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HP 40gs User Manual

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HP 40gs graphing calculator
user's guide
Part Number F2225AA-90001


Table of Contents

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  • Page 1 HP 40gs graphing calculator user's guide Edition1 Part Number F2225AA-90001...
  • Page 3: Table Of Contents

    Contents Preface Manual conventions .............. P-1 Notice ................. P-2 1 Getting started On/off, cancel operations............1-1 The display ................1-2 The keyboard ...............1-3 Menus .................1-8 Input forms ................1-9 Mode settings ..............1-10 Setting a mode...............1-11 Aplets (E-lessons)..............1-12 Aplet library ..............1-16 Aplet views..............1-16 Aplet view configuration..........1-18 Mathematical calculations ............1-19 Using fractions..............1-25 Complex numbers ...............1-29...
  • Page 4 Function aplet interactive analysis........... 3-9 Plotting a piecewise-defined function ........ 3-12 4 Parametric aplet About the Parametric aplet ............ 4-1 Getting started with the Parametric aplet......4-1 5 Polar aplet Getting started with the Polar aplet ......... 5-1 6 Sequence aplet About the Sequence aplet............
  • Page 5 About the Inference aplet .............11-1 Getting started with the Inference aplet ......11-1 Importing sample statistics from the Statistics aplet ....11-4 Hypothesis tests ..............11-8 One-Sample Z-Test............11-8 Two-Sample Z-Test ............11-9 One-Proportion Z-Test............11-10 Two-Proportion Z-Test ............11-11 One-Sample T-Test ............11-12 Two-Sample T-Test ............11-14 Confidence intervals ............11-15 One-Sample Z-Interval...........11-15 Two-Sample Z-Interval ...........11-16 One-Proportion Z-Interval..........11-17...
  • Page 6 Symbolic calculations............13-20 Finding derivatives ............13-21 Program constants and physical constants ......13-24 Program constants............13-25 Physical constants ............13-25 14 Computer Algebra System (CAS) What is a CAS? ..............14-1 Performing symbolic calculations .......... 14-1 An example ..............14-2 CAS variables..............
  • Page 7 Accessing CAS functions............15-12 Equation Writer variables ..........15-16 Predefined CAS variables ..........15-16 The keyboard in the Equation Writer ......15-17 16 Step-by-Step Examples Introduction ...............16-1 17 Variables and memory management Introduction ................17-1 Storing and recalling variables ..........17-2 The VARS menu ..............17-4 Memory Manager ...............17-9 18 Matrices Introduction ................18-1 Creating and storing matrices..........18-2...
  • Page 8 Sorting items in the aplet library menu list ......22-6 Reference information Glossary................R-1 Resetting the HP 40gs ............R-3 To erase all memory and reset defaults ....... R-3 If the calculator does not turn on ........R-4 Operating details ..............R-4 Batteries .................
  • Page 9 Solve aplet variables............R-11 Statistics aplet variables ..........R-12 MATH menu categories ............R-13 Math functions ............... R-13 Program constants ............R-15 Physical Constants ............R-16 CAS functions ..............R-17 Program commands ............R-19 Status messages..............R-20 Limited Warranty Service................W-3 Regulatory Notices ............
  • Page 11: Preface

    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.
  • Page 12: Notice

    © Copyright 1994-1995, 1999-2000, 2003, 2006 Hewlett-Packard Development Company, L.P. The programs that control your HP 40gs are copyrighted and all rights are reserved. Reproduction, adaptation, or translation of those programs without prior written permission from Hewlett-Packard Company is also...
  • Page 13: Getting Started

    Getting started 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 to turn the calculator off. To save power, the calculator turns itself off after several minutes of inactivity.
  • Page 14: The Display

    The display To adjust the Simultaneously press ) to increase (or contrast decrease) the contrast. To clear the display • Press to clear the edit line. CANCEL • Press to clear the edit line and the CLEAR display history. Parts of the display Title History...
  • Page 15: The Keyboard

    α Alpha in effect for next keystroke. To cancel, press again. ((•)) Low battery power. Busy. Data is being transferred. The keyboard P 40gs Graphing Calculator Menu Key Labels Menu Keys Aplet Control Cursor Keys Keys Alpha Key Shift Key...
  • Page 16 Menu keys • On the calculator keyboard, the top row of keys are called menu keys. Their meanings depend on the context—that’s why they are blank. The menu keys are sometimes called “soft keys”. • The bottom line of the display shows the labels for the menu keys’...
  • Page 17 Entry/Edit keys The entry and edit keys are: Meaning Cancels the current operation if the CANCEL calculator is on by pressing Pressing , then turns the calculator off. Accesses the function printed in blue above a key. Returns to the HOME view, for performing calculations.
  • Page 18 Meaning (Continued) Displays a menu of all available CHARS characters. To type one, use the arrow keys to highlight it, and press . To select multiple characters, select each and press , then press Shifted keystrokes There are two shift keys that you use to access the operations and characters printed above the keys: Description Press the...
  • Page 19 HELPWITH The HP 40gs built-in help is available in HOME only. It provides syntax help for built-in math functions. Access the HELPWITH command by pressing and then the math key for which you require SYNTAX syntax help. Example Press SYNTAX...
  • Page 20: Menus

    H I N T When using the MATH menu, or any menu on the HP 40gs, pressing an alpha key takes you straight to the first menu option beginning with that alpha character. With this method, you do not need to press first.
  • Page 21: Input Forms

    To search a menu • Press to scroll through the list. If you press , you’ll go all the way to the end or the beginning of the list. Highlight the item you want to select, then press • If there are two columns, the left column shows general categories and the right column shows specific contents within a category.
  • Page 22: Mode Settings

    Mode settings You use the Modes input form to set the modes for HOME. H I N T 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.
  • Page 23: Setting A Mode

    Setting Options (Continued) Engineering. Displays result with an exponent that is a multiple of 3, and the specified number of significant digits beyond the first one. Example: 123.456E7 becomes 1.23E9 in Engineering 2 format. Fraction. Displays results as fractions based on the specified number of decimal places.
  • Page 24: Aplets (e-lessons)

    You select the aplet that you want to work with. Aplets come from a variety of sources: • Built-in the HP 40gs (initial purchase). • Aplets created by saving existing aplets, which have been modified, with specific configurations. See “Creating new aplets based on existing aplets”...
  • Page 25 Quad Explorer and Trig Explorer. You cannot modify configuration settings for these aplets. A great many more teaching aplets can be found at HP’s web site and other web sites created by educators, together with accompanying documentation, often with student work sheets.
  • Page 26 HP 40gs using the provided Connectivity Kit. Quad Explorer The Quad Explorer aplet is used to investigate the aplet behaviour of as the values of a, h and v change, both by manipulating the equation and seeing the change in the graph, and by manipulating the graph and seeing the change in the equation.
  • Page 27 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.
  • Page 28: Aplet Library

    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 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 display it in different views.
  • Page 29: Overlay Plot

    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 for further information. Plot-Table view The VIEWS menu contains the Plot-Table view. Select Plot-Table Splits the screen into the plot and the data table.
  • Page 30: Aplet View Configuration

    Note view Press to display the aplet’s note view. NOTE 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 page 20-1 for further information.
  • Page 31: Mathematical Calculations

    To save aplet You can save an aplet configuration that you have used, configuration and transfer the aplet to other HP 40gs calculators. See “Creating new aplets based on existing aplets” on page 22-1. Mathematical calculations The most commonly used math operations are available from the keyboard.
  • Page 32 – 14 8 Example Calculate --------------------------- - 3 – 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 Type to start a negative number or to insert a...
  • Page 33 The calculator inserts it automatically. Parentheses are also important in specifying the order of operation. Without parentheses, the HP 40gs calculates according to the order of algebraic precedence (the next topic). Following are some examples using parentheses.
  • Page 34 7. AND and NOT. 8. OR and XOR. 9. Left argument of | (where). 10.Equals, =. Largest and The smallest number the HP 40gs can represent is –499 1 × 10 (1E–499). A smaller result is displayed as smallest zero. The largest number is 9.99999999999 × 10 numbers (1E499).
  • Page 35 When you highlight a previous input or result (by pressing ), the menu labels appear. To copy a previous Highlight the line (press ) and press . The line number (or expression) is copied into the edit line. To reuse the last Press (last answer) to put the last result from the result...
  • Page 36 H I N T When you retrieve a number from , 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 copies the last result (as into the edit line.
  • Page 37: Using Fractions

    Accessing the Pressing enables the highlight bar in the display display history history. While the highlight bar is active, the following menu and keyboard keys are very useful: Function Scrolls through the display history. Copies the highlighted expression to the position of the cursor in the edit line. Displays the current expression in standard mathematical form.
  • Page 38 Setting fraction The fraction precision setting determines the precision in which the HP 40gs converts a decimal value to a fraction. precision The greater the precision value that is set, the closer the fraction is to the decimal value.
  • Page 39 • Precision set to 1: • Precision set to 2: • Precision set to 3: • Precision set to 4 Fraction When entering fractions: calculations • You use the key to separate the numerator part and the denominator part of the fraction. •...
  • Page 40 2. Enter the calculation. Note: Ensure you are in the HOME view. 3. Evaluate the calculation. 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 To convert a decimal value to a fraction: decimals to 1.
  • Page 41: Complex Numbers

    6. Complex numbers Complex results The HP 40gs can return a complex number as a result for some math functions. A complex number appears as an ordered pair (x, y), where x is the real part and y is the 1 –...
  • Page 42: Catalogs And Editors

    Catalogs and editors The HP 40gs has several catalogs and editors. You use them to create and manipulate objects. They access features and stored values (numbers or text or other items) that are independent of aplets. • A catalog lists items, which you can delete or transmit, for example an aplet.
  • Page 43: Aplets And Their Views

    Aplets and their views Aplet views This section examines the options and functionality of the three main views for the Function, 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 Sequence aplets.
  • Page 44 – For a Function definition, enter an expression to define F(X). The only independent variable in the expression is X. – For a Parametric definition, enter a pair of expressions to define X(T) and Y(T). The only independent variable in the expressions is T. –...
  • Page 45: Evaluating Expressions

    Evaluating expressions In aplets In the Symbolic view, a variable is a symbol only, and does not represent one specific 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 variable.
  • Page 46 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 Symbolic view.
  • Page 47: About The Plot View

    Meaning (Continued) Displays the menu for entering math operations. Displays special characters. To enter CHARS one, place the cursor on it and press . To remain in the CHARS menu and enter another special character, press Deletes the highlighted expression or the current character in the edit line.
  • Page 48 Plot view The plot view settings are: settings Field Meaning XRNG, YRNG Specifies the minimum and maximum horizontal (X) and vertical (Y) values for the plotting window. For function plots: Resolution; “Faster” plots in alternate pixel columns; “Detail” plots in every pixel column.
  • Page 49: Exploring The Graph

    Field Meaning (Continued) CONNECT Connect the plotted points. (The Sequence aplet always connects them.) LABELS Label the axes with XRNG and YRNG values. AXES Draw the axes. GRID Draw grid points using XTICK and YTICK spacing. Reset plot To reset the default values for all plot settings, press settings in the Plot Setup view.
  • Page 50 Meaning (Continued) Turns menu-key labels on and off. When the labels are off, pressing turns them back on. • Pressing once displays the full row of labels. • Pressing a second time removes the row of labels to display only the graph. •...
  • Page 51 To jump directly to To jump straight to a value rather than using the Trace a value 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 not displayed, press first.
  • Page 52 Option Meaning (Continued) Y-Zoom In Divides 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-Zoom, or Y-Zoom.) Sets the X-Zoom and Y-Zoom factors Factors...
  • Page 53 Option Meaning (Continued) Un-zoom Returns the display to the previous zoom, or if there has been only one zoom, un-zoom displays the graph with the original plot settings. ZOOM examples The following screens show the effects of zooming options on a plot of Plot of Zoom In: Un-zoom:...
  • Page 54 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.
  • Page 55: Other Views For Scaling And Splitting The Graph

    To set zoom factors 1. In the Plot view, press 2. Press 3. Select Set Factors... and press 4. Enter the zoom factors. There is one zoom factor for the horizontal scale (XZOOM) and one for the vertical scale (YZOOM). Zooming out multiplies the scale by the factor, so that a greater scale distance appears on the screen.
  • Page 56 Option Meaning (Continued) Auto Scale Rescales the vertical axis so that the display shows a representative piece of the plot, for the supplied x axis settings. (For Sequence and Statistics aplets, autoscaling rescales both axes.) The autoscale process uses the first selected function only to determine the best scale to use.
  • Page 57 – moves the leftmost cursor to the screen’s left edge and moves the rightmost cursor to the screen’s right edge. – menu key copies the right plot to the left plot. 3. To un-split the screen, press . The left side takes over the whole screen.
  • Page 58: About The Numeric View

    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.
  • Page 59: Exploring The Table Of Numbers

    Field Meaning (Continued) NUMTYPE Type of numeric table: Automatic or Build Your Own. To build your own table, you must type each independent value into the table yourself. NUMZOOM Allows you to zoom in or out on a selected value of the independent variable.
  • Page 60 Zoom within a Zooming redraws the table of numbers in greater or table lesser detail. ZOOM options The following table lists the zoom options: Option Meaning Decreases the intervals for the independent variable so a narrower range is shown. Uses the NUMZOOM factor in Numeric Setup.
  • Page 61: Building Your Own Table Of Numbers

    Automatic You can enter any new value in the X column. When you recalculation 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 data for regular intervals of the independent (X, T, θ, or N) variable.
  • Page 62: Build Your Own" Menu Keys

    “Build Your Own” menu keys Meaning Puts the highlighted independent value (X, T, θ, or N) into the edit line. Pressing replaces this variable with its current value. Inserts a zero value at the position of the highlight. Replace a zero by typing the number you want and pressing Sorts the independent variable...
  • Page 63 Select Function 2. Reset the graph setup to the default settings. SETUP PLOT CLEAR 3. Plot the two functions and hide the menu so that you can see all the circle. 4. Reset the numeric setup to the default settings. SETUP CLEAR 5.
  • Page 65: Function Aplet

    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: • create graphs to find roots, intercepts, slope, signed area, and extrema •...
  • Page 66 Define the 2. There are 10 function definition fields on the Function expressions aplet’s Symbolic view screen. They are labeled F1(X) to F0(X). Highlight the function definition field you want to use, and enter an expression. (You can press to delete an existing line, or CLEAR clear all lines.) Set up the plot...
  • Page 67 Change the 6. You can change the scale to see more or less of your scale graphs. In this example, choose Auto Scale. (See “VIEWS menu options” on page 2-13 for a description of Auto Scale). Select Auto Scale Trace a graph 7.
  • Page 68 Analyse graph 9. Display the Plot view menu. with FCN functions 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 aplets).
  • Page 69 12.Choose the linear function whose intersection with the quadratic function you wish to find. The coordinates of the intersection point are displayed at the bottom of the screen. Note: If there is more than one intersection (as in our example), the coordinates of the intersection point closest to the current cursor position are displayed.
  • Page 70 to accept using F2(x) = (x + 3) – 2 as the 16.Press other boundary for the integral. 17. Choose the end value for x. The cursor jumps to x = –1 on the linear function. 18.Display the numerical value of the integral. Note: See “Shading area”...
  • Page 71 H I N T The Root and Extremum functions return one value only even if the function 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.
  • Page 72 To navigate around 24.Move to X = –5.9. a table 6 times To go directly to a 25. Move directly to X = 10. value To access the zoom 26. Zoom in on X = 10 by a factor of 4. Note: NUMZOOM options has a setting of 4.
  • Page 73: Function Aplet Interactive Analysis

    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.
  • Page 74: Signed Area

    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 found, but only an extremum, then the result is labeled EXTR: instead of ROOT:. (The root-finder is also used in the Solve aplet.
  • Page 75 Function Description (Continued) Intersection Select Intersection to find the intersection of two graphs nearest the cursor. (You need to have at least two selected expressions in Symbolic view.) Displays the coordinate values and moves the cursor to the intersection. (Uses Solve function.) The resulting x- value is saved in a variable named ISECT.
  • Page 76: Plotting A Piecewise-defined Function

    Plotting a piecewise-defined function Suppose you wanted to plot the following piecewise- defined function. ⎧ ≤ 1 – ⎪ f x ( ) ⎨ < ≤ 1 – ⎪ ≥ ⎩ – 1. Open the Function aplet. Select Function 2. Highlight the line you want to use, and enter the expression.
  • Page 77: Parametric Aplet

    Parametric aplet About the Parametric aplet The Parametric aplet allows you to explore parametric equations. These are equations in which both x and y are f t ( ) defined as functions of t. They take the forms g t ( ) Getting started with the Parametric aplet The following example uses the parametric equations x t ( )
  • Page 78 Set angle 3. Set the angle measure to degrees. measure MODES Select Degrees Set up the plot 4. Display the graphing options. PLOT The Plot Setup input form has two fields not included in the Function aplet, TRNG and TSTEP. TRNG specifies the range of t values.
  • Page 79 Overlay plot 8. Plot a triangle graph over the existing circle graph. PLOT Select Overlay Plot A triangle is displayed rather than a circle (without changing the equation) because the changed value of TSTEP ensures that points being plotted are 120° apart instead of nearly continuous.
  • Page 81: Polar Aplet

    Polar aplet Getting started with the Polar aplet Open the Polar 1. Open the Polar aplet. aplet Select Polar Like the Function aplet, the Polar aplet opens in the Symbolic view. Define the θ 2 ⁄ θ ( ) 2π 2.
  • Page 82 Explore the 5. Display the Plot view menu key labels. graph The Plot view options available are the same as those found in the Function aplet. See “Exploring the graph” on page 2-7 for further information. Display the 6. Display the table of values for θ and R1. numbers The Numeric view options available are...
  • Page 83: Sequence Aplet

    Sequence aplet About the Sequence aplet The Sequence aplet allows you to explore sequences. You can define a sequence named, for example, U1: • in terms of n • in terms of U1(n–1) • in terms of U1(n–2) • in terms of another sequence, for example, U2(n) •...
  • Page 84 Open the 1. Open the Sequence aplet. Sequence aplet Select Sequence The Sequence aplet starts in the Symbolic view. Define the 2. Define the Fibonacci sequence, in which each term (after the first two) is the sum of the preceding two expression terms: >...
  • Page 85 Plot the 4. Plot the Fibonacci sequence sequence. 5. In Plot Setup, set the SEQPLOT option to Cobweb. SETUP PLOT Select Cobweb Display the table 6. Display the table of values for this example. Sequence aplet...
  • Page 87: Solve Aplet

    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.
  • Page 88: Getting Started With The Solve Aplet

    Getting started with the Solve aplet Suppose you want to find the acceleration needed to increase the speed of a car from 16.67 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 1.
  • Page 89 4. Enter the values for the known variables. 1 0 0 H I N T If the Decimal Mark setting in the Modes input form ) is set to Comma, use instead of MODES Solve the 5. Solve for the unknown variable (A). unknown variable Therefore, the acceleration needed to increase the...
  • Page 90 6. Plot the equation for variable A. Select Auto Scale 7. Trace along the graph representing the left side of the equation until the cursor nears the intersection. 20 times Note the value of A displayed near the bottom left corner of the screen.
  • Page 91: Use An Initial Guess

    Meaning (Continued) Clears highlighted variable to zero or deletes current character in edit line, if edit line is active. Resets all variable values to zero or CLEAR clears the edit line, if cursor is in edit line. Use an initial guess You can usually obtain a faster and more accurate solution if you supply an estimated value for the unknown variable before pressing...
  • Page 92: Interpreting Results

    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. Message Condition Zero The Solve aplet found a point where both sides of the equation were equal, or where the expression was zero (a root), within the calculator's...
  • Page 93: Plotting To Find Guesses

    If Solve could not find a solution, you will see one of the following two messages. Message Condition Bad Guess(es) The initial guess lies outside the domain of the equation. Therefore, the solution was not a real number or it caused an error. Constant? The value of the equation is the same at every point sampled.
  • Page 94 where X is distance, V is initial velocity, T is time, and A is acceleration. This is actually two equations, Y = X and Y = V T + (AT ) / 2. Since this equation is quadratic for T, there can be both a positive and a negative solution.
  • Page 95 5. Move the cursor near the positive (right-side) intersection. This cursor value will be an initial guess for T. Press until the cursor is at the intersection. The two points of intersection show that there are two solutions for this equation. However, only positive values for X make sense, so we want to find the solution for the intersection on the right side of the y-axis.
  • Page 96: Using Variables In Equations

    Using variables in equations You can use any of the real variable names, A to Z and θ. Do not use variable names defined for other types, such as M1 (a matrix variable). Home variables All home variables (other than those for aplet settings, like Xmin and Ytick) are global, which means they are shared throughout the different aplets of the calculator.
  • Page 97: Linear Solver Aplet

    Linear Solver aplet 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 .
  • Page 98 example in the previous step). To solve a three- equation set, press . Now the input form displays three equations. If the three-equation input form 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-equation input form.
  • Page 99 As you enter each of the remaining known values, the solution changes. The example at the right shows the final solution once all the co-efficients and constants are entered for the set of equations we set out to solve. Linear Solver aplet...
  • Page 101: Triangle Solve Aplet

    In each case, the solver will calculate the remaining lengths or angles. The HP 40gs will alert you if no solution can be found, or if you have provided insufficient data. If you are determining the properties of a right-angled triangle, a simpler input form is available by pressing the menu key.
  • Page 102 Open the 1. Open the Triangle Solver aplet. Triangle Select Solver aplet Triangle Solver The Triangle Solver aplet opens. Note: if you have already used the Triangle Solver, the entries and results from the previous use will still be displayed. To start the Triangle Solver afresh, clear the previous entries and results by pressing CLEAR.
  • Page 103 lengths as B and C, we would need to specify the α angle as . The illustration on the display will help you determine where to enter the known values. Note: if you need to change the angle neasure mode, press MODES, change the mode, and then press...
  • Page 104 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.
  • Page 105: Statistics Aplet

    Statistics aplet About the Statistics aplet The Statistics aplet can store up to ten data sets at one time. It can perform one-variable or two-variable statistical analysis of one or more sets of data. The Statistics aplet starts with the Numeric view which is used to enter data.
  • Page 106 Open the 1. Open the Statistics aplet and clear existing data by Statistics aplet pressing Select Statistics The Statistics aplet 1VAR/2VAR starts in the Numerical menu key label view. At any time the Statistics aplet is configured for only one of two types of statistical explorations: one- variable ( ) or two-variable ( ).
  • Page 107 Choose fit and 4. Select a fit in the Symbolic setup view. data columns SETUP SYMB Select Linear You can create up to five explorations of two-variable data, named S1 to S5. In this example, we will create just one: S1. 5.
  • Page 108 Setup plot 8. Change the plotting range to ensure all the data points are plotted (and select a different point mark, if you wish). SETUP PLOT 4000 Plot the graph 9. Plot the graph. Draw the 10.Draw the regression curve (a curve to fit the data points).
  • Page 109 Predict values 13.To find the predicted sales figure if advertising were to go up to 6 minutes: S (to highlight Stat-Two) (to highlight PREDY) 14.Return to the Plot view. 15.Jump to the indicated point on the regression line. Observe the predicted y-value in the left bottom corner of the screen.
  • Page 110: Entering And Editing Statistical Data

    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 ( H I N T A data column must have at least four data points to provide valid two-variable statistics, or two data points...
  • Page 111 Meaning (Continued) Deletes the currently highlighted value. Clears the current column or all CLEAR columns of data. Pregss to display a menu list, CLEAR then select the current column or all columns option, and press Moves to the first or last row, or first cursor key or last column.
  • Page 112 3. Find the mean of the sample. Ensure the menu key label reads . Press to see the statistics calculated from the sample data in C1. Note that the title of the column of statistics is H1. There are 5 data set definitions available for one-variable statistics: H1–H5.
  • Page 113 Meaning (Continued) Displays the current variable expression in standard mathematical form. Press when done. Evaluates the variables in the highlighted column (C1, etc.) expression. Displays the menu for entering variable names or contents of variables. Displays the menu for entering math operations.
  • Page 114 5. Move the highlight bar into the right column of the H1 definition and replace the frequency value of 1 with the name C2. 6. Return to the numeric view. 7. Enter the frequency data shown in the above table. 8.
  • Page 115 Edit a data set In the Numeric view of the Statistics aplet, highlight the data value to change. Type a new value and press , or press to copy the value to the edit line for modification. Press after modifying the value on the edit line.
  • Page 116: Defining A Regression Model

    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: •...
  • Page 117 Fit model Meaning (Continued) Quadratic Fits to a quadratic curve, y = ax +bx+c. Needs at least three points. Cubic Fits to a cubic curve, y = ax +cx+d. Needs at least four points. Logistic Fits to a logistic curve, ------------------------- - –...
  • Page 118: Computed Statistics

    Computed statistics One-variable Statistic Definition NΣ Number of data points. TOTΣ Sum of data values (with their frequencies). MEANΣ Mean value of data set. PVARΣ Population variance of data set. SVARΣ Sample variance of data set. PSDEV Population standard deviation of data set.
  • Page 119: Plotting

    Two-variable Statistic Definition Mean of x- (independent) values. MEANX Σ Sum of x-values. Σ Sum of x -values. Mean of y- (dependent) values. MEANY Σ Sum of y-values. Σ Sum of y -values. Σ Sum of each xy. SCOV Sample covariance of independent and dependent data columns.
  • Page 120: Plot Types

    To plot statistical 1. In Symbolic view ( ), select ( ) the data data sets you want to plot. 2. For one-variable data ( ), select the plot type in ). Highlight STATPLOT, Plot Setup ( SETUP PLOT press , select either Histogram or BoxWhisker, and press 3.
  • Page 121: Fitting A Curve To 2var Data

    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.
  • Page 122: Setting Up The Plot (plot Setup View)

    Relative Error The relative error is a measure of the error between predicted values and actual values 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 of fit accuracy for all fits, and it does depend on the Fit model you have chosen.
  • Page 123: Trouble-shooting A Plot

    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 correct menu label on (Numeric view).
  • Page 124: Calculating Predicted Values

    Meaning (Continued) Displays ZOOM menu. Turns trace mode on/off. The white box appears next to the option when Trace mode is active. Turns fit mode on or off. Turning on draws a curve to fit the data points according to the current regression model.
  • Page 125 • Enter PREDY(x-value) to find the predicted value of the dependent variable given a hypothetical independent variable. 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. H I N T In cases where more than one fit curve is displayed, the PREDY function uses the most recently calculated curve.
  • Page 127: Inference Aplet

    Inference aplet About the Inference aplet The Inference capabilities include calculation of confidence intervals and hypothesis tests based on the Normal Z-distribution or Student’s t-distribution. Based on the statistics from one or two samples, you can test hypotheses and find confidence intervals for the following quantities: •...
  • Page 128: Confidence Intervals

    Inference aplet’s SYMB view keys The table below summarizes the options available in Symbolic view. Hypothesis Confidence Intervals Tests Z: 1 μ, the Z-Test Z-Int: 1 μ, the confidence on 1 mean interval for 1 mean, based on the Normal distribution Z: μ...
  • Page 129 Select the 2. Select the Hypothesis Test inferential method. 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 current Z-Test: 1 μ...
  • Page 130: Importing Sample Statistics From The Statistics Aplet

    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 6. To display the on-line help help, press 7. To close the on-line help, press Display test 8.
  • Page 131 A calculator produces the following 6 random numbers: 0.529, 0.295, 0.952, 0.259, 0.925, and 0.592 Open the 1. Open the Statistics aplet and reset the current settings. Statistics aplet Select Statistics The Statistics aplet opens in the Numeric view. Enter data 2.
  • Page 132 Open Inference 6. Open the Inference aplet and clear current settings. aplet Select Inference Select inference 7. Select an inference method. method and type Select CONF INTERVAL 8. Select a distribution statistic type. Select T-Int: 1 μ Set up the 9.
  • Page 133 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 based on the Statistics aplet, you are prompted to choose one.
  • Page 134: 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 based 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. One-Sample Z-Test Z-Test: 1 μ...
  • Page 135: Two-sample Z-test

    Results The results are: Result Description Test Z Z-test statistic. Prob Probability associated with the Z-Test statistic. Critical Z Boundary values of Z associated with the α level that you supplied. Boundary values of required Critical by the α value that you supplied.
  • Page 136: One-proportion Z-test

    Field name Definition σ2 Population 2 standard deviation. α Significance level. Results The results are: Result Description Test Z Z-Test statistic. Prob Probability associated with the Z-Test statistic. Critical Z Boundary value of Z associated with the α level that you supplied.
  • Page 137: Two-proportion Z-test

    Inputs The inputs are: Field name Definition Number of successes in the sample. Sample size. π Population proportion of successes. α Significance level. Results The results are: Result Description Test P Proportion of successes in the sample. Test Z Z-Test statistic. Prob Probability associated with the Z-Test statistic.
  • Page 138: One-sample T-test

    Inputs The inputs are: Field name Definition Sample 1 mean. Sample 2 mean. Sample 1 size. Sample 2 size. α Significance level. Results The results are: Result Description Test π1–π2 Difference between the proportions of successes in the two samples. Test Z Z-Test statistic.
  • Page 139 Inputs The inputs are: Field name Definition Sample mean. Sample standard deviation. Sample size. μ0 Hypothetical population mean. α Significance level. Results The results are: Result Description Test T T-Test statistic. Prob Probability associated with the T-Test statistic. Critical T Boundary value of T associated with the α...
  • Page 140: Two-sample T-test

    Two-Sample T-Test T-Test: μ1 – μ2 Menu name The Two-sample T-Test is used when the population standard deviation is not known. 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 null hypothesis.
  • Page 141: Confidence Intervals

    Critical T Boundary values of T associated with the α level that you supplied. Confidence intervals The confidence interval calculations that the HP 40gs can perform are based on the Normal Z-distribution or Student’s t-distribution. One-Sample Z-Interval Z-INT: μ 1...
  • Page 142: Two-sample Z-interval

    Results The results are: Result Description Critical Z Critical value for Z. μ min Lower bound for μ. μ max Upper bound for μ. Two-Sample Z-Interval Z-INT: μ1– μ2 Menu name This option uses the Normal Z-distribution to calculate a confidence interval for the difference between the means of two populations, μ...
  • Page 143: One-proportion Z-interval

    One-Proportion Z-Interval Z-INT: 1 π Menu name 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: Field...
  • Page 144: One-sample T-interval

    Field Definition (Continued) name Sample 1 size. Sample 2 size. Confidence level. Results The results are: Result Description Critical Z Critical value for Z. Lower bound for the difference between Δ π Min the proportions of successes. Δ π Max Upper bound for the difference between the proportions of successes.
  • Page 145: Two-sample T-interval

    Results The results are: Result Description Critical T Critical value for T. μ Min Lower bound for μ. μ Max Upper bound for μ. Two-Sample T-Interval T-INT: μ1 – μ2 Menu name This option uses the Student’s t-distribution to calculate a confidence interval for the difference between the means of two populations, μ1 –...
  • Page 146 Results The results are: Result Description Critical T Critical value for T. Lower bound for μ 1 – μ 2 . Δ μ Min Upper bound for μ 1 – μ 2 . Δ μ Max 11-20 Inference aplet...
  • Page 147: Using The Finance Solver

    The resulting screen shows the different elements involved in the solution of financial problems with your HP 40gs calculator. Background information on and applications of financial calculations are provided next. Background...
  • Page 148 combined amount earns interest at a certain rate. Financial calculations involving compound interest include savings accounts, mortgages, 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 sometime in the future.
  • Page 149 flow diagram shows lease payments at the beginning of each period. Capitalized value of lease 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: The total number of compounding periods or payments.
  • Page 150: Performing Tvm Calculations

    The periodic payment amount. The payments are the same amount each period and the TVM calculation assumes that no payments are skipped. Payments can occur at the beginning or the end of each compounding period -- an option you control by setting the Payment mode to Beg or End.
  • Page 151 Example 1 - Loan calculations Suppose you finance the purchase of a car with a 5-year loan at 5.5% annual interest, compounded monthly. The purchase price of the car is $19,500, and the down payment is $3,000. What are the required monthly payments? What is the largest loan you can afford if your maximum monthly payment is $300? Assume that the payments start at the end of the first period.
  • Page 152 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, repaying the loan in a balloon payment. Find the size of the balloon payment, the value of the mortgage after 10 years of payment.
  • Page 153: Calculating Amortizations

    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 Finance Solver as indicated at the beginning of this section. 2.
  • Page 154 3. Press the soft menu key to amortize the new batch of payments. Repeat steps 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.
  • Page 155: Using Mathematical Functions

    Using mathematical functions Math functions The HP 40gs contains many math functions. The functions 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 probability.
  • Page 156: Math Functions By Category

    To select a function 1. Press to display the MATH menu. The categories appear in alphabetical order. 2. Press to scroll through the categories. To jump directly to a category, press the first letter of the category’s name. Note: You do not need to press first.
  • Page 157: Keyboard Functions

    Functions common to keyboard and menus These functions are common to the keyboard and MATH menu. For a description, see “p” on π page 13-8. For a description, see “ARG” on page 13-7. ∂ For a description, see “ ” on page 11-7.
  • Page 158 Add, Subtract, Multiply, Divide. Also accepts complex numbers, lists and matrices. value1+ value2, etc. Natural exponential. Also accepts complex numbers. e^value Example e^5 returns 148.413159103 Natural logarithm. Also accepts complex numbers. LN(value) Example LN(1) returns 0 Exponential (antilogarithm). Also accepts complex numbers.
  • Page 159 Example ASIN(1) returns 90 (Degrees mode). –1 Arc cosine: cos x. Output range is from 0° to 180°, 0 to ACOS 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 ≤...
  • Page 160: Calculus Functions

    Example 2^8 returns 256 Absolute value. For a complex number, this is ABS(value) ABS((x,y)) Example ABS(–1) returns 1 ABS((1,2)) returns 2.2360679775 Takes the nth root of x. root NTHROOT value Example 3 NTHROOT 8 returns 2 Calculus functions The symbols for differentiation and integration are available directly form the keyboard—...
  • Page 161: Complex Number Functions

    Example ∫ (0,s1,2*X+3,X) finds the indefinite result 3*s1+2*(s1^2/2) See “To find the indefinite integral using formal variables” on page 13-23 for more information on finding indefinite integrals. TAYLOR Calculates the nth order Taylor polynomial of expression at the point where the given variable = 0. TAYLOR (expression, variable, n) Example TAYLOR(1 + sin(s1)
  • Page 162: Constants

    Constants The constants available from the MATH FUNCTIONS menu are mathematical constants. These are described in this section. The HP 40gs has two other menus of constants: program constants and physical constants. These are described in “Program constants and physical constants”...
  • Page 163: Hyperbolic Trigonometry

    →C Convert from Fahrenheit to Celcius. Example →C(212) returns 100 →F Convert from Celcius to Fahrenheit. Example →F(0) returns 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.
  • Page 164: List Functions

    COSH Hyperbolic cosine COSH(value) SINH Hyperbolic sine. SINH(value) TANH Hyperbolic tangent. TANH(value) ALOG Antilogarithm (exponential). This is more accurate than 10^x due to limitations of the power function. ALOG(value) Natural exponential. This is more accurate than to limitations of the power function. EXP(value) EXPM1 –...
  • Page 165: Matrix Functions

    RECURSE Provides a method of defining a sequence without using the Symbolic view of the Sequence aplet. If used with | (“where”), RECURSE will step through the evaluation. RECURSE(sequencename, term , term , term Example RECURSE(U,U(N-1)*N,1,2) U1(N) Stores a factorial-calculating function named U1. When you enter U1(5), for example, the function calculates 5! (120).
  • Page 166: Probability Functions

    Example For x –25x –26x+120: POLYEVAL([1,2,-25,-26,120],8) returns 3432. POLYFORM Polynomial form. Creates a polynomial in variable1 from expression. POLYFORM(expression, variable1) Example POLYFORM((X+1)^2+1,X) returns X^2+2*X+2. POLYROOT Polynomial roots. Returns the roots for the nth-order polynomial with the specified n+1 coefficients. POLYROOT([coefficients]) Example For x –25x...
  • Page 167 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) returns 20. That is, there are 20 different permutations of five things taken two at a time.
  • Page 168: Real-number Functions

    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 value. UTPT(degrees, value) Real-number functions Some real-number functions can also take complex arguments. CEILING Smallest integer greater than or equal to value. CEILING(value) Examples CEILING(3.2) returns 4...
  • Page 169 → Hours-minutes-seconds to decimal. 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) returns 8.5 → Decimal to hours-minutes-seconds.
  • Page 170 Example 9 MOD 4 returns 1 x percent of y; that is, x/100*y. %(x, y) Example %(20,50) returns 10 %CHANGE Percent change from x to y, that is, 100(y–x)/x. %CHANGE(x, y) Example %CHANGE(20,50) returns 150 %TOTAL Percent total : (100)y/x. What percentage of x, is y. %TOTAL(x, y) Example %TOTAL(20,50) returns 250...
  • Page 171: Two-variable Statistics

    Examples SIGN (–2) returns –1 SIGN((3,4)) returns (.6,.8) TRUNCATE Truncates value to decimal places. Accepts complex numbers. TRUNCATE(value, places) Example TRUNCATE(2.3678,2) returns 2.36 XPON Exponent of value. XPON(value) Example XPON(123.4) returns 2 Two-variable statistics These are functions for use with two-variable statistics. See “Two-variable”...
  • Page 172 Examples ISOLATE(2*X+8,X) returns -4 ISOLATE(A+B*X/C,X) returns -(A*C/B) LINEAR? Tests whether expression is linear for the specified variable. Returns 0 (false) or 1 (true). LINEAR?(expression, variable) Example LINEAR?((X^2-1)/(X+1),X) returns 0 QUAD Solves quadratic expression=0 for variable and returns a new expression, where variable=newexpression. The result is a general solution that represents both positive and negative solutions by including the formal variable S1 to represent any sign: + or –...
  • Page 173: Test Functions

    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. value1<value2 ≤ Less than or equal to. Returns 1 if true, 0 if false. value1≤value2 Equals (logical test).
  • Page 174: Trigonometry Functions

    Exclusive OR. Returns 1 if either value1 or value2—but not both of them—is non-zero, otherwise returns 0. value1 XOR value2 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 cotangent.
  • Page 175: Finding Derivatives

    Symbolic view” on page 13-22 for an example. Finding derivatives The HP 40gs can perform symbolic differentiation on some functions. There are two ways of using the HP 40gs to find derivatives. • You can perform differentiations in HOME by using the formal variables, S1 to S5.
  • Page 176 differentiation function substitutes the value that X holds, and returns a numeric result. For example, consider the function: x ( ) ) 1. Enter the differentiation function onto the command line, substituting S1 in place of X. 2. Evaluate the function. 3.
  • Page 177 3. Select F2(X) and evaluate it. 4. Press to display the result. Note: Use the arrow keys to view the entire function. You could also just define F1 x ( ) x ( ) To find the For example, to find the indefinite integral of indefinite integral ∫...
  • Page 178: Program Constants And Physical Constants

    4. Copy the result and evaluate. Thus, substituting X for S1, it can be seen that: ⎛ ⎞ ---- - ⎜ ⎟ ∫ --------------- ⎜ ⎟ – – ∂ ⎜ ⎟ X ( ) ⎝ ⎠ ∂ This result is derived from substituting X=S1 and X=0 into the original expression found in step 1.
  • Page 179: Program Constants

    Program constants The program constants are numbers that have been assigned to various calculator settings to enable you to test for or specify such a setting in a program. For example, the various display formats are assigned the following numbers: 1 Standard 2 Fixed 3 Scientific...
  • Page 180 3. Use the arrow keys to navigate through the options. 4. To see the symbol and value of a selected constant, press . (Click to close the information window that appears.) The following example shows the information available about the speed of light (one of the physics constants).
  • Page 181 3. Select light s...from the Physics menu. 4. Press . The menu closes and the value of the selected constant is copied to the edit line. 5. Complete the equation as you would normally and press to get the result. Using mathematical functions 13-27...
  • Page 183: Computer Algebra System (cas)

    –12 the precision of the calculator (to 10 in the case of the HP 40gs). For example, with Standard as your numerical format, 1/2 + 1/6 returns 0.6666666666667 if you are working in the HOME screen; however, 1/2 + 1/6 returns 2/3 if you are working with CAS.
  • Page 184: An Example

    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.
  • Page 185 3. Press select just the 20 in the term. 4. Press the menu key and choose FACTOR. Then press Note that the FACTOR function is added to the selected term. 5. Press to factor the selected term. 6. Press to select the entire second term, and then press simplify it.
  • Page 186: Cas Variables

    10.Press 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, you are working with symbolic variables (variables that do not contain a permanent value). In the HOME screen, a variable of this kind must have a name like S1…S5, s1…s5, n1…n5, but not X, which is assigned to a real value.
  • Page 187: Cas Modes

    CAS modes The modes that determine how CAS operates can be set on CAS MODES screen. 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 correct setting is displayed (indicated by a check mark in the field).
  • Page 188 calculated as closed-form algebraic expressions, whenever possible. [Default: unselected.] Num. Factor mode When the setting is selected, approximate NUM FACTOR roots are used when factoring. For example, is irreducible over the integers but has approximate roots over the reals. With set, the approximate roots NUM FACTOR are returned.
  • Page 189: Using Cas Functions In Home

    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 that take matrices as an argument work only from HOME. CAS functions can be accessed by pressing when MATH menu is displayed.
  • Page 190: Online Help

    Symbolic matrices are stored as a list of lists and therefore must be stored in L0, L1…L9 (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}}) then you have:...
  • Page 191: Cas Functions In The Equation Writer

    HELP and press . The menu of help topics 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 T I P If you highlight a CAS command and then press 2, help about the highlighted command is displayed.
  • Page 192: Algb Menu

    For example, suppose you have stored the expression x in G, and have defined the function F(x) as x . Suppose now you want to calculate INTVX(X ). You could: • enter INTVX(X ) directly, or • enter INTVX(G), or •...
  • Page 193 Typing: DEF(U(N) = 2N+1) produces the result: U(N) = 2N+1 Typing: U(3) then returns: Example Calculate the first six Fermat numbers F1...F6 and determine whether they are prime. So, you want to calculate: F k ( ) for k = 1...6 Typing the formula: gives a result of 17.
  • Page 194 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 simplifies an expression. Example Typing: ⋅...
  • Page 195 In real mode, the result is: 2 x ⋅ ⋅ 2 x ⋅ – In complex mode (using CFG), the result is: ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ----- - 2x 1 – 1 i – ⋅ ⋅ 2x 1 i – –...
  • Page 196 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 -4,ABC) or type: then select it and call STORE, then type ABC, then press ENTER to confirm the definition of the variable ABC.
  • Page 197 SUBST Substitute a value for a variable SUBST has two parameters: an expression dependent on a parameter, and an equality (parameter=substitute value). SUBST substitutes the specified value for the variable in the expression. Typing: SUBST(A +1,A=2) gives: TEXPAND Develop in terms of sine and cosine TEXPAND has a trigonometric expression or transcendental function as an argument.
  • Page 198: Diff Menu

    DIFF menu DERIV Derivative and partial derivative DERIV has two arguments: an expression (or a function) and a variable. DERIV returns the derivative of the expression (or the function) with respect to the variable given as the second parameter (used for calculating partial derivatives). Example Calculate: ∂...
  • Page 199 DERVX(F) Or, if you have defined F(X) using DEF, that is, if you have typed: ⎛ ⎞ ⎞ -------------- - ------------ - DEF(F(X) ⎝ ⎠ ⎠ – – then type: DERVX(F(X)) Simplify the result to get: ⋅ – -------------------------------- - –...
  • Page 200 and with period T (T being equal to the contents of the variable PERIOD). If f(x) is a discrete series, then: 2 iNxπ ∞ --------------- - ∑ f x ( ) ∞ – Example Determine the Fourier coefficients of a periodic function f with period 2π...
  • Page 201 u x ( ) v x ( ) ⋅ v – x ( ) u' x ( ) ⋅ 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 u x ( ) v' x ( ) ⋅...
  • Page 202 Example Given: ⎛ ⎞ f x ( ) ------------- - ----------- - ⎝ ⎠ – – calculate a primitive of f. Type: ⎛ ⎛ ⎞ ⎞ -------------- - ------------ - NTVX ⎝ ⎝ ⎠ ⎠ – Or, if you have stored f(x) in F, that is, if you have already typed: ⎛...
  • Page 203 ⎛ ⎞ -------------------------------------- NTVX ⎝ ⎠ ⋅ gives a primitive: ⋅ x ( ) -- - ------------- - 3 – atan – – ∫ -------------------------------------- X N o t e You can also type which gives the ⋅ primitive which is zero for x = 1 3 π...
  • Page 204 QUOTE(expression), to avoid rewriting the expression in normal form (i.e., not to have a rational simplification of the arguments) during the execution of the LIMIT command. Example Typing: ⎛ ⎞ ⎞ X ⋅ ∞ ----------- - lim QUOTE 2X 1 –...
  • Page 205 Typing: N TAN X ( ) TAN N X ) ⋅ ⋅ – ⎛ ⎞ ---------------------------------------------------------------- - 0 , ⎝ ⎠ ⋅ ) N SIN X ( ) ⋅ SIN N X – gives: – NOTE: To find the limit as x approaches a (resp a ), the second argument is written:...
  • Page 206 PREVAL is used for calculating an integral defined from a primitive: it evaluates this primitive between the two limits of the integral. Typing: PREVAL(X +X,2,3) gives: RISCH Primitive and defined integral RISCH has two parameters: an expression and the name of a variable.
  • Page 207 Typing: π ⎛ ⎞ ⋅ -- - SERIES COS 2 X ⎝ ⎠ gives: ⎛ ⎞ π 〈 〉 -- - --------- - h -- - h ---- - – – ⎝ ⎠ -- - – • Example — Expansion in the vicinity of x=+∞ or x=–∞...
  • Page 208 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 changing modes. Example 1 Give a 3rd-order expansion of in the vicinity of x = 0 Typing: 0 3.0 SERIES...
  • Page 209 1 – ⋅ ⋅ ⋅ ----- - h ----- - h -- - h TABVAR Variation table TABVAR has as a parameter an expression with a rational derivative. TABVAR returns the variation table for the expression in terms of the current variable. Typing: TABVAR(3X -8X-11)
  • Page 210: Rewri Menu

    Typing: ⋅ ) SIN P X ⋅ – TAN P X ⎛ ⎞ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - TAYLOR0 ⎝...
  • Page 211 Typing: DISTRIB((X+1)·(X+2)·(X+3)) gives: ⋅ ⋅ ⋅ ⋅ EPSX0 Disregard small values EPSX0 has as a parameter an expression 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...
  • Page 212 Typing: EXP2POW(EXP(N · LN(X))) gives: FDISTRIB Distributivity FDISTRIB has an expression as argument. FDISTRIB enables you to apply the distributivity of multiplication with respect 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 + x·2·1 + 3·2·1 After simplification (by pressing ENTER): + 6·x...
  • Page 213 ⋅ – ( ⋅ ⋅ ⋅ ⋅ ⋅ -- - -- - -- - 2 i x 2 i x Example 3 Typing: LIN(SIN(X)) gives: ⋅ i x ⋅ ⋅ – ( i x ⋅ -- - -- - – LNCOLLECT Regroup the logarithms LNCOLLECT has as an argument an expression...
  • Page 214 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: ⋅ ⋅ SIN 3 X SIN 7 X ⎛ ⎞ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - SIMPLIFY ⎝...
  • Page 215: Solv Menu

    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 having constant coefficients, it is better to use LDEC.) DESOLVE has two arguments: 1.
  • Page 216 To produce the solutions for y(0) = 1, type: SUBST Y X ( ) ⋅ cC0 COS X ( ) ⋅ ⋅ SIN X ( ) cC0 - - - - - - - - - - - - - - - - which gives: ⋅...
  • Page 217 LDEC Linear differential equations having constant coefficients LDEC enables you to directly solve linear differential equations having constant coefficients. The parameters are the second member and the characteristic equation. Solve: 3·x y” − 6 · y’ + 9 · y = x · e Typing: −6·X+9) LDEC(X·EXP(3·X),X...
  • Page 218 L1=2L1+L2 1 1 3 – – ENTER Reduction Result 2 0 4 – – 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.
  • Page 219 then press ENTER. The following is then written to the Equation Writer: ⎛ ⎞ AND y ⎛ ⎞ AND z ⎛ ⎞ -- - -- - -- - – – ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ SOLVE Solve equations 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.
  • Page 220: Trig Menu

    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 -1=3)
  • Page 221 Typing: ACOS2S(ACOS(X) + ASIN(X)) gives, when simplified: π -- - ASIN2C Transform the arcsin into arccos ASIN2C has as a trigonometric expression as an argument. ASIN2C transforms the expression by replacing arcsin(x) π − arccos(x). with ---- - Typing: ASIN2C(ACOS(X) + ASIN(X)) gives, when simplified: π...
  • Page 222 Typing: ATAN2S(ATAN(X)) gives: ⎛ ⎞ ⎜ ------------------ ⎟ asin ⎝ ⎠ 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) + COS(X)
  • Page 223 TAN2CS2 transforms this expression by replacing tan(x) 2 x ⋅ – -------------------------------- with 2 x ⋅ Typing: TAN2CS2(TAN(X)) gives: 2 x ⋅ – -------------------------------- 2 x ⋅ TAN2SC Replace tan(x) with sin(x)/cos(x) TAN2SC has a trigonometric expression as an argument. TAN2SC transforms this expression by replacing tan(x) x ( ) --------------- -...
  • Page 224 TCOLLECT linearizes this expression 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: π ⎛ ⎞ ⋅ -- - – ⎝ ⎠ TEXPAND Develop transcendental expressions TEXPAND has as an argument a transcendental expression (that is, an expression with trigonometric,...
  • Page 225 gives: 4·cos(x) –3·cos(x) TLIN Linearize a trigonometric 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: ⋅ ⋅ -- - -- - –...
  • Page 226 Typing: TRIG(SIN(X) + COS(X) + 1) gives: TRIGCOS Simplify using the cosines TRIGCOS has as an argument a trigonometric expression. TRIGCOS simplifies this expression, using the identity sin(x) +cos(x) = 1 to rewrite it in terms of cosines. Typing: TRIGCOS(SIN(X) + COS(X) + 1) gives:...
  • Page 227: Cas Functions On The Math Menu

    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 menus in the Equation Writer;...
  • Page 228: Constant Menu

    returns: Y = X –1 + 2 Pressing simplifies this to: Y = X + 1 See “IM” on page 13-7. – Specifies the negation of the argument. See “RE” on page 13-8. SIGN Determines the quotient of the argument divided by its modulus.
  • Page 229 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.
  • Page 230 In step-by-step mode, there are a number of intermediate results: 18 mod 15 = 3 15 mod 3 = 0 Result: 3 Pressing 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.
  • Page 231 [48,1,0] [30,0,1]*–1 [18,1,–1]*–1 [12,–1,2]*–1 [6,2,–3]*–2 Result: [6,2,–3] Pressing 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 the Euclidean quotient of the integers at the beginning of the vector, these integers being the sequence of remainders).
  • Page 232 IREMAINDER works with integers and with Gaussian integers. This is what distinguishes it from MOD. Example 2 Typing: IREMAINDER(2 + 3·i, 1 + i) gives: ISPRIME? Returns a value indicating whether an integer is a prime number. ISPRIME?(n) returns 1 (TRUE) if n is a prime or pseudo-prime, and 0 (FALSE) if n is not prime.
  • Page 233: Modular Menu

    NEXTPRIME NEXTPRIME(n) returns the smallest prime or pseudo-prime greater than n. Example Typing: NEXTPRIME(75) gives: PREVPRIME PREVPRIME(n) returns the greatest prime or pseudo-prime less than n. Example Typing: PREVPRIME(75) gives: Modular menu All the examples of this section assume that p =13; that is, you have entered MODSTO(13) or STORE(13,MODULO), or have specified 13 for Modulo in CAS MODES screen (as explained on page 15-16).
  • Page 234 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: DIVMOD(5, 3) gives: Example 2 In Z/pZ[X], the arguments are two polynomials: A[X] and B[X].
  • Page 235 Factors a polynomial in Z/pZ[X], providing that p ≤ 97, FACTORMOD p is prime and the order of the multiple factors is less than the modulo. Example Typing: FACTORMOD(–(3X – 5X + 5X – 4)) gives: ⋅ – 3x 5 –...
  • Page 236 MULTMOD Performs a multiplication in Z/pZ or in Z/pZ[X]. Example 1 Typing: MULTMOD(11, 8) gives: –3 Example 2 Typing: MULTMOD(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].
  • Page 237: Polynomial Menu

    SUBTMOD Performs a subtraction in Z/pZ or Z/pZ[X]. Example 1 Typing: SUBTMOD(29, 8) gives: –5 Example 2 Typing: SUBTMOD(11X + 5, 8X + 6) gives: 3x 1 – 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 such that D(X) = U(X)·A(X) + V(X)·B(X).
  • Page 238 FACTOR Factors a polynomial. Example 1 Typing: FACTOR(X – 2) gives: ⋅ – Example 2 Typing: FACTOR(X + 2·X + 1) gives: Returns the GCD (Greatest Common Divisor) of two polynomials. Example Typing: GCD(X + 2·X + 1, X – 1) gives: HERMITE Returns the Hermite polynomial of degree n (where n is a...
  • Page 239 Returns the LCM (Least Common Multiple) of two polynomials. Example Typing: LCM(X + 2·X + 1, X – 1) gives: ⋅ – LEGENDRE Returns the polynomial L , a non-null solution of the differential equation: ) y″ ⋅ ⋅ ⋅ ) y ⋅...
  • Page 240 PROPFRAC PROPFRAC rewrites a rational fraction so as to bring out its whole number part. PROPFRAC(A(X)/ B(X)) writes the rational fraction A(X)/ B(X) in the form: R X ( ) Q X ( ) ----------- - B X ( ) where R”(X) = 0, or 0 ≤...
  • Page 241 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), divided in decreasing order by exponent.
  • Page 242: Real Menu

    Example 1 Typing: TCHEBYCHEFF(4) gives: – Example 2 Typing: TCHEBYCHEFF(–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. See “INT” on page 13-15. See “MAX” on page 13-15. See “MIN”...
  • Page 243: Tests Menu

    Tests menu ASSUME Use this function to make a hypothesis about a specified argument or variable. Example Typing: ASSUME(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 ≥...
  • Page 244: Cas Functions On The Cmds Menu

    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;...
  • Page 245 Example Find the solutions P(X) of: P(X) = X (mod X + 1) P(X) = X – 1 (mod X – 1) Typing: CHINREM((X) AND (X + 1), (X – 1) AND (X – 1)) gives: – – ------------------------- - ------------- - –...
  • Page 246 Example 1 Typing: EXP2HYP(EXP(A)) gives: sinh(a) + cosh(a) Example 2 Typing: EXP2HYP(EXP(–A) + EXP(A)) gives: 2 · cosh(a) Returns the values of the Γ function at a given point. GAMMA The Γ function is defined as: + ∞ t – –...
  • Page 247 Example Typing: IABCUV(48, 30, 18) gives: 6 AND –9 IBERNOULLI Returns the nth Bernoulli’s number B(n) where: + ∞ B n ( ) ∑ ------------ - ----------- t – 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.
  • Page 248 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 transform (ILAP) are useful in solving linear differential equations with constant coefficients, for example: ⋅...
  • Page 249 Typing: ⎛ ⋅ ⎞ --------------------------- - X 6 – ⎜ ⎟ – ------------------------------------------------------------------ - ⎜ ⎟ ILAP ⎜ ⎟ – ⎝ ⎠ gives: ⎛ ⎞ e ⋅ ⋅ ---- - – 3a b – ⎝ ⎠ See ILAP above. PA2B2 Decomposes a prime integer p congruent to 1 modulo 4, as follows: p = a...
  • Page 250 gives: -- - π ⋅ -- - – 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: Psi(3) and pressing gives: .922784335098 REORDER Reorders the input expression following the order of variables given in the second argument.
  • Page 251 Example Typing: SIGMA(X · X!, X) gives: because (X + 1)! – X! = X · X!. SIGMAVX Returns the discrete antiderivative 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.
  • Page 252 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: SIN 3X SIN 7X ⎛ ⎞ -------------------------------------------------- - TSIMP ⎝ ⎠ SIN 5X gives: EXP i x ⋅...
  • Page 253: Equation Writer

    Equation Writer Using CAS in the Equation Writer The Equation Writer enables you to type expressions that you want to simplify, factor, differentiate, integrate, and so on, and then work them through as if on paper. key on the HOME screen menu bar opens the Equation Writer, and the key closes it.
  • Page 254: Cursor Mode

    Cursor mode Enables you to go into cursor mode, for quicker selection of expressions and subexpressions (see page 15-10). Edit expr. Enables you to edit the highlighted expression on the edit line, just as you do in the HOME screen (see page 15-11).
  • Page 255: Configuration Menus

    REWRI menu menu contains functions that enable you to rewrite an expression in another form. SOLV menu menu contains functions that enable you to solve equations, linear systems, and differential equations. TRIG menu menu contains functions that enable you to transform trigonometric expressions.
  • Page 256 • The fourth symbol, S, in the above example, indicates that you are in step-by-step mode. If you were not in step-by-step mode, this symbol would be D (which stands for Direct). The first line of an Equation Writer menu only indicates some of the mode settings.
  • Page 257: Entering Expressions And Subexpressions

    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, the operator that you are typing always carries over to the adjacent or selected expression.
  • Page 258 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 Equation Writer. It illustrates a tree view of the expression: ⋅ – ---------------------------------------- - ÷...
  • Page 259 • Press again and again to progressively select more of the top-most branch, and then lower branches (5x, 5x + 3, and then the entire numerator and finally the entire expression). More Examples Example1 If you enter: 2 + X × 3– X and press entire expression is selected.
  • Page 260 (– 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 evaluate it by pressing . The result is: –(X –X–6) Example2 To enter X –3X+1, press: –...
  • Page 261 Select the fifth branch by pressing . At this point, the desired expression is in the Equation Writer, as shown at the right. Suppose that you want to select the second and third -- - -- - -- - branches, that is: .
  • Page 262 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.
  • Page 263: How To Modify An Expression

    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 selection without deleting the expression by pressing . The cursor moves to the end of the deselected portion.
  • Page 264: Accessing Cas Functions

    Accessing CAS functions While you are in the Equation Writer, you can access all CAS functions, and you can access 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 selected).
  • Page 265 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 An infix function is one that is typed between its functions arguments.
  • Page 266 First option: function first, then arguments In the Equation Writer, press , select FACTOR and then press 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.
  • Page 267 Press to obtain the an intermediate result (4 – 4) 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 earlier. The entire expression is now selected. Now press and select FACTOR.
  • Page 268: Equation Writer Variables

    Press to obtain an intermediate result, (4– 2)(4 + 2), and again to evaluate the intermediate result. The final answer, as before, is 12. N o t e If you call a CAS function while you’re writing an expression, whatever is currently selected is copied to the function’s first or main argument.
  • Page 269: The Keyboard In The Equation Writer

    • MODULO contains the value of p for performing symbolic calculations in Z/pZ or in Z/pZ[X]. You can change the value of p either with the MODSTO command on the MODULAR menu, (by typing, for example, MODSTO(n) to give p a value of n), or from CAS MODES screen (see page 14-5).
  • Page 270 Diff&Int ( ), Rewrite ( ), Solve ) and Trig ( • The Complex menu, providing functions specific to manipulating with complex numbers. The Constant menu, containing e, i,∞ and π. • • The Hyperb. menu, containing hyperbolic functions. • The Integer menu, containing functions that enable you to perform integer arithmetic.
  • Page 271 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.
  • Page 272 N O T E This operation supposes that the current 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.
  • Page 273 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 ≥ Equation Writer 15-21...
  • Page 275: Step-by-step Examples

    Step-by-Step Examples 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: -- - 1...
  • Page 276 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 2 45 3 12 – – Given that write C in the form , where d is a whole number. Solution: In the Equation Writer, enter C by typing: Press –...
  • Page 277 Press to factor 5 ⋅ 20 into Press to select 5 ⋅ simplify it. Press to select – 3 12 to exchange with – Press to select 2 45 to select 45. Press , select FACTOR and press Press to factor 5 ⋅...
  • Page 278 Press evaluate the selection. It remains to transform 3 12 and combine it – with . Follow the same procedure as undertaken a number of 3 12 times above. You will find that is equal to , and so the final two terms cancel each other out.
  • Page 279 Press to select the entire equation, then press to reduce it to – – Press , select FACTOR, press then . The result is as shown at the right. Now press , select SOLVEVX, press press . The result is shown at the right.
  • Page 280 Press , select LINSOLVE and press Enter 17 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...
  • Page 281 1. Find the exact length of AB in centimetres. 2. Determine the equation of the line AB. Type: First method 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...
  • Page 282 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 determi1ne the equation of the line typing: DROITE(( -1,3), (-3,-1))
  • Page 283 4. Show that for every integer n > 0, b × c 5. Deduce the prime factor decomposition of a 6. Show that GCD(b ) = GCD(c ,2). Deduce that b and c are prime together. Solution: Begin by entering the three definitions. Type: DEF(A(N) = 4 ·...
  • Page 284 Show that the whole numbers k such that: ≤ < k 10 have digits in decimal notation. We have: < ⋅ < < ⋅ < 3 10 4 10 < < ⋅ < 2 10 < ⋅ < < ⋅ <...
  • Page 285 Now consider the product of two of the definitions entered above: 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 prime factors.
  • Page 286 GCD c GCD c GCD b Part 2 Given the equation: ⋅ ⋅ where the integers x and y are unknown and b and c are defined as in part 1 above: 1. Show that [1] has at least one solution. 2.
  • Page 287 × – , or × × – ( 1000 The calculator is not needed for finding the general solution to equation [1]. x ⋅ y ⋅ We started with × × – ( 1000 and have established that So, by subtraction we have: ⋅...
  • Page 288 the circle C, M will move on a curve Γ. In this exercise we will study and plot Γ. ∉ – π π 1. Let and m be the point on C of affix i t ⋅ . Find the coordinates of M in terms of t. 2.
  • Page 289 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 menu). The result, after accepting the switch to complex mode, is shown at the right: Now store the result in variable M.
  • Page 290 DEF command to it. Press to complete the definition. To calculate the real part 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...
  • Page 291 Then press produce the result at the right: In other words, y t ( ) y t – – x t ( ) y t ( ) Γ ( ) y t – x t – is part of , then is also Γ...
  • Page 292 y′ t ( ) 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 y′ t ( ) function (in the same way that you defined x′...
  • Page 293 Now press to see the graphs. π -- - 2 π ⋅ x t ( ) y t ( ) ---------- π Part 6 To find the values of return to CAS, type each function in turn and press . (You may need to press twice for further simplification).
  • Page 294 The example at the right shows the case for t = 0. Select the entire expression and press to get the answer: 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.
  • Page 295 π 2 π π ----- - -- - x' t ( ) – ↓ ↑ ↑ x t ( ) 1 – 3 – ----- - ----- - -- - -- - ↓ ↓ ↑ y t ( ) – –...
  • Page 296 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: -- - ∫ -------------- - e Define g over [0,2] where: g x ( ) -------------- - 1.
  • Page 297 Solution 1 Start by defining G(X): Now press Press to select the numerator and denominator, and then press . This leaves G(X) displayed: Finally, apply the TABVAR function: TABVAR and press number of times until the variation table appears (shown above). The first line of the variation table gives the sign of g′...
  • Page 298 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 calculate g(0) by -- - typing G(0) and pressing...
  • Page 299 We can now see that: -- - -- - ⎛ ⎞ ⎛ ⎞ ≤ ≤ -- - ne -- - ne ⎜ ⎟ ⎜ ⎟ – – ⎝ ⎠ ⎝ ⎠ To justify the preceding calculation, we must assume that -- - -- - ⋅...
  • Page 300 The variable VX is now set to N. Reset it to X by NOTE pressing (to display CAS MODES screen) and change the INDEP VAR setting. To check the result, we can say that: – ------------- → and that therefore: -- - –...
  • Page 301 Solution 1 Start by defining the g x ( ) ----------- - – following: Now type PROPFRAC(G(X)). Note that PROPFRAC can be found on the POLYNOMIAL submenu of the MATH menu. Pressing yields the result shown at the right. Solution 2 Enter the integral: ∫...
  • Page 302 Solution 3 The calculator is not needed here. Simply stating that -- - ∈ [ , ] increases for is sufficient to yield the inequality: -- - -- - ≤ ≤ Solution 4 g x ( ) is positive over [0, 2], through multiplication Since we get: -- -...
  • Page 303 Step-by-Step Examples 16-29...
  • Page 305: Variables And Memory Management

    The calculator uses this memory to store variables, perform computations, and store history. A variable is an object that you create in memory to hold data. The HP 40gs has two types of variables, home variables and aplet variables. •...
  • Page 306: Storing And Recalling Variables

    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 precision in the display, however, depends on the display mode (Standard, Fixed, Scientific, Engineering, or Fraction).
  • Page 307 5. Enter a name for the variable. 6. Press to store the result. The results of a calculation can also be stored directly to a variable. For example: To recall a value To recall a variable’s value, type the name of the variable and press To use variables in You can use variables in calculations.
  • Page 308: The Vars Menu

    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 column. You select a variable category and then select a variable in the category.
  • Page 309 5. Choose whether to place the variable name or the variable value on the command line. – Press to indicate that you want the variable’s contents to appear on the command line. – Press to indicate that you want the variable’s name to appear on the command line.
  • Page 310 4. Enter data for L2. 5. Press to access HOME. 6. Open the variable menu and select L1. 7. Copy it to the command line. Note: Because the option is highlighted, the variable’s name, rather than its contents, is copied to the command line.
  • Page 311 Home variables It is not possible to store data of one type in a variable of another type. For example, you use the Matrix catalog to create matrices. You can create up to ten matrices, and you can store these in variables M0 to M9. You cannot store matrices in variables other than M0 to M9.
  • Page 312 Aplet variables Most aplet variables store values that are 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 information about aplet variables.
  • Page 313: Memory Manager

    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.
  • Page 315: Matrices

    Matrices Introduction You can perform matrix calculations in 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: 1 2 3 4 5 6 is displayed in the history as: [[1,2,3],[4,5,6]]...
  • Page 316: Creating And Storing Matrices

    Transmits the highlighted matrix to another HP 40gs or a disk drive. See. Receives a matrix from another HP 40gs or a disk drive. See . Clears the highlighted matrix. Clears all matrices. CLEAR Moves to the end or the beginning of the catalog.
  • Page 317 2. Highlight the matrix variable name you want to use and press 3. Select the type of matrix to create. – For a vector (one-dimensional array), select Real vector or Complex vector. Certain operations (+, –, CROSS) do not recognize a one-dimensional matrix as a vector, so this selection is important.
  • Page 318: Working With Matrices

    To transmit a You can send matrices between calculators just as you matrix can send aplets, programs, lists, and notes. 1. Connect the calculators using an appropriate cable. 2. Open the Matrix catalogs on both calculators. 3. Highlight the matrix to send. 4.
  • Page 319 Meaning (Continued) Moves to the first row, last row, first column, or last column respectively. To display a matrix • In the Matrix catalog ( ), highlight the MATRIX matrix name and press • In HOME, enter the name of the matrix variable and press To display one In HOME, enter matrixname(row,column).
  • Page 320: Matrix Arithmetic

    To store one In HOME, enter, value matrixname(row,column). element For example, to change the element in the first row and second column of M5 to 728, then display the resulting matrix: An attempt to store an element to a row or column beyond the size of the matrix results in an error message.
  • Page 321 To multiply and For division by a scalar, enter the matrix first, then the divide by a scalar operator, then the scalar. For multiplication, the order of the operands does not matter. The matrix and the scalar can be real or complex. For example, to divide the result of the previous example by 2, press the following keys: To multiply two...
  • Page 322: Solving Systems Of Linear Equations

    To divide by a For division of a matrix or a vector by a square matrix, 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.
  • Page 323 3. Return to the Matrix Catalog. MATRIX In this example, the vector you created is listed as M1. 4. Create a new matrix. Select Real matrix 5. Enter the equation coefficients. In this example, the matrix you created is listed as 6.
  • Page 324: Matrix Functions And Commands

    Matrix functions and commands About functions • Functions can be used in any aplet or in HOME. They are listed in the MATH menu under the Matrix category. They can be used in mathematical expressions—primarily in HOME—as well as in programs.
  • Page 325 COND Condition Number. Finds the 1-norm (column norm) of a square matrix. COND(matrix) CROSS Cross Product of vector1 with vector2. CROSS(vector1, vector2) Determinant of a square matrix. DET(matrix) Dot Product of two arrays, matrix1 matrix2. DOT(matrix1, matrix2) EIGENVAL Displays the eigenvalues in vector form for matrix. EIGENVAL(matrix) EIGENVV Eigenvectors and Eigenvalues for a square matrix.
  • Page 326 LU Decomposition. Factors a square matrix into three matrices: {[[lowertriangular]],[[uppertriangular]],[[permutation]]} The uppertriangular has ones on its diagonal. LU(matrix) MAKEMAT Make Matrix. Creates a matrix of dimension rows × columns, using expression to calculate each element. If expression contains the variables I and J, then the calculation for each element substitutes the current row number for I and the current column number for J.
  • Page 327: Examples

    SPECNORM Spectral Norm of matrix. SPECNORM(matrix) SPECRAD Spectral Radius of a square matrix. SPECRAD(matrix) Singular Value Decomposition. Factors an m × n matrix into two matrices and a vector: {[[m × m square orthogonal]],[[n × n square orthogonal]], [real]}. SVD(matrix) Singular Values.
  • Page 328 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 x 2y – The following set of equations – – Echelon Form – can be written as the augmented matrix 1 2 –...
  • Page 329 The final row of zeros in the reduced-row echelon form of the augmented matrix indicates an inconsistent system with infinite solutions. Matrices 18-15...
  • Page 331: Lists

    Lists You can do list operations in HOME and in programs. A list consists of comma-separated real or complex numbers, expressions, or matrices, 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 represent a convenient way to group related objects.
  • Page 332 Meaning Opens the highlighted list for editing. Transmits the highlighted list to another HP 40gs or a PC. See “Sending and receiving aplets” on page 22-4 for further information. Receives a list from another HP 40gs or a PC. See “Sending and receiving aplets”...
  • Page 333 List edit keys When you press to create or change a list, the following keys are available to you: Meaning Copies the highlighted list item into the edit line. Inserts a new value before the highlighted item. Deletes the highlighted item from the list.
  • Page 334: Displaying And Editing Lists

    Displaying and editing lists To display a list • In the List catalog, highlight the list name and press • In HOME, enter the name of the list and press To display one In HOME, enter listname(element#). For example, if L2 is element {3,4,5,6}, then L2(2) returns 4.
  • Page 335 To insert an element 1. Open the List catalog. in a list LIST 2. Press highlight the name of the list you want to edit (L1, etc.) and press to display the list contents. New elements are inserted above the highlighted position.
  • Page 336: Deleting Lists

    Deleting lists To delete a list In the List catalog, highlight the list name and press You are prompted to confirm that you want to delete the contents of the highlighted list variable. Press delete the contents. To delete all lists In the List catalog, press CLEAR Transmitting lists...
  • Page 337 variable name (such as L1) or the actual list. For example, REVERSE({1,2,3}). • If Decimal Mark in Modes is set to Comma, use periods to separate arguments. For example, CONCAT(L1.L2). Common operators like +, –, ×, and / can take lists as arguments.
  • Page 338 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,variable,begin,end, increment) The MAKELIST function generates a series by automatically producing a list from the repeated evaluation of an expression. Example In HOME, generate a series of squares from 23 to 27.
  • Page 339: Finding Statistical Values For List Elements

    SIZE Calculates the number of elements in a list. SIZE(list) Also works with matrices. ΣLIST Calculates the sum of all elements in list. ΣLIST(list) Example ΣLIST({2,3,4}) returns 9. SORT Sorts elements in ascending order. SORT(list) Finding statistical values for list elements To find values such as the mean, median, maximum, and minimum values of the elements in a list, use the Statistics aplet.
  • Page 340 3. Start the Statistics aplet, and select 1-variable mode (press , if necessary, to display Select Statistics Note: Your list values are now in column 1 (C1). 4. In the Symbolic view, define H1 (for example) as C1 (sample) and 1 (frequency). 5.
  • Page 341: Notes And Sketches

    Notes and sketches Introduction The HP 40gs has text and picture editors for entering notes and sketches. • Each aplet has its own independent Note view and Sketch view. Notes and sketches that you create in these views are associated with the aplet. When you save the aplet, or send it to another calculator, the notes and sketches are saved or sent as well.
  • Page 342 Note edit keys Meaning Space key for text entry. Displays next page of 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. Erases the entire note. CLEAR Menu for entering variable names, and contents of variables.
  • Page 343: Aplet Sketch View

    Aplet sketch view You can attach pictures to an aplet in its Sketch view ). Your work is automatically saved with the SKETCH aplet. Press any other view key or to exit the Sketch view Sketch keys Meaning Stores the specified portion of the current sketch to a graphics variable (G1 through G0).
  • Page 344 To draw a box 1. In Sketch view, press and move the cursor to where you want any corner of the box to be. 2. Press 3. Move the cursor to mark the opposite corner for the box. You can adjust the size of the box by moving the cursor.
  • Page 345 To label parts of a 1. Press and type the text on the edit line. To lock sketch the Alpha shift on, press (for uppercase) or (for lowercase). To make the label a smaller character size, turn off before pressing is a toggle between small and large font size).
  • Page 346: The Notepad

    To import a You can copy the contents of a graphics variable into the graphics variable Sketch view of an aplet. 1. Open the Sketch view of the aplet ( SKETCH The graphic will be copied here. 2. Press 3. Highlight Graphic, then press and highlight the name of the variable (G1, etc.).
  • Page 347 Opens the selected note for editing. Begins a new note, and asks for a name. Transmits the selected note to another HP 40gs or PC. Receives a note being transmitted from another HP 40gs or PC. Deletes the selected note.
  • Page 348 To import a note You can import a note from the Notepad into an aplet’s Note view, and vice versa. Suppose you want to copy a note named “Assignments” from the Notepad into the Function Note view: 1. In the Function aplet, display the Note view NOTE 2.
  • Page 349: Programming

    • programming variables. H I N T More information on programming, including examples and special tools, can be found at HP’s calculators web site: The Contents of a An HP 40gs program contains a sequence of numbers,...
  • Page 350: Program Catalog

    HOME, or the last data you entered in an input form. (If you press from HOME without entering any data, the HP 40gs runs the contents of Editline.) Before starting to work with programs, you should take a few minutes to become familiar with the Program catalog menu keys.
  • Page 351 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 highlighted program.
  • Page 352: Creating And Editing Programs

    4. Enter your program. When done, start any other activity. Your work is saved automatically. Enter commands Until you become familiar with the HP 40gs commands, the easiest way to enter commands is to select them from the Commands menu from the Program editor. You can also type in commands using alpha characters.
  • Page 353 2. Use the arrow keys to highlight the program you want to edit, and press . The HP 40gs opens the Program Editor. The name of your program appears in the title bar of the display. You can use the following keys to edit your program.
  • Page 354 Editing keys The editing keys are: Meaning Inserts the character at the editing point. Inserts space into text. Displays previous 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.
  • Page 355: Using Programs

    HOME, the HP 40gs displays the contents of Ans (Home variable containing the last result), when the program has finished. If you start the program from the Program catalog, the HP 40gs returns you to the Program catalog when the program ends. Debug a If you run a program that contains errors, the program will stop and you will see an error message.
  • Page 356 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. Press to open the Program catalog. PROGRM 2.
  • Page 357: Customizing An Aplet

    Delete a To delete a program: program 1. Press to open the Program catalog. PROGRM 2. Highlight a program to delete, then press Delete all You can delete all programs at once. programs 1. In the Program catalog, press CLEAR 2.
  • Page 358: Aplet Naming Convention

    4. Develop a program that uses the SETVIEWS command to modify the aplet’s VIEWS menu. The menu options provide links to associated programs. You can specify any other programs that you want transferred with the aplet. See “SETVIEWS” on page 21-14 for information on the command.
  • Page 359 Save the aplet 1. Open the Function aplet and save it as “EXPERIMENT”. The new aplet appears in the Aplet library. Select Function EXPERIMENT 2. Create a program called EXP.ME1 with contents as shown. This program configures the plot ranges, then runs a program that allows you to set the angle format.
  • Page 360 6. Open the Program catalog and create a program named “EXP.SV”. Include the following code in the program. Each entry line after the command SETVIEWS is a trio that consists of a VIEWS menu text line (a space indicates none), a program name, and a number that defines the view to go to after the program has run its course.
  • Page 361: Programming Commands

    9. You can now return to the Aplet library and press to run your new aplet. Programming commands This section describes the commands for programming with HP 40gs. You can enter these commands in your program by typing them or by accessing them from the Commands menu. Programming...
  • Page 362: Aplet Commands

    Aplet commands CHECK Checks (selects) the corresponding function in the current aplet. For example, Check 3 would check F3 if the current aplet is Function. Then a checkmark would appear next to F3 in Symbolic view, F3 would be plotted in Plot view, and evaluated in Numeric view.
  • Page 363 options use, or the program that defines the aplet’s VIEWS menu. • You can include a “Start” option in the VIEWS menu to specify a program that you want to run automatically when the aplet starts. This program typically sets up the aplet’s initial configuration. The START option on the menu is also useful for resetting the aplet.
  • Page 364 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 SETVIEWS command are transferred when the aplet is transmitted. ViewNumber ViewNumber is the number of a view to start after the program finishes running.
  • Page 365: Branch Commands

    View numbers The Function aplet views are numbered as follows: HOME List Catalog Plot Matrix Catalog Symbolic Notepad Catalog Numeric Program Catalog Plot-Setup Plot-Detail Symbolic-Setup Plot-Table Numeric-Setup Overlay Plot Views Auto scale Note Decimal Sketch view Integer Aplet Catalog Trig View numbers from 15 on will vary according to the parent aplet.
  • Page 366 . Execution with the CASE structure continues until a true-clause is executed (or until all the test-clauses evaluate to false). IFERR... Many conditions are automatically recognized by the HP THEN... 40gs as error conditions and are automatically treated as errors in programs. ELSE… END... 21-18 Programming...
  • Page 367: Drawing Commands

    Xmin, Xmax, Ymin, and Ymax values. The following examples assume the HP 40gs default settings with the Function aplet as the current aplet. Draws a circular arc, of given radius, whose centre is at (x,y) The arc is drawn from start_angle_measurement to end_angle_measurement.
  • Page 368 Example ARC 0;0;2;0;2π: FREEZE: Draws a circle centered at (0,0) of radius 2. The FREEZE command causes the circle to remain displayed on the screen until you press a key. Draws a box with diagonally opposite corners (x1,y1) and (x2,y2). BOX x1;y1;x2;y2: Example BOX -1;-1;1;1:...
  • Page 369: Graphic Commands

    Creates a graphic from expression, using font_size, and stores the resulting graphic in graphicname. Font sizes are 1, 2, or 3. If the fontsize argument is 0, the HP 40gs creates a graphic display like that created by the SHOW operation.
  • Page 370 GROBXOR Using the logical XOR, superimposes graphicname2 onto graphicname1. The upper left corner of graphicname2 is placed at position. GROBXOR graphicname1;(position);graphicname2: MAKEGROB Creates graphic with given width, height, and hexadecimal data, and stores it in graphicname. MAKEGROB graphicname;width;height;hexdata: PLOT→ Stores the Plot view display as a graphic in graphicname. PLOT→...
  • Page 371: Loop Commands

    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 …END Do ... Until ... End is a loop command that executes the loop-clause repeatedly until test-clause returns a true (nonzero) result.
  • Page 372: Matrix Commands

    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 values must be the same as the number of rows in the matrix name.
  • Page 373: Print Commands

    SWAPCOL name;column1;column2: SWAPROW Swap Rows. Exchanges row1 and row2 in the specified matrix. SWAPROW name;row1;row2: Print commands These commands print to an HP infrared printer, for example the HP 82240B printer. PRDISPLAY Prints the contents of the display. PRDISPLAY: PRHISTORY Prints all objects in the history.
  • Page 374: Prompt Commands

    PRVAR Prints name and contents of variablename. PRVAR variablename: You can also use the PRVAR command to print the contents of a program or a note. PRVAR programname;PROG: PRVAR notename;NOTE: Prompt commands BEEP Beeps at the frequency and for the time you specify. BEEP frequency;seconds: CHOOSE Creates a choose box, which is a box containing a list of...
  • Page 375 Example If you have stored {1,2,3,4} in variable L1, entering CLVAR L1 will 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.
  • Page 376 Examples 5.152000 DATE(sets the date to May 15, 2000). 10.1500 TIME (sets the time to 10:15 am). EDITMAT Matrix Editor. Opens the Matrix editor for the specified matrix. Returns to the program when user presses EDITMAT matrixname: The EDITMAT command can also be used to create matrices.
  • Page 377: Stat-one And Stat-two Commands

    Example INPUT R; "Circular Area"; "Radius"; "Enter Number";1: MSGBOX Displays a message box containing textitem. A text item consists of any number of expressions and quoted strings of text. The expressions are evaluated and turned into strings of text. For example, "AREA IS:" 2 +2 becomes AREA IS: 4. to type the quote marks "...
  • Page 378: Stat-two Commands

    Stat-One commands DO1VSTATS Calculates STATS using datasetname and stores the results in the corresponding variables: NΣ, TotΣ, MeanΣ, PVarΣ, SVarΣ, PSDev, SSDev, MinΣ, Q1, Median, Q3, and MaxΣ. Datasetname can be H1, H2, ..., or H5. Datasetname must include at least two data points. DO1VSTATS datasetname: SETFREQ Sets datasetname frequency according to column or...
  • Page 379: Storing And Retrieving Variables In Programs

    Storing and retrieving variables in programs The HP 40gs has both Home variables and Aplet variables. Home variables are used for real numbers, complex numbers, graphics, lists, and matrices. Home variables keep the same values in HOME and in aplets.
  • Page 380 Coord Turns the coordinate-display mode in Plot view on or off. Function From Plot view, use the Menu mean key to toggle Parametric coordinate display on an off. Polar In a program, type Sequence Solve Coord—to turn coordinate display on (default). Statistics Coord—to turn coordinate display off.
  • Page 381 Hwidth Sets the width of histogram bars. Statistics From Plot Setup in 1VAR stats set a value for Hwidth In a program, type Hwidth Indep Defines the value of the independent variable used in tracing mode. All Aplets In a program, type Indep InvCross Toggles between solid crosshairs or inverted crosshairs.
  • Page 382 Nmin / Nmax Defines the minimum and maximum independent variable values. Appears as the NRNG fields in the Plot Setup input Sequence form. From Plot Setup, enter values for NRNG. In a program, type Nmin Nmax > where Recenter Recenters at the crosshairs locations when zooming. All Aplets From Plot-Zoom-Set Factors, check (or uncheck) Recenter...
  • Page 383 Simult Enables you to choose between simultaneous and sequential graphing of all selected expressions. Function Parametric From Plot Setup, check (or uncheck) _SIMULT Polar Sequence In a program, type Simult—for simultaneous graphing (default). Simult—for sequential graphing. Slope Contains the last value found by the Slope function in the Plot-FCN menu.
  • Page 384 Tmin / Tmax Sets the minimum and maximum independent variable values. Appears as the TRNG field in the Plot Setup input Parametric form. From Plot Setup, enter values for TRNG. In a program, type Tmin Tmax > where Tracing Turns the tracing mode on or off in Plot view. All Aplets In a program, type Tracing—to turn Tracing mode on (default).
  • Page 385 Xtick Sets the distance between tick marks for the horizontal AAll Aplets axis. From the Plot Setup input form, enter a value for Xtick. In a program, type > Xtick where Ytick Sets the distance between tick marks for the vertical axis. All Aplets From the Plot Setup input form, enter a value for Ytick.
  • Page 386: Symbolic-view Variables

    Xzoom Sets the horizontal zoom factor. All Aplets From Plot-ZOOM-Set Factors, enter the value for XZOOM. In a program, type XZOOM > where The default value is 4. Yzoom Sets the vertical zoom factor. All Aplets From Plot-ZOOM-Set Factors, enter the value for YZOOM. In a program, type YZOOM The default value is 4.
  • Page 387 X1, Y1...X9,Y9 Can contain any expression. Independent variable is T. X0,Y0 Example Parametric 'SIN(4*T)' Y1(T):'2*SIN(6*T)' X1(T) Can contain any expression. Independent variable is θ. R1...R9, R0 Polar Example '2*SIN(2*θ)' R1(θ) U1...U9, U0 Can contain any expression. Independent variable is N. Sequence Example RECURSE (U,U(N-1)*N,1,2)
  • Page 388: Numeric-view Variables

    Example Cubic S2fit S2fit Numeric-view variables The following aplet variables control the Numeric view. The value of the variable applies to the current aplet only. C1...C9, C0 C0 through C9, for columns of data. Can contain lists. Statistics Enter data in the Numeric view In a program, type LIST Cn where n = 0, 1, 2, 3 ...
  • Page 389: Scientific Format

    1 Standard 2 Fixed 3 Sci 4 Eng 5 Fraction 6 MixFraction Note: if Fraction or Mixed Fraction is chosen, the setting will be disregarded when labeling axes in the Plot view. A setting of Scientific will be used instead. Example Scientific Format...
  • Page 390 NumStart Sets the starting value for a table in Numeric view. Function From Num Setup, enter a value for NUMSTART. Parametric Polar Sequence In a program, type NumStart NumStep Sets the step size (increment value) for an independent variable in Numeric view. Function Parametric From Num Setup, enter a value for NUMSTEP.
  • Page 391: Note Variables

    Example 1VAR StatMode StatMode Note variables The following aplet variable is available in Note view. NoteText Use NoteText to recall text previously entered in Note view. All Aplets Sketch variables The following aplet variables are available in Sketch view. Page Sets a page in a sketch set.
  • Page 393: Extending Aplets

    Extending aplets Aplets are the application environments where you explore different classes of mathematical operations. You can extend the capability of the HP 40gs in the following ways: • Create new aplets, based on existing aplets, with specific configurations such as angle measure, graphical or tabular settings, and annotations.
  • Page 394 1. Open the Solve aplet and save it under the new name. Solve T R I A N G L E S 2. Enter the four formulas: θ θ θ 3. Decide whether you want the aplet to operate in Degrees, Radians, or Grads.
  • Page 395: Using A Customized Aplet

    Using a customized aplet To use the “Triangles” aplet, simply select the appropriate 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 with the horizontal and extends 5 metres up the wall.
  • Page 396: Annotating An Aplet With Notes

    A convenient way to distribute or share problems in class and to turn in homework is to transmit (copy) aplets directly from one HP 40gs to another. This can take place via a suitable cable. ( You can use a serial cable with a 4-pin mini-USB connector, which plugs into the RS232 port on the calculator.
  • Page 397 To transmit 1. Connect the PC or aplet disk drive to the calculator by an appropriate cable. an aplet 2. Sending calculator: Open the Library, highlight the aplet to send, and press – menu appears with the following SEND TO options: = to send via the USB port HP39/40 (USB)
  • Page 398: Sorting Items In The Aplet Library Menu List

    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 many items as you would like 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.
  • Page 399: Glossary

    Reference information 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.
  • Page 400 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.
  • Page 401: Resetting The Hp 40gs

    Setup, Symbolic, Symbolic Setup, Sketch, Note, and special views like split screens. 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 certain conditions, and clears temporary memory locations.
  • Page 402: If The Calculator Does Not Turn On

    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.
  • Page 403 To install the main a. Slide up the battery compartment cover as illustrated. batteries b. Insert 4 new AAA (LR03) batteries into the main compartment. Make sure each battery is inserted in the indicated direction. To install the a. Press down the holder. Push the plate to the shown backup battery direction and lift it.
  • Page 404: Variables

    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 L1...L9, L0 List Matrix M1...M9, M0 Modes Date HAngle HDigits HFormat Ierr Time Notepad User-named Editline Program User-named A...Z, θ...
  • Page 405: Function Aplet Variables

    Function aplet variables The function aplet variables are: Category Available name Plot Axes Xcross Connect Ycross Coord Xtick FastRes Ytick Grid Xmin Indep Xmax InvCross Ymin Labels Ymax Recenter Xzoom Simult Yxoom Tracing Plot-FCN Area Root Extremum Slope Isect Angle Symbolic Numeric Digits...
  • Page 406: Parametric Aplet Variables

    Parametric aplet variables The parametric aplet variables are: Category Available name Plot Axes Tracing Connect Tstep Coord Xcross Grid Ycross Indep Xtick InvCross Ytick Labels Xmin Recenter Xmax Simult Ymin Tmin Ymax Tmax Xzoom Yzoom Symbolic Angle Numeric Digits NumRow Format NumStart NumCol...
  • Page 407: Polar Aplet Variables

    Polar aplet variables The polar aplet variables are: Category Available names Plot Axes Connect Xcross Coord Ycross Grid Xtick Indep Ytick InvCross Xmin Labels Xmax Recenter Ymin Simult Ymax Umin Xzoom Umax Yxoom θstep Tracing Symbolic Angle Numeric Digits NumRow Format NumStart NumCol...
  • Page 408: Sequence Aplet Variables

    Sequence aplet variables The sequence aplet variables are: Category Available name Plot Axes Tracing Coord Xcross Grid Ycross Indep Xtick InvCross Ytick Labels Xmin Nmin Xmax Nmax Ymin Recenter Ymax SeqPlot Xzoom Simult Yzoom Symbolic Angle Numeric Digits NumRow Format NumStart NumCol NumStep...
  • Page 409: Solve Aplet Variables

    Solve aplet variables The solve aplet variables are: Category Available name Plot Axes Xcross Connect Ycross Coord Xtick FastRes Ytick Grid Xmin Indep Xmax InvCross Ymin Labels Ymax Recenter Xzoom Tracing Yxoom Symbolic Angle Numeric Digits NumCol Format NumRow Note NoteText Sketch Page...
  • Page 410: Statistics Aplet Variables

    Statistics aplet variables The statistics aplet variables are: Category Available name Plot Axes S4mark Connect S5mark Coord StatPlot Grid Tracing Hmin Xcross Hmax Ycross Hwidth Xtick Indep Ytick InvCross Xmin Labels Xmax Recenter Ymin S1mark Ymax S2mark Xzoom S3mark Yxoom Symbolic Angle S3fit...
  • Page 411: Math Menu Categories

    MATH menu categories Math functions The math functions are: Category Available name Calculus ∂ ∫ TAYLOR Complex CONJ Constant MAXREAL MINREAL π Hyperb. ACOSH TANH ASINH ALOG ATANH COSH EXPM1 SINH LNP1 List CONCAT REVERSE ΔLIST SIZE ΣLIST MAKELIST πLIST SORT Loop ITERATE...
  • Page 413: Program Constants

    Category Available name (Continued) Tests < IFTE ≤ ≠ > ≥ Trig ACOT ACSC ASEC Program constants The program constants are: Category Available name Angle Degrees Grads Radians Format Standard Fixed Fraction SeqPlot Cobweb Stairstep S1...5fit Linear QuadFit LogFit Cubic ExpFit Logist Power...
  • Page 414: Physical Constants

    Physical Constants The physical constants are: Category Available Name Chemist • Avogadro (Avagadro’s Number, • Boltz. (Boltmann, k) • mol. vo... (molar volume, Vm) • univ gas (universal gas, R) • std temp (standard temperature, St dT) • std pres (standard pressure, St dP) •...
  • Page 415: Cas Functions

  • Page 417: Program Commands

  • Page 418: Status Messages

    Category Command (Continued) Stat-Two DO2VSTATS SETDEPEND SETINDEP Status messages Message Meaning Bad Argument Incorrect input for this Type operation. Bad Argument The value is out of range for this Value operation. Infinite Result Math exception, such as 1/0. Insufficient You must recover some memory Memory to continue operation.
  • Page 419 Message Meaning (Continued) 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.
  • Page 421: Limited Warranty

    Replacement products may be either new or like-new. 2. HP warrants to you that HP software will not fail to execute its programming instructions after the date of purchase, for the period specified above, due to defects in material and workmanship when properly installed and used.
  • Page 422 8. The only warranties for HP products and services are set forth in the express warranty statements accompanying such products and services . HP shall not be liable for technical or editorial errors or omissions contained herein.
  • Page 423: Service

    Service Europe Country : Telephone numbers Austria +43-1-3602771203 Belgium +32-2-7126219 Denmark +45-8-2332844 Eastern Europe +420-5-41422523 countries Finland +35-89640009 France +33-1-49939006 Germany +49-69-95307103 Greece +420-5-41422523 Holland +31-2-06545301 Italy +39-02-75419782 Norway +47-63849309 Portugal +351-229570200 Spain +34-915-642095 Sweden +46-851992065 Switzerland +41-1-4395358 (German) +41-22-8278780 (French) +39-02-75419782 (Italian)
  • Page 424 Guatemala 1-800-999-5105 Puerto Rico 1-877-232-0589 Costa Rica 0-800-011-0524 N.America Country : Telephone numbers U.S. 1800-HP INVENT Canada (905) 206-4663 or 800- HP INVENT ROTC = Rest of the country Please logon to for the latest service and support information.h...
  • Page 425: Regulatory Notices

    Regulatory Notices Federal Commu- This equipment has been tested and found to comply with the limits for a Class B digital device, pursuant to Part 15 nications of the FCC Rules. These limits are designed to provide Commission reasonable protection against harmful interference in a Notice residential installation.
  • Page 426 Hewlett-Packard Company P. O. Box 692000, Mail Stop 530113 Houston, Texas 77269-2000 Or, call 1-800-474-6836 For questions regarding this FCC declaration, contact: Hewlett-Packard Company P. O. Box 692000, Mail Stop 510101 Houston, Texas 77269-2000 Or, call 1-281-514-3333 To identify this product, refer to the part, series, or model number found on the product.
  • Page 427 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 ン受信機に近接 し て使用 さ れる と 、 受信障害を引き起 こ す こ と があ り ます。 取...
  • Page 429: Index

    Index CHECK 21-14 SELECT 21-14 ABCUV 14-62 SETVIEWS 21-17 ABS 14-45 UNCHECK 21-17 absolute value 13-6 aplet variables ACOS2S 14-38 definition 17-1 17-8 add 13-4 in Plot view 21-31 ADDTMOD 14-51 new 17-1 ALGB menu 14-10 aplet views algebraic entry 1-19 canceling operations in 1-1 alpha characters changing 1-19...
  • Page 430 bad guesses error message 7-7 graphic 21-21 loop 21-23 batteries R-4 print 21-25 Bernoulli’s number 14-65 program 21-4 R-19 box-and-whisker plot 10-16 stat-one 21-29 branch commands stat-two 21-30 CASE...END 21-18 with matrices 18-10 IF...THEN...ELSE...END 21-18 complex number functions 13-6 IFERR...THEN...ELSE 21-18 13-17 branch structures 21-17 conjugate 13-7...
  • Page 431 cosine 13-4 differentiation 13-6 14-33 inverse hyperbolic 13-9 digamma function 14-67 14-68 cotangent 13-20 display 21-21 covariance adjusting contrast 1-2 statistical 10-15 annunciator line 1-2 creating capture 21-21 clearing 1-2 aplet 22-1 date and time 21-27 lists 19-1 element 18-5 matrices 18-2 elements 19-4 notes in Notepad 20-6...
  • Page 432 notes 20-2 extremum 3-10 programs 21-5 Editline Program catalog 21-2 FACTOR 14-12 14-47 14-56 editors 1-30 factorial 13-13 EGCD 14-55 factorization 14-12 eigenvalues 18-11 FACTORMOD 14-53 eigenvectors 18-11 FastRes variable 21-32 element FDISTRIB 14-30 storing 18-6 E-lessons 1-12 a curve to 2VAR data 10-17 engineering number format 1-11 choosing 10-12 EPSX0 14-29...
  • Page 433 glossary R-1 adjusting 10-16 range 10-18 graph setting min/max values for bars analyzing statistical data in 10-19 21-32 auto scale 2-14 width 10-18 box-and-whisker 10-16 history 1-2 14-8 21-25 capture current display 21-21 Home 1-1 cobweb 6-1 comparing 2-5 calculating in 1-19 connected points 10-17 display 1-2 defining the independent variable...
  • Page 434 using symbolic variables 13-23 ISPRIME? 14-50 independent values adding to table 2-19 independent variable keyboard defined for Tracing mode 21-33 editing keys 1-5 inference entry keys 1-5 confidence intervals 11-15 inactive keys 1-8 hypothesis tests 11-8 list keys 19-2 One-Proportion Z-Interval 11-17 math functions 1-7 One-Sample Z-Interval 11-15 menu keys 1-4...
  • Page 435 finding statistical values in list ele- logical operators 13-19 ments 19-9 menu 1-7 generate a series 19-8 polynomial 13-11 list function syntax 19-6 probability 13-12 list variables 19-1 real-number 13-14 returning position of element in symbolic 13-17 19-8 trigonometry 13-20 reversing order in 19-8 MATH menu 13-1 sending and receiving 19-6...
  • Page 436 redimension 21-24 searching 1-9 replacing portion of matrix or vec- minimum real number 13-8 tor 21-25 mixed fraction format 1-11 sending or receiving 18-4 modes singular value decomposition angle measure 1-10 18-13 CAS 14-5 singular values 18-13 decimal mark 1-11 size 18-12 number format 1-10 spectral norm 18-13...
  • Page 437 mixed fraction 1-11 permutations 13-13 scientific 1-10 pictures Standard 1-10 attaching in Sketch view 20-3 numeric precision 17-9 plot Numeric view analyzing statistical data in 10-19 adding values 2-19 auto scale 2-14 automatic 2-16 box-and-whisker 10-16 build your own table 2-19 cobweb 6-1 display defining function for col- comparing 2-5...
  • Page 438 labels 21-34 ! 13-13 recenter 21-34 COMB 13-12 root 21-34 RANDOM 13-13 s1mark-s5mark 21-34 UTPC 13-13 statplot 21-35 UTPF 13-13 tracing 21-33 UTPN 13-13 umin/umax 21-35 UTPT 13-14 ustep 21-35 program polar variables commands 21-4 axes 21-31 copying 21-8 connect 21-31 creating 21-4 grid 21-32 debugging 21-7...
  • Page 439 RE 13-8 rigorous 14-6 real number RISCH 14-24 maximum 13-8 root minimum 13-8 interactive 3-10 real part 13-8 nth 13-6 real-number functions 13-14 variable 21-34 root-finding % 13-16 %CHANGE 13-16 displaying 7-7 %TOTAL 13-16 interactive 3-9 CEILING 13-14 operations 3-10 DEGtoRAD 13-14 variables 3-10 FNROOT 13-14...
  • Page 440 date 21-27 aplets in chronological order 22-6 time 21-27 elements in a list 19-9 SEVAL 14-68 spectral norm 18-13 SIGMA 14-68 spectral radius 18-13 SIGMAVX 14-69 square root 13-5 SIGN 14-46 stack history sign reversal 7-6 printing 21-25 stairsteps graph 6-1 SIMPLIFY 14-32 standard number format 1-10 simplify 14-68...
  • Page 441 Labels 21-34 TABVAR 14-27 Recenter 21-34 TAN2CS2 14-40 S1mark-S5mark 21-34 TAN2SC 14-41 Ycross 21-37 TAN2SC2 14-41 step size of independent variable tangent 13-4 21-36 inverse hyperbolic 13-9 step-by-step 14-6 Taylor polynomial 13-7 STORE 14-14 TAYLOR0 14-27 storing TCHEBYCHEFF 14-59 list elements 19-1 19-4 19-5 TCOLLECT 14-41...
  • Page 442 COT 13-20 CAS 14-4 CSC 13-20 categories 17-7 HALFTAN 14-40 clearing 17-3 SEC 13-20 definition 17-1 17-7 SINCOS 14-40 in equations 7-10 TAN2CS2 14-40 in Symbolic view 2-3 TAN2SC 14-41 independent 14-6 21-36 TAN2SC2 14-41 local 17-1 TRIGCOS 14-44 previous result (Ans) 1-23 TRIGSIN 14-44 printing 21-26 TRIGTAN 14-44...
  • Page 443 in 2-9 square 2-10 options 2-9 un-zoom 2-11 options within a table 2-18 within Numeric view 2-18 out 2-9 X-zoom 2-9 redrawing table of numbers op- Y-zoom 2-10 tions 2-18 I-15...

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