format. See “About the numeric view” on page 2-16 f or further information. Plot-Table view The VIEWS menu contains the Plot-Table view. Select Plot-Table Splits the screen into the plot and t[...]

1-18 Getting started Note view Press NOTE to display the aplet’s note view. This note is transferred with the aplet if it is sent to another calculator or to a PC. A note view contains text to supplement an aplet. See “Notes and sketches” on pa ge 20-1 for further information. Sketch view Press SKETCH to disp lay the ap let’s sket ch view. [...]

Getting started 1-19 To change views Each view is a separate environment. To change a view, select a different view by pressing , , keys or select a view from the VIEWS menu. To change to HOME, press . You do not explicitly close the current view, you just ente r another one—like passing from one room into another in a house. Data that you enter [...]

1-20 Getting started Example Calculate : Long results If the result is too long to fit on the display line, or if you want to see an expression in textbook format, press to highlight it and then press . Negative numbers Type to start a negative number or to insert a negative sign. To raise a negative number to a power, enclose it in parentheses. Fo[...]

Getting started 1-21 However, for clarity, it is better to include the multiplication sign where you expect multiplication in an expression. It is clearest to enter AB as A*B . HINT Implied multiplication will not always work as expected. For example, entering A(B+4) will not give A*(B+4) . Instead an error message is displayed: “Invalid User Fun[...]

1-22 Getting started Algebraic precedence order of evaluation Functions within an expression are evaluated in the following order of precedence. Functions with the same precedence are evaluated in order from left to right. 1. Ex pressions within p arentheses. Neste d parenth eses are e valuated fr om inner to outer . 2 . Pr efi x func tions, suc h [...]

Getting started 1-23 When you highlight a previous input or result (by pressing ), the and menu labels appear. To copy a previous line Highlight the line (press ) and press . The number (or expression) is co pied into the ed it line. To reuse the last result Press ANS (last answer) to put the last result from the HOME display into an expression. AN[...]

1-24 Getting started HINT When you retrieve a number from ANS , you obtain the result to its full precision. When you retrieve a number from the HOME’s display history, you obtain exactly what was displayed. Pressing evaluates (or re-evaluates) the last input, whereas pressing ANS copies the last result (as ANS ) into the edit line. Storing a val[...]

Getting started 1-25 Accessing the display history Pressing enables the highlight bar in the display history. While the highlight bar is active, the following menu and keyboard keys are very useful: Clearing the display history It’s a good habit to cl ear the display history ( CLEAR ) whenever you have finish ed working in HOME. It saves calculat[...]

1-26 Getting started 2 . Select Number Format , press to display the options , and highlight Fraction or Mixed Fraction . 3 . Press to selec t the Number F ormat option, then mov e to the precisi on value field . 4. Enter the prec ision v alue that you w ant to use , and pre ss to set the pr ec ision . Pr ess to r etur n to HOME . See “Setting fr[...]

Getting started 1-27 • Prec ision set to 1: • Prec ision set to 2 : • Prec ision set to 3: • Prec ision set to 4 Fraction calculations When entering fractions: • Y ou use the ke y to separate the numerator part and the denominator par t of the fr action . • T o enter a mi xed f rac tion , fo r ex ample , 1 1 / 2 , you enter it in the fo[...]

1-28 Getting started 2. E n t e r t h e c a l c u l a t i o n . 32 3 45 7 8 Note: Ensur e y ou ar e in the HOME v ie w . 3 . Ev aluate the calc ulation. Note that if you had selected Mixed Fraction instead of Fraction as the Number format, the answer would have been expressed as 25+7/8. Converting decimals to fractions To convert a decimal value to[...]

Getting started 1-29 In this ex ample , the fr action pr ec ision is set to 6. Complex numbers Complex results The HP 40gs can return a complex number as a result for some math functions. A comp lex number appears as an ordered pair ( x, y ), where x is the real part and y is the imaginary part. For example, entering returns (0,1). To enter complex[...]

1-30 Getting started Catalogs and editors The HP 40gs has several catalogs and editors. You use them to create and manipulate objects. They acc ess features and stored values (numbe rs or text or other items) that are independent of aplets. • A catalog lists items, w hich y ou can delete or trans mit , for e xam ple an aplet . • An editor lets [...]

Aplets and their views 2-1 2 Aplets and th eir vie ws Aplet views This section examines the options and functionality of the three main views for the Functio n, Polar, Parametric, and Sequence aplets: Symbolic, Plot, and Numeric views. About the Symbolic view The Symbolic view is the defining view for the Function, Parametric, Polar, and Seque nce [...]

2-2 Aplets and their views – For a Function definition , en ter an ex pre ssion t o def ine F(X) . The only independent variab le in th e exp re ss io n i s X. – For a P arametric definition , en ter a pair of expr essi ons to def ine X(T) and Y(T) . The onl y independent var iable in the e xpre ssions is T . – For a Pol ar definition , en te[...]

Aplets and their views 2-3 Evaluating expressions In aplets In the Symbolic view, a variable is a symbol only, and does not represent one specif ic value. To evaluate a function in Symbolic view, press . If a function calls another function, then resolves all references to other functions in terms of their independent variab le. 1. Choos e the F un[...]

2-4 Aplets and their views In HOME You can also evaluate any expression in HOME by entering it into the edit line and pressing . For example, define F4 as below. In HOME, type F4(9) and press . This evaluates the expression, substituting 9 in place of X into F4 . SYMB view keys The following table details the menu keys that you use to work with the[...]

Aplets and their views 2-5 About the Plot view After entering and selecting (check marking) the expression in the Symbolic view, press . To adjust the appearance of the graph or the interval that is displayed, you can change the Plot view settings. You can plot up to ten expressions at the same time. Select the expressions you want to be plotted to[...]

2-6 Aplets and their views Plot view settings The plot view settings are: Those items with space for a checkmark are settings you can turn on or off. Press to display the second page. Field Meaning XRNG, YRNG Specifies the minimum and maximum horizontal ( X ) and vertical ( Y ) values for the plotting window. RES For function plots: Resolution; “[...]

Aplets and their views 2-7 Reset plot settings To reset the default values for all plot settings, press CLEAR in the Plot Setup view. To reset the default value for a field, highlight the field, and press . Exploring the graph Pl ot v ie w gi v es yo u a s e le c ti on o f k e ys a nd me n u k e ys t o explore a graph further. The options vary from[...]

2-8 Aplets and their views Trace a graph You can trace along a function using the or key which moves the cursor along the graph. The display also shows the current coordinate position ( x, y ) of the cursor. Trace mode and the coordinate disp lay are automatically set when a plot is drawn. Note: Tracing might not appe ar to exactly follow your plot[...]

Aplets and their views 2-9 To jump directly to a value To jump straight to a value rather than using the Trace function, use the menu key. Press , then enter a value. Press to jump to the value. To turn trace on/off If the menu labels are no t displayed, press first. • T urn off trace mode b y pres sing . • T urn on trace mode by pr essing . ?[...]

2-10 Aplets and their views Y-Zoom In Di vides vertical scale only, using Y-factor. Y-Zoom Out Multiplies vertical scale only, using Y-factor. Square Changes the vertical scale to match the horizontal scale. (Use this after doing a Box Zoom, X-Zoo m, or Y-Zoom.) Set Factors... Sets the X-Zoom and Y-Zoom factors for zooming in or zooming out. Includ[...]

Aplets and their views 2-11 ZOOM examples The following screens show the effects of zooming options on a plot of . Plot of Zoom In : In Un-zoom : Un-zoom Note: Press to move to the bottom of the Zoom list. Zoom Out : Out Now un- zoom. X-Zoom In : X-Zoom In Now un- zoom. X-Zoom Out : X-Zoom Out Now un- zoom. Un-zoom Returns the display to the previo[...]

2-12 Aplets and their views Y-Zoom In: Y-Zoom In Now un-zoom. Y-Zoom Out: Y-Zoom Out Zoom Square: Square To box zoom The Box Zoom option lets you draw a box around the area you want to zoom in on by selecting the endpoints of one diagonal of the zoom rectangle. 1. If necessary , press to tur n on the menu-ke y labels . 2. P r e s s a n d s e l e c [...]

Aplets and their views 2-13 To set zoom factors 1. In the P lot vi ew , pre ss . 2. P r e s s . 3. Se l e c t Set Factors... and pre ss . 4. Enter the z oom fac tors . Ther e is one z oom facto r for the horiz onta l scal e ( XZOOM ) and one for the v ertical scal e ( YZOOM ). Z ooming out m ultiplies the scale b y the fa ctor , so that a greater s[...]

2-14 Aplets and their views Split the screen The Plot-Detail view can give you two simultaneous views of the plot. 1. Pres s . Select Plot-Detai l and press . The gr aph is plotted t wi ce. Y ou can now z o om in on the ri ght side . 2. P r e s s , select the z oom method and press or . This z ooms the ri ght side . Her e is an ex ample o f split s[...]

Aplets and their views 2-15 – mov es the leftmost cur sor to the scr een’s le ft edge and mov es the ri ghtmost c urs or to the scr een’s r ight edge . – The menu k ey cop ies the r ight plo t to the left plot . 3 . T o un -split the sc reen , pr ess . T he left side tak es ov er the whole scr een. The Plot-Table view gives you two simultan[...]

2-16 Aplets and their views About the numeric view After entering and selecting (check marking) the expression or expressions that you want to explore in the Symbolic view, press to view a table of data values for the independent variable ( X , T, θ , or N ) and dependent variables. Setting up the table (Numeric view setup) Press NUM to define any[...]

Aplets and their views 2-17 Reset numeric settings To reset the default values for all table settings, press CLEAR . Exploring the table of numbers NUM view menu keys The following table details the menu keys that you use to work with the table of numbers. NUMTYPE Type of numeric table: Automatic or Build Your Own. To build your own table, you must[...]

2-18 Aplets and their views Zoom within a table Zooming redraws the table of numbers in greater or lesser detail. ZOOM options The following table lists the zoom options: The display on the right is a Zoom In of the display on the left. The ZOOM factor is 4. HINT To jump to an independent va riable value in the table, use the arrow keys to place th[...]

Aplets and their views 2-19 Automatic recalculation You can enter any new value in the X column. Wh en you press , the values for the dependent variables are recalculated, and the entire table is regenerated with the same interval between X values. Building your own table of numbers The default NUMTYPE is “Automatic”, which fills the table with[...]

2-20 Aplets and their views “Build Your Own” menu keys Example: plotting a circle Plot the circle, x 2 + y 2 = 9 . First rearrange it to read . To plot both the positive and negative y values, you need to define two equations as follows: and 1. In the F unctio n aplet, s pecify the f uncti ons. Ke y Meaning Puts the highlighted independent valu[...]

Aplets and their views 2-21 Select Function 9 9 2 . R eset the gr aph se tup to the def ault settings . SETUP - PLOT CLEAR 3 . P lot the two func tions and hide the menu so that yo u can see all the ci rcl e. 4. Re set the numer ic setu p to the default s ettings. SETUP - NUM CLEAR 5 . Display the functi ons in numer ic for m. hp40g+.book Page 21 F[...]

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Function aplet 3-1 3 Function aplet About the Function aplet The Function aplet enables you to explore up to 10 real-valued, rectangular functions y in terms of x . For example . Once you have defined a function you can: • cr eate gr aphs to find r oots, in ter cepts, slope , signed area , and extr ema • cr eate tables to ev aluate functi ons a[...]

3-2 Function aplet Define the expressions 2 . T her e are 10 f unction de finiti on fie lds on the F unction aplet’s S ymbolic v ie w scr een . The y ar e labeled F1(X) to F0(X). Highlight the f unction de finiti on fi eld yo u wan t to us e , and ente r an ex pre ssio n. ( Y ou can pr ess to delete an ex isting line , or CLEAR to clear all lines[...]

Function aplet 3-3 Change the scale 6. Y ou ca n change the sc ale to see more or less of your gra phs. In this e xample , choose Auto Scale . (See “VIEW S menu options ” on page 2-13 f or a descrip tio n of Auto Sc al e) . Select Auto Scale Trace a graph 7 . T race the linear fu nction . 6 times Note: B y def ault, the trace r is activ e . 8. [...]

3-4 Function aplet Analyse graph with FCN functions 9. Display the Plot view menu. From the Plot view menu, you can use the functions on the FCN menu to find roots, intersections, slopes, and areas for a function defined in the Function aplet (and any Function-based ap lets). The FCN functions act on the currently selected graph. See “FCN functi [...]

Function aplet 3-5 12 . Choos e the linear functi on wh ose int ersec tion w ith the quadr atic func tion y ou w ish to find . The coo rdinate s of the intersec tion po int are display ed at the bottom of the screen . Note: If ther e is mor e than one inters ection (as in our e xam ple), the coordinates o f the inters ection po int clo sest to the [...]

3-6 Function aplet 16. Pr ess to accept using F2(x) = (x + 3) 2 – 2 as the other boundar y for the integr al. 17 . Choos e the end v alue for x . 1 The cu rso r jum ps to x = – 1 on the linear functi on. 18. Display the numerical value of the integral. Note: See “Shading ar ea” o n page 3-11 for another method of calculating ar ea. To find [...]

Function aplet 3-7 HINT The Root and Extremum functions return one value only even if the fun ction has more than one root or extremum. The function finds the value closest to the position of the cursor. You need to re-locate the cursor to find other roots or extrema that may exist. Display the numeric view 20. Display the numer ic v iew . Set up t[...]

3-8 Function aplet To navigate around a table 2 4. Mov e to X = –5 .9 . 6 times To go directly to a value 2 5 . Mov e direc tly to X = 10. 1 0 To access the zoom options 2 6. Zoom in on X = 10 by a factor of 4. No te: NUMZOOM has a setting of 4 . In To change font size 2 7 . Display table n umbers in lar ge fo nt . To display the symbolic definit[...]

Function aplet 3-9 Function aplet interactive analysis From the Plot view ( ), you can use the functions on the FCN menu to find roots, intersections, slopes, and areas for a function defined in the Function aplet (and any Function-based aplets). See “FCN functions” on page 3- 10. The FCN operations act on the currently selected graph. The resu[...]

3-10 Function aplet FCN functions The FCN functions are: Function Description Root Select Root to find the root of the current function nearest the cursor. If no root is fo und, but only an extremum, then the result is labeled EXTR: instead of ROOT: . (The root-finder is also used in the Solve aplet. See also “Interpreting results” on page 7-6.[...]

Function aplet 3-11 Shading area You can shade a selected area between functions. This process also gives yo u an approximate measurement o f the area shaded. 1. Open the Function aplet . The F unction aplet opens in the S ymboli c vi ew . 2 . Se lect the ex pre ssio ns who se c urves y ou w ant to study . 3 . Pre ss to plot the functio ns. 4. Pr e[...]

3-12 Function aplet Plotting a piecewise-defined function Suppose you wanted to plot the following piecewise- defined function. 1. Open the F unctio n aplet. Select Function 2 . Highli ght the line y ou wan t to use , and enter the expr ession . (Y ou c an press to d elete an ex isting line , or CLEAR to clear all line s.) 2 CHARS ≤ 1 CHARS > [...]

Parametric aplet 4-1 4 Pa r a m e t r i c a p l e t About the Parametric aplet The Parametric aplet allows you to explore parametric equations. These are equations in which both x and y are defined as functions of t . They take the forms and . Getting started with the Parametric aplet The following example uses the parametric equations Note: This e[...]

4-2 Parametric aplet Set angle measure 3 . Set the ang le measure to degrees. MODES Select Degrees Set up the plot 4. Display the graphing options. PLOT The P lot Se tup input f orm has tw o fi elds not inc luded in the Functi on aplet, TRNG and TSTEP . TRNG spec if ies the r ange of t val u es . TSTEP specifi es the step value between t values. 5 [...]

Parametric aplet 4-3 Overlay plot 8. Plot a triangle graph over the existing circle graph. PLOT 120 Select Overlay Plot A triangle is dis play ed ra ther than a cir cle (w ithout ch anging the equation) because the c hanged value of TSTEP ensur es that points being plotted are 120 ° apart ins tead of near ly continuous . Y ou ar e able to explor e[...]

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Polar aplet 5-1 5 Po l a r a p l e t Getting started with the Polar aplet Open the Polar aplet 1. Open the P olar aplet. Se lect Polar Li ke the F uncti on aplet , the P olar aplet opens in the S ymbo lic v ie w . Define the expression 2 . Def ine the polar equati on . 2 π 2 Specify plot settings 3 . Spec ify the plot settings . In this exam ple, [...]

5-2 Polar aplet Explore the graph 5 . Displa y the Plot v ie w menu k e y labels. Th e Pl o t view o p t io n s av ailable ar e the same as those fo und in the F unction aplet . See “Explor ing the gr aph ” on page 2 - 7 for f urther information . Display the numbers 6. Displa y the table o f values f or θ and R1. The N u me ric view options a[...]

Sequence aplet 6-1 6 Sequence aplet About the Sequence aplet The Sequence aplet allows you to explore s equences. You can define a sequence named, for example, U1: • in terms of n • in terms of U1 ( n –1) •i n ter ms of U1 ( n –2) • in terms o f another sequence , for e xample , U2 ( n ) • in an y combination of the a bov e . The Sequ[...]

6-2 Sequence aplet Open the Sequence aplet 1. Open t he Sequence aplet. Select Sequence The Sequence apl et starts in the S ymboli c view . Define the expression 2 . Def ine the Fibonacc i sequence, in w hic h each term (after the fir st two) is the sum of the pr eceding two terms: , , for . In the S ymboli c v ie w of the Sequ ence aplet, highligh[...]

Sequence aplet 6-3 Plot the sequence 4. P lot the F ibonacci seque nce. 5. In Plot Setup, set the SEQPLOT option to Cobweb . SETUP - PLOT Select Cobweb Display the table 6. Di splay the table of va lues for this ex ample. hp40g+.book Page 3 Friday, December 9, 2005 1:03 AM[...]

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Solve aplet 7-1 7 Solve aplet About the Solve aplet The Solve aplet solves an equation or an expression for its unknown variable . You define an equation or expression in the symbolic view, then supply values for all the variables except one in the numeric view. Solve works only with real numbers. Note the differences between an equation and an exp[...]

7-2 Solve apl et Getting started with the Solve aplet Suppose you want to find th e acceleration needed to increase the speed of a car from 1 6.67 m/sec (60 kph) to 27.78 m/sec (100 kph) in a distance of 100 m. The equation to solve is: Open the Solve aplet 1. Open the Solve aplet . Select Solve The S olv e aplet starts in the s ymboli c vie w . De[...]

Solve aplet 7-3 4. Enter the value s for the kno wn v ari ables . 2 7 7 8 1 6 6 7 1 0 0 HINT If the Decimal Mark setting in the Modes input form ( MODES ) is set to Comma, use instead of . Solve the unknown variable 5. Sol ve f or the unkno wn v aria ble ( A ). Ther efor e, the accele rati on needed to incr ease the speed of a car fr om 16.6 7 m/se[...]

7-4 Solve apl et 6. Plo t the equation f or var iable A . Sele ct Auto Scale 7 . T r ace along the graph repr ese nting the left side o f the equation until the c ursor near s the inters ectio n. 20 times Note the v alue of A displa yed near the bottom left corner of the scr een. The P lot vi ew pr ovi des a conv enient w ay t o find an appr ox ima[...]

Solve aplet 7-5 Use an initial guess You can usually obtain a fa ster and more accurate solution if you supply an estimated value for the unknown variable before pressing . Solve starts looking for a solution at the initial guess. Bef ore plo tting, mak e sure the unknow n var iable is highligh ted in the numer ic v ie w . Plo t the equation t o he[...]

7-6 Solve apl et Interpreting results After Solve has returned a solution, press in the Numeric view for more information. You will see one of the following three messages. Press to clear the message. Messag e Condition Zero The Solve aplet found a point where both sides of the equation were equal, o r wher e the expression was zero (a root), withi[...]

Solve aplet 7-7 If Solve could not find a solution, you will see one of the following two messages. HINT It is important to check the information relating to the solve process. For example, the solution that the Solve aplet finds is not a solution, but the closest that the function gets to zero. Only by checking the i nformation will you know that [...]

7-8 Solve apl et where X is distance, V 0 is initial velocity, T is time, and A is acceleration. This is actually two equations, Y = X and Y = V 0 T + (AT 2 ) / 2 . Since this equation is quadratic for T , there can be both a positive and a negative solution. However, we are concerned only with positive so lutions, since only positive distance make[...]

Solve aplet 7-9 5. Move the cursor near the positive (right-side) intersection. This cursor value will be an initial guess for T . Pres s until the cur sor is at the intersec tion. The t wo po in ts o f inters ection sh ow that ther e are tw o solutio ns for this equati on. Ho w ev er , on ly p os i tive va lu es fo r X m ake s ense , so we w ant t[...]

7-10 Solve apl et Using variables in equations You can use any of the real variable names, A to Z and θ . Do not use variable nam es defined for other types, such as M 1 (a matrix variable). Home variables All home variables (othe r than those for aplet settings, like Xmin and Ytick ) are global , which means they are shared throughout the differe[...]

Linear Solver ap let 8-1 8 Li n e a r S ol ve r a p l e t About the Linear Solver aplet The Linear Solver aplet allows you to solve a set of linear equations. The set can contain two or three linear equations. In a two-equation set, each equation must be in the form . In a three-equation set , each equation must be in the form . You provide values [...]

8-2 Linear Solver aplet ex ample in the pr ev io us step). T o sol ve a thr ee - equation s et, pr ess . No w the input for m displa ys thr ee eq uations . If the three-equation input fo rm is displayed and you want to solve a two-equation set, press . In this example, we are going to solve the following equation set: Hence we need the three-equati[...]

Linear Solver ap let 8-3 As you enter each of the re maining know n value s, the soluti on change s. T he e x ample at the ri ght sho ws the final so lution once all the co -ef ficients a nd constants ar e enter e d for the s et of equati ons we se t out to solve. hp40g+.book Page 3 Friday, December 9, 2005 1:03 AM[...]

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Triangle S olve aplet 9-1 9 T riangle Solv e aplet About the Triangle Solver aplet The Triangle Solver aplet a llows you to determine the length of a side of a triangle, or the angle at the vertex of a triangle, from information you supply about the other lengths and/or other angles. You need to specify at leas t three of the six possible values—[...]

9-2 Triangle Solve aple t Open the Triangle Solver aplet 1. Open the T riangle Sol ver a plet. Select Triangle Solver The T riangle Solver aplet o pens. Note : if y ou hav e alr eady u sed the T r iangle Sol ver , the entries and results from the pre v ious use will still be displayed . T o start the T riangle Sol ver afr esh, c lear the pre v ious[...]

Triangle S olve aplet 9-3 lengths as B and C, w e wo uld need to spec ify the angle as α . The illus trati on on the displa y will help yo u determine where to enter the known values . Note: if you need to c hange the angle neasure mode , pres s MOD ES , change the mode , and then pres s to r eturn to t he aplet. 4. Pres s . The solv er calculat e[...]

Not enough data If you are using the general input form, you need to specify at least three values for the Triangle Solver to be able to calculate the remaining attributes of the triangle. If you specify less than three, Not enough data appears on the screen. If you are using the simplified input form (for a right- angled triangle), you must specif[...]

Statistics aplet 10-1 10 Statist ic s apl et About the Statistics aplet The Statistics aplet can store up to ten data sets at one time. It can perform one- variable or two-vari able statistical analysis of one or more sets of data. The Statistics aplet starts with the Numeric view which is used to enter data. The Symbol ic view is used to specify w[...]

10-2 Statistics aplet Open the Statistics aplet 1. Open the Statis tics aplet and c lear e xis ting data by pres sing . Select Statistics The St at i st ic s ap le t starts in the Numer ical view . At an y time the Statisti cs aplet is conf igur ed for onl y one of t wo types of stat istical explorations: on e - var iable ( ) or t wo- var iable ( )[...]

Statistics aplet 10-3 Choose fit and data columns 4. Select a fit in the S ymbolic setup v ie w . SETUP - SYMB Select Linear Y o u c an cre a t e u p t o f ive exp l o rat i on s of t wo - va ri ab l e data, named S1 to S5 . I n t h is exa m pl e, we wil l cre a te just on e : S1 . 5 . Spec ify the columns that hold the data y ou w ant to analyz e [...]

10-4 Statistics aplet Setup plot 8. Change the plotting range to ensur e all the data points ar e plotted (and select a differ ent point mark , if yo u wi s h ) . SETUP - PLOT 7 100 400 0 Plot the graph 9 . Plot the gr aph . Draw the regression curve 10. Dr aw the r egr essio n curve (a c ur ve to f it the data points). This dr aw s the regr ession[...]

Statistics aplet 10-5 Predict values 13 . T o f ind the predi cted s ales f igur e if adv ertising w er e to go up to 6 minute s: S ( to highlight Stat-Two ) (to highli ght PREDY ) 6 14. Retur n to the P lot vie w . 15 . Jump to the indi cated poin t on the regr essi on line . 6 Observe the pr edic ted y -value in the left bottom corner of the scre[...]

10-6 Statistics aplet Entering and editing statistical data The Numeric view ( ) is used to enter data into the Statistics aplet. Each column represents a variable named C0 to C9 . After entering the data, you must define the data set in the Symbolic view ( ). HINT A data column must have at least four data points to provide valid two-vari able sta[...]

Statistics aplet 10-7 Example You are measuring the height of students in a classroom to find the mean height. The first five students have the following measurements 160cm, 165cm, 170cm, 175cm, 180cm. 1. Open the Statistics apl et. Select Statistics 2 . Enter the measurement data. 160 16 5 17 0 17 5 180 Deletes the currently highlighted value. CLE[...]

10-8 Statistics aplet 3 . F ind the mean of the sample. Ensur e the / menu ke y label reads . Pr ess to see the statistic s calculated fr om the sample data in C1 . Note that the title o f the column of st atist ics is H1 . Ther e are 5 data set de finitions av ailable for one- var iable stat ist ics: H1–H5 . If data is entered in C1 , H1 i s aut[...]

Statistics aplet 10-9 To continue our example, supp ose that the heights of the rest of the students in the class are measured, but each one is rounded to the nearest of the five values first recorded. Instead of entering all the new data in C1 , we shall simply add another column, C2 , that holds the frequencies of our five data points in C1 . Dis[...]

10-10 Statistics aplet 5 . Mov e the highli ght bar into the ri ght column of the H1 definiti on and replace the frequency value o f 1 w ith the name C2 . 2 6. Re turn to the numer ic v ie w . 7 . Enter the fr equency data sho wn in the abo ve ta ble. 5 3 8 2 1 8. Displa y the computed stat ist ics. The mean height is approximately 167.63cm. 9 . Se[...]

Statistics aplet 10-11 Edit a data set In the Numeric view of the Statistics aplet , highlight the data value to change. Type a new va lue and press , or press to copy the value to the e dit line for modification. Press after modifying the value on the edit line. Delete data • T o delete a single data item, highli ght it and pres s . The v alues [...]

10-12 Statistics aplet Defining a regression model The Symbolic view includes an expression (Fit1 through Fit5) that defines the regression model, or “fit”, to use for the regression analysis of each two-variable data set. There are three ways to select a regression model: • Accept the default option to fit the d ata to a straight line. • S[...]

Statistics aplet 10-13 To define your own fit 1. In Numeri c vi ew , make sur e is set . 2 . Display the S ymbolic v iew . 3 . Highli ght the F it e xpres sion ( Fit1 , etc .) for the desired data set . 4. T ype in an e xpr ess ion and pr ess . The independent variable must be X , and the expr ession mus t not contain any unkn ow n var iables . Exa[...]

10-14 Statistics aplet Computed statistics One-variable When the data set contains an odd number of values, the data set’s median value is no t used when calculating Q1 and Q3 in the table abo ve. For example, for the following data set: { 3,5,7,8,15,16,17 } only the first three items, 3, 5, and 7 are used to calculate Q1, and only the last three[...]

Statistics aplet 10-15 Two-variable Plotting You can plot : • histogr ams ( ) • box -a nd-whisk er plots ( ) • scat ter p lots ( ). Once you have entere d your data ( ), defined your data set ( ), and defined your Fit mod el for two- variable statistics ( SETUP - SYMB ), you can plot your data. You can plot up to five scatter or box-and-whisk[...]

10-16 Statistics aplet To plot statistical data 1. In S ymbo lic v ie w ( ) , select ( ) the data sets y ou w ant to plot . 2 . F or one-vari able data ( ) , select the plo t type in Plot Setup ( SETUP - PLOT ). Highlight ST A TPLOT , pres s , select either Histogram or BoxWhisker , and pr ess . 3 . F or any plot , but espec iall y for a histogr am[...]

Statistics aplet 10-17 Scatter Plot Two-variable statistics . The numbers below the plot indicate that the cursor is at the first data point for S2, at (1, 6). Press to move to the next data point and display information about it. To connect the data points as they are plotted, checkmark CONNECT in the second page of the Plot Setup. This is not a r[...]

10-18 Statistics aplet Relative Er ror The relative error is a measure of the error between predicted values and actual va lues based on the specified Fit. A smaller number means a better fit. The relative error is stored in a variable named RELERR . The relative error provides a measure o f fit accuracy for all fits, and it does depend on the Fit [...]

Statistics aplet 10-19 For instance, the data set (1,1 ), (3,9), (4,16), (2,4) would be plotted and traced in the order (1,1), (2,4), (3,9), (4,16). Trouble-shooting a plot If you have problems plotting, check that you have the following: • The co rr ect or menu label o n (Numeri c view ) . • The cor rec t fit (r egre ssion model), if the data [...]

10-20 Statistics aplet Calculating predicted values The functions PREDX and PREDY estimate (predict ) values for X or Y given a hypothetical value for the other. The estimation is made based on the curve that has been calculated to fit the data a ccording to the specified fit. Find predicted values 1. In P lot v ie w , dra w the regr essi on c urve[...]

Statistics aplet 10-21 • Enter P RED Y( x-value ) to find the pr edic ted value o f the dependent var iable gi ven a h ypothetical independent vari ab le. You can type PREDX and PREDY into the edit line, or you can copy these function names from the MATH menu under the Stat-Two category. HINT In cases where more than one fit curve is displayed, t[...]

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Inference aplet 11-1 11 Inference aplet About the Inference aplet The Inference capabilities include calculation of confidence intervals and hy pothesis tests based on the Normal Z-distribution or Student’s t-distribution. Based on the statistics from one or two samples, y ou can test hypotheses and find confidence intervals for the following qua[...]

11-2 Inference ap let Inference aplet’s SYMB view keys The table below summarizes the options available in Symbolic view. If you choose one of the hypoth esis tests, you can choose the alternative hypothesis to test against the null hypothesis. For each test, th ere are three possible choices for an alternative hypothesis based on a quantitative [...]

Inference aplet 11-3 Select the inferential method 2. Select the Hypothesis Test inferential method. Select HYPOTH TEST 3. Define the type of test. Z–Test: 1 μ 4. Select an alternative hypothesis. μ< μ0 Enter data 5. Enter the sample statistics and population parameters. setup-NUM The table below lists the fields in this view for our curren[...]

11-4 Inference ap let By default, each field already contains a value. These values constitute the example database and are explained in the feature of this aplet. Display on-line help 6. To display the on-line help, press 7. To close the on-line help, press . Display test results in numeric format 8. Display the test results in numeric format. The[...]

Inference aplet 11-5 A calculator produces the following 6 random numbers: 0.529, 0.295, 0.952, 0.2 59, 0.925, and 0.592 Open the Statistics aplet 1. Open the Statistics aplet and reset the current settings. Select Statistics The Statistics aplet opens in the Numeric view. Enter data 2. In the C1 column, enter the random numbers produced by the cal[...]

11-6 Inference ap let Open Inference aplet 6. Open the Infere nce aplet and cle ar current settings . Select Inference Select inference method and type 7. Select an inference method. Select CONF INTERVAL 8. Select a distribution statistic type. Select T-Int: 1 μ Set up the interval calculation 9. Set up the interval calculation. Note: The default [...]

Inference aplet 11-7 Import the data 10. Import the data from the Statistics aplet. Note: The data from C1 is displayed by default. Note: Press to see the statistics before importing them into the Numeric Setup view. Also, if there is more than one aplet base d on the Statistics aplet, you are prompted to choose one. 11. Specify a 90% confidence in[...]

11-8 Inference ap let Hypothesis tests You use hypothesis tests to test the validity of hypotheses that relate to the statistical parameters of one or two populations. The tests are base d on statistics of samples of the populations. The HP 40gs hypothesis tests use the Normal Z-distribution or Student’s t-distribution to calculate probabilities.[...]

Inference aplet 11-9 Results The results are: Two-Sample Z-Test Menu name Z-Test: μ 1– μ 2 On the basis of two samples, each from a separate population, this test measures t he strength of the evidence for a selected hypothesis against the null hypothesis. The null hypothesis is that the mean of the two populations are equal (H 0 : μ 1= μ 2).[...]

11-10 Inference ap let Results The results are: One-Proportion Z-Test Menu name Z-Test: 1π On the basis of statistics from a single sample, this test measures the strength of the evidence for a selected hypothesis against the null hypothesis. The null hypothesis is that the proportion of su ccesses in the two populations is equal: H 0 : π = π 0 [...]

Inference aplet 11-11 Inputs The inputs are: Results The results are: Two-Proportion Z-Test Menu name Z-Test: π 1 – π 2 On the basis of statistics from two samples, each from a different population, the Two-Proportion Z-Test measures the strength of the evide nce for a selected hypothesis against the null hypothesis. The null hypothesis is that[...]

11-12 Inference ap let Inputs The inputs are: Results The results are: One-Sample T-Test Menu name T-Test: 1 μ The One-sample T-Test is used when the population standard deviation is not know n. On the basis of statistics from a single sample, this test measures the strength of the evidence for a se lected hypothesis against the null hypothesis. T[...]

Inference aplet 11-13 Inputs The inputs are: Results The results are: Field name Definiti on Sample mean. Sx Sample standard deviat ion. n Sample size. μ0 Hypothetical population mean. α Significance level. x Result Description Test T T-Test statistic. Prob Probability associated with the T-Test statistic. Critical T Boundary value of T associate[...]

11-14 Inference ap let Two-Sample T-Test Menu name T-Test: μ 1 – μ 2 The Two-sample T-Test is used when the population standard deviation is not know n. On the basis of statistics from two samples, each sample from a different population, this test measures the strength of the evidence for a selected hypothesis against the nu ll hypothesis. The[...]

Inference aplet 11-15 Results The results are: Confidence intervals The confidence interval calc ulations that the HP 40gs can perform are based on the Normal Z-distribution or Student’s t-distribution. One-Sample Z-Interval Menu name Z-INT: μ 1 This option uses the Normal Z-distribution to calculate a confidence interval for m, the true mean of[...]

11-16 Inference ap let Results The results are: Two-Sample Z-Interval Menu name Z-INT: μ1 – μ2 This option uses the Normal Z- distribution to calculate a confidence interval for the difference between the means of two populations, μ 1 – μ 2 , when the population standard deviations, σ 1 and σ 2 , are known. Inputs The inputs are: Results [...]

Inference aplet 11-17 One-Proportion Z-Interval Menu name Z-INT: 1 π This option uses the Normal Z-distribution to calculate a confidence interval for the proportion of successes in a population for the case in which a sample of size, n , has a number of successes, x . Inputs The inputs are: Results The results are: Two-Proportion Z-Interval Menu [...]

11-18 Inference ap let Results The results are: One-Sample T-Interval Menu name T-INT: 1 μ This option uses the Student’s t-distribution to calculate a confidence interval for m, the true mean o f a population, for the case in which the true population standard deviation, s, is unknown. Inputs The inputs are: n1 Sample 1 size . n2 Sample 2 size [...]

Inference aplet 11-19 Results The results are: Two-Sample T-Interval Menu name T-INT: μ 1 – μ 2 This option uses the Student’s t-distribution to calculate a confidence interval for the difference between the means of two populations, μ 1 – μ 2, when the population standard deviations, s1and s2, are unk nown. Inputs The inputs are: Result [...]

11-20 Inference ap let Results The results are: Result Description Critical T Critical value for T. μ Min Lower bound for μ 1 – μ 2 . μ Max Upper bound for μ 1 – μ 2 . Δ Δ hp40g+.book Page 20 Friday, December 9, 2005 1:03 AM[...]

Using the Fin ance Solver 12-1 12 Using the Finance Solver The Finance Solver, or Finance a plet , is available by using the APLET key in your calculator. Use the up and down arrow keys to select the Finance aplet. Your screen should look as follows: Press the key or the soft menu key to activate the aplet. The result ing screen shows the diff eren[...]

12-2 Using the Fina nce Solver combined amount earns interest at a certain rate. Financial calculations involving compound interest include savings accounts, mo rtgages, pension funds, leases, and annuities. Time Value of Money (TVM) calculations, as the name implies, make use of the notion that a dollar today will be worth more than a dollar somet[...]

Using the Fin ance Solver 12-3 flow diagram shows lease payments at the beginning of each period. The following cash flow diagram shows deposits into an account at the end of each period. As these cash-flow diagrams imply, there are five TVM variables: PV 1 23 4 5 FV Capitalized value o f lease } PM T PMT PMT PM T PM T PV 1 23 4 5 FV PM T PMT PMT P[...]

12-4 Using the Fina nce Solver Performing TVM calculations 1. Launc h the F inanc ial Sol ver as indi cated at the beginning of this secti on. 2 . Use the arr o w ke y s to highligh t the differ ent f ields and enter the kno wn v aria bles in the T VM calculati ons, pres sing the soft-menu ke y after enter ing each kno wn va lue. Be sur e that v al[...]

Using the Fin ance Solver 12-5 Example 1 - Loan calculations Suppose you finance the purcha se of a car with a 5-year loan at 5.5% annual intere st, compounded monthly. The purchase price of the car is $19,500 , and the down payment is $3,000. What are the required month ly payments? What is the largest loan you can afford if your maximum monthly p[...]

12-6 Using the Fina nce Solver Example 2 - Mortgage with balloon payment Suppose you have taken out a 30-year, $150, 000 house mortgage at 6.5% annual interest. You expect to sell the house in 10 years, repay ing the loan in a balloon payment. Find the size of the balloon payment, the value of the mortgage after 10 years of payment. Solution. The f[...]

Using the Fin ance Solver 12-7 Calculating Amortizations Amortization calculations, which also use the TVM variables, determine the amounts applied towards principal and interest in a payment or series of payments. To calculate amortizations: 1. Start the F inance Solv er as indicated at the beginning of t his sec tion. 2 . Set the f ollo wing TVM [...]

12-8 Using the Fina nce Solver 3 . Pre ss the soft menu k ey to amorti z e the ne w batch of pa yments . Repeat step s 1 through 3 as often as needed. Example 4 - Amortization for home mortgage For the results of Example 3, show the amortization of the next 10 years of the mortgage loan. First, press the soft menu key. Then, keeping 120 in the PAYM[...]

Using mathematical fun ctions 13-1 13 Using mathematical func tions Math functions The HP 40gs contains many math functions. The function s are grouped in categories. For example, the Matrix category contains functions for manipulating matrices. The Probability category (shown as Prob. on the MATH menu) contains functions for working with pro babil[...]

13-2 Using mathematical functio ns To select a function 1. Pres s to displa y the MA TH menu. T he categorie s appear in alph abetical or der . 2 . Pr ess or to sc ro ll thro ugh the categori es. T o jump dir ectly to a category , pr ess the f irst letter o f the category’s name. No te: Y ou do not need to pr ess fi rst . 3 . The lis t of func ti[...]

Using mathematical fun ctions 13-3 Functions common to keyboard and menus These functions are common to the keyboard and MATH menu. Keyboard functions The most frequently used functions are available directly from the keyboard. Many of the keyboard functions also accept complex numbers as arguments. π F or a descr iption , see “ p ” on page 13[...]

13-4 Using mathematical functio ns ,, , Add, Subtract, Multiply, Di vide. Also accepts complex numbers, lists and matrices. val u e1 + va lu e 2 , etc. e x Natural exponential. Also accepts complex numbers. e^ val u e Example e^5 ret u rn s 148.413159103 Natural logarithm. Also accepts complex numbers. LN ( val ue ) Example LN(1) re t u rn s 0 10 x[...]

Using mathematical fun ctions 13-5 Example ASIN(1) r eturns 90 (Degr ees mode) . ACOS Arc co sine: cos –1 x . Output range is from 0° to 180°, 0 to π , or 0 to 200 grads. Inputs and outputs depend on the current angle format. Also accepts complex numbers. Output will be complex for values outside the normal COS domain of . ACOS ( valu e ) Exam[...]

13-6 Using mathematical functio ns Example 2^8 r eturns 256 ABS Absolute value. For a co mplex number, this is . ABS ( val ue ) ABS (( x , y )) Example ABS(–1 ) r eturns 1 ABS((1,2)) r eturns 2.2360679775 Takes the n th root of x . roo t NTHROOT valu e Example 3 NTHROOT 8 r eturns 2 Calculus functions The symbols for differentiation and integrati[...]

Using mathematical fun ctions 13-7 Example (0,s1,2*X+3,X) finds the inde finite r esult 3*s1+2* (s1^2/2) See “T o find the indef inite integral u sing for mal var iables ” on page 13- 2 3 for more inf ormation o n finding indef inite integr als. TAYLOR Calculates the n th order Taylor polynomial of expression at the point where the given variab[...]

13-8 Using mathematical functio ns Example IM((3,4)) r eturns 4 RE Real part x , of a complex number, ( x, y ). RE (( x, y )) Example RE((3,4)) r eturns 3 Constants The constants available from the MATH FUNCTIONS menu are mathematical constants. These are describe d in this section. The HP 40gs has two other menus of constants: program constant s a[...]

Using mathematical fun ctions 13-9 → C Convert from Fahrenheit to Celcius. Example → C(212) r eturns 100 → F Convert from Celcius to Fahrenheit. Example → F(0) r eturns 32 → CM Convert from inches to centimeters. → IN Convert from centimeters to inches. → L Convert from US gallons to liters. → LGAL Convert from liters to US gallons.[...]

13-10 Using mathematical fun ctions COSH Hyperbolic cosine COSH ( val ue ) SINH Hyperbolic sine. SINH ( val ue ) TANH Hyperbolic tangent. TANH ( val ue ) ALOG Antilogarithm (exponential). Th is is more accurate than 10^x due to limitations of the power function. ALOG ( val ue ) EXP Natural exponential. This is more accurate than due to limitations [...]

Using mathematical fun ctions 13-11 RECURSE Provides a method of defining a sequence without using the Symbolic view of the Sequ ence aplet. If used with | (“where”), RECURSE will step through the evaluation. RECURSE( sequencename , term n , term 1 , term 2 ) Example RECURSE(U,U(N-1)*N,1,2) U1(N) Stor es a factori al-calculating f unction named[...]

13-12 Using mathematical fun ctions Example For x 4 +2x 3 –25x 2 –26x+120 : POLYEVAL([1,2,-25,-26,120],8) ret u r n s 3432 . POLYFORM Polynomial form. Creates a polynomial in variable1 from expression. POLYFORM ( expr ession , vari ab le 1 ) Example POLYFORM((X+1)^2+1,X) ret u r n s X^2+2*X+2 . POLYROOT Polynomial roots. Return s the roots for [...]

Using mathematical fun ctions 13-13 Factorial of a positive integer. For non-integers, ! = Γ (x + 1) . This calculates the gamma function. value! PERM Number of permutations (with regard to order) of n things taken r at a time: n!/(r!(n-r)! PERM ( n, r ) Example PERM(5,2) r eturns 20 . T hat is, there are 20 differ ent perm utations of f i ve thin[...]

13-14 Using mathematical fun ctions UTPT Upper-Tail Student’s t-Probability given degrees of freedom, evaluated at value. Returns the probability that the Student's t- random variable is greater than va lue. UTPT ( degr ees , valu e ) Real-number functions Some real-number functions can also take complex arguments. CEILING Smallest integer g[...]

Using mathematical fun ctions 13-15 HMS → Hours-minutes-seconds to deci mal. Converts a number or expression in H.MMSSs format (time or angle that can include fractions of a second) to x.x format (number of hours or degrees with a decimal fraction). HMS → ( H.MMSSs ) Example HMS → ( 8.30) r eturns 8. 5 → HMS Decimal to hours-minutes-seconds[...]

13-16 Using mathematical fun ctions Example 9 MOD 4 retur ns 1 % x percent of y ; that is, x /100* y . % ( x , y ) Example % (20,50) r eturns 10 %CHANGE Percent change from x to y , that is, 100( y–x )/ x . % CHANGE ( x , y ) Example % CHANGE(20,50) r eturns 150 %TOTAL Percent total : (100) y/ x . What percentage of x , is y . % TOTAL ( x , y ) E[...]

Using mathematical fun ctions 13-17 Examples SIGN (–2) ret u rn s –1 SIGN((3,4)) r eturns (.6,.8) TRUNCATE Truncates value to decimal places . Accepts complex numbers. TRUNCATE ( valu e , places ) Example TRUNCATE(2.3678,2) r eturns 2.36 XPON Exponent of value . XPON ( valu e ) Example XPON(123.4) r eturns 2 Two-variable statistics These are fu[...]

13-18 Using mathematical fun ctions Examples ISOLATE(2*X+8,X) r eturns -4 ISOLATE(A+B*X/C,X) r eturns -(A* C/B) LINEAR? Tests whethe r expression is linear for the specified variable . Returns 0 (false) or 1 (true). LINEAR?( expr ession , vari ab le ) Example LINEAR?((X^2-1)/(X+1),X) r etur ns 0 QUAD Solves quadratic expression= 0 for va riable and[...]

Using mathematical fun ctions 13-19 Test functions The test functions are logical operators that always return either a 1 ( true ) or a 0 ( false ). < Less than. Returns 1 if true, 0 if false. val u e1 < va l ue 2 ≤ Less than or equal to. Returns 1 if true, 0 if false. val u e1 ≤ va l ue 2 = = Equals (logical test). Returns 1 if true, 0 i[...]

13-20 Using mathematical fun ctions XOR Exclusive OR. Returns 1 if either value1 or value2 —but not both of them—is non-zero, otherwise returns 0. val u e1 XOR val ue2 Trigonometry functions The trigonometry functions can also take complex numbers as arguments. For SIN, COS, TAN, ASIN, ACOS, and ATAN, see the Keyboard category. ACOT Arc cotange[...]

Using mathematical fun ctions 13-21 names. The HP 40gs has six formal names available for use in symbolic calculations. These are S1 to S5. When you perform a calculation that contains a formal name, the HP 40gs does not carry out any substitutions. You can mix formal names and real variables. Evaluating (A+B+S1) 2 will evaluate A+B , but not S1 . [...]

13-22 Using mathematical fun ctions differentiation function substi tutes the value that X holds, and returns a numeric result. For example, consider the function: 1. Enter the differ entiati on functi on onto the command line , substituting S1 in place of X . S1 S1 2 S1 2 . Ev aluate the func tion. 3 . Sho w the re sult . To find derivatives in th[...]

Using mathematical fun ctions 13-23 F1 3. Se l e c t F 2 ( X ) and eval u a te i t. 4. Pres s to display the re sult . Note: Use the arr o w ke y s to vi ew the entir e functi on . | Y ou could also ju st de fine . To find the indefinite integral using formal variables F or ex ample, to find the indefinite integral of use: 1. Enter the f unction . [...]

13-24 Using mathematical fun ctions 4. Cop y the r esult and eva lu a te. Thu s, sub stituting X for S1, it can be seen that: This result is derived from substituting X =S 1 and X =0 into the original expression found in step 1. However, substituting X =0 will not always evaluate to zero and may result in an unwanted constant. To see this, consider[...]

Using mathematical fun ctions 13-25 Program constants The program constants are numbers that have been assigned to various calculat or settings to enable you to test for or specify such a setting in a program. For example, the various displa y formats are assigned the following numbers: 1 Standar d 2 F ix ed 3 Scient ific 4 Engineering 5 Fraction 6[...]

13-26 Using mathematical fun ctions 3 . Use the ar r ow k e y s to nav igate thr ough the opti ons. 4. T o see the sy mbol and v alue of a selec ted constant , pre ss . (Cli ck to c lose the inf ormati on w indow that appears .) The f ollo wing e xample sho ws the inf ormati on av ailabl e about the speed of light ( one of the phy sics constants). [...]

Using mathematical fun ctions 13-27 3. Se l e c t light s... from the Ph ysics me nu . 4. Pr ess . T he menu clo ses and the v alue of the select ed constant is copied t o the edit line. 5 . Co mplete the equati on as y ou w ould nor mally and pre ss to get the r esult . hp40g+.book Page 27 Friday, December 9, 2005 1:03 AM[...]

hp40g+.book Page 28 Friday, December 9, 2005 1:03 AM[...]

Computer Al gebra System ( CAS) 14-1 14 Computer Algebra S y stem (CAS) What is a CAS? A computer al gebra system ( hereafter C AS) enables y ou to perform symbolic calculations. With a CAS you manipulate mathematical equations and expressions in symbolic form, rather than manipulating approximations of the numerical quantities re presented by thos[...]

14-2 Computer Algebra System (CAS) using vectors and matrices. (Vectors and matrices cannot be entered using the Equation Writer). To open the Equation Writer, press the soft- key on the menu bar of the HOME screen. The illustration at the right shows an expression being written in the Equation Writer. The soft keys on the menu bar provide access t[...]

Computer Al gebra System ( CAS) 14-3 3. P re s s a nd to select j ust the 20 in the term . 4. Pres s the menu ke y and choose FACTOR . Then pr ess . Note that the FACTOR functi on is added to the sele cted t erm. 5. Press to factor the selected term. 6 . Pr ess to select the entire second term, and then press to simplify it. 7 . P r e s s to select[...]

14-4 Computer Algebra System (CAS) 10. Pres s three times to select the entire expression and then press to simplify it to the form required. CAS variables When you use the symbolic calculation functions, y ou are working with symbolic variab les (variables that do not contain a permanent value). In the HOME screen, a variable of this kind must hav[...]

Computer Al gebra System ( CAS) 14-5 CAS modes The modes that determine how CAS operates can be set on CAS MODES scre en. To display CAS MODES screen, press: ·To navigate through the options in CAS MODES screen, press the arrow keys. To select or deselect a mode , navigate to the appropriate field and press until the co rrect setting is displayed [...]

14-6 Computer Algebra System (CAS) calculated as closed-form algebraic expressions, whenever possible. [Default: u nselected.] Num. Factor mode When the NUM FACTOR setting is selected, approximate roots are used when factoring . For example, is irreducible over the intege rs but has approximate roots over the reals. With NUM FACTOR set, the approxi[...]

Computer Al gebra System ( CAS) 14-7 Using CAS functions in HOME You can use many computer algebra functions directly in the HOME screen, as long as you take certain precautions. CAS functions th at take matrices as an argument work only from HOME. CAS functions can be accessed by pressing when MATH menu is displayed. Yo u can also directly type a [...]

14-8 Computer Algebra System (CAS) Symbolic matrices are stored as a list of lists and therefore must be stored in L0, L1…L 9 (whereas numeric matrices are stored in M0, M1,…M9). CAS linear algebra instructions accept lists of lists as input. For example, if you type in HOME: XQ({{S2 + 1, 1}, { , 1}}) L1 then you have: TRAN(L1) = {{S2 + 1, }, {[...]

Computer Al gebra System ( CAS) 14-9 HELP and press . The menu of help topi cs appears. Each help topic includes the required syntax, along with real sample values. You can copy the syntax, with the sample values, to the HOME screen or to the Equation Writer, by pressing . TIP If you highlight a CAS command and then press 2, help about the highligh[...]

14-10 Computer Alge bra System (CAS) For example, suppose you have stored the expression x 2 in G, and have defined the function F(x) as x 2 . Suppose now you want to calculate INTVX(X 2 ). You could: • enter INTVX(X 2 ) direc tly , or • enter INTVX(G) , or • enter INTVX(F(X)) . Note that you can apply the command directly to an expression or[...]

Computer Algebra Syst em (CAS) 14-11 Typing: DEF(U(N) = 2N+1) produces the result: U(N) = 2N+1 Typing: U(3) then returns: 7 Example Calculate the first six Fermat numbers F1...F6 and determine whether they are prime. So, you want to calculate: for k = 1...6 Typing the formula: gives a result of 17. You can then invoke the ISPRIME?() command, which [...]

14-12 Computer Alge bra System (CAS) which gives 4294967297 You can factor F(5) with FACTOR , which you’ll find in the ALGB menu on the menu bar. Typing: FACTOR(F(5)) gives: 641·6700417 Typing: F(6) gives: 18446744073709551617 Using FACTOR to factor it, then yields: 274177·67280421310721 EXPAND Distributivity EXPAND expands and simp lifies an e[...]

Computer Algebra Syst em (CAS) 14-13 In real mode, the result is: In complex mode (using CFG ), the result is: PARTFRAC Partial fraction expansion PARTFRAC has a rational fraction as an argument. PARTFRAC returns the partial fraction decomposition of this rational fraction. Example To perform a partial fraction decomposition of a rational function,[...]

14-14 Computer Alge bra System (CAS) Example 2 Typing: SUBST(QUOTE(CONJ(Z)),Z=1+i) gives: CONJ(1+i) STORE Store an object in a variable STORE stores an object in a variable. STORE is found in the ALGB menu or the Equation Writer menu bar. Example Type: STORE(X 2 -4,ABC) or type: X 2 -4 then select it and call STORE , then type ABC , then press ENTE[...]

Computer Algebra Syst em (CAS) 14-15 SUBST Substi tute a value for a variable SUBST has two parameters: an expression depe ndent on a parameter, and an equality (parameter=substitute value). SUBST substitutes the specifie d value for the variable in the expression. Typing: SUBST(A 2 +1,A=2) gives: TEXPAND Develop in terms of sine and cosine TEXPAND[...]

14-16 Computer Alge bra System (CAS) DIFF menu DERIV Derivative and partial derivative DERIV has two arguments: an expression (or a functi on) and a variable. DERIV returns the derivative of the expression (or the function) with respect to th e variable given as the second parameter (used for calculating partial derivatives). Example Calculate: Typ[...]

Computer Algebra Syst em (CAS) 14-17 DERVX(F) Or, if you have defined F(X) using DEF , that is, if you have typed: then type: DERVX(F(X)) Simplify the result to get: DIVPC Division in increasing order by exponent DIVPC has three ar guments: two polynomials A(X) and B(X) (where B(0) ≠ 0), and a whole number n. DIVPC returns the quotient Q(X) of th[...]

14-18 Computer Alge bra System (CAS) and with period T ( T being equal to the contents of the variable PERIOD ). If f(x) is a discrete series, then: Example Determine the Fourier coefficients of a periodic function f with period 2 π and defined over interval [0, 2 π ] by f(x)=x 2 . Typing: STORE(2 π ,PERIOD) FOURIER(X 2 ,N) The calculator does n[...]

Computer Algebra Syst em (CAS) 14-19 IBP returns the AND of and of that is, the terms that are calculated when performing a partial integration. It remains then to calculate the integral of the second term of the AND, then add it to the first term of the AND to obtain a primitive of . Typing: IBP(LN(X),X) gives: X·LN(X) AND - 1 The integration is [...]

14-20 Computer Alge bra System (CAS) Example Given: calculate a primitive of f . Type: Or, if you have stored f(x) in F, that is, if you have already typed: then type: INTVX(F) Or, if you have used DEF to define f( x ), that is, if you have already typed: then type: INTVX(F(X)) The result in all cases is equivalent to: You will obtain absolute valu[...]

Computer Algebra Syst em (CAS) 14-21 gives a primitive: Note You can also type which gives the primitive which is zero for x = 1 Example Calculate: Typing: gives the result: NOTE: If the argument to INTVX is the AND of two elements, INTVX concerns itself only with the se cond element of the AND, and adds the result to the first argument. lim Calcul[...]

14-22 Computer Alge bra System (CAS) QUOTE(expression), to avoid rewriting the expression i n normal form (i.e., not to have a rational simplification of the arguments) during the execution of the LIMIT command. Example Typing: gives: + ∞ To find a right limit, for example, type: gives (if X is the current variable): + ∞ To find a left limit, f[...]

Computer Algebra Syst em (CAS) 14-23 Typing: gives: 2 NOTE: To find the limit as x approaches a + (resp a – ), the second argument is written: X=A+0(resp X=A-0) For the following expression, find the limi t as x approaches + ∞ : Typing: produces (after a short wait): NOTE: the symbol ∞ is obtained by typing SHIFT 0. To obtain – ∞ : (–) [...]

14-24 Computer Alge bra System (CAS) PREVAL is used for calculatin g an integral defined from a primitive: it evaluates this pr imitive between the two limits of the integral. Typing: PREVAL(X 2 +X,2,3) gives: 6 RISCH Primitive and defined integral RISCH has two parameters: an expression and the name of a variable. RISCH returns a primitive of the [...]

Computer Algebra Syst em (CAS) 14-25 Typing: gives: • Ex ampl e — Expansion in the vic inity of x=+ ∞ or x=– ∞ Example 1 Give a 5th-order expansion of arctan(x) in the vic inity of x =+ ∞ , taking as infinitely small . Typing: SERIES(ATAN(X),X =+ ∞ ,5) gives: Example 2 Give a 2nd-order expan sion of in the vicinity of x =+ ∞, taking[...]

14-26 Computer Alge bra System (CAS) You must be in Rigorous (not Sloppy) mode to apply SERIES with unidirectional expansion. (See “CAS modes” on page 14-5 for instructions on setting and c hanging modes. Example 1 Give a 3rd-order expansion of in the vicinity of x = 0 + . Typing: gives: Example 2 Give a 3rd-order expansion of in the vicinity o[...]

Computer Algebra Syst em (CAS) 14-27 TABVAR Variation table TABVAR has as a parameter an expression wi th a rational derivative. TABVAR returns the variation table for the expression in terms of the current variable. Typing: TABVAR(3X 2 -8X-11) gives, in step-by-step mode: Variation table: The arrows indicate whether th e function is increasing or [...]

14-28 Computer Alge bra System (CAS) Typing: gives: Note ‘th-order’ means that the numerator and the denomi nator are expanded to the 4th relative order (here, the 5th absolute order for the numerator, and fo r the denominator, which is given in the end, the 2nd order (5 − 3), seeing that the exponent of the denominator is 3). TRUNC Truncate [...]

Computer Algebra Syst em (CAS) 14-29 Typing: DISTRIB((X+1)·(X+2)·(X+3)) giv es: EPSX0 Disregard small values EPSX0 has as a parameter an ex pres sion in X, and returns the same expression with the values less than EPS replaced by zeroes. Typing: EPSX0(0.001 + X) gives, if EPS=0.01: 0 + x or, if EPS=0.0001: . 001 + x EXPLN Transform a trigonometri[...]

14-30 Computer Alge bra System (CAS) Typing: EXP2POW(EXP(N · LN(X))) gives: FDISTRIB Distributivity FDISTRIB has an expression as argument. FDISTRIB enables you to appl y the distributivity of multiplication with respec t to addition all at once. Typing: FDISTRIB((X+1)·(X+2)·(X+3)) gives: x·x·x + 3·x·x + x·2·x + 3·2·x + x·x·1 + 3·x·1[...]

Computer Algebra Syst em (CAS) 14-31 Example 3 Typing: LIN(SIN(X)) gives: LNCOLLECT Regroup the logarithms LNCOLLECT has as an argument an expression containing logarithms. LNCOLLECT regroups the terms in the logarithms. It is therefore preferable to use an expression that has already been factored (using FACTOR ). Typing: LNCOLLECT(LN(X+1)+LN(X-1)[...]

14-32 Computer Alge bra System (CAS) Typing: SINCOS(EXP(i·X)) gives after turning on complex mode, if necessary: cos(x) + i · sin(x) SIMPLIFY Simplify SIMPLIFY simplifies an expression automatically. Typing: gives, after simplification: 4 · cos(x) 2 − 2 XNUM Evaluation of reals XNUM has an expression as a parameter. XNUM puts the calculator in[...]

Computer Algebra Syst em (CAS) 14-33 Typing: XQ(1.414213562) gives: √ 2 SOLV menu The SOLV menu contains functions that enable you to solve equations, linear systems, and differential equations. DESOLVE Solve differential equations DESOLVE enables you to solve differential equations. (For linear differential equations ha ving constant coefficient[...]

14-34 Computer Alge bra System (CAS) To produce the solutions for y(0) = 1, type: which gives: Example 2 Solve: y” + y = cos(x) y(0) = 1 y’(0) = 1 It is possible to solve for the constants from the outset. Typing: DESOLVE((d1d1Y(X)+Y(X)=COS(X)) AND (Y(0)=1) AND (d1Y(0)=1),Y(X)) gives: ISOLATE The zeros of an expression ISOLATE returns the value[...]

Computer Algebra Syst em (CAS) 14-35 LDEC Linear diffe rential equations having constant coefficients LDEC enables you to directly solve linear differential equations having cons tant coefficients. The parameters are the second member and the characteristic equation. Solve: y” − 6 · y’ + 9 · y = x · e 3·x Typing: LDEC(X·EXP(3·X),X 2 −[...]

14-36 Computer Alge bra System (CAS) L1=2L1+L2 ENTER Reduction Result then press ENTER. The following is then written to the Equation Writer: (x = − 2) AND (y = − 1) Example 2 Type: (2·X+Y+Z=1)AND(X+Y+2·Z=1)AND(X+2·Y+Z=4) Then, invoke LINSOLVE and type the unknowns: X AND Y AND Z and press the ENTER key. The following result is produced if y[...]

Computer Algebra Syst em (CAS) 14-37 then press ENTER. The following is then written to the Equation Writer: SOLVE Solve equat ions SOLVE has as two parameters: (1) either an equality between two expressions, or a single expression (in which case = 0 is implied), and (2) the name of a variable. SOLVE solves the equation in R in real mode and in C i[...]

14-38 Computer Alge bra System (CAS) SOLVEVX Solve equations SOLVEVX has as a parameter either: (1) an equality between two expressions in the variable contained in VX, or (2) a single such expression (in which case = 0 is implied). SOLVEVX solves the equation. Example 1 Typing: SOLVEVX(X 4 -1=3) gives, in real mode: (x = −√ 2) OR (x = √ 2) o[...]

Computer Algebra Syst em (CAS) 14-39 Typing: ACOS2S(ACOS(X) + ASIN(X)) gives, when simplified: ASIN2C Transform the arcsin into arccos ASIN2C has as a trigonometric expression as an argument. ASIN2C transforms the express ion by re placing arcsin (x) with − arccos(x). Typing: ASIN2C(ACOS(X) + ASIN(X)) gives, when simplified: ASIN2T Transform the [...]

14-40 Computer Alge bra System (CAS) Typing: ATAN2S(ATAN(X)) gives: HALFTAN Transform in terms of tan(x/2) HALFTAN has a trigonometric expression as an argument. HALFTAN transforms sin(x), cos(x) and tan(x) in the expression, rewriting them in terms of tan(x/2). Typing: HALFTAN(SIN(X) 2 + COS(X) 2 ) gives (SQ(X) = X 2 ): or, after simplification: 1[...]

Computer Algebra Syst em (CAS) 14-41 TAN2CS2 transforms this expr ession by replacing tan(x) with . Typing: TAN2CS2(TAN(X)) gives: TAN2SC Replace tan(x) with sin(x)/cos(x) TAN2SC has a trigonometric expression as an argument. TAN2SC transforms this expr ession by replacing tan(x) with . Typing: TAN2SC(TAN(X)) gives: TAN2SC2 Transform tan(x) with si[...]

14-42 Computer Alge bra System (CAS) TCOLLECT linearizes this ex pression in terms of sin( n x ) and cos( n x ), then (in Real mode) reconstructs the sine and cosine of the same angle. Typing: TCOLLECT(SIN(X) + COS(X)) gives: TEXPAND Develop transcenden tal expressions TEXPAND has as an argument a transcendental expression (that is, an expression w[...]

Computer Algebra Syst em (CAS) 14-43 gives: 4·cos(x) 3 –3·cos(x) TLIN Lineariz e a trigonometri c expression TLIN has as an argument a trigonometric expression. TLIN linearizes this expression in terms of sin( n x ) and cos( n x ). Example 1 Typing: TLIN(COS(X) · COS(Y)) gives: Example 2 Typing: TLIN(COS(X) 3 ) gives: Example 3 Typing: TLIN(4?[...]

14-44 Computer Alge bra System (CAS) Typing: TRIG(SIN(X) 2 + COS(X) 2 + 1) gives: 2 TRIGCOS Simplify using the cosines TRIGCOS has as an argument a trigonometric expression. TRIGCOS simplifies this expression, using the identity sin(x) 2 +cos(x) 2 = 1 to re write it in terms of cosines. Typing: TRIGCOS(SIN(X) 4 + COS(X) 2 + 1) gives: TRIGSIN Simpli[...]

Computer Algebra Syst em (CAS) 14-45 CAS Functions on the MATH menu When you are in the Equation Writer and press , a menu of additional CAS functions available to you is displayed. Many of the functions in this menu match the functions available from the soft-key men us in the Equation Writer; but there are other functions that are only available [...]

14-46 Computer Alge bra System (CAS) returns: Y = X –1 + 2 Pressing simplifies this to: Y = X + 1 IM See “IM” on page 13-7. – Specifies the negation of the argument. RE See “RE” on page 13-8. SIGN Determines the quotient of the argument divided by its modulus. Example Typing SIGN(7 + 4i) or SIGN(7,4) yields . Constant menu e, i, π See [...]

Computer Algebra Syst em (CAS) 14-47 DIVIS Gives the divisors of an integer. Example Typing: DIVIS(12) gives: 12 OR 6 OR 3 OR 4 OR 2 OR 1 Note: DIVIS(0) returns 0 OR 1. EULER Returns the Euler index of a whole number. The Euler index of n is the number of whole numbers less than n that are prime with n . Example Typing: EUL E R (2 1 ) gives: 12 Exp[...]

14-48 Computer Alge bra System (CAS) In step-by-step mode, there ar e a number of intermediate results: 18 mod 15 = 3 15 mod 3 = 0 Res ul t : 3 Pressing or then causes 3 to be written to the Equation Writer. Note that the last non-zero remainder in the sequence of remainders shown in the intermediate steps is the GCD. IDIV2 Returns the quotient and[...]

Computer Algebra Syst em (CAS) 14-49 [48, 1 ,0 ] [30, 0,1]*–1 [18,1,–1]*–1 [12 ,–1 ,2]*–1 [6,2 ,–3]*–2 Re sult: [6,2 ,–3] Pressing or then causes 2 AND –3 = 6 to be written to the Equation Writer. The intermediate steps shown are the combination of lines. For example, to get line L( n + 2), take L( n ) – q *L( n + 1) where q is [...]

14-50 Computer Alge bra System (CAS) IREMAINDER works with integers and with Gaussian integers. This is what distingui shes it from MOD. Example 2 Typing: IREMAINDER(2 + 3·i, 1 + i) gives: i ISPRIME? Returns a value indicating whe ther an integer is a prime number. ISPRIME?( n ) returns 1 (TRUE) if n is a prime or pseudo-prime, and 0 (FALSE) if n [...]

Computer Algebra Syst em (CAS) 14-51 NEXTPRIME NEX TPRIM E( n ) returns the smallest prime or pseudo-prime greater than n . Example Typing: NEXTPRIME( 7 5) gives: 79 PREVPRIME PREVPRIME( n ) returns the greatest prime or pseudo-prime less than n . Example Typing: PRE VP RI ME (7 5) gives: 73 Modular menu All the examples of this section assume that[...]

14-52 Computer Alge bra System (CAS) DIVMOD Division in Z/pZ or Z/pZ[X]. Example 1 In Z/pZ, the arguments are two integers: A and B. When B has an inverse in Z/pZ, the result is A/B simplified as Z/pZ. Typing: DIVM OD(5 , 3) gives: 6 Example 2 In Z/pZ[X], the arguments are two polynomials: A[X] and B[X]. The result is a rational fraction A[X]/B[X] [...]

Computer Algebra Syst em (CAS) 14-53 FACTORMOD Factors a polynomial i n Z/pZ[X], providing that p ≤ 97, p is prime and the order of the multiple factors is less than the modu lo. Example Typing: FA C TO R M O D ( – ( 3 X 3 – 5X 2 + 5X – 4)) gives: GCDMOD Calculates the GCD of the two polynomials in Z/pZ[X]. Example Typing: GCDMOD(2X 2 + 5, [...]

14-54 Computer Alge bra System (CAS) MULTMOD Performs a multiplication in Z/pZ or i n Z/pZ[X]. Example 1 Typing: MUL TMOD(11, 8) gives: –3 Example 2 Typing: MUL TMOD(11X + 5, 8X + 6) gives: POWMOD Calculates A to the power of N in Z/pZ[X], and A(X) to the power of N in Z/pZ[X]. Example 1 If p = 13, typing: POWMO D (1 1 , 1 95) gives: 5 In effect:[...]

Computer Algebra Syst em (CAS) 14-55 SUBTMOD Performs a subtraction in Z/pZ or Z/pZ[X]. Example 1 Typing: SU BTM O D ( 29 , 8 ) gives: –5 Example 2 Typing: S UB TMOD(11X + 5, 8X + 6) gives: Polynomial menu EGCD Returns Bézout’s Identity, the Extended Greatest Common Divisor (EGCD). EGCD(A(X), B(X)) returns U(X) AND V(X) = D(X), with D, U, V su[...]

14-56 Computer Alge bra System (CAS) FACTOR Factors a polynomial. Example 1 Typing: F ACT OR(X 2 – 2) gives: Example 2 Typing: F ACT OR(X 2 + 2·X + 1) gives: GCD Returns the GCD (Greatest Common Divisor) of two polynomials. Example Typing: GCD(X 2 + 2·X + 1, X 2 – 1) gives: HERMITE Returns th e Hermite polynomia l of degree n (where n is a wh[...]

Computer Algebra Syst em (CAS) 14-57 LCM Returns the LCM (Least Common Multiple) of two polynomials. Example Typing: LC M ( X 2 + 2·X + 1, X 2 – 1) gives: LEGENDRE Returns the polynomial L n , a non-null solution of the differential equation: where n is a whole number. Example Typing: LEG ENDRE(4) gives: PARTFRAC Returns the partial fraction dec[...]

14-58 Computer Alge bra System (CAS) PROPFRAC PROPFRAC rewrites a rational fr action so as to bring out its whole number part. PROPFRAC(A(X)/ B(X)) writes th e rational fraction A(X)/ B(X) in the form: where R”(X) = 0, or 0 ≤ deg (R(X) < deg (B(X). Example Typing: gives: PTAYL PTAYL rewrites a polynomial P(X) in order of its powers of X – [...]

Computer Algebra Syst em (CAS) 14-59 Note that in step-by-step mode, synthetic division is shown, with each polynomial represented as the list of its coefficients in descending order of power. REMAINDER Returns the remainder from the division of the two polynomials, A(X) and B(X), div ided in decreasing order by exponent. Example Typing: REMAINDER([...]

14-60 Computer Alge bra System (CAS) Example 1 Typing: T CHEB Y CHEFF(4) gives: Example 2 Typing: T CHEB Y CHEFF(–4) gives: Real menu CEILING See “CEILING” on page 13-14. FLOOR See “FLOOR” on page 13-14. FRAC See “FRAC” on page 13-14. INT See “INT” on page 13-15. MAX See “MAX” on page 13-15. MIN See “MIN” on page 13-15. Re[...]

Computer Algebra Syst em (CAS) 14-61 Tests menu ASSUME Use this function to make a hypothesis about a specified argument or variable. Example Typing: ASSUM E( X > Y) sets an assumption that X is greater than Y. In fact, the calculator works only with large not strict relations, and thus ASSUME(X>Y) will actually set the assumption that X ≥ [...]

14-62 Computer Alge bra System (CAS) CAS Functions on the CMDS menu When you are in the Equation Writer and press , a menu of the full set of CAS functions available to you is displayed. Many of the functions in this menu match the functions available from the soft-key menus in the Equation Writer; but there are other functions that are only availa[...]

Computer Algebra Syst em (CAS) 14-63 Example Find the solutions P(X) of: P(X) = X (mod X 2 + 1) P(X) = X – 1 (mod X 2 – 1) Typing: CHINREM((X) AND (X 2 + 1) , (X – 1) AND (X 2 – 1)) gives: That is: CYCLOTOMIC Returns the cyclotomic polynomial of order n . This is a polynomial having the n th primitive roots of unity as zeros. CYCLOTOMIC has[...]

14-64 Computer Alge bra System (CAS) Example 1 Typing: EXP 2HY P (EXP (A)) gives: sinh( a) + co sh(a ) Example 2 Typing: EXP 2HY P( EXP (– A) + EXP(A) ) gives: 2 · cosh( a) GAMMA Returns the values of the Γ function at a given point. The Γ function is defined as: We have: Γ (1) = 1 Γ ( x + 1) = x · Γ ( x ) Example 1 Typing: GA M M A ( 5 ) [...]

Computer Algebra Syst em (CAS) 14-65 Example Typing: I AB CU V (48, 3 0, 1 8 ) gives: 6 AND –9 IBERNOULLI Returns the n th Bernoulli’s number B( n ) where: Example Typing: IBERNOULLI(6) gives: ICHINREM Chinese Remainders: ICHINREM(A AND P,B AND Q) returns C AND R, where A, B, P and Q are whole numbers. The numbers X = C + k · R where k is an i[...]

14-66 Computer Alge bra System (CAS) ILAP is the inverse Laplace transform of a given expression. Again, the expression is the value of a function of the variable stored in VX. Laplace transform (LAP ) and inverse Laplace transfo rm (ILAP) are useful in solving linear differential equations with constant coefficients, for example: The following rel[...]

Computer Algebra Syst em (CAS) 14-67 Typing: gives: LAP See ILAP above. PA2B2 Decomposes a prime integer p congruent to 1 modulo 4, as follows: p = a 2 + b 2 . The calculator gives the result as a + b · i. Example 1 Typing: P A2B2(17) gives: 4 + i that is, 17 = 4 2 + 1 2 Example 2 Typing: P A2B2(2 9) gives: 5 + 2 · i that is, 29 = 5 2 + 2 2 PSI R[...]

14-68 Computer Alge bra System (CAS) gives: Psi Returns the value of the Digamma function at a . The Digamma function is defined as the derivative of ln( Γ (x)), so we have PSI( a ,0) = Psi( a ). Example Typing: Ps i ( 3 ) and pressing gives: . 922 7 84335 098 REORDER Reorders the input expression following the order of variables given in the seco[...]

Computer Algebra Syst em (CAS) 14-69 Example Typing: SIGMA(X · X!, X) gives: X! because (X + 1)! – X! = X · X!. SIGMAVX Returns the discrete antideriva tive of the input function, that is a function, G, that satisfies the relation: G( x + 1) – G( x ) = f( x ). SIGMAVX has as its argument a function f of the current variable VX. Example Typing[...]

14-70 Computer Alge bra System (CAS) TSIMP Simplifies a given expression by rewriting it as a function of complex exponentials, and then reducing the number of variables (enabling complex mode in the process). Example Typing: gives: VER Returns the version number of your CAS. Example Typing: VER might give: 4.200 5 0 219 This particular result mean[...]

Equation Writer 15-1 15 Equation W riter Using CAS in the Equation Writer The Equation Writer enables yo u to type expre ssions that you want to simplify, factor, differentiate, integrate, and so on, and then work them through as if on paper. The key on the HOME screen menu bar opens the Equation Writer, and the key closes it. This chapter explains[...]

15-2 Equation Writer ALGB menu The menu contains functions that enable you to perform algebra, such as factoring, expansion, simplification, substitution, and so on. DIFF menu The menu contains functions that enable you to perform differential calculus, such as differentiation, integration, series expansion, limits, and so on. Cursor mode Enables y[...]

Equation Writer 15-3 REWRI menu The menu contains functions that enable you to rewrite an expression in another form. SOLV menu The menu contains functions that enable you to solve equation s, linear systems, and diffe rential equations. TRIG menu The menu contains functions that enable you to transform trigonometric expressions. NOTE You can get o[...]

15-4 Equation Writer • The fo u r th sym bo l, S , in the abov e e xample , indicates that y ou ar e in step-b y-step mode . If you w ere not in step-b y-step mode, this s y mbol wo uld be D (whi ch stands for Direct ). The first line of an Equation Writer me nu only indica tes some of the mode settings. To see more settings, highlight the first [...]

Equation Writer 15-5 Entering expressions and subexpressions You type expressions in the Equation Writer is much the same way as you type them in the HOME screen, using the keys to directly enter numbers, letters and operators, and menus to select various functions and commands. When you type an expression in the Equation Writer, th e operator that[...]

15-6 Equation Writer this case, you have to press to select elements in the expression. The following illustration shows how an expression can be viewed as a tree in the Eq uation W riter. It illustrates a tree view of the expression: Suppose that the cursor is positioned to the right of 3: • If you pr ess once, the 3 component is selected. • I[...]

Equation Writer 15-7 • Pres s again an d again to progre ssiv ely select mor e of the top-most br anch , and then low er branc hes (5 x , 5 x + 3, and then the entire numerator and finall y the entir e expr essi on). More Examples Example1 If you enter: 2 + X × 3– X and press the entire expres sion is select ed. Pressing evaluates what is sele[...]

15-8 Equation Writer (– X) apply to it. As a result, the entered expression is interpreted, and displayed, as (2 + X)(3 – X). Select the entire expression by pressing and evaluate it by pressing . The result is: –(X 2 –X–6) Example2 To enter X 2 –3X+1, press: 2 – 3 +1 If, instead, you had to enter –x 2 –3X+1, you would need to pre[...]

Equation Writer 15-9 Select the fifth branch by pressing . At this point, the desired expression is in the Equation Writer, as shown at t he right. Suppose that you want to select the second and third branches, that is: . Firs t press . This select s , the second term. Now press . This key combination enables you to select two contiguous branches, [...]

15-10 Equation Writer Pressing produces the result of the partial calculation. Summing up Pressing enables you to select the current element and its neighbour to the right. enables you to exchange the selected element with its neighbour to the left. The selected element remains selected after you move it. Cursor mode In cursor mode you can select a[...]

Equation Writer 15-11 How to modify an expression If you’re typing an expression, the key enables you to erase what you’ve typed. If you’re selecting, you can: • Cancel the sele ction w ithout dele ting the expr essi on by pre ssing . The cur sor mo v es to the end o f the deselected portion . • Replace the selection with an e xpression ,[...]

15-12 Equation Writer Accessing CAS functions While you are in the Equation Writer, you can access all CAS functions, and you can ac cess them in various ways. General principle: When you have written an expression in the Equation Writer, all you have to do is press to evaluate whatever you have selected (or the entire expression, if nothing is sel[...]

Equation Writer 15-13 select the entire expression and press , you obtain: However, if you type: select the entire expression and press , you obtain 1. How to enter infix function s An infix function is one that is typed between its arguments. For example, AND , | and MOD are infix functions.You can either: • type them in Alpha mode and then ente[...]

15-14 Equation Writer First option: function first, then arguments In the Equation Writer, press , select FACTOR and then press or . FACTOR() is displayed in the Equation Writer, with the cursor between the parentheses (as shown at the right). Enter your expression, using the rules of selection described earlier. 2 4 The entire expression is now se[...]

Equation Writer 15-15 Press to obtain the an intermediate result (4 2 – 4) and again to evaluate the intermediate result. The final answer is 12. Second option: arguments first, then function Enter your expression, using the rules of selection described earlie r. 2 4 The entire expression is now selected. Now press and select FACTOR . Notice that[...]

15-16 Equation Writer Press to obtain an intermediate result, (4– 2)(4 + 2), and again to evaluate the intermediate result. The final answer, as before, is 12. Note If you call a CAS function while you’re writing an expression, whatever is currentl y selected is copied to the function’s first or main argument. If nothing is selected, the curs[...]

Equation Writer 15-17 • MODULO contains the v alue of p fo r performing symbolic c alcu lat ion s i n Z/pZ or in Z/pZ [ X ]. Y ou can change the value of p either with the MODSTO command on the MODULAR menu , (by typ ing, f or ex ample , MODS T O( n ) to gi ve p a va lue of n ), o r f r o m CAS M ODE S scr een (see page 14- 5). • PERIOD mus t c[...]

15-18 Equation Writer Diff&Int () , Rewrite () , Solve () and Trig () . • The Complex menu , pr ov iding f unctions spec ific to manipulating with com plex numbers . • The Constant menu , containing e , i, ∞ and π . • The Hyperb . menu , containing hy perboli c functions . • The Integer men u, con taining functions that enable you to[...]

Equation Writer 15-19 Press to clear the value of the highlighted variable. Press to change the name of the highlighted variable. Press to define a new variable (which you do by specifying an object and a name for the object. SYMB key Pressing the key in the Equation Writer gives you access to CAS history. As in the HOME screen history, the calcula[...]

15-20 Equation Writer NOTE This operation supposes that the c urrent variable is also the variable of the function or curve you want to graph. When the expression is copied, it is evaluated, and the current variable (contained in VX) is changed to X, T, or θ , depending on the type of plot you chose. If the function depends on a pa rameter, it is [...]

Equation Writer 15-21 Short-cut keys In the Equation Writer, the following are short-cut keys to the symbols indicated: 0 for ∞ 1 for i 3 for π 5 for < 6 for > 8 for ≤ 9 for ≥ hp40g+.book Page 21 Friday, December 9, 2005 1:03 AM[...]

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Step-by-Step Examples 16-1 16 Step-b y-Step Ex ampl es Introduction This chapter illustrates the power of CAS, and the Equation Writer, by working though a number of examples. Some of these examples are variations on questions from senior math examination papers. The examples are given in order of increasing difficulty. Example 1 If A is: calculate[...]

16-2 Step-by-Step Examples Press to simplify the numerator. Press to select the entire fraction. Press to simplify the selected fraction, giving the result shown at the right. Example 2 Given that write C in the form , where d is a whole number. Solution: In the Equation Writer, enter C by typing: 2 45 3 12 20 6 3 Pres s to select . Pres s to selec[...]

Step-by-Step Examples 16-3 Pres s to factor 20 into . Pre ss to selec t and to simplify it. Pre ss to selec t and to e xc hange with . Pre ss to selec t and to select 45 . Pres s , sele ct FACTOR and pres s . Press to factor 45 i n t o . Pre ss to selec t and to simplify the selecti on. Pre ss to selec t , and to select . 2 2 5 ⋅ 2 2 5 ⋅ 25 –[...]

16-4 Step-by-Step Examples Pres s to ev aluate the selection . It re mains to transf orm and combine it w ith . Fo llow the same procedur e as undertaken a number of times abo ve . Y ou w ill find that is equal to , and so the final tw o terms cancel each other out. Hence the r esult is Example 3 Given the expression : • expand and r educe D • [...]

Step-by-Step Examples 16-5 Press to select the entire equation, then press to reduce it to . Press , select FACTOR, press and then . The r esult is as shown at the right. Now press , select SOLVEVX, press and press . The result is shown at t he right. Press to display CAS history, select D or a version of it, and press . Press , select SUBST , pres[...]

16-6 Step-by-Step Examples Pres s , select LINSOLVE and pr ess . Enter 17 X 20 Y 90 10 X 25 Y 90 X Y If you are working in step by step mode, pressing produces the result at the right. Press again to produce the next step in the solution: Press again to produce the reduction result: Pressing again produces the final resul t: If you select , and pre[...]

Step-by-Step Examples 16-7 1. F ind the ex act le ngth of AB in cen timetr es. 2 . Deter mine the equation o f the line AB . First method Type: STORE((-1,3),A) and press . Accept the change to Complex mode, if necessary. Note that pressing returns the coordinates in complex form: –1+3i. Now type: STORE((-3,-1),B) and press . The coordinates this [...]

16-8 Step-by-Step Examples Press again to simplify the result to Y = 2X+5. Second method Type: (-3,-1 )-(-1,3) The answer is –(2+4i). With the answer still selected, apply the ABS command by pressing . Pressing gives , the same answer as with method 1 above. You can also deter mi1ne the equati on of the line b y ty pi ng: DROITE(( -1,3), (-3,-1))[...]

Step-by-Step Examples 16-9 4. Show that for e very integer n > 0, b n × c n = a 2n . 5 . Deduce the prime factor decompositi on of a 6 . 6. S h ow t h a t GC D ( b n , c n ) = GCD( c n ,2) . Deduce that b n and c n are prime together . Solution: Begin by entering the three definitions. Type: DEF(A(N) = 4 · 10 N –1) DEF(B(N) = 2 · 10 N –1)[...]

16-10 Step-by-Step Examples Show that the whole numbers k such that: have digits in decimal notation. We have: so have digits in decimal notation. Moreover, is divisible by 9, since its decimal notation can only end in 9. We also have: and so and are both di visible by 3. Let’s consider whether B(3) is a prime number. Type ISPRIME?(B(3)) and pres[...]

Step-by-Step Examples 16-11 Now consider the product of two of the definitions entered above: B(N) × C(N): B N C N . Press , to select EXP2POW and press . Press to evaluate the expression, yielding the result of B(N) × C(N). Consider now the decomposition of A(6) into its pr ime factors. Press , to select FACTOR and press . Now press A 6. Finally[...]

16-12 Step-by-Step Examples Part 2 Given the equation: [1] where the integers x and y are unknown and b 3 and c 3 are defined as in part 1 abo ve: 1. Sho w that [1] has at least one so lution . 2 . Appl y Euc lid’s algor ithm to b 3 and c 3 and f ind a solution to [1]. 3 . F ind all solutio ns of [1]. Solution : Equation [1] must have at least on[...]

Step-by-Step Examples 16-13 so , , or The calculator is not needed for finding the general solution to equation [1]. We started with and have established that . So, by subtraction we have: or According to Gauss’s Theorem, is prime with , so is a divisor of . Hence there ex ists such that: and Solving for x and y , we get: and for . This gives us:[...]

16-14 Step-by-Step Examples the circle C , M will move on a curve Γ . In this exercise we will study and plot Γ . 1. Let and m be the point o n C of affix . F ind the coor dinates of M in ter ms of t . 2 . Co mpare x(–t) w ith x(t) and y(–t) with y(t). 3. Com p u t e x ′ ( t ) and f ind the var iations o f x ov er [0, π ]. 4. Repeat step 3[...]

Step-by-Step Examples 16-15 Selecting the entire expression and pressing gives the result at the right: Now linearize the result by applying the LIN command (which can be found on the me nu). The result, after accepting the switch to complex mode, is shown at the right: Now store th e result in variable M. Note that STORE is on the menu. To calcula[...]

16-16 Step-by-Step Examples DEF command to it. Press to complete the definition. To calculate the real pa rt of the expression, apply the IM command (available on the COMPLEX submenu of the MATH menu) to the stored variable M. Press to get the result at the right: Finally, define the result as Y(t) in the same way that you defined X(t): by firstly [...]

Step-by-Step Examples 16-17 Then press to produce the result at t he right: In other words, . If is part of , then is also part of . Since and are symmetrical with respect to the x- axis, we can deduce that the x-axis is an axis of symmetry for . Part 3 Calculate b y typing: DERVX X t . P ress to highli ght the exp res s io n. Pressing returns the [...]

16-18 Step-by-Step Examples Part 4 To calculate , begin by typing: DERVX(Y(t)) . Pressing returns: Press again to simplify the result: Select FACTOR and press . You can now define the function (in the same way that you defined ). Part 5 To show the variations of and , we will trace and on the same graph. The independent variable must be t which it [...]

Step-by-Step Examples 16-19 Now press to see the graphs. Part 6 To find the values of and for return to CAS, type each function in turn and press . (You may need to press twice for further simplification). For example, pressing X 0 gives the result at the right: Likewise, pressing X 3 gives this answer at the right: The other results are: The slope[...]

16-20 Step-by-Step Examples The example at the right shows the case for t = 0. Select the entire expression and press to get the answer: 0 The example at the right shows the case for t = π /3. Selecting the entire expression and pressing displays the message shown at the right. Accept YES and press . Press again to get the result: ∞ The next exa[...]

Step-by-Step Examples 16-21 Now we will graph Γ , which is a parametric curve. In the Equation Writer, type X(t) + i × Y(t) . Select the entire expression and press . Now press , select Parametric and press . Select X1,Y1 as the destination and press . To make the graph of Γ , quit CAS and choose the Parametric aplet. Check X1(T) and Y1(T) . Now[...]

16-22 Step-by-Step Examples Exercise 8 For this exercise, make sure that the calculator is in exact real mode with X as the current variable. Part 1 For an integer, n , define the following: Define g over [0,2] where: 1. F ind the var iati ons of g o ver [0,2]. Sho w that for ev ery real x in [0,2]: 2 . Sho w that for e v ery real x in [0,2]: 3 . A[...]

Step-by-Step Examples 16-23 Solution 1 Start by defining G(X): DEF G X = 2 X 3 X 2 Now press : Press and to select the numerator and denominator, and then press . This leaves G(X) displayed: Finally, apply the TABVAR function: TABVAR and pres s a number of times until the var iation table appears (sho w n abov e) . The first line of the variation t[...]

16-24 Step-by-Step Examples Now press and scroll down the screen to: Now press to obtain the table of variations. If you are not in step-by -step mode, you can also get the calculation of the derivative by typing: DERVX(G(X)) which produces the preceding result. To prove the stated inequality, first calcu late g (0) by typing G(0) and pressing . Th[...]

Step-by-Step Examples 16-25 We can now see that: To justify the preceding calculation, we must assume that is a primitive of . If you are not s ure, you can use the INTVX function as illustrated at the right: Note that the INTVX command is on the menu. The simplified result, got by pressing twice, is shown at t he right: Solution 4 To find the limi[...]

16-26 Step-by-Step Examples NOTE : The variable VX is now set to N . Reset it to X by pressing (to display CAS MODES screen) and change the INDEP VAR s etting. To check the result, we can say that: and that therefore: or, simplifying: If the limit of exists as approaches + in the inequalities in solution 2 above, we get: Part 2 1. Sho w that for e [...]

Step-by-Step Examples 16-27 Solution 1 Start by defining the following: Now type PROPFRAC(G(X)) . Note that PROPFRAC can be found on the POLYNOMIAL submenu of th e MATH menu. Pressing yields the result shown at the right. Solution 2 Enter the integral: . Pressing yields the result shown at the right: Pressing again yields: Working by hand: , so: Th[...]

16-28 Step-by-Step Examples Solution 3 The calculator is not needed here. Simply stating that increases for is sufficient to yield the inequality: Solution 4 Since is positive over [0, 2 ], through multiplication we get: and then, integrating: Solution 5 First find the limit of when → + . Note: pressing after you have selected the infinity sign f[...]

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Variables and me mory management 17-1 17 V ariables and memory manag ement Introduction The HP 40gs has approximately 200K of user memory. The calculator uses this memory to store variables, perform computations, and store history. A v a r i a b l e i s a n o b j e c t t h a t y o u c r e a t e i n m e m o r y t o h o l d data. The HP 40gs has two [...]

17-2 Variables and memory management Storing and recalling variables You can store numbers or expressions from a previous input or result into variables. Numeric Precision A number stored in a variable is always stored as a 12- digit mantissa with a 3-digit exponent. Numeric precisio n in the display, however, de pends on the display mode (Standard[...]

Variables and me mory management 17-3 5 . Enter a name f or the var iable . A 6 . Pr ess to stor e the re sult . The results of a calculation can also be stored directly to a variable. For example: 2 5 3 B To recall a value To recall a variable’s value, type the name of the variable and press . A To use variables in calculations You can use varia[...]

17-4 Variables and memory management The VARS menu You use the VARS menu to access all variables in the calculator. The VARS menu is organised by category. For each variable category in the left column, there is a list of variables in the right colu mn. You select a variable category and then select a variable in the category. 1. Open the V ARS men[...]

Variables and me mory management 17-5 5 . Choos e whether to place the v ari able name or the var iable v alue on the command line . – Pres s to indicate that y ou w ant the var iable ’s contents to appear on the command line. – Pres s to indicate that y ou w ant the var iable ’s name to appear on the co mmand line. 6. Pr ess to place the v[...]

17-6 Variables and memory management 4. Enter data for L2 . 55 48 86 90 77 5 . Pres s to access HOME . 6. Open the var iab le menu and select L1. 7 . Copy it to the command line . Note: Because th e option is highli ghted, the v ar iable ’s name, rather than its conten ts, is copied to the command line . 8. Insert the + oper ator and select the L[...]

Variables and me mory management 17-7 Home variables It is not possible to store data of one type in a variable of another type. For example, yo u use the Matrix catalog to create matrices. You can crea te up to ten matrices, and you can store these in variables M0 to M9. You cannot store matric es in variab les other than M0 to M9. Cate- gory Av a[...]

17-8 Variables and memory management Aplet variables Most aplet va riables stor e values that a re unique to a particular aplet. These include symbolic expressions and equations (see below), settings for the Plot and Numeric views, and the results of some calculations such as roots and intersections. See the Reference Information chapter for more i[...]

Variables and me mory management 17-9 Memory Manager You can use the Memory Manager to determine the amount of available memory on the calculator. You can also use Memory Manager to organize memory. For example, if the available memory is low, you can use the Memory Manager to determine which aplets or variables consume large amounts of memory. You[...]

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Matrices 18-1 18 Matrices Introduction You can perform matrix calc ulations i n HOME and in programs. The matrix and each row of a matrix appear in brackets, and the elements and rows are separated by commas. For example, the following matrix: is displayed in the history as: [[1,2,3],[4,5,6]] (If the Decimal Mark mode is set to Comma , then separat[...]

18-2 Matri ces Creating and storing matrices You can create, edit, delete, send, and receive matrices in the Matrix catalog. To open the Matrix catalog, press MATRIX . You can also create and store matrices—named or unnamed—-in HOME. For example, the command: POLYROOT([1,0,–1,0]) X M1 stores the root of the complex vector of length 3 into the[...]

Matrices 18-3 2 . Highli ght the matr ix v aria ble name you w ant to use and pres s . 3 . Select the ty pe of matr ix t o cr eate . – For a v ector (on e-d imensional array) , select Real vector or Complex vector . Certain oper ations ( + , – , CRO SS ) do not r ecogni ze a one-dimensi onal matr i x as a v ect or , so th is se lect ion i s imp[...]

18-4 Matri ces To transmit a matrix You can send matrices between calculators just as you can send aplets, programs, lists, and notes. 1. Connec t the calculat ors using an appr opri ate cable . 2 . Open the Matr ix catalogs on both calc ulators. 3 . Highlig ht the matri x to send . 4. Pres s and choose the method of sending . 5 . Press on the rece[...]

Matrices 18-5 To display a matrix • In the Matri x catalog ( MATRIX ) , highlight the matri x name and pr ess . • In HOME , enter the name of the matri x var iable and pres s . To display one element In HOME, enter matrixname ( row,column ). For example, if M2 is [[3,4],[5,6]] , then M2(1,2) returns 4 . To create a matrix in HOME 1. Enter the m[...]

18-6 Matri ces To store one element In HOME, enter, value matrixname ( row, column ). For example, to change the element in the first row and second column of M5 to 728, then display the resulting matrix: 728 M 512 M5 . An attempt to store an element to a row or column beyond the size of the matrix results in an error message. Matrix arithmetic You[...]

Matrices 18-7 M1 M2 To multiply and divide by a scalar For division by a scalar, enter the matrix first, then the operator, then the scalar. For multiplication, the order of the operands does not matter. The matrix and the sc alar can be real or complex. For example, to divide the result of the previous example by 2, press the following keys: 2 To [...]

18-8 Matri ces To divide by a square matrix For division of a matrix or a vector by a square matrix, the number of rows of the dividend (or the number of elements, if it is a vector) must equal the number of rows in the divisor. This operation is not a mathematical division: it is a left- multiplication by the inverse of the divi sor. M1/M2 is equi[...]

Matrices 18-9 3 . Retu rn to the Matri x Cat al og. MATRIX In this ex ample , the vec tor you c reated is listed a s M1. 4. Creat e a new matr ix . Select Real matrix 5 . Enter the eq uation coeffi ci ents. 23 4 11 1 4 12 In this ex ample , the matr ix y ou c reat ed is listed as M2 . 6. Re turn to HOME and ent er the calculati on to left-multipl y[...]

18-10 Matri ces Matrix functions and commands About functions • Fu n c t io n s c an b e u s e d i n a ny a p l et o r i n H O M E. Th e y are lis ted in the MA TH menu under the Matr i x category . The y can be used in mathematical expr essions —primar ily in HOME—as w ell as in progr ams. • F unctio ns alw ay s pr oduce and displa y a r e[...]

Matrices 18-11 COND Condition Number. Finds the 1-norm (column norm) of a square matrix . COND ( matri x ) CROSS Cross Product of vector1 with vector2 . CROSS ( vect or 1 , vec to r2 ) DET Determinant of a square matrix . DET ( matri x ) DOT Dot Product of two arrays, matrix1 matrix2 . DOT ( matri x1, matri x2 ) EIGENVAL Displays the eigenvalue s i[...]

18-12 Matri ces LU LU Decomposition. Factors a squar e matrix into three matrices: {[[ lowertriangular ]],[[ uppertriangular ]],[[ permutation ]]} The uppertriangular has ones on its diagonal. LU ( matri x ) MAKEMAT Make Matrix. Creates a matrix of dimension rows × columns , using expression to calculate each element. If expression contains the va[...]

Matrices 18-13 SPECNORM Spectral No rm of matrix . SPECNORM ( matr ix ) SPECRAD Spectral R adius of a s quare matrix . SPECRAD ( matr ix ) SVD Singular Value Decomp osition. Factors an m × n matrix into two matrices and a vector: {[[ m × m square orthogonal ]],[[ n × n square orthogonal ]], [ real ]}. SVD ( matri x ) SVL Singular Values. Returns[...]

18-14 Matri ces column 2) is swapped with element 2,1; element 2,3 is swapped with element 3,2; and so on. For example, TRN([[1,2],[3,4]]) creates the matrix [[1,3],[2,4]] . Reduced-Row Echelon Form The following set of equations can be written as the augmented matrix which can then stored as a real matrix in any matrix variable. M1 is used in this[...]

Matrices 18-15 The final row of zeros in the reduced-row echelon form of the augmented matrix indicates an inconsistent system with infinite solutions . hp40g+.book Page 15 Friday, December 9, 2005 1:03 AM[...]

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Lists 19-1 19 L ists You can do list operations in HOME and in programs. A list consists of comma-separated real or complex numbers, expressions, or matr ices, all enclosed in braces. A list may, for example, contain a sequence of real numbers such as {1,2,3} . (If the Decimal Mark mode is set to Comma , then the separators are periods.) Lists repr[...]

19-2 Lists 3. En ter t he va lues you want i n th e l ist, pressin g after each one. V alues can be r eal or comple x number s (or an expr ession). If you enter a calculati on, it is ev aluated and the re sult is inserted in the list . 4. When done , pr ess LIST to see th e List catalog, or pres s to re turn to HOME . List catalog keys The list cat[...]

Lists 19-3 List edit keys When you press to create or change a list, the following keys are available to you: Create a list in HOME 1. Enter the list on the edit line. Start and end t he list w ith brace s (the shifted and ke y s) and separate each element w ith a comma. 2 . Pres s to evaluate and display the li st. Immediatel y after typ ing in th[...]

19-4 Lists Displaying and editing lists To display a list • In the List catalog , highligh t the list name and pres s . • In HOME , enter the name of the list and pr ess . To display one element In HOME, enter listname ( element# ). For example, if L2 is {3,4,5,6}, then L2(2) returns 4 . To edit a list 1. Open the List catalog. LIST . 2 . Pr es[...]

Lists 19-5 To insert an element in a list 1. Open t he List catalog. LIST . 2. P r e s s o r t o highligh t the name of the list y ou wan t to edit (L1, etc.) and pr ess to display the lis t contents . New elements are inserted above the highlighted positio n. In this example, an element, with the value of 9, is inserted between the first and secon[...]

19-6 Lists Deleting lists To delete a list In the List catalog, highli ght the list name and press . You are prompted to confirm that you want to delete the contents of the highlighted list variable. Press to delete the contents. To delete all lists In the List catalog, press CLEAR . Transmitting lists You can send lists to calculators or PCs just [...]

Lists 19-7 var iable name (su ch as L1) or the actual list . F or ex ample , REVERSE({1,2,3}) . • If Dec imal Mark in Modes is set to C omma, use peri ods to separ ate arguments . F or e xample , CONCAT(L1.L2) . Common operators like +, –, ×, and / can take lists as arguments. If th ere are two ar guments a nd both are lists, then the lists mu[...]

19-8 Lists MAKELIST Calculates a sequence of elements for a new list. Evaluates expression with variable from begin to end values, taken at increment steps. MAKELIST( expression , va riab l e , begin , end , incr ement ) The MAKELIST function generates a series by automatically producing a list from the repeated evaluation of an expression. Example[...]

Lists 19-9 SIZE Calculates the number of elements in a list. SIZE( list ) Also works with matrices. Σ LIST Calculates the sum of all elements in list. Σ LIST( list ) Example Σ LIST({2,3,4}) ret ur n s 9 . SORT Sorts eleme nts in ascending o rder. SORT( list ) Finding statistical values for list elements T o f i n d v a l u e s s u c h a s t h e [...]

19-10 Lists 3 . Start the S tatistic s aplet, and se lect 1-var iable mode (pre ss , if necessary , to displa y ). Select Statistics Note: Y our list values are n ow in column 1 (C1). 4. In the S ymbo lic vi ew , define H1 (fo r ex ample) as C1 (sample) and 1 (f req uency). 5 . Go to the Numeri c vie w to displa y calc ulated statistic s. See “On[...]

Notes and sketches 20-1 20 Notes and sk etch es Introduction The HP 40gs has text and pi cture editors for entering notes and sk etches. • E ach aplet has its o wn independent No te vie w and Sk etch vi e w . Notes and sk etc hes that y ou c reat e in these vi ews ar e associ ated with the aplet. When you sav e the aplet , or send it to another c[...]

20-2 Notes and sketches Note edit keys Key Me a n in g Space key for text entry. Displays next page o f a multi-page note. Alpha-lock for letter entry. Lower-case alpha-lock for letter entry. Backspaces cursor and deletes character. Deletes current character. Starts a new line. CLEAR Erases the entire note. Menu for entering variable names, and con[...]

Notes and sketches 20-3 Aplet sketch view You can attach pictures to an aplet in its Sketch view ( SKETCH ). Y our wo rk is automati cally s av ed with the aplet . Pres s any other v ie w ke y or to ex it the Sketc h vie w Sketch keys To draw a line 1. In an aplet, pr ess SKETCH for the Sk etch v iew . 2 . In Sk etch v ie w , pres s and mo ve the c[...]

20-4 Notes and sketches To draw a box 1. In Sketch v ie w , pres s and mov e the c ursor to wher e you w ant any corner of the box to be . 2. P r e s s . 3 . Mov e the cur sor to mark the oppo site corner for the box . Y ou can adjust the siz e of the bo x by mov ing the cu rs or. 4. Pre ss to f inish the box . To draw a circle 1. In Sketc h vi ew [...]

Notes and sketches 20-5 To label parts of a sketch 1. Pres s and type the te xt on the edit line . T o lock the Alpha shift on, pr ess (f or uppercas e) or (fo r low er case). T o make the label a smaller c har acter si ze , turn o ff befor e pr essing . ( i s a toggle betwee n small and large f ont si z e). The smaller char acte r siz e cannot dis[...]

20-6 Notes and sketches To import a graphics variable You can copy the contents of a graphics varia ble into the Sketch view of an aplet. 1. Open the Sketch v iew o f the aplet ( SKETCH ). The gr aphic w ill be copied her e. 2 . Pr ess , . 3 . Highli ght Graphic , then pr ess and highlight the name of the var iable ( G1 , etc .) . 4. Pr ess to reca[...]

Notes and sketches 20-7 4. W rite your note . See “Note edit k e ys ” on page 20 - 2 for more infor mation on the entry and editing of notes. 5 . When you ar e finished, press or an aplet key to e xit Not epad. Y our wor k is automati cally sa ved . Notepad Catalog keys Key M e a n i n g Opens the selected note for editing. Begins a new note, a[...]

20-8 Notes and sketches To import a note You can import a note from the Notepad i nto an aplet’s Note view, and vice versa. Suppose you wan t to copy a note named “Assignments” fr om the Notepad into the Function Note view: 1. In the F unction aplet , displa y the Note v ie w ( NOTE ). 2 . Pr ess , highligh t Notepad in the left column, then [...]

Programming 21-1 21 Pr ogramming Introduction This chapter describes how to program using the HP 40gs. In this chapter you’ll learn about: • using the Pr ogram catalog to c reat e and edit progr ams • progr amming commands • stor ing and re trie v ing var iables in pr ograms • progr amming v ariables . HINT More information on programming[...]

21-2 Programming Example RUN GETVALUE: RUN CALCULATE: RUN " SHOW ANSWER " : This program is separated into three main tasks, each an individual program. Within each program, the task can be simple—or it can be di vided further into other programs that perform smaller tasks. Program catalog The Program catalog is wher e you create, edit,[...]

Programming 21-3 Program catalog keys The program catalog keys are: Key M e a n i n g Opens the highlighted program for editing. Prompts for a new program name, then opens an empty program. Transmits the highlighted program to another HP 40gs or to a disk drive. Receives the highlighted program from another HP 40gs or from a disk drive. Runs the hi[...]

21-4 Programming Creating and editing programs Create a new program 1. Pres s PROGRM t o open the Progr am catalog . 2. P r e s s . The HP 40gs prompts you f o r a n a m e. A progr a m name can contain spec ial c harac ters, such as a space . How ev er , if you use spec ial char acter s and then run the pr ogram b y typing it in HOME , you mu st en[...]

Programming 21-5 2 . On the left , use or t o highlight a command category , then press to access the commands in the category . Select the command that y ou want . 3 . Pres s to paste the command into the pr ogr am editor . Edit a program 1. Press PROGRM to open the Progr am catalog. 2 . Us e the arr o w ke y s to highligh t the progr am y ou wan [...]

21-6 Programming Editing keys The editing keys are: Key M e a n i n g Inserts the character at the editing point. Inserts space into text. Displays pre vious page of the program. Displays next page of the program. Moves up or down one line. Moves right or left one character. Alpha-lock for letter entry. Press A...Z to lock lower case. Backspaces cu[...]

Programming 21-7 Using programs Run a program From HOME, type RUN program_name. or From the Program catalog, highlight the program you want to run and press Regardless of where you star t the program, all programs run in HOME. What you see will differ slightly depending on where you started the program. If you start the program from HOME, the HP 40[...]

21-8 Programming Copy a program You can use the following procedure if you want to make a copy of your work before editing—or if you want to use one program as a template for another. 1. Pres s PROGRM t o open the Progr am catalog . 2. P r e s s . 3 . T ype a ne w file name , then choose . The Pr ogram E ditor opens with a new pr ogr am. 4. Pres [...]

Programming 21-9 Delete a program To delete a program: 1. Pres s PROGRM to open the Progr am catalog. 2 . Hi ghlight a pr ogr am to delete , then pr ess . Delete all programs You can delete all programs at once. 1. In the Pr ogram catalog , pr ess CLEAR . 2. P r e s s . Delete the contents of a program You can clear the contents of a program withou[...]

21-10 Programming 4. Dev elop a progr am that use s the SETVIEWS command to modify the aplet’s VIEW S menu . The menu options pr ovid e links to ass oci ated pr ograms . Y ou can spec ify any ot her progr ams that y ou want trans ferr ed w ith the aplet . See “SETVIEWS” on page 21-14 for info rmation on the command . 5 . Ensur e that the cu s[...]

Programming 21-11 Save the aplet 1. Open the Functi on aplet and sa ve it as “EXPERI MENT ” . The ne w aplet appear s in the Aplet library . Select Function EXP ERIMENT 2 . Cr eate a progr am called EXP .ME1 w ith contents as shown . This progr am conf igur es the plot ranges , then runs a progr am that allo w s y ou to set the angle f ormat . [...]

21-12 Programming 6. Open the Pr ogram cat alog and cr eate a pr ogram named “EXP .S V” . Include the follo w ing code in the progr am . E ach entry line after the command SE T VIEW S is a tri o that consists of a VIEW S menu text line (a space indicate s none), a progr am name, and a number that def ines the vi ew t o go to after the pr ogram [...]

Programming 21-13 ’ ’ ’ ’ ;’ ’ EXP.ANG’ ’ ;0; The pr ogram EXP .ANG is a small routine that is called by other pr ograms that the aplet use s. T his entry specif ies that the progr am EXP.ANG is transferr ed when the aplet is tr ansfer red , but the space in the fir st quote s ensure s that no entry appears on the menu . ’ ’ Sta[...]

21-14 Programming Aplet commands CHECK Checks (selects) the correspon ding function in the current aplet. For example, Check 3 would check F3 if the current aplet is Function. Then a checkmark would app ear next to F3 in Symbolic view, F3 would be plotted in Plot view, and evaluated in Numeric view. CHECK n : SELECT Selects the named aplet and ma k[...]

Programming 21-15 options us e , or the pr ogram that def ines the aplet ’s VIEW S menu . • Y ou can inclu de a “Start” opti on in the VIEW S menu to spec ify a progr am that y ou want to r un automati cally when the aplet s tarts. This pr ogr am typically sets up the aplet’s initial configur ation. T he S T ART optio n on the menu is als[...]

21-16 Programming ProgramName ProgramName is the name of the program that runs when the corresponding menu entry is selected. All programs that are identified in the aplet’s SETVIEW S command are transferred when the aplet is transmitted. ViewNumber V iewNumber is the number of a view to start after the program finishes running. For example, if y[...]

Programming 21-17 View numbers The Function aplet views are numbered as fo llows: View numbers from 15 on will vary according to the parent aplet. The list shown above is for the Function aplet. Whatever the normal VIEWS menu for the parent aplet, the first entry will become number 15, the second number 16 and so on. UNCHECK Unchecks (unselects) th[...]

21-18 Programming Example 1 X A : IF A==1 THEN MSGBOX " A EQUALS 1" : END: IF... THEN... ELSE... END Executes the true-clause sequence of commands if the test- clause is true, or the false-clause sequence of commands if the test-clause is false. IF test-clause THEN true-clause ELSE false-clause END Example 1 X A : IF A==1 THEN MSGBOX &quo[...]

Programming 21-19 IFERR...THEN...ELSE...END allows a program to intercept error conditions that otherwise would cause the program to abort. Its syntax is: IFERR tr ap-claus e THEN clause _1 ELSE clause_ 2 END : Example IFERR 60/X X Y: THEN MSGBOX "Error: X is zero.": ELSE MSGBOX "Value is "Y: END: RUN Runs the named program. If [...]

21-20 Programming Example ARC 0;0;2;0;2 π : FREEZE: Dra w s a c irc le cente red at (0, 0) of r adius 2 . The FREEZE command causes the cir c le to remain display ed on the screen until yo u press a k ey . BOX Draws a box with diagonally opposite corners ( x1,y1 ) and ( x2,y2 ). BOX x1 ; y1 ; x2 ; y2 : Example BOX -1;-1;1;1: FREEZE: Dra w s a bo x[...]

Programming 21-21 Example TLINE 0;0;3;3: Era ses pr ev iously dr a wn 4 5 degr ee line fr om (0, 0) to (3, 3), or draw s that line if it doesn ’t alread y e xist . Graphic commands The graphic commands use th e graphics variables G0 through G9—or the Page variable from Sketch—as graphicname arguments. The position argument takes the form ( x,[...]

21-22 Programming GROBXOR Using the logical XOR, superimposes graphicname2 onto graphicname1 . The upper left corner of graphicname2 is placed at position . GROBXOR gra phicname1 ; ( position ) ; gr aphicname2 : MAKEGROB Creates graphic with given width, height, and hexadecimal data, and stores it in graphicname . MAKEGROB gr aphicname ; widt h ; h[...]

Programming 21-23 Loop commands Loop hp allow a program to execute a routine repeatedly. The HP 40gs has three loop structures. The example programs below illustrate each of these structures incrementing the variable A from 1 to 12. DO…UNTIL …E ND Do ... Until ... End is a loop command that executes the loop-clause repeatedly until test-clause [...]

21-24 Programming Matrix commands The matrix commands take variables M0–M9 as arguments. ADDCOL Add Column. Inserts values into a column before column_number in the specified matrix . You enter the values as a vector. The values must be separated by commas and the number of valu es must be the same as the number of rows in the matrix name . ADDCO[...]

Programming 21-25 REPLACE Replaces portion of a matrix or vector stored in name with an object starting at position start . start for a matrix is a list containing two numbers; for a vector, it is a single number. Replace also works with lists and graphics. REPLACE name ; start ; object : SCALE Multiplies the specified row_number of the specifie d [...]

21-26 Programming PRVAR Prints name and contents o f variablename . PRVAR var iablename : You can also use the PRVAR command to print th e contents of a program or a note. PRVAR progr amname ;PROG: PRVAR notename ; NOTE: Prompt commands BEEP Beeps at the frequency and for the time you specify. BEEP frequen cy ; second s : CHOOSE Creates a choose bo[...]

Programming 21-27 Example If you have stored {1,2,3,4} in variable L1, entering CLVAR L1 w ill clear L1. DISP Displays textitem in a row of the display at the line_number . A text item consists of any number of expressions and quoted strings of text. The expressions are evaluated and turned into strings. Lines are numbered from the top of the scree[...]

21-28 Programming Examples 5.152000 X DATE( sets the date to May 15, 2000) . 10.1500 X TIME (sets the time to 10:15 am). EDITMAT Matrix Editor. Opens the Matr ix editor for the specified matrix. Returns to the program when user presses EDITMAT matr ixname : The EDITMAT command can also be used to create matrices. 1. Pres s CMDS 2. P r e s s M 1, an[...]

Programming 21-29 Example INPUT R; "Circular Area"; "Radius"; "Enter Number";1: MSGBOX Displays a mess age box containing textitem. A text ite m consists of any number of expressions and quoted strings of text. The expressions are evaluate d and turned into strings of text. For example , "AREA IS:" 2+ 2 becom[...]

21-30 Programming Stat-One commands DO1VSTATS Calculates STATS using datasetname and stores the results in the corresponding variables: N Σ , Tot Σ , Mean Σ , PVar Σ , SVar Σ , PSDev, SSDev, Min Σ , Q1, Median, Q3, and Max Σ . Datasetname can be H1, H2, ..., or H5. Datasetname must include at least two data points. DO1VSTATS datase tname : S[...]

Programming 21-31 Storing and retrieving variables in programs The HP 40gs has both Ho me variables and Aplet variables. Home variables ar e used for real numbers, complex numbers, graphics, lists, and matrices. Home variables keep the same values in HOME and in aplets. Aplet variables are those whose values depend on the current aplet. The aplet v[...]

21-32 Programming Coord Function Parametric Polar Sequence Solve Statistics Turns the coordinate-display mode in Plot view on o r off. From Plot view, use the Menu mean key to toggle coordinate display on an off. In a program, type 1 X Coord —to turn coor dinate displa y on (defa ult) . 0 X Coord —to turn coor dinate display off . Extremum Func[...]

Programming 21-33 Hwidth Statistics Sets the width of histogram bars. From Plot Setup in 1VAR stats set a value for Hwidth or In a program, type n X Hwidth Indep All Aplets Defines the value of the independent variable used in tracing mode. In a program, type n X Indep InvCross All Aplets Toggles between solid crosshai rs or inverted crosshairs. (I[...]

21-34 Programming Nmin / Nmax Sequence Defines the minimum and maxi mum independent variable values. Appears as the NRNG fields in the Plot Setup input form. From Plot Setup, enter values for NRNG . or In a program, type X Nmin X Nmax whe re Recenter All Aplets Recenters at the crosshairs locations when zooming. From Plot-Zoom-Set Factors, c heck ([...]

Programming 21-35 Simult Function Parametric Polar Sequence Enables you to choose between simultaneous and sequential graphing of all selected expressions. From Plot Setup, check (or uncheck) _ SIMULT or In a program, type 1 X Simult —fo r simultaneous gr aphing (defa ult). 0 X Simult —fo r sequenti al gra phing. Slope Function Contains the las[...]

21-36 Programming Tmin / Tmax Parametric Sets the minimum and maxi mum independent variable values. Appears as the TRNG field in the Plot S etup input form. From Plot Setup, enter values for TRNG . or In a pr ogram , type X Tmin X Tmax wher e Tracing All Aplets Turns the tracing mode on or off in Plot view. In a program, type 1 X Tracing —to turn[...]

Programming 21-37 Xtick AAll Aplets Sets the distance between tick marks for the horizontal axis. From the Plot Setup input form, enter a value for Xtick . or In a program, type n X Xtick whe re Ytick All Aplets Sets the distance between tick marks for the vertical axis. From the Plot Setup input form, enter a value for Ytick . or In a program, typ[...]

21-38 Programming Xzoom All Aplets Sets the horizontal zoom factor. From Plot-ZOOM-Set Factors, enter the value for XZOOM . or In a program, type n X XZOOM wher e The default value is 4. Yzoom All Aplets Sets the vertical zoom factor. From Plot-ZOOM-Set Factors, enter the value for YZOOM . or In a program, type n X YZOOM The default value is 4. Sym[...]

Programming 21-39 X1, Y1...X9,Y9 X0,Y0 Parametric Can contain any expression. Independent variable is T. Example 'SIN(4*T)' X Y1(T):'2*SIN(6*T)' X X1(T) R1...R9, R0 Polar Can contain any expression. Independent variable is θ . Example '2*SIN(2* θ )' X R1( θ ) U1...U9, U0 Sequence Can contain any expression. Independ[...]

21-40 Programming Example Cubic X S2fit or 6 X S2fit Numeric-view variables The following aplet variable s control the Numeric view. The value of the variable applies to the current aplet only. C1...C9, C0 Statistics C0 through C9 , for columns of data. Can contain lists. Enter data in the Numeric view or In a program, type LIST X C n wher e n = 0,[...]

Programming 21-41 1 Standard 2 Fixed 3 Sci 4 Eng 5 Fraction 6 MixFraction Note: if Fraction or Mixed Fracti on is chosen, the setting will be disregarded when labeling axes in the Plot view. A setting of Scientific will be used instead. Example Scientific X Format or 3 X Format NumCol All Aplets except Statistics aplet Sets the column to be highlig[...]

21-42 Programming NumStart Function Parametric Polar Sequence Sets the starting value for a table in Numeric view. From Num Setup, enter a value for NUMSTART . or In a program, type n X NumStart NumStep Function Parametric Polar Sequence Sets the step size (increment value) for an independ ent variable in Numeric view. From Num Setup, enter a value[...]

Programming 21-43 Example 1VAR X StatMode or 1 X StatMode Note variables The following aplet variable is availa ble in Note view. NoteText All Aplets Use NoteText to recall text previously entered in Note view. Sketch variables The following aplet variables are available in Sketch view. Page All Aplets Sets a pa ge in a sketch set. The graphics can[...]

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Extending aplets 22-1 22 Extending aplets Aplets are the application environments where you explore different classes of m athematical operations. You can extend the capabili ty of the HP 40gs i n the following ways: • Creat e new aplets , based on ex isting aplets, with spec ifi c confi gurati ons suc h as angle measur e, gra phical or tabular s[...]

22-2 Exten ding aple ts 1. Open the Solve aplet and sav e it under th e new name . Solve | T R I A N G L E S 2 . Ent er th e fou r fo rmu la s: θ O H θ A H θ OA AB C 3 . Deci de whether y ou w ant the aplet to oper ate in Degree s, R adians, or Gr ads. MODES Degrees 4. Vi ew the A plet L ibrary . The “ TRIANGLE S” aplet is listed in the Aple[...]

Extending aplets 22-3 Using a customized aplet To use the “Triangles” aplet, simply select the appropria te formula, change to the Numeric view and solve for the missing variable. Find the length of a ladder leaning against a vertical wall if it forms an angle of 35 o with the horizontal and extends 5 metres up the wall. 1. Select the a plet. T[...]

22-4 Exten ding aple ts Annotating an aplet with notes The Note view ( NOTE ) attaches a note to the current aplet. See Chapter 2 0, “Notes and sketches” . Annotating an aplet with sketches The Sketch view ( SKETCH ) attaches a picture to the current aplet. See chapter 20, “Notes and sketches”. HINT Notes and sketches that you attach to an [...]

Extending aplets 22-5 To transmit an aplet 1. Connect the P C or aple t disk dri ve t o the calculat or by an appropr iate cable. 2 . Sending calc ulator : Open the L ibr ary , highli ght the aplet to send , and pres s . – The SEND TO menu appears w ith the follo w ing options: HP39/40 (USB) = to send via the U SB port HP39/40 (SER) = to send via[...]

22-6 Exten ding aple ts If you are using the PC Connectivity Kit to download aplets from a PC, you will see a list of aplets in the PC’s current directory. Check as ma ny items a s you would li ke to receive. Sorting items in the aplet library menu list Once you have entered information into an aplet, you have defined a new version of an aplet. T[...]

R-1 R Re ference inf ormation Glossary aplet A small application, limited to one topic. The built-in aplet types are Function, Parametric, Polar, Sequence, Solve, Statistics, Inference, Finance, Trig Explorer, Quad Explorer, Linear Explorer and Triangle Solve. An aplet can be filled with the data and solutions for a specific problem. It is reusable[...]

R-2 list A set of values separated by commas (periods if the Decimal Mark mode is set to Comma ) and enclosed in braces. Lists are commonly used to enter statistical data and to evaluate a function with multiple values. Created and manipulated by the List editor and catalog. matrix A two-dimensional array of values separated by commas (periods if t[...]

R-3 Resetting the HP 40gs If the calculator “locks up” and seems to be stuck, you must reset it. This is much like resetting a PC. It cancels certain operations, restores ce rtain conditions, and clears temporary memory locations. However, it does not clear stored data (variables, aplet datab ases, programs) unless you use th e procedur e, “T[...]

R-4 If the calculator does not turn on If the HP 40gs does not turn on follow the steps below until the calculator turns on. You may find that the calculator turns on before you have completed the procedure. If the calculator still does not turn on, please contact Customer Support for further information. 1. Pres s and hold the ke y fo r 10 seconds[...]

R-5 To install the main batteries a. Slide up the battery compartment cover as illustrated. b. Insert 4 new AAA (LR03) batteries into the main compartment. Make sure each batte ry is inserted in the indicated direction. To install the backup battery a. Pres s down the holder. Push the plate to the shown direction and lift it. b. Insert a new CR2032[...]

R-6 Variables Home variables The home variables are: Category Available name Complex Z1 ... Z9 , Z0 Graphic G1 ... G9 , G0 Library Function Parametric Polar Sequence Solve Statistics User-named List L1 ... L9 , L0 Matrix M1 ... M9 , M0 Modes Ans Date HAngle HDigits HFormat Ierr Time Notepad User-named Program Editline User-named Real A...Z, θ hp40[...]

R-7 Function aplet variables The function aplet variables are: Category Availa ble name Plot Axes Connect Coord FastRes Grid Indep InvCross Labels Recenter Simult Tracing Xcross Ycross Xtick Ytick Xmin Xmax Ymin Ymax Xzoom Yxoom Plot-FCN Area Extremum Isect Root Slope Symbolic Angle F1 F2 F3 F4 F5 F6 F7 F8 F9 F0 Numeri c Digits Format NumCol NumFon[...]

R-8 Parametric aplet variables The parametric aplet variables are: Category Available name Plot Axes Connect Coord Grid Indep InvCross Labels Recenter Simult Tmin Tmax Tracing Tstep Xcross Ycross Xtick Ytick Xmin Xmax Ymin Ymax Xzoom Yzoom Symbolic Angle X1 Y1 X2 Y2 X3 Y3 X4 Y4 X5 Y5 X6 Y6 X7 Y7 X8 Y8 X9 Y9 X0 Y0 Numeric Digits Format NumCol NumFon[...]

R-9 Polar aplet variables The polar aplet variables are: Category Available names Plot Axes Connect Coord Grid Indep InvCross Labels Recenter Simult Umin Umax θ step Tracing Xcross Ycross Xtick Ytick Xmin Xmax Ymin Ymax Xzoom Yxoom Symbolic Angle R1 R2 R3 R4 R5 R6 R7 R8 R9 R0 Numeri c Digits Format NumCol NumFont NumIndep NumRow NumStart NumStep N[...]

R-10 Sequence aplet variables The sequence aplet variables are: Category Available name Plot Axes Coord Grid Indep InvCross Labels Nmin Nmax Recenter SeqPlot Simult Tracing Xcross Ycross Xtick Ytick Xmin Xmax Ymin Ymax Xzoom Yzoom Symbolic Angle U1 U2 U3 U4 U5 U6 U7 U8 U9 U0 Numeric Digits Format NumCol NumFont NumIndep NumRow NumStart NumStep NumT[...]

R-11 Solve aplet variables The solve aplet variables are: Category Availa ble name Plot Axes Connect Coord FastRes Grid Indep InvCross Labels Recenter Tracing Xcross Ycross Xtick Ytick Xmin Xmax Ymin Ymax Xzoom Yxoom Symbolic Angle E1 E2 E3 E4 E5 E6 E7 E8 E9 E0 Numeri c Digits Format NumCol NumRow Note NoteText Sketch Page PageNum hp40g+.book Page [...]

R-12 Statistics aplet variables The statistics aplet variables are: Category Available name Plot Axes Connect Coord Grid Hmin Hmax Hwidth Indep InvCross Labels Recenter S1mark S2mark S3mark S4mark S5mark StatPlot Tracing Xcross Ycross Xtick Ytick Xmin Xmax Ymin Ymax Xzoom Yxoom Symbolic Angle S1fit S2fit S3fit S4fit S5fit Numeric C0,...C9 Digits Fo[...]

R-13 MATH menu categories Math functions The math functions are: Category Availa ble name Calculus TAYLOR Complex ARG CONJ IM RE Constant e i MAXREAL MINREAL π Hyperb . ACOSH ASINH ATANH COSH SINH TANH ALOG EXP EXPM1 LNP1 List CONCAT Δ LIST MAKELIST π LIST POS REVERSE SIZE Σ LIST SORT Loop ITERATE RECURSE Σ ∂ ∫ hp40g+.book Page 13 Friday, [...]

R-14 Matrix COLNORM COND CROSS DET DOT EIGENVAL EIGENVV IDENMAT INVERSE LQ LSQ LU MAKEMAT QR RANK ROWNORM RREF SCHUR SIZE SPECNORM SPECRAD SVD SVL TRACE TRN Polynom. POLYCOEF POLYEVAL POLYFORM POLYROOT Prob. COMB ! PERM RANDOM UTPC UTPF UTPN UTPT Real CEILING DEG → RAD FLOOR FNROOT FRAC HMS → → HMS INT MANT MAX MIN MOD % %CHANGE %TOTAL RAD ?[...]

R-15 Program constants The program constants are: Tests < ≤ = = ≠ > ≥ AND IFTE NOT OR XOR Trig ACOT ACSC ASEC COT CSC SEC Category Av ailable name (Continued) Category Availa ble name Angle Degrees Grads Radians Format Standard Fixed Sci Eng Fraction SeqPlot Cobweb Stairstep S1...5fit Linear LogFit ExpFit Power Trigonometric QuadFit Cub[...]

R-16 Physical Constants The physical constants are: Category Available Nam e Chemist • Avogadro (A vagadr o ’s Number , NA) • Boltz . (Boltmann, k) • mol. vo... (molar v olume , Vm) • univ gas (uni ver sal gas , R) • std temp (standard temper ature , St d T) • std pres (standar d pressur e, St d P) Phyics • StefBolt (Stef an -Boltzm[...]

R-17 CAS functions CAS functions are: Category Function Algebra COLLECT DEF EXPAND FACTOR PARTFRAC QUOTE STORE | SUBST TEXPAND UNASSIGN Complex i ABS ARG CONJ DROITE IM – RE SIGN Constant e i ∞ π Diff & Int DERIV DERVX DIVPC FOURIER IBP INTVX lim PREVAL RISCH SERIES TABVAR TAYLOR0 TRUNC Hyperb . ACOSH ASINH ATANH COSH SINH TANH Integer DIV[...]

R-18 Polynom. EGCD FACTOR GCD HERMITE LCM LEGENDRE PARTFRAC PROPFRAC PTAYL QUOT REMAINDER TCHEBYCHEFF Real CEILING FLOOR FRAC INT MAX MIN Rewrite DISTRIB EPSX0 EXPLN EXP2POW FDISTRIB LIN LNCOLLECT POWEXPAND SINCOS SIMPLIFY XNUM XQ Solve DESOLVE ISOLATE LDEC LINSOLVE SOLVE SOLVEVX Tests ASSUME UNASSUME > ≥ < ≤ = = ≠ AND OR NOT IFTE Trig [...]

R-19 Program commands The program commands are: Category Command Aplet CHECK SELECT SETVIEWS UNCHECK Branch IF THEN ELSE END CASE IFERR RUN STOP Drawing ARC BOX ERASE FREEZE LINE PIXOFF PIXON TLINE Graphic DISPLAY → → DISPLAY → GROB GROBNOT GROBOR GROBXOR MAKEGROB PLOT → → PLOT REPLACE SUB ZEROGROB Loop FOR = TO STEP END DO UNTIL END WHIL[...]

Status messages Stat-Two DO2VSTATS SETDEPEND SETINDEP Category Command (Continued) Message Meaning Bad Argument Type Incorrect input for this operation. Bad Argument Value The value is out of range for this operation. Infinite Result Math exception, such as 1/0. Insufficient Memory You must recover some memory to continue operation. Delete one or m[...]

R-21 Invalid Syntax The function or command you entered does not include the proper arguments or order of arguments. The delimiters (parentheses, commas, periods, and semi-colons) must also be correct. Look up the function name in the index to find its proper syntax. Name Conflict The | (where) function attempted to assign a value to the variable o[...]

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W-1 Limited W arr anty HP 40gs Graphing Calculator; Warranty period: 1 2 months 1. HP warr ants to y ou, the end-user c ustomer , that HP hard war e, accessor ies and supplies w i ll be fr ee fr om defec ts in materi als and wo rkmanship after the date of pur chase , for the per iod spec ifi ed abov e. If HP recei ves notice of such defects during [...]

W-2 6. HP MAKE S NO O THER EXP RE S S W ARRANTY OR CONDIT ION WHE THER WRITTEN OR ORAL. T O THE EXTENT ALL OWED B Y L OCAL L A W , ANY IMPLIED W ARR ANTY OR CONDIT ION OF MERCHANT ABILITY , SA TISF ACT OR Y QUALI TY , OR FITNE SS FOR A P ARTICULAR PURP OSE IS LIMI TED T O THE DURA TION OF THE EXPRE SS W A RRANTY SET F ORTH AB OVE . Some countri es,[...]

W-3 Service Europe Country : Telephone numbers Austr ia +4 3-1-36 0 2 7 71203 Belgium +3 2 - 2 - 712 6 219 D e n m a r k + 45 - 8 -2 332 84 4 Ea st e r n Eu ro p e countr ies +4 20 -5- 414 2 2 5 2 3 Fi n l a n d + 35 - 89 6 40 0 09 Fr ance +33-1- 4 9 9 3 9006 German y +4 9-6 9-95 30 7103 Gr eece +4 20 -5- 414 2 25 2 3 Holland +31- 2-06 54 5 301 Ita[...]

W-4 P lease logon to http://www .hp.com for the la test se r vice and suppo rt informati on .h L.Ame ric a Country: Telephone numbers Ar gentina 0 -810 -55 5-5 5 20 Bra zil Sao P aulo 3 7 4 7 - 77 99; RO T C 0 -800 -15 77 51 M e xi c o M x Ci t y 5258 - 9922; RO T C 01-800 - 4 7 2 -6 68 4 Ven e z u e l a 0 8 0 0 - 4 7 4 6 -8368 Chil e 800 - 3609 99[...]

W-5 Regulatory Notices Federal Commu- nications Commission Notice This equipment has been tested and found to comply with the limits for a Class B digital device , pursuant to Part 15 of the FCC Rules. These limi ts are designed to provide reasonable protection agains t harmful interference in a residential installation. This equipment generates, u[...]

W-6 Hewlett-Packard Company P. O. Box 692000, Mail Sto p 530113 Houston, Texas 77269-2000 Or, call 1-800-474-6836 For questions regarding this FCC declaration, co ntact: Hewlett-Packard Company P. O. Box 692000, Mail Sto p 510101 Houston, Texas 77269-2000 Or, call 1-281-514-3333 To identify this product, refer to the part, series, or model number f[...]

W-7 Japanese Notice こ の装置は、 情報処理装 置等電波障害自主規制協議会 （VCCI） の基準 に 基 づ く ク ラ ス B 情報技術装置 で す。 こ の装 置は、 家庭環 境 で 使用 す る こ と を 目的 と し て い ま す が、 こ の 装置が ラ ジ オや テ レ ビ ジ ョ ン 受信機 ?[...]

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I-1 Index A ABCUV 14-62 ABS 14-45 absolute value 13-6 ACOS2S 14-38 add 13-4 ADDTMOD 14-51 ALGB menu 14-10 algebraic entry 1-19 alpha characters typing 1-6 alphabetical sorting 22-6 angle measure 1-10 in statistics 10-12 setting 1-11 animation 20-5 creating 20-5 annunciators 1-3 Ans (last answer) 1-24 antiderivative 14-68 , 14-69 antilogarithm 13-4 [...]

I-2 bad guesses error message 7-7 batteries R-4 Bernoulli’s number 14-65 box-and-whisker plot 10-16 branch commands CASE...END 21-18 IF...THEN...ELSE...END 21-18 IFERR...THEN...ELSE 21-18 branch structu res 21-17 build your own table 2-19 C calculus operation s 13-7 CAS 14-1 , 15-1 configurat ion 15-3 help 15-4 history 14-8 in HOME 14-7 list of f[...]

I-3 cosine 13-4 inverse hyperbolic 13-9 cotangent 13-20 covariance statistical 10-15 creating aplet 22-1 lists 19-1 matrices 18-2 notes in Notepad 20-6 programs 21-4 sketches 20-3 critical value(s) displayed 11-4 cross product vector 18-11 curve fitting 10-12 , 10-17 CYCLOTOMIC 14-63 D data set definition 10-8 date, setting 21-27 debugging programs[...]

I-4 notes 20-2 programs 21-5 Editline Program catalog 21-2 editors 1-30 EGCD 14-55 eigenvalues 18-11 eigenvectors 18-11 element storing 18-6 E-lessons 1-12 engineering number format 1-11 EPSX0 14-29 equals for equations 13-17 logical test 13-19 Equation Writer 14-2 , 15-1 , 16-1 selecting terms 15-5 equations solving 7-1 erasing a line in Sketch vi[...]

I-5 glossary R-1 graph analyzing statistical data in 10-19 auto scale 2-14 box-and-wh isker 10-16 capture current display 21-21 cobweb 6-1 comparing 2-5 connected points 10-17 defining the independent variable 21-36 drawing axes 2-7 expressions 3-3 grid points 2-7 histogram 10-15 in Solve aplet 7-7 one-variable statistics 10-18 overlaying 2-15 scat[...]

I-6 using symbolic variables 13-23 independent values adding to table 2-19 independent variable defined for Tracing mode 21-33 inference confidence intervals 11-15 hypothesis tests 11-8 One-Proportion Z- Interval 11-17 One-Sampl e Z-Interva l 11-15 One-Sampl e Z-Test 11-8 Two-Proportion Z-Interval 11-17 Two-Proportion Z-T est 11-11 Two-Sample T-Int[...]

I-7 finding statistical values in list ele- ments 19-9 generate a series 19-8 list function syntax 19-6 list variables 19-1 returning position of element in 19-8 reversing order in 19-8 sending and receiving 19-6 sorting elements 19-9 storing elements 19-1 , 19-4 , 19-5 storing one element 19-6 LNCOLLECT 14-31 logarithm 13-4 logarithmic fit 10-13 f[...]

I-8 redimension 21-24 replacing portion of matrix or vec- tor 21-25 sending or receiving 18-4 singular value decomposition 18-13 singular values 18-13 size 18-12 spectral norm 18-13 spectral radius 18 -13 start Matrix Editor 21-24 storing elements 18-3 , 18-5 storing matrix elements 18-6 swap column 21-25 swap row 21-25 transposing 18-13 variables [...]

I-9 mixed fraction 1-11 scientific 1-10 Standard 1-10 numeric precision 17-9 Numeric view adding values 2-19 automatic 2-16 build your own table 2-19 display defining function for col- umn 2-17 recalculating 2-19 setup 2-16 , 2-19 O off automatic 1-1 power 1-1 on/cancel 1-1 One-Proportion Z-Interval 11-17 One-Sample T-Interval 11-18 One-Sample T-Te[...]

I-10 labels 21-34 recenter 21-34 root 21-34 s1mark-s5mark 21-34 statplot 21-35 tracing 21-33 umin/umax 21-35 ustep 21-35 polar variables axes 21-31 connect 21-31 grid 21-32 in menu map R-9 indep 21-33 labels 21-34 recenter 21-34 ycross 21-37 polynomial coefficients 13-11 evaluation 13-11 form 13-12 roots 13-12 Taylor 13-7 polynomial functions POLYC[...]

I-11 RE 13-8 real number maximum 13-8 minimum 13-8 real part 13-8 real-number functions 13-14 % 13-16 %CHANGE 13-16 %TOTAL 13-16 CEILING 13 -14 DEGtoRAD 13-14 FNROOT 13-14 HMSto 13-15 INT 13-15 MANT 13-15 MAX 13-15 MIN 13-15 MOD 13-15 RADtoDEG 13-16 ROUND 13-16 SIGN 13-16 TRUNCATE 13-17 XPON 13-17 reatest common divisor 14-47 recalculation for tabl[...]

I-12 date 21-27 time 21-27 SEVAL 14-68 SIGMA 14-68 SIGMAVX 14-69 SIGN 14-46 sign revers al 7-6 SIMPLIFY 14-32 simplify 14-68 , 14-70 SINCOS 14-31 , 14-40 sine 13-4 inverse hyper bolic 13-9 singular value decomposition matrix 18-13 singular values matrix 18-13 sketches creating 20 -5 creating a blank graph ic 21-22 creating a set of 20-5 erasing a l[...]

I-13 Labels 21-34 Recenter 21-34 S1mark-S5mark 21-34 Ycross 21-37 step size of independent variable 21-36 step-by-step 14-6 STORE 14-14 storing list elements 19-1 , 19 -4 , 19-5 , 19-6 matrix elemen ts 18-3 , 18-5 , 18-6 results of calculation 17-2 value 17-2 strings literal in symbolic operations 13-18 STURMAB 14-69 SUBST 14-15 substitution 14-14 [...]

I-14 COT 13-20 CSC 13-20 HALFTAN 14-40 SEC 13-20 SINCOS 14-40 TAN2CS2 14-40 TAN2SC 14-41 TAN2SC2 14-41 TRIGCOS 14-44 TRIGSIN 14-44 TRIGTAN 14-44 TRIGSIN 14-44 TRIGTAN 14-4 4 TRUNC 14-28 truncating values to decimal places 13-17 TSIMP 14-70 tstep 21-36 Two-Prop ortion Z-Interv al 11-17 Two-Prop ortion Z-Test 11-11 Two-Sample T-Interva l 11-19 Two-Sa[...]

I-15 in 2-9 options 2-9 , 3-8 options within a table 2-18 out 2-9 redrawing table of numbers op- tions 2-18 square 2-10 un-zoom 2-11 within Numeric view 2-18 X-zoom 2-9 Y-zoom 2-10 hp40g+.book Page 15 Friday, December 9, 2005 1:03 AM[...]

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