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hp 49g+ graphing calculator
user's guide
H
Edition 4
HP part number F2228-90006

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Table of Contents

   Related Manuals for HP 49g+

   Summary of Contents for HP 49g+

  • Page 1 49g+ graphing calculator user’s guide Edition 4 HP part number F2228-90006...
  • Page 2 Notice REGISTER YOUR PRODUCT AT: www.register.hp.com THIS MANUAL AND ANY EXAMPLES CONTAINED HEREIN ARE PROVIDED “AS IS” AND ARE SUBJECT TO CHANGE WITHOUT NOTICE. HEWLETT-PACKARD COMPANY MAKES NO WARRANTY OF ANY KIND WITH REGARD TO THIS MANUAL, INCLUDING, BUT LIMITED...
  • Page 3 49g+ should be thought of as a graphics/programmable hand-held computer. The hp 49g+ can be operated in two different calculating modes, the Reverse Polish Notation (RPN) mode and the Algebraic (ALG) mode (see page 1-11 in user’s guide for additional details).
  • Page 4 (numerical) mode. The display can be adjusted to provide textbook-type expressions, which can be useful when working with matrices, vectors, fractions, summations, derivatives, and integrals. The high-speed graphics of the calculator are very convenient for producing complex figures in very little time.
  • Page 5: Table Of Contents

    Table of contents Chapter 1 - Getting started , 1-1 Basic Operations, 1-1 Batteries, 1-1 Turning the calculator on and off, 1-2 Adjusting the display contrast, 1-2 Contents of the calculator’s display, 1-2 Menus, 1-3 SOFT menus vs. CHOOSE boxes, 1-4 Selecting SOFT menus or CHOOSE boxes, 1-5 The TOOL menu, 1-6 Setting time and date, 1-7...
  • Page 6 Editing algebraic expressions, 2-8 Using the Equation Writer (EQW) to create expressions, 2-10 Creating arithmetic expressions, 2-11 Editing arithmetic expressions, 2-16 Creating algebraic expressions, 2-19 Editing algebraic expressions, 2-20 Creating and editing summations, derivatives, and integrals, 2-28 Organizing data in the calculator, 2-32 Functions for manipulation of variables, 2-33 The HOME directory, 2-34 The CASDIR sub-directory, 2-35...
  • Page 7 Addition, subtraction, multiplication, division, 3-3 Using parentheses, 3-4 Absolute value function, 3-4 Squares and square roots, 3-5 Powers and roots, 3-5 Base-10 logarithms and powers of 10, 3-5 Using powers of 10 in entering data, 3-6 Natural logarithms and exponential function, 3-6 Trigonometric functions, 3-6 Inverse trigonometric functions, 3-6 Differences between functions and operators, 3-7...
  • Page 8 Chapter 4 - Calculations with complex numbers , 4-1 Definitions, 4-1 Setting the calculator to COMPLEX mode, 4-1 Entering complex numbers, 4-2 Polar representation of a complex number, 4-3 Simple operations with complex numbers, 4-4 Changing sign of a complex number, 4-4 Entering the unit imaginary number, 4-5 The CMPLX menus, 4-5 CMPLX menu through the MTH menu, 4-5...
  • Page 9 LGCD, 5-10 PROPFRAC, 5-10 SIMP2, 5-10 INTEGER menu, 5-10 POLYNOMIAL menu, 5-11 MODULO menu, 5-12 Applications of the ARITHMETIC menu, 5-12 Modular arithmetic, 5-12 Finite arithmetic in the calculator, 5-15 Polynomials, 5-18 Modular arithmetic with polynomials, 5-19 The CHINREM function, 5-19 The EGCD function, 5-19 The GCD function, 5-19 The HERMITE function, 5-20...
  • Page 10 BASE convert menu, 5-28 TRIGONOMETRIC convert menu, 5-28 MATRICES convert menu, 5-28 REWRITE convert menu, 5-28 Chapter 6 - Solution to single equations , 6-1 Symbolic solution of algebraic equations, 6-1 Function ISOL, 6-1 Function SOLVE, 6-2 Function SOLVEVX, 6-4 Function ZEROS, 6-4 Numerical solver menu, 6-5 Polynomial equations, 6-6...
  • Page 11 Chapter 8 - Operations with Lists , 8-1 Definitions, 8-1 Creating and storing lists, 8-1 Composing and decomposing lists, 8-2 Operations with lists of numbers, 8-3 Changing sign, 8-3 Addition, subtraction, multiplication, division, 8-3 Real number functions from the keyboard, 8-5 Real number functions from the MTH menu, 8-5 Examples of functions that use two arguments, 8-6 Lists of complex numbers, 8-7...
  • Page 12 Building a vector with ARRY, 9-6 Identifying, extracting, and inserting vector elements, 9-7 Simple operations with vectors, 9-9 Changing sign, 9-9 Addition, subtraction, 9-9 Multiplication and division by a scalar, 9-10 Absolute value function, 9-10 The MTH/VECTOR menu, 9-10 Magnitude, 9-11 Dot product, 9-11 Cross product, 9-11 Decomposing a vector, 9-12...
  • Page 13 Functions GETI and PUTI, 10-6 Function SIZE, 10-7 Function TRN, 10-8 Function CON, 10-8 Function IDN, 10-9 Function RDM, 10-10 Function RANM, 10-11 Function SUB, 10-11 Function REPL, 10-12 Function DIAG, 10-12 Function DIAG , 10-13 Function VANDERMONDE, 10-14 Function HILBERT, 10-14 A program to build a matrix out of a number of lists, 10-15 Lists represent columns of the matrix, 10-15 Lists represent rows of the matrix, 10-17...
  • Page 14 Function ABS, 11-7 Function SNRM, 11-7 Functions RNRM and CNRM, 11-8 Function SRAD, 11-9 Function COND, 11-9 Function RANK, 11-10 Function DET, 11-11 Function TRACE, 11-13 Function TRAN, 11-14 Additional matrix operations (the matrix OPER menu), 11-14 Function AXL, 11-15 Function AXM, 11-15 Function LCXM, 11-15 Solution of linear systems, 11-16...
  • Page 15 Matrix Quadratic Forms, 11-51 The QUADF menu, 11-52 Linear Applications, 11-54 Function IMAGE, 11-54 Function ISOM, 11-54 Function KER, 11-55 Function MKISOM, 11-55 Chapter 12 - Graphics , 12-1 Graphs options in the calculator, 12-1 Plotting an expression of the form y = f(x) , 12-2 Some useful PLOT operations for FUNCTION plots, 12-5 Saving a graph for future use, 12-8 Graphics of transcendental functions, 12-10...
  • Page 16 Gridmap plots, 12-42 Pr-Surface plots, 12-43 The VPAR variable, 12-44 Interactive drawing, 12-44 DOT+ and DOT-, 12-45 MARK, 12-46 LINE, 12-46 TLINE, 12-46 BOX, 12-47 CIRCL, 12-47 LABEL, 12-47 DEL, 12-47 ERASE, 12-48 MENU, 12-48 SUB, 12-48 REPL, 12-48 PICT , 12-48 X,Y , 12-48 Zooming in and out in the graphics display, 12-49 ZFACT, ZIN, ZOUT, and ZLAST, 12-49...
  • Page 17 Derivatives, 13-3 Function DERIV and DERVX,13-3 The DERIV&INTEG menu, 13-3 Calculating derivatives with ∂,13-4 The chain rule,13-6 Derivatives of equations,13-6 Implicit derivatives,13-7 Application of derivatives,13-7 Analyzing graphics of functions,13-7 Function DOMAIN, 13-9 Function TABVAL, 13-9 Function SIGNTAB, 13-10 Function TABVAR, 13-10 Using derivatives to calculate extreme points, 13-12 Higher-order derivatives, 13-13 Anti-derivatives and integrals, ,13-14...
  • Page 18 Total differential of a function z = z(x,y) , 14-5 Determining extrema in functions of two variables, 14-5 Using function HESS to analyze extrema, 14-6 Multiple integrals, 14-8 Jacobian of coordinate transformation, 14-9 Double integral in polar coordinates, 14-9 Chapter 15 - Vector Analysis Applications , 15-1 Definitions, 15-1 Gradient and directional derivative, 15-1...
  • Page 19 Function FOURIER, 16-28 Fourier series for a quadratic function, 16-29 Fourier series for a triangular wave, 16-35 Fourier series for a square wave, 16-39 Fourier series applications in differential equations, 16-42 Fourier Transforms, 16-43 Definition of Fourier transforms, 16-46 Properties of the Fourier transform, 16-48 Fast Fourier Transform (FFT) , 16-49 Examples of FFT applications, 16-50 Solution to specific second-order differential equations, 16-53...
  • Page 20 Discrete probability distributions, 17-4 Binomial distribution, 17-4 Poisson distribution, 17-5 Continuous probability distributions, 17-6 The gamma distribution, 17-6 The exponential distribution, 17-7 The beta distribution, 17-7 The Weibull distribution, 17-7 Functions for continuous distributions, 17-7 Continuous distributions for statistical inference, 17-9 Normal distribution pdf, 17-9 Normal distribution cdf, 17-10 The Student-t distribution, 17-10...
  • Page 21 population variance is known, 18-23 Confidence intervals for the population mean when the population variance is unknown, 18-24 Confidence interval for a proportion, 18-24 Sampling distributions of differences and sums of statistics, 18-25 Confidence intervals for sums and differences of mean values, 18-26 Determining confidence intervals, 18-27 Confidence intervals for the variance, 18-33 Hypothesis testing, 18-34...
  • Page 22 Wordsize, 19-4 Operations with binary integers, 19-4 The LOGIC menu, 19-5 The BIT menu, 19-6 The BYTE menu, 19-6 Hexadecimal numbers for pixel references, 19-7 Chapter 20 - Customizing menus and keyboard , 20-1 Customizing menus, 20-1 The PRG/MODES/MENU, 20-1 Menu numbers (RCLMENU and MENU functions), 20-2 Custom menus (MENU and TMENU functions), 20-2 Menu specification and CST variable, 20-4...
  • Page 23 Interactive input in programs, 21-19 Prompt with an input string, 21-21 A function with an input string, 21-22 Input string for two or three input values, 21-24 Input through input forms, 21-27 Creating a choose box, 21-31 Identifying output in programs, 21-33 Tagging a numerical result, 21- 33 Decomposing a tagged numerical result into number and tag, 21-33 “De-tagging”...
  • Page 24 Generating plots with programs, 22-14 Two-dimensional graphics, 22-14 Three-dimensional graphics, 22-15 The variable EQ, 22-15 Examples of interactive plots using the PLOT menu, 22-15 Examples of program-generated plots, 22-17 Drawing commands for use in programming, 22-19 PICT, 22-20 PDIM, 22-20 LINE, 22-20 TLINE, 22-20 BOX, 22-21...
  • Page 25 The characters list, 23-3 Chapter 24 - Calculator objects and flags , 24-1 Description of calculator objects, 24-1 Function TYPE, 24-2 Function VTYPE, 24-2 Calculator flags, 24-3 System flags, 24-3 Functions for setting and changing flags, 24-3 User flags, 24-4 Chapter 25 - Date and Time Functions , 25-1 The TIME menu, 25-1...
  • Page 26 Installing and attaching a library, 26-8 Library numbers, 26-9 Deleting a library, 26-9 Creating libraries, 26-9 Backup battery, 26-9 Appendices Appendix A - Using input forms , A-1 Appendix B - The calculator’s keyboard , B-1 Appendix C - CAS settings , C-1 Appendix D - Additional character set , D-1...
  • Page 27: Chapter 1 - Getting Started

    Chapter 1 Getting started This chapter is aimed at providing basic information in the operation of your calculator. The exercises are aimed at familiarizing yourself with the basic operations and settings before actually performing a calculation. Basic Operations The following exercises are aimed at getting you acquainted with the hardware of your calculator.
  • Page 28: Turning The Calculator On And Off

    b. Insert a new CR2032 lithium battery. Make sure its positive (+) side is facing up. c. Replace the plate and push it to the original place. After installing the batteries, press [ON] to turn the power on. Warning: When the low battery icon is displayed, you need to replace the batteries as soon as possible.
  • Page 29: Menus

    At the top of the display you will have two lines of information that describe the settings of the calculator. The first line shows the characters: RAD XYZ HEX R= 'X' For details on the meaning of these specifications see Chapter 2. The second line shows the characters: { HOME } indicating that the HOME directory is the current file directory in the calculator’s memory.
  • Page 30 pressing the L (NeXT menu) key. This key is the third key from the left in the third row of keys in the keyboard. Press L once more to return to the main TOOL menu, or press the I key (third key in second row of keys from the top of the keyboard).
  • Page 31 using the up and down arrow keys, —˜, or by pressing the number corresponding to the function in the CHOOSE box. After the function name is selected, press the @@@OK@@@ soft menu key (F). Thus, if you wanted to use function R B (Real to Binary), you could press 6F.
  • Page 32 To navigate through the functions of this menu, press the L key to move to the next page, or „«(associated with the L key) to move to the previous page. The following figures show the different pages of the BASE menu accessed by pressing the L key twice: Pressing the L key once more will takes us back to the first menu page.
  • Page 33: Setting Time And Date

    @VIEW VIEW the contents of a variable @@ RCL @@ ReCaLl the contents of a variable @@STO@ STOre the contents of a variable ! PURGE PURGE a variable CLEAR CLEAR the display or stack The calculator has only six soft menu keys, and can only display 6 labels at any point in time.
  • Page 34 As indicated above, the TIME menu provides four different options, numbered 1 through 4. Of interest to us as this point is option 3. Set time, date... Using the down arrow key, ˜, highlight this option and press the !!@@OK#@ F soft menu key.
  • Page 35 25 !!@@OK#@ . Let’s change the minute field to 25, by pressing: seconds field is now highlighted. Suppose that you want to change the 45 !!@@OK#@ seconds field to 45, use: The time format field is now highlighted. To change this field from its current setting you can either press the W key (the second key from the left in the fifth row of keys from the bottom of the keyboard), or press the @CHOOS soft menu key ( B).
  • Page 36: Introducing The Calculator's Keyboard

    To set the date, first set the date format. The default format is M/D/Y (month/day/year). To modify this format, press the down arrow key. This will highlight the date format as shown below: Use the @CHOOS soft menu key ( B), to see the options for the date format: Highlight your choice by using the up and down arrow keys,—...
  • Page 37 The figure shows 10 rows of keys combined with 3, 5, or 6 columns. Row 1 has 6 keys, rows 2 and 3 have 3 keys each, and rows 4 through 10 have 5 keys each. There are 4 arrow keys located on the right-hand side of the keyboard in the space occupied by rows 2 and 3.
  • Page 38: Selecting Calculator Modes

    combined with some of the other keys to activate the alternative functions shown in the keyboard. For example, the key, key(4,4), has the following six functions associated with it: Main function, to activate the SYMBolic menu „´ Left-shift function, to activate the MTH (Math) menu …...
  • Page 39: Operating Mode

    Reverse Polish Notation (RPN) mode. The default mode is the Algebraic mode (as indicated in the figure above), however, users of earlier HP calculators may be more familiar with the RPN mode. To select an operating mode, first open the CALCULATOR MODES input form by pressing the H button.
  • Page 40 The equation writer is a display mode in which you can build mathematical expressions using explicit mathematical notation including fractions, derivatives, integrals, roots, etc. To use the equation writer for writing the expression shown above, use the following keystrokes: ‚OR3*!Ü5- 1/3*3 ———————...
  • Page 41 Notice that the display shows several levels of output labeled, from bottom to top, as 1, 2, 3, etc. This is referred to as the stack of the calculator. The different levels are referred to as the stack levels, i.e., stack level 1, stack level 2, etc.
  • Page 42 Enter 3 in level 1 Enter 5 in level 1, 3 moves to y Enter 3 in level 1, 5 moves to level 2, 3 to level 3 Place 3 and multiply, 9 appears in level 1 1/(3×3), last value in lev. 1; 5 in level 2; 3 in level 3 5 - 1/(3×3) , occupies level 1 now;...
  • Page 43: Number Format And Decimal Dot Or Comma

    line will execute the DUP function which copies the contents of stack level 1 of the stack onto level 2 (and pushes all the other stack levels one level up). This is extremely useful as showed in the previous example. To select between the ALG vs.
  • Page 44 The number is rounded to the maximum 12 significant figures, and is displayed as follows: In the standard format of decimal display, integer numbers are shown with no decimal zeros whatsoever. Numbers with different decimal figures will be adjusted in the display so that only those decimal figures that are necessary will be shown.
  • Page 45 • Fixed format with decimals: This mode is mainly used when working with limited precision. For example, if you are doing financial calculation, using a FIX 2 mode is convenient as it can easily represent monetary units to a 1/100 precision. Press the H button.
  • Page 46 • Scientific format The scientific format is mainly used when solving problems in the physical sciences where numbers are usually represented as a number with limited precision multiplied by a power of ten. To set this format, start by pressing the H button. Next, use the down arrow key, ˜, to select the option Number format.
  • Page 47 Press the !!@@OK#@ soft menu key return to the calculator display. number now is shown as: Because this number has three figures in the integer part, it is shown with four significative figures and a zero power of ten, while using the Engineering format.
  • Page 48: Angle Measure

    Angle Measure Trigonometric functions, for example, require arguments representing plane angles. The calculator provides three different Angle Measure modes for working with angles, namely: • Degrees: There are 360 degrees (360 ) in a complete circumference, or 90 degrees (90 ) in a right angle.
  • Page 49 The coordinate system selection affects the way vectors and complex numbers are displayed and entered. To learn more about complex numbers and vectors, see Chapters 4 and 9, respectively. Two- and three-dimensional vector components and complex numbers can be represented in any of 3 coordinate systems: The Cartesian (2 dimensional) or Rectangular (3 dimensional), Cylindrical (3 dimensional) or Polar (2 dimensional), and Spherical (only 3 dimensional).
  • Page 50: Beep, Key Click, And Last Stack

    • Press the H button. Next, use the down arrow key, ˜, three times. Select the Angle Measure mode by either using the \ key (second from left in the fifth row from the keyboard bottom), or pressing the @CHOOS soft menu key ( B).
  • Page 51: Selecting Cas Settings

    • Use the down arrow key, ˜, four times to select the _Last Stack option. Use the soft menu key (i.e., the B key) to change the selection. @ CHK@ • Press the left arrow key š to select the _Key Click option. Use the @ CHK@ soft menu key (i.e., the B key) to change the selection.
  • Page 52: Selecting The Display Font

    • To navigate through the many options in the DISPLAY MODES input form, use the arrow keys: š™˜—. • To select or deselect any of the settings shown above, that require a check mark, select the underline before the option of interest, and toggle the soft menu key until the right setting is achieved.
  • Page 53: Selecting Properties Of The Line Editor

    additional fonts that you may have created (see Chapter 23) or downloaded into the calculator. Practice changing the display fonts to sizes 7 and 6. Press the OK soft menu key to effect the selection. When done with a font selection, press the @@@OK@@@ soft menu key to go back to the CALCULATOR MODES input form.
  • Page 54: Selecting Properties Of The Equation Writer (eqw)

    To illustrate these settings, either in algebraic or RPN mode, use the equation writer to type the following definite integral: ‚O…Á0™„虄¸\x™x` In Algebraic mode, the following screen shows the result of these keystrokes with neither _Small nor _Textbook are selected: With the _Small option selected only, the display looks as shown below: With the _Textbook option selected (default value), regardless of whether the _Small option is selected or not, the display shows the following result:...
  • Page 55: Selecting The Size Of The Header

    ∞ ∫ − For the example of the integral , presented above, selecting the _Small Stack Disp in the EQW line of the DISPLAY MODES input form produces the following display: Selecting the size of the header First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key (D) to display the DISPLAY MODES input form.
  • Page 56: Chapter 2 - Introducing The Calculator

    Chapter 2 Introducing the calculator In this chapter we present a number of basic operations of the calculator including the use of the Equation Writer and the manipulation of data objects in the calculator. Study the examples in this chapter to get a good grasp of the capabilities of the calculator for future applications.
  • Page 57 If the approximate mode (APPROX) is selected in the CAS (see Appendix C), integers will be automatically converted to reals. If you are not planning to use the CAS, it might be a good idea to switch directly into approximate mode. Refer to Appendix C for more details.
  • Page 58: Editing Expressions In The Screen

    An algebraic object, or simply, an algebraic (object of type 9), is a valid algebraic expression enclosed between apostrophes. Binary integers, objects of type 10, are used in some computer science applications. Graphics objects, objects of type 11, store graphics produced by the calculator.
  • Page 59 5.*„Ü1.+1./7.5™/ „ÜR3.-2.Q3 The resulting expression is: 5.*(1.+1./7.5)/(ƒ3.-2.^3). Press ` to get the expression in the display as follows: Notice that, if your CAS is set to EXACT (see Appendix C) and you enter your expression using integer numbers for integer values, the result is a symbolic quantity, e.g., 5*„Ü1+1/7.5™/ „ÜR3-2Q3...
  • Page 60 The result will be shown as follows: To evaluate the expression we can use the EVAL function, as follows: µ„î` As in the previous example, you will be asked to approve changing the CAS setting to Approx. Once this is done, you will get the same result as before. An alternative way to evaluate the expression entered earlier between quotes is by using the option …ï.
  • Page 61: Editing Arithmetic Expressions

    This latter result is purely numerical, so that the two results in the stack, although representing the same expression, seem different. To verify that they are not, we subtract the two values and evaluate this difference using function EVAL: Subtract level 1 from level 2 Evaluate using function EVAL µ...
  • Page 62: Creating Algebraic Expressions

    The editing cursor is shown as a blinking left arrow over the first character in the line to be edited. Since the editing in this case consists of removing some characters and replacing them with others, we will use the right and left arrow keys, š™, to move the cursor to the appropriate place for editing, and the delete key, ƒ, to eliminate characters.
  • Page 63: Editing Algebraic Expressions

    We set the calculator operating mode to Algebraic, the CAS to Exact, and the display to Textbook. To enter this algebraic expression we use the following keystrokes: ³2*~l*R„Ü1+~„x/~r™/ „ Ü ~r+~„y™+2*~l/~„b Press ` to get the following result: Entering this expression when the calculator is set in the RPN mode is exactly the same as this Algebraic mode exercise.
  • Page 64 The editing cursor is shown as a blinking left arrow over the first character in the line to be edited. As in an earlier exercise on line editing, we will use the right and left arrow keys, š™, to move the cursor to the appropriate place for editing, and the delete key, ƒ, to eliminate characters.
  • Page 65: Using The Equation Writer (eqw) To Create Expressions

    • Press „˜ to activate the line editor once more. The result is now: • Pressing ` once more to return to normal display. To see the entire expression in the screen, we can change the option _Small Stack Disp in the DISPLAY MODES input form (see Chapter 1). After effecting this change, the display will look as follows: Note: To use Greek letters and other characters in algebraic expressions use the CHARS menu.
  • Page 66: Creating Arithmetic Expressions

    The Equation Writer is launched by pressing the keystroke combination … ‚O (the third key in the fourth row from the top in the keyboard). The resulting screen is the following: The six soft menu keys for the Equation Writer activate the following functions: @EDIT: lets the user edit an entry in the line editor (see examples above) @CURS: highlights expression and adds a graphics cursor to it @BIG: if selected (selection shown by the character in the label) the font used in...
  • Page 67 in “textbook” style instead of a line-entry style. Thus, when a division sign (i.e., /) is entered in the Equation Writer, a fraction is generated and the cursor placed in the numerator. To move to the denominator you must use the down arrow key.
  • Page 68 The expression now looks as follows: Suppose that now you want to add the fraction 1/3 to this entire expression, i.e., you want to enter the expression: π First, we need to highlight the entire first term by using either the right arrow (™) or the upper arrow (—) keys, repeatedly, until the entire expression is highlighted, i.e., seven times, producing: NOTE: Alternatively, from the original position of the cursor (to the right of the...
  • Page 69 To recover the larger-font display, press the @BIG C soft menu key once more. Evaluating the expression To evaluate the expression (or parts of the expression) within the Equation Writer, highlight the part that you want to evaluate and press the @EVAL D soft menu key.
  • Page 70 Use the function UNDO ( …¯) once more to recover the original expression: Evaluating a sub-expression Suppose that you want to evaluate only the expression in parentheses in the denominator of the first fraction in the expression above. You have to use the arrow keys to select that particular sub-expression.
  • Page 71: Editing Arithmetic Expressions

    Then, press the @EVAL D soft menu key to obtain: Let’s try a numerical evaluation of this term at this point. Use …ï to obtain: Let’s highlight the fraction to the right, and obtain a numerical evaluation of that term too, and show the sum of these two decimal values in small-font format by using:™...
  • Page 72 And will use the editing features of the Equation Editor to transform it into the following expression: In the previous exercises we used the arrow keys to highlight sub-expressions for evaluation. In this case, we will use them to trigger a special editing cursor. After you have finished entering the original expression, the typing cursor (a left-pointing arrow) will be located to the right of the 3 in the denominator of the second fraction as shown here:...
  • Page 73 Next, press the down arrow key (˜) to trigger the clear editing cursor highlighting the 3 in the denominator of π /3. Press the left arrow key (š) once to highlight the exponent 2 in the expression π /3. Next, press the delete key (ƒ) once to change the cursor into the insertion cursor.
  • Page 74: Creating Algebraic Expressions

    down arrow key (˜) in any location, repeatedly, to trigger the clear editing cursor. In this mode, use the left or right arrow keys (š™) to move from term to term in an expression. When you reach a point that you need to edit, use the delete key (ƒ) to trigger the insertion cursor and proceed with the edition of the expression.
  • Page 75: Editing Algebraic Expressions

    The expression tree The expression tree is a diagram showing how the Equation Writer interprets an expression. See Appendix E for a detailed example. The CURS function The CURS function (@CURS) in the Equation Writer menu (the B key) converts the display into a graphical display and produces a graphical cursor that can be controlled with the arrow keys (š™—˜) for selecting sub- expressions.
  • Page 76 • At an editing point, use the delete key (ƒ) to trigger the insertion cursor and proceed with the edition of the expression. To see the clear editing cursor in action, let’s start with the algebraic expression that we entered in the exercise above: Press the down arrow key, ˜, at its current location to trigger the clear editing cursor.
  • Page 77 Enters the factorial for the 3 in the square root ™ ~‚2 (entering the factorial changes the cursor to the selection cursor) Selects the µ in the exponential function ˜˜™™ /3*~‚f Modifies exponential function argument Selects ∆y ™™™™ Places a square root symbol on ∆y (this operation also changes the cursor to the selection cursor) ˜˜...
  • Page 78 This expression does not fit in the Equation Writer screen. We can see the entire expression by using a smaller-size font. Press the @BIG C soft menu key to get: Even with the larger-size font, it is possible to navigate through the entire expression by using the clear editing cursor.
  • Page 79 Factoring an expression In this exercise we will try factoring a polynomial expression. To continue the previous exercise, press the ` key. Then, launch the Equation Writer again by pressing the ‚O key. Type the equation: XQ2™+2*X*~y+~y Q2™- ~‚a Q2™™+~‚b Q2 resulting in Let’s select the first 3 terms in the expression and attempt a factoring of this sub-expression: ‚—˜‚™‚™...
  • Page 80 Press ‚¯to recover the original expression. Note: Pressing the @EVAL or the @SIMP soft menu keys, while the entire original expression is selected, produces the following simplification of the expression: Using the CMDS menu key With the original polynomial expression used in the previous exercise still selected, press the L key to show the @CMDS and @HELP soft menu keys.
  • Page 81 Next, press the L key to recover the original Equation Writer menu, and press the @EVAL@ soft menu key (D) to evaluate this derivative. The result is: Using the HELP menu Press the L key to show the @CMDS and @HELP soft menu keys. Press the @HELP soft menu key to get the list of CAS commands.
  • Page 82 2 / R3 ™™ * ~‚m + „¸\ ~‚m ™™ * ‚¹ ~„x + 2 * ~‚m * ~‚c ~„y ——— / ~‚t Q1/3 The original expression is the following: We want to remove the sub-expression x+2⋅λ⋅∆y from the argument of the LN function, and move it to the right of the λ...
  • Page 83: Creating And Editing Summations, Derivatives, And Integrals

    To select the sub-expression of interest, use: ™™™™™™™™‚¢ ™™™™™™™™™™‚¤ The screen shows the required sub-expression highlighted: We can now copy this expression and place it in the denominator of the LN argument, as follows:‚¨™™… (27 times) … ™ ƒƒ… (9 times) … ƒ ‚¬ The line editor now looks like this: Pressing ` shows the expression in the Equation Writer (in small-font format, press the @BIG C soft menu key):...
  • Page 84 Press ‚O to activate the Equation Writer. Then press ‚½to enter the summation sign. Notice that the sign, when entered into the Equation Writer screen, provides input locations for the index of the summation as well as for the quantity being summed. To fill these input locations, use the following keystrokes: ~„k™1™„è™1/~„kQ2 The resulting screen is:...
  • Page 85 Derivatives We will use the Equation Writer to enter the following derivative: α β δ Press ‚O to activate the Equation Writer. Then press ‚¿to enter the (partial) derivative sign. Notice that the sign, when entered into the Equation Writer screen, provides input locations for the expression being differentiated and the variable of differentiation.
  • Page 86 α β δ α β Second order derivatives are possible, for example: which evaluates to: ∂ Note: The notation is proper of partial derivatives. The proper x ∂ notation for total derivatives (i.e., derivatives of one variable) is . The calculator, however, does not distinguish between partial and total derivatives.
  • Page 87: Organizing Data In The Calculator

    This indicates that the general expression for a derivative in the line editor or in the stack is: ∫(lower_limit, upper_limit,integrand,variable_of_integration) Press ` to return to the Equation Writer. The resulting screen is not the definite integral we entered, however, but its symbolic value, namely, To recover the derivative expression use ‚¯.
  • Page 88: Functions For Manipulation Of Variables

    This screen gives a snapshot of the calculator’s memory and of the directory tree. The screen shows that the calculator has three memory ports (or memory partitions), port 0:IRAM, port 1:ERAM, and port 2:FLASH . Memory ports are used to store third party application or libraries, as well as for backups. size of the three different ports is also indicated.
  • Page 89: The Home Directory

    @RENAM To rename a variable @NEW To create a new variable @ORDER To order a set of variables in the directory @SEND To send a variable to another calculator or computer @RECV To receive a variable from another calculator or computer If you press the L key, the third set of functions is made available: @HALT To return to the stack temporarily...
  • Page 90: The Casdir Sub-directory

    subdirectories, in a hierarchy of directories similar to folders in modern computers. The subdirectories will be given names that may reflect the contents of each subdirectory, or any arbitrary name that you can think of. The CASDIR sub-directory The CASDIR sub-directory contains a number of variables needed by the proper operation of the CAS (Computer Algebraic System, see appendix C).
  • Page 91 GNAME means a global name, and REAL means a real (or floating-point) numeric variable. • The fourth and last column represents the size, in bytes, of the variable truncated, without decimals (i.e., nibbles). Thus, for example, variable PERIOD takes 12.5 bytes, while variable REALASSUME takes 27.5 bytes (1 byte = 8 bits, 1 bit is the smallest unit of memory in computers and calculators).
  • Page 92: Typing Directory And Variable Names

    variable, but one created by a previous exercise CASINFO a graph that provides CAS information MODULO Modulo for modular arithmetic (default = 13) REALASSUME List of variable names assumed as real values PERIOD Period for trigonometric functions (default = 2π) Name of default independent variable (default = X) Value of small increment (epsilon), (default = 10 These variables are used for the operation of the CAS.
  • Page 93: Creating Subdirectories

    ³~~math` ³~~m„a„t„h` ³~~m„~at„h` The calculator display will show the following (left-hand side is Algebraic mode, right-hand side is RPN mode): Note: if system flag 60 is set, you can lock the alphabetical keyboard by just pressing ~. See Chapter 1 for more information on system flags. Creating subdirectories Subdirectories can be created by using the FILES environment or by using the command CRDIR.
  • Page 94 showing that only one object exists currently in the HOME directory, namely, the CASDIR sub-directory. Let’s create another sub-directory called MANS (for MANualS) where we will store variables developed as exercises in this manual. To create this sub-directory first enter: L @@NEW@@ (C) . This will produce the following input form: The Object input field, the first input field in the form, is highlighted by default.
  • Page 95 Next, we will create a sub-directory named INTRO (for INTROduction), within MANS, to hold variables created as exercise in subsequent sections of this chapter. Press the $ key to return to normal calculator display (the TOOLS menu will be shown). Then, press J to show the HOME directory contents in the soft menu key labels.
  • Page 96 Use the down arrow key (˜) to select the option 2. MEMORY… , or just press 2. Then, press @@OK@@. This will produce the following pull-down menu: Use the down arrow key (˜) to select the 5. DIRECTORY option, or just press 5.
  • Page 97: Moving Among Subdirectories

    Command CRDIR in RPN mode To use the CRDIR in RPN mode you need to have the name of the directory already available in the stack before accessing the command. For example: ~~„~chap2~` Then access the CRDIR command by either of the means shown above, e.g., through the ‚N key: Press the @@OK@ soft menu key to activate the command, to create the sub- directory:...
  • Page 98 key to list the contents of the directory in the screen. Select the sub-directory (or variable) that you want to delete. Press L@PURGE. A screen similar to the following will be shown: The ‘S2’ string in this form is the name of the sub-directory that is being deleted.
  • Page 99 Use the down arrow key (˜) to select the option 2. MEMORY… Then, press @@OK@@. This will produce the following pull-down menu: Use the down arrow key (˜) to select the 5. DIRECTORY option. Then, press @@OK@@. This will produce the following pull-down menu: Use the down arrow key (˜) to select the 6.
  • Page 100 Press @@OK@@, to get: Then, press ) @ @S3@@ to enter ‘S3’ as the argument to PGDIR. Press ` to delete the sub-directory: Command PGDIR in RPN mode To use the PGDIR in RPN mode you need to have the name of the directory, between quotes, already available in the stack before accessing the command.
  • Page 101: Variables

    Using the PURGE command from the TOOL menu The TOOL menu is available by pressing the I key (Algebraic and RPN modes shown): The PURGE command is available by pressing the @PURGE soft menu key (E). In the following examples we want to delete sub-directory S1: •...
  • Page 102 sub-directory {HOME MANS INTRO}, created in an earlier example, we want to store the following variables with the values shown: Name Contents Type 12.5 real α -0.25 real 3×10 real ‘r/(m+r)' algebraic [3,2,1] vector 3+5i complex << → r 'π*r^2' >> program Using the FILES menu We will use the FILES menu to enter the variable A.
  • Page 103 To enter variable A (see table above), we first enter its contents, namely, the number 12.5, and then its name, A, as follows: 12.5 @@OK@@ ~a@@OK@@. Resulting in the following screen: Press @@OK@@ once more to create the variable. The new variable is shown in the following variable listing: The listing indicates a real variable ( ), whose name is A, and that occupies...
  • Page 104 Name Contents Type α -0.25 real 3×10 real ‘r/(m+r)' algebraic [3,2,1] vector 3+5i complex << → r 'π*r^2' >> program • Algebraic mode Use the following keystrokes to store the value of –0.25 into variable α: 0.25\ K ~‚a. AT this point, the screen will look as follows: This expression means that the value –0.25 is being stored into α...
  • Page 105 You will see six of the seven variables listed at the bottom of the screen: p1, z1, R, Q, A12, α. • RPN mode Use the following keystrokes to store the value of –0.25 into variable α: 0.25\` ~‚a`. At this point, the screen will look as follows: This expression means that the value –0.25 is ready to be stored into α.
  • Page 106: Checking Variable Contents

    Checking variables contents As an exercise on peeking into the contents of variables we will use the seven variables entered in the exercise above. We showed how to use the FILES menu to view the contents of a variable in an earlier exercise when we created the variable A.
  • Page 107 The « » symbols indicate a program in User RPL language (the original programming language of the HP 28/48 calculators, and available in the HP 49G series). The characters → r indicate that an input, to be referred to as r, is to be provided to the program.
  • Page 108: Replacing The Contents Of Variables

    Using the right-shift key ‚ followed by soft menu key labels This approach for viewing the contents of a variable works the same in both Algebraic and RPN modes. Try the following examples in either mode: J‚@@p1@@ ‚ @@z1@@ ‚ @@@R@@ ‚@@@Q@@ ‚ @@A12@@ This produces the following screen (Algebraic mode in the left, RPN in the right) Notice that this time the contents of program p1 are listed in the screen.
  • Page 109: Copying Variables

    Check the new contents of variable A12 by using ‚@@A12@@ . Using the RPN operating mode: ³~‚b/2` ³@@A12@@ ` K or, in a simplified way, ³~‚b/2™ ³@@A12@@ K Using the left-shift „ key followed by the variable’s soft menu key (RPN) This is a very simple way to change the contents of a variable, but it only works in the RPN mode.
  • Page 110 variables p1, z1, R, Q, A12, α, and A. Suppose that we want to copy variable A and place a copy in sub-directory {HOME MANS}. Also, we will copy variable R and place a copy in the HOME directory. Here is how to do it: Press „¡@@OK@@ to produce the following list of variables: Use the down-arrow key ˜...
  • Page 111 Using the history in Algebraic mode Here is a way to use the history (stack) to copy a variable from one directory to another with the calculator set to the Algebraic mode. Suppose that we are within the sub-directory {HOME MANS INTRO}, and want to copy the contents of variable z1 to sub-directory {HOME MANS}.
  • Page 112: Reordering Variables In A Directory

    ‚@@ @Q@@ K@@@Q@@ ` „§` ƒ ƒ ƒ` ƒ ƒ ƒ ƒ ` To verify the contents of the variables, use ‚@@ @R@ and ‚@@ @Q. This procedure can be generalized to the copying of three or more variables. Copying two or more variables using the stack in RPN mode The following is an exercise to demonstrate how to copy two or more variables using the stack when the calculator is in RPN mode.
  • Page 113: Moving Variables Using The Files Menu

    Next, we’ll list the new order of the variables by using their names typed between quotes: „ä ³) @ INTRO ™‚í³@@@@A@@@ ™‚í³@@@z1@@™‚í³@@@Q@@@™ ‚í³@@@@R@@@ ™‚í³@@A12@@ ` The screen now shows the new ordering of the variables: RPN mode In RPN mode, the list of re-ordered variables is listed in the stack before applying the command ORDER.
  • Page 114: Deleting Variables

    Notice that variable A12 is no longer there. If you now press „§, the screen will show the contents of sub-directory MANS, including variable A12: Note: You can use the stack to move a variable by combining copying with deleting a variable. Procedures for deleting variables are demonstrated in the next section.
  • Page 115 Using function PURGE in the stack in Algebraic mode We start again at subdirectory {HOME MANS INTRO} containing now only variables p1, z1, Q, R, and α. We will use command PURGE to delete variable p1. Press I @PURGE@ J@@p1@@ `. The screen will now show variable p1 removed: You can use the PURGE command to erase more than one variable by placing their names in a list in the argument of PURGE.
  • Page 116: Undo And Cmd Functions

    UNDO and CMD functions Functions UNDO and CMD are useful for recovering recent commands, or to revert an operation if a mistake was made. These functions are associated with the HIST key: UNDO results from the keystroke sequence ‚¯, while CMD results from the keystroke sequence „®.
  • Page 117: Flags

    Pressing „® produces the following selection box: As you can see, the numbers 3, 2, and 5, used in the first calculation above, are listed in the selection box, as well as the algebraic ‘SIN(5x2)’, but not the SIN function entered previous to the algebraic. Flags A flag is a Boolean value, that can be set or cleared (true or false), that specifies a given setting of the calculator or an option in a program.
  • Page 118: Example Of Flag Setting: General Solution Vs. Principal Value

    Example of flag setting: general solutions vs. principal value For example, the default value for system flag 01 is General solutions. What this means is that, if an equation has multiple solutions, all the solutions will be returned by the calculator, most likely in a list. By pressing the soft @ CHK@ menu key you can change system flag 01 to Principal value.
  • Page 119: Other Flags Of Interest

    ‚O~ „t Q2™+5*~ „t+6—— ‚Å0` ` (keeping a second copy in the RPN stack) ³~ „t` Use the following keystroke sequence to enter the QUAD command: ‚N~q (use the up and down arrow keys, —˜ , to select command QUAD) , press @@OK@@ . The screen shows the principal solution: Now, change the setting of flag 01 to General solutions: H@FLAGS@ @ CHK@ @@OK@@ @@OK@@ .
  • Page 120: Choose Boxes Vs. Soft Menus

    CHOOSE boxes vs. Soft MENU In some of the exercises presented in this chapter we have seen menu lists of commands displayed in the screen. This menu lists are referred to as CHOOSE boxes. For example, to use the ORDER command to reorder variables in a directory, we used: „°˜...
  • Page 121: Selected Choose Boxes

    Press twice to return to normal calculator display. Now, we’ll try to find the ORDER command using similar keystrokes to those used above, i.e., we start with „°. Notice that instead of a menu list, we get soft menu labels with the different options in the PROG menu, i.e., Press B to select the MEMORY soft menu () @ @MEM@@).
  • Page 122 • The HELP menu, activated with I L @HELP • The CMDS (CoMmanDS) menu, activated within the Equation Writer, i.e., ‚O L @CMDS Page 2-67...
  • Page 123: Checking Calculator Settings

    Chapter 3 Calculation with real numbers This chapter demonstrates the use of the calculator for operations and functions related to real numbers. Operations along these lines are useful for most common calculations in the physical sciences and engineering. The user should be acquainted with the keyboard to identify certain functions available in the keyboard (e.g., SIN, COS, TAN, etc.).
  • Page 124: Checking Calculator Mode

    The symbol ∠ 2. Coordinate system specification (XYZ, R∠Z, R∠∠). stands for an angular coordinate. XYZ: Cartesian or rectangular (x,y,z) R∠Z: cylindrical Polar coordinates (r,θ,z) R∠∠: Spherical coordinates (ρ,θ,φ) 3. Number base specification (HEX, DEC, OCT, BIN) HEX: hexadecimal numbers (base 16) DEC: decimal numbers (base 10) OCT: octal numbers (base 8) BIN: binary numbers (base 2)
  • Page 125: Changing Sing Of A Number, Variable, Or Expression

    specification. Real number calculations will be demonstrated in both the Algebraic (ALG) and Reverse Polish Notation (RPN) modes. Changing sign of a number, variable, or expression Use the \ key. In ALG mode, you can press \ before entering the number, e.g., \2.5`.
  • Page 126: Using Parentheses

    3.7#5.2 + 6.3#8.5 - 4.2#2.5 * 2.3#4.5 / Using parentheses Parentheses can be used to group operations, as well as to enclose arguments of functions. The parentheses are available through the keystroke combination „Ü. Parentheses are always entered in pairs. For example, to calculate (5+3.2)/(7-2.2): In ALG mode: „Ü5+3.2™/„Ü7-2.2`...
  • Page 127: Squares And Square Roots

    Squares and square roots The square function, SQ, is available through the keystroke combination: „º. When calculating in the stack in ALG mode, enter the function before the argument, e.g., „º\2.3` In RPN mode, enter the number first, then the function, e.g., 2.3\„º...
  • Page 128: Using Powers Of 10 In Entering Data

    Using powers of 10 in entering data Powers of ten, i.e., numbers of the form -4.5×10 , etc., are entered by using the V key. For example, in ALG mode: \4.5V\2` Or, in RPN mode: 4.5\V2\` Natural logarithms and exponential function Natural logarithms (i.e., logarithms of base e = 2.7182818282) are calculated by the keystroke combination ‚¹...
  • Page 129: Differences Between Functions And Operators

    these functions will be given in the selected angular measure (DEG, RAD, GRD). Some examples are shown next: In ALG mode: „¼0.25` „¾0.85` „À1.35` In RPN mode: 0.25`„¼ 0.85`„¾ 1.35`„À All the functions described above, namely, ABS, SQ, √, ^, XROOT, LOG, ALOG, LN, EXP, SIN, COS, TAN, ASIN, ACOS, ATAN, can be combined with the fundamental operations (+-*/) to form more complex expressions.
  • Page 130 for system flag 117 (see Chapter 2), the MTH menu is shown as the following menu list: As they are a great number of mathematic functions available in the calculator, the MTH menu is sorted by the type of object the functions apply on. For example, options 1.
  • Page 131: Hyperbolic Functions And Their Inverses

    2. To quickly select one of the numbered options in a menu list (or CHOOSE box), simply press the number for the option in the keyboard. For example, to select option 4. HYPERBOLIC.. in the MTH menu, simply press 4. Hyperbolic functions and their inverses Selecting Option 4.
  • Page 132 4 @@OK@@ Select the 4. HYPERBOLIC.. menu 5 @@OK@@ Select the 5. TANH function The result is: The operations shown above assume that you are using the default setting for system flag 117 (CHOOSE boxes). If you have changed the setting of this flag (see Chapter 2) to SOFT menu, the MTH menu will show as labels of the soft menu keys, as follows (left-hand side in ALG mode, right –hand side in RPN mode):...
  • Page 133: Real Number Functions

    Note: To see additional options in these soft menus, press the L key or the „«keystroke sequence. For example, to calculate tanh(2.5), in the ALG mode, when using SOFT menus over CHOOSE boxes, follow this procedure: „´ Select MTH menu ) @ @HYP@ Select the HYPERBOLIC..
  • Page 134 Option 19. MATH.. returns the user to the MTH menu. The remaining functions are grouped into six different groups described below. If system flag 117 is set to SOFT menus, the REAL functions menu will look like this (ALG mode used, the same soft menu keys will be available in RPN mode): The very last option, ) @ @MTH@, returns the user to the MTH menu.
  • Page 135 The result is shown next: In RPN mode, recall that argument y is located in the second level of the stack, while argument x is located in the first level of the stack. This means, you should enter x first, and then, y, just as in ALG mode. Thus, the calculation of %T(15,45), in RPN mode, and with system flag 117 set to CHOOSE boxes, we proceed as follows: Enter first argument...
  • Page 136: Special Functions

    As an exercise, verify that 15 MOD 4 = 15 mod 4 = residual of 15/4 = 3 Absolute value, sign, mantissa, exponent, integer and fractional parts ABS(x) : calculates the absolute value, |x| SIGN(x) : determines the sign of x, i.e., -1, 0, or 1. MANT(x): determines the mantissa of a number based on log XPON(x): determines the power of 10 in the number IP(x)
  • Page 137 The Gamma function Γ(α) GAMMA: PSI: N-th derivative of the digamma function Psi: Digamma function, derivative of the ln(Gamma) ∞ ∫ α − − α The Gamma function is defined by . This function has applications in applied mathematics for science and engineering, as well as in probability and statistics.
  • Page 138: Calculator Constants

    Calculator constants The following are the mathematical constants used by your calculator: • the base of natural logarithms. • the imaginary unit, i = -1. • π: the ratio of the length of the circle to its diameter. • MINR: the minimum real number available to the calculator. •...
  • Page 139: Operation With Units

    Operations with units Numbers in the calculator can have units associated with them. Thus, it is possible to calculate results involving a consistent system of units and produce a result with the appropriate combination of units. The UNITS menu The units menu is launched by the keystroke combination ‚Û(associated with the 6 key).
  • Page 140: Available Units

    For extensive operations with units SOFT menus provide a more convenient way of attaching units. Change system flag 117 to SOFT menus (see Chapter 1), and use the keystroke combination ‚Û to get the following menus. Press L to move to the next menu page. Pressing on the appropriate soft menu key will open the sub-menu of units for that particular selection.
  • Page 141 chain (chain), rd (rod), fath (fathom), ftUS (survey foot), Mil (Mil), µ (micron), Å (Angstrom), fermi (fermi) AREA m^2 (square meter), cm^2 (square centimeter), b (barn), yd^2 (square yard), ft^2 (square feet), in^2 (square inch), km^2 (square kilometer), ha (hectare), a (are), mi^2 (square mile), miUS^2 (square statute mile), acre (acre) VOLUME m^3 (cubic meter), st (stere), cm^3 (cubic centimeter), yd^3 (cubic yard), ft^3...
  • Page 142 POWER W (watt), hp (horse power), PRESSURE Pa (pascal), atm (atmosphere), bar (bar), psi (pounds per square inch), torr (torr), mmHg (millimeters of mercury), inHg (inches of mercury), inH20 (inches of water), TEMPERATURE C (degree Celsius), F (degree Fahrenheit), K (Kelvin),...
  • Page 143: Converting To Base Units

    MENU(117.02) in ALG mode, or 117.02 ` MENU in RPN mode. The menu will show in the screen as follows (use ‚˜to show labels in display): These units are also accessible through the catalog, for example: gmol: ‚N~„g lbmol: ‚N~„l rpm: ‚N~„r ‚N~„d...
  • Page 144: Attaching Units To Numbers

    In RPN mode, system flag 117 set to CHOOSE boxes: Enter 1 (no underline) Select the UNITS menu ‚Û — @@OK@@ Select the VISCOSITY option @@OK@@ Select the unit P (poise) Select the UNITS menu ‚Û @@OK@@ Select the TOOLS menu ˜...
  • Page 145 Select units of force (8. Force..) 8@@OK@@ @@OK@@ Select Newtons (N) Enter quantity with units in the stack The screen will look like the following: Note: If you forget the underscore, the result is the expression 5*N, where N here represents a possible variable name and not Newtons. To enter this same quantity, with the calculator in RPN mode, use the following keystrokes: Enter number (do not enter underscore)
  • Page 146 The same quantity, entered in RPN mode uses the following keystrokes: Enter number (no underscore) Access the UNITS menu ‚Û L @) @ FORCE Select units of force @ @@N@@ Select Newtons (N) Note: You can enter a quantity with units by typing the underline and units with the ~keyboard, e.g., 5‚Ý~n will produce the entry: 5_N Unit prefixes You can enter prefixes for units according to the following table of prefixes...
  • Page 147: Operations With Units

    Using UBASE to convert to the default unit (1 m) results in: Operations with units Once a quantity accompanied with units is entered into the stack, it can be used in operations similar to plain numbers, except that quantities with units cannot be used as arguments of functions (say, SQ or SIN).
  • Page 148 which transformed to SI units, with function UBASE, produces: Addition and subtraction can be performed, in ALG mode, without using parentheses, e.g., 5 m + 3200 mm, can be entered simply as 5_m + 3200_mm `: More complicated expression require the use of parentheses, e.g., (12_mm)*(1_cm^2)/(2_s) `: Stack calculations in the RPN mode, do not require you to enclose the different terms in parentheses, e.g.,...
  • Page 149: Units Manipulation Tools

    5_m ` 3200_mm ` + 12_mm ` 1_cm^2 `* 2_s ` / These last two operations produce the following output: Note: Units are not allowed in expressions entered in the equation writer. Units manipulation tools The UNITS menu contains a TOOLS sub-menu, which provides the following functions: CONVERT(x,y): convert unit object x to units of object y UBASE(x):...
  • Page 150: Physical Constants In The Calculator

    Examples of UVAL: UVAL(25_ft/s) ` UVAL(0.021_cm^3) ` Examples of UFACT UFACT(1_ha,18_km^2) ` UFACT(1_mm,15.1_cm) ` Examples of UNIT UNIT(25,1_m) ` UNIT(11.3,1_mph) ` Physical constants in the calculator Following along the treatment of units, we discuss the use of physical constants that are available in the calculator’s memory. These physical constants are contained in a constants library activated with the command CONLIB.
  • Page 151 ~~conlib~` or, you can select the command CONLIB from the command catalog, as follows: First, launch the catalog by using: ‚N~c. Next, use the up and down arrow keys —˜ to select CONLIB. Finally, press the F(@@OK@@) soft menu key. Press `, if needed. The constants library screen will look like the following (use the down arrow key to navigate through the library): The soft menu keys corresponding to this CONSTANTS LIBRARY screen...
  • Page 152 VALUE when selected, constants are shown without units STK copies value (with or without units) to the stack QUIT exit constants library (*) Active only if the function VALUE is active. This is the way the top of the CONSTANTS LIBRARY screen looks when the option VALUE is selected (units in the SI system): To see the values of the constants in the English (or Imperial) system, press the @ENGL option:...
  • Page 153: Special Physical Functions

    The display shows what is called a tagged value, Vm:359.0394. In here, Vm, is the tag of this result. Any arithmetic operation with this number will ignore the tag. Try, for example: ‚¹2*„î`, which produces: The same operation in RPN mode will require the following keystrokes (after the value of Vm was extracted from the constants library): 2`*‚...
  • Page 154: Function Zfactor

    TINC: temperature increment command Out of all the functions available in this MENU (UTILITY menu), namely, ZFACTOR, FANNING, DARCY, F0λ, SIDENS, TDELTA, and TINC, functions FANNING and DARCY are described in Chapter 6 in the context of solving equations for pipeline flow. The remaining functions are described following. Function ZFACTOR Function ZFACTOR calculates the gas compressibility correction factor for nonideal behavior of hydrocarbon gas.
  • Page 155: Function Tdelta

    Function TDELTA Function TDELTA(T ) yields the temperature increment T – T . The result is returned with the same units as T , if any. Otherwise, it returns simply the difference in numbers. For example, The purpose of this function is to facilitate the calculation of temperature differences given temperatures in different units.
  • Page 156 << x ‘LN(x+1) + EXP(x)’ >> This is a simple program in the default programming language of the HP 48 G series, and also incorporated in the HP 49 G series. This programming language is called UserRPL. The program shown above is relatively simple and consists of two parts, contained between the program containers <<...
  • Page 157: Functions Defined By More Than One Expression

    In this section we discuss the treatment of functions that are defined by two or more expressions. An example of such functions would be The HP 49 G provides the function IFTE (IF-Then-Else) to describe such functions. The IFTE function...
  • Page 158: Combined Ifte Functions

    (associated with the Y key). To define this function in ALG mode use the command: DEF(f(x) = IFTE(x>0, x^2-1, 2*x-1)) then, press `. In RPN mode, type the function definition between apostrophes: ‘f(x) = IFTE(x>0, x^2-1, 2*x-1)’ then press „à. The function @@@f@@@ should be Press J to recover your variable menu.
  • Page 159: Chapter 4 - Calculations With Complex Numbers

    Chapter 4 Calculations with complex numbers This chapter shows examples of calculations and application of functions to complex numbers. Definitions A complex number z is a number written as z = x + iy, where x and y are real numbers, and i is the imaginary unit defined by i = -1.
  • Page 160: Entering Complex Numbers

    Press @@OK@@ , twice, to return to the stack. Entering complex numbers Complex numbers in the calculator can be entered in either of the two Cartesian representations, namely, x+iy, or (x,y). The results in the calculator will be shown in the ordered-pair format, i.e., (x,y). For example, with the calculator in ALG mode, the complex number (3.5,-1.2), is entered as: „Ü3.5‚í\1.2` A complex number can also be entered in the form x+iy.
  • Page 161: Polar Representation Of A Complex Number

    Polar representation of a complex number The result shown above represents a Cartesian (rectangular) representation of the complex number 3.5-1.2i. A polar representation is possible if we change the coordinate system to cylindrical or polar, by using function CYLIN. You can find this function in the catalog (‚N). Changing to polar shows the result: For this result the angular measure is set to radians (you can always change to radians by using function RAD).
  • Page 162: Simple Operations With Complex Numbers

    Simple operations with complex numbers Complex numbers can be combined using the four fundamental operations (+-*/). The results follow the rules of algebra with the caveat that = -1. Operations with complex numbers are similar to those with real numbers. For example, with the calculator in ALG mode and the CAS set to Complex, we’ll attempt the following sum: (3+5i) + (6-3i): Notice that the real parts (3+6) and imaginary parts (5-3) are combined together and the result given as an ordered pair with real part 9 and...
  • Page 163: Entering The Unit Imaginary Number

    Entering the unit imaginary number To enter the unit imaginary number type : „¥ Notice that the number i is entered as the ordered pair (0,1) if the CAS is set to APPROX mode. In EXACT mode, the unit imaginary number is entered as i. Other operations Operations such as magnitude, argument, real and imaginary parts, and complex conjugate are available through the CMPLX menus detailed later.
  • Page 164 RE(z) : Real part of a complex number IM(z) : Imaginary part of a complex number →R(z) : Takes a complex number (x,y) and separates it into its real and imaginary parts →C(x,y): Forms the complex number (x,y) out of real numbers x and y ABS(z) : Calculates the magnitude of a complex number or the absolute value of a real number.
  • Page 165: Cmplx Menu In Keyboard

    Also, the result of function ARG, which represents an angle, will be given in the units of angle measure currently selected. In this example, ARG(3.+5. i) = 1.0303… is given in radians. In the next screen we present examples of functions SIGN, NEG (which shows up as the negative sign - ), and CONJ.
  • Page 166: Functions Applied To Complex Numbers

    Functions applied to complex numbers Many of the keyboard-based functions defined in Chapter 3 for real numbers, e.g., SQ, ,LN, e , LOG, 10 , SIN, COS, TAN, ASIN, ACOS, ATAN, can be applied to complex numbers. The result is another complex number, as illustrated in the following examples.
  • Page 167: Function Droite: Equation Of A Straight Line

    The following screen shows that functions EXPM and LNP1 do not apply to complex numbers. However, functions GAMMA, PSI, and Psi accept complex numbers: Function DROITE: equation of a straight line Function DROITE takes as argument two complex numbers, say, x , and returns the equation of the straight line, say, y = a+bx, that contains the points (x ) and (x...
  • Page 168: Chapter 5 - Algebraic And Arithmetic Operations

    Chapter 5 Algebraic and arithmetic operations An algebraic object, or simply, algebraic, is any number, variable name or algebraic expression that can be operated upon, manipulated, and combined according to the rules of algebra. Examples of algebraic objects are the following: •...
  • Page 169: Simple Operations With Algebraic Objects

    Simple operations with algebraic objects Algebraic objects can be added, subtracted, multiplied, divided (except by zero), raised to a power, used as arguments for a variety of standard functions (exponential, logarithmic, trigonometry, hyperbolic, etc.), as you would any real or complex number. To demonstrate basic operations with algebraic objects, let’s create a couple of objects, say ‘π*R^2’...
  • Page 170: Functions In The Alg Menu

    @@A1@@ * @@A2@@ ` @@A1@@ / @@A2@@ ` ‚¹@@A1@@ „¸@@A2@@ The same results are obtained in RPN mode if using the following keystrokes: @@A1@@ ` @@A2@@ + @@A1@@ `@@A2@@ - @@A1@@ ` @@A2@@ * @@A1@@ `@@A2@@ / @@A1@@ ` ‚¹ @@A2@@ `„¸...
  • Page 171 We notice that, at the bottom of the screen, the line See: EXPAND FACTOR suggests links to other help facility entries, the functions EXPAND and FACTOR. To move directly to those entries, press the soft menu key @SEE1! for EXPAND, and @SEE2! for FACTOR. Pressing @SEE1!, for example, shows the following information for EXPAND: The help facility provides not only information on each command, but also provides an example of its application.
  • Page 172: Factor,

    The help facility will show the following information on the commands: COLLECT: EXPAND: FACTOR: LNCOLLECT: LIN: PARTFRAC: SOLVE: SUBST: TEXPAND: Page 5-5...
  • Page 173: Other Forms Of Substitution In Algebraic Expressions

    Note: Recall that, to use these, or any other functions in the RPN mode, you must enter the argument first, and then the function. For example, the example for TEXPAND, in RPN mode will be set up as: ³„¸+~x+~y` At this point, select function TEXPAND from menu ALG (or directly from the catalog ‚N), to complete the operation.
  • Page 174: Operations With Transcendental Functions

    In ALG mode, substitution of more than one variable is possible as illustrated in the following example (shown before and after pressing `) In RPN mode, it is also possible to substitute more than one variable at a time, as illustrated in the example below. Recall that RPN mode uses a list of variable names and values for substitution.
  • Page 175: Expansion And Factoring Using Log-exp Functions

    hyperbolic functions in terms of trigonometric identities or in terms of exponential functions. The menus containing functions to replace trigonometric functions can be obtained directly from the keyboard by pressing the right-shift key followed by the 8 key, i.e., ‚Ñ. The combination of this key with the left-shift key, i.e., ‚...
  • Page 176: Functions In The Arithmetic Menu

    These functions allow to simplify expressions by replacing some category of trigonometric functions for another one. For example, the function ACOS2S allows to replace the function arccosine (acos(x)) with its expression in terms of arcsine (asin(x)). Description of these commands and examples of their applications are available in the calculator’s help facility (IL@HELP).
  • Page 177: Divis

    of functions that apply to specific mathematical objects. This distinction between sub-menus (options 1 through 4) and plain functions (options 5 through 9) is made clear when system flag 117 is set to SOFT menus. Activating the ARITHMETIC menu („Þ ), under these circumstances, produces: Following, we present the help facility entries for the functions of options 5 through 9 in the ARITHMETIC menu:...
  • Page 178: Polynomial Menu

    IABCUV Solves au + bv = c, with a,b,c = integers IBERNOULLI n-th Bernoulli number ICHINREM Chinese reminder for integers IDIV2 Euclidean division of two integers IEGCD Returns u,v, such that au + bv = gcd(a,b) IQUOT Euclidean quotient of two integers IREMAINDER Euclidean remainder of two integers ISPRIME? Test if an integer number is prime...
  • Page 179: Modulo Menu

    MODULO menu ADDTMOD Add two expressions modulo current modulus DIVMOD Divides 2 polynomials modulo current modulus DIV2MOD Euclidean division of 2 polynomials with modular coefficients EXPANDMOD Expands/simplify polynomial modulo current modulus FACTORMOD Factorize a polynomial modulo current modulus GCDMOD GCD of 2 polynomials modulo current modulus INVMOD inverse of integer modulo current modulus (not entry available in the help facility)
  • Page 180 Operations in modular arithmetic Addition in modular arithmetic of modulus n, which is a positive integer, follow the rules that if j and k are any two nonnegative integer numbers, both smaller than n, if j+k≥ n, then j+k is defined as j+k-n. For example, in the case of the clock, i.e., for n = 12, 6+9 “=”...
  • Page 181 6 does not show the result 5 in modulus 12 arithmetic. This multiplication table is shown below: 6*0 (mod 12) 6*6 (mod 12) 6*1 (mod 12) 6*7 (mod 12) 6*2 (mod 12) 6*8 (mod 12) 6*3 (mod 12) 6*9 (mod 12) 6*4 (mod 12) 6*10 (mod 12) 6*5 (mod 12)
  • Page 182: Finite Arithmetic In The Calculator

    Finite arithmetic rings in the calculator All along we have defined our finite arithmetic operation so that the results are always positive. The modular arithmetic system in the calculator is set so that the ring of modulus n includes the numbers -n/2+1, …,-1, 0, 1,…,n/2-1, n/2, if n is even, and –(n-1)/2, -(n-3)/2,…,-1,0,1,…,(n-3)/2, (n-1)/2, if n is odd.
  • Page 183 ADDTMOD examples 6+5 ≡ -1 (mod 12) 6+6 ≡ 0 (mod 12) 6+7 ≡ 1 (mod 12) 11+5 ≡ 4 (mod 12) 8+10 ≡ -6 (mod 12) SUBTMOD examples 5 - 7 ≡ -2 (mod 12) 8 – 4 ≡ 4 (mod 12) 5 –10 ≡...
  • Page 184 before operating on them. You can also convert any number into a ring number by using the function EXPANDMOD. For example, EXPANDMOD(125) ≡ 5 (mod 12) EXPANDMOD(17) ≡ 5 (mod 12) EXPANDMOD(6) ≡ 6 (mod 12) The modular inverse of a number Let a number k belong to a finite arithmetic ring of modulus n, then the modular inverse of k, i.e., 1/k (mod n), is a number j, such that j⋅k ≡...
  • Page 185: Polynomials

    Note: Refer to the help facility in the calculator for description and examples on other modular arithmetic. Many of these functions are applicable to polynomials. For information on modular arithmetic with polynomials please refer to a textbook on number theory. Polynomials Polynomials are algebraic expressions consisting of one or more terms containing decreasing powers of a given variable.
  • Page 186: The Chinrem Function

    The CHINREM function CHINREM stands for CHINese REMainder. The operation coded in this command solves a system of two congruences using the Chinese Remainder Theorem. This command can be used with polynomials, as well as with integer numbers (function ICHINREM). The input consists of two vectors [expression_1, modulo_1] and [expression_2, modulo_2].
  • Page 187: The Hermite Function

    The HERMITE function The function HERMITE [HERMI] uses as argument an integer number, k, and returns the Hermite polynomial of k-th degree. A Hermite polynomial, He is defined as − − ,... An alternate definition of the Hermite polynomials is −...
  • Page 188: The Lagrange Function

    The LAGRANGE function The function LAGRANGE requires as input a matrix having two rows and n columns. The matrix stores data points of the form [[x , …, x ] [y , …, ]]. Application of the function LAGRANGE produces the polynomial expanded from ∏...
  • Page 189: The Legendre Function

    The LEGENDRE function A Legendre polynomial of order n is a polynomial function that solves the differential equation To obtain the n-th order Legendre polynomial, use LEGENDRE(n), e.g., LEGENDRE(3) = ‘(5*X^3-3*X)/2’ LEGENDRE(5) = ‘(63*X ^5-70*X^3+15*X)/8’ The PCOEF function Given an array containing the roots of a polynomial, the function PCOEF generates an array containing the coefficients of the corresponding polynomial.
  • Page 190: The Epsx0 Function And The Cas Variable Eps

    The QUOT and REMAINDER functions The functions QUOT and REMAINDER provide, respectively, the quotient Q(X) and the remainder R(X), resulting from dividing two polynomials, P (X) and (X). In other words, they provide the values of Q(X) and R(X) from (X)/P (X) = Q(X) + R(X)/P (X).
  • Page 191: The Tchebycheff Function

    The TCHEBYCHEFF function The function TCHEBYCHEFF(n) generates the Tchebycheff (or Chebyshev) polynomial of the first kind, order n, defined as T (X) = cos(n⋅arccos(X)). If the integer n is negative (n < 0), the function TCHEBYCHEFF(n) generates the (X) = Tchebycheff polynomial of the second kind, order n, defined as sin(n⋅arccos(X))/sin(arccos(X)).
  • Page 192: The Partfrac Function

    PROPFRAC(‘5/4’) = ‘1+1/4’ PROPFRAC(‘(x^2+1)/x^2’) = ‘1+1/x^2’ The PARTFRAC function The function PARTFRAC decomposes a rational fraction into the partial fractions that produce the original fraction. For example: PARTFRAC(‘(2*X^6-14*X^5+29*X^4-37*X^3+41*X^2-16*X+5)/(X^5- 7*X^4+11*X^3-7*X^2+10*X)’) = ‘2*X+(1/2/(X-2)+5/(X-5)+1/2/X+X/(X^2+1))’ This technique is useful in calculating integrals (see chapter on calculus) of rational fractions.
  • Page 193: The Froots Function

    If you press µ you will get: ‘(X^6+8*X^5+5*X^4-50*X^3)/(X^7+13*X^6+61*X^5+105*X^4-45*X^3- 297*X^2-81*X+243)’ The FROOTS function The function FROOTS obtains the roots and poles of a fraction. As an example, applying function FROOTS to the result produced above, will result in: [1 –2. –3 –5. 0 3. 2 1. –5 2.]. The result shows poles followed by their multiplicity as a negative number, and roots followed by their multiplicity as a positive number.
  • Page 194: The Convert Menu And Algebraic Operations

    The CONVERT Menu and algebraic operations The CONVERT menu is activated by using „Ú key (the 6 key). This menu summarizes all conversion menus in the calculator. The list of these menus is shown next: The functions available in each of the sub-menus are shown next. UNITS convert menu (Option 1) This menu is the same as the UNITS menu obtained by using ‚Û.
  • Page 195: Base Convert Menu

    BASE convert menu (Option 2) This menu is the same as the UNITS menu obtained by using ‚ã. The applications of this menu are discussed in detail in Chapter 19. TRIGONOMETRIC convert menu (Option 3) This menu is the same as the TRIG menu obtained by using ‚Ñ. The applications of this menu are discussed in detail in this Chapter.
  • Page 196 NUM has the same effect as the keystroke combination ‚ï Function (associated with the ` key). Function NUM converts a symbolic result into its floating-point value. Function Q converts a floating-point value into Qπ converts a floating-point value into a fraction of π, a fraction.
  • Page 197 LNCOLLECT POWEREXPAND SIMPLIFY Page 5-30...
  • Page 198: Chapter 6 - Solution To Single Equations

    Chapter 6 Solution to single equations In this chapter we feature those functions that the calculator provides for solving single equations of the form f(X) = 0. Associated with the 7 key there are two menus of equation-solving functions, the Symbolic SOLVer („Î), and the NUMerical SoLVer (‚Ï).
  • Page 199: Function Solve

    Using the RPN mode, the solution is accomplished by entering the equation in the stack, followed by the variable, before entering function ISOL. Right before the execution of ISOL, the RPN stack should look as in the figure to the left.
  • Page 200 The following examples show the use of function SOLVE in ALG and RPN modes: The screen shot shown above displays two solutions. In the first one, β -5β =125, SOLVE produces no solutions { }. In the second one, β - 5β...
  • Page 201: Function Solvevx

    Function SOLVEVX The function SOLVEVX solves an equation for the default CAS variable contained in the reserved variable name VX. By default, this variable is set to ‘X’. Examples, using the ALG mode with VX = ‘X’, are shown below: In the first case SOLVEVX could not find a solution.
  • Page 202: Numerical Solver Menu

    To use function ZEROS in RPN mode, enter first the polynomial expression, then the variable to solve for, and then function ZEROS. The following screen shots show the RPN stack before and after the application of ZEROS to the two examples above: The Symbolic Solver functions presented above produce solutions to rational equations (mainly, polynomial equations).
  • Page 203: Polynomial Equations

    Notes: 1. Whenever you solve for a value in the NUM.SLV applications, the value solved for will be placed in the stack. This is useful if you need to keep that value available for other operations. 2. There will be one or more variables created whenever you activate some of the applications in the NUM.SLV menu.
  • Page 204 Press ` to return to stack. The stack will show the following results in ALG mode (the same result would be shown in RPN mode): To see all the solutions, press the down-arrow key (˜) to trigger the line editor: All the solutions are complex numbers: (0.432,-0.389), (0.432,0.389), (- 0.766, 0.632), (-0.766, -0.632).
  • Page 205 Press ` to return to stack, the coefficients will be shown in the stack. Press ˜ to trigger the line editor to see all the coefficients. Note: If you want to get a polynomial with real coefficients, but having complex roots, you must include the complex roots in pairs of conjugate numbers.
  • Page 206: Financial Calculations

    To generate the algebraic expression using the roots, try the following example. Assume that the polynomial roots are [1,3,-2,1]. Use the following keystrokes: ‚Ϙ˜@@OK@@ Select Solve poly… ˜„Ô1‚í3 Enter vector of roots ‚í2\‚í 1@@OK@@ ˜@SYMB@ Generate symbolic expression Return to stack. The expression thus generated is shown in the stack as:' (X-1)*(X-3)*(X+2)*(X-1) To expand the products, you can use the EXPAND command.
  • Page 207 Definitions Often, to develop projects, it is necessary to borrow money from a financial institution or from public funds. The amount of money borrowed is referred to as the Present Value (PV). This money is to be repaid through n periods (typically multiples or sub-multiples of a month) subject to an annual interest rate of I%YR.
  • Page 208 The screen now shows the value of PMT as –39,132.30, i.e., the borrower must pay the lender US $ 39,132.30 at the end of each month for the next 60 months to repay the entire amount. The reason why the value of PMT turned out to be negative is because the calculator is looking at the money amounts from the point of view of the borrower.
  • Page 209 This means that at the end of 60 months the US $ 2,000,000.00 principal amount has been paid, together with US $ 347,937.79 of interest, with the balance being that the lender owes the borrower US $ 0.000316. Of course, the balance should be zero. The value shown in the screen above is simply round-off error resulting from the numerical solution.
  • Page 210 2. The values calculated in the financial calculator environment are copied to the stack with their corresponding tag (identifying label). Deleting the variables When you use the financial calculator environment for the first time within the HOME directory, or any sub-directory, it will generate the variables @@@N@@ @I©YR@ @@PV@@ @@PMT@@ @@PYR@@ @@FV@@ to store the corresponding terms in the calculations..
  • Page 211: Solving Equations With One Unknown Through Num.slv

    J „ä Prepare a list of variables to be purged @@@n@@ Enter name of variable N @I©YR@ Enter name of variable I%YR @@PV@@ Enter name of variable PV @@PMT@@ Enter name of variable PMT @@PYR@@ Enter name of variable PYR @@FV@@ Enter name of variable FV Enter list of variables in stack...
  • Page 212 Press J to see the newly created EQ variable: Then, enter the SOLVE environment and select Solve equation…, by using: ‚Ï@@OK@@. The corresponding screen will be shown as: The equation we stored in variable EQ is already loaded in the Eq field in the SOLVE EQUATION input form.
  • Page 213 • The user then highlights the field corresponding to the unknown for which to solve the equation, and presses @SOLVE@ • The user may force a solution by providing an initial guess for the solution in the appropriate input field before solving the equation. The calculator uses a search algorithm to pinpoint an interval for which the function changes sign, which indicates the existence of a root or solution.
  • Page 214 At this point follow the instructions from Chapter 2 on how to use the Equation Writer to build an equation. The equation to enter in the Eq field should look like this (notice that we use only one sub-index to refer to the variables, i.e., is translated as ex, etc.
  • Page 215 The solution can be seen from within the SOLVE EQUATION input form by pressing @EDIT while the ex: field is highlighted. The resulting value is 2.470833333333E-3. Press @@@OK@@ to exit the EDIT feature. Suppose that you now, want to determine the Young’s modulus that will produce a strain of e = 0.005 under the same state of stress, neglecting thermal expansion.
  • Page 216 We can type in the equation for E as shown above and use auxiliary variables for A and V, so that the resulting input form will have fields for the fundamental variables y, Q, g, m, and b, as follows: •...
  • Page 217 The result is 0.149836.., i.e., y = 0.149836. • It is known, however, that there are actually two solutions available for y in the specific energy equation. The solution we just found corresponds to a numerical solution with an initial value of 0 (the default value for y, i.e., whenever the solution field is empty, the initial value is zero).
  • Page 218 . The quantity f is known as the friction factor of written as the flow and it has been found to be a function of the relative roughness of the pipe, ε/D, and a (dimensionless) Reynolds number, Re. The Reynolds number is defined as Re = ρVD/µ...
  • Page 219 Example 3 – Flow in a pipe You may want to create a separate sub-directory (PIPES) to try this example. The main equation governing flow in a pipe is, of course, the Darcy-Weisbach equation. Thus, type in the following equation into EQ: Also, enter the following variables (f, A, V, Re): In this case we stored the main equation (Darcy-Weisbach equation) into EQ, and then replaced several of its variables by other expressions through the...
  • Page 220 Thus, the equation we are solving, after combining the different variables in the directory, is: ε π DARCY π The combined equation has primitive variables: h , Q, L, g, D, ε, and Nu. Launch the numerical solver (‚Ï@@OK@@) to see the primitive variables listed in the SOLVE EQUATION input form: Suppose that we use the values hf = 2 m, ε...
  • Page 221 Example 4 – Universal gravitation Newton’s law of universal gravitation indicates that the magnitude of the attractive force between two bodies of masses m and m separated by a distance r is given by the equation Here, G is the universal gravitational constant, whose value can be obtained through the use of the function CONST in the calculator by using: We can solve for any term in the equation (except G) by entering the equation as:...
  • Page 222 Solve for F, and press to return to normal calculator display. The solution is F : 6.67259E-15_N, or F = 6.67259×10 Note: When using units in the numerical solver make sure that all the variables have the proper units, that the units are compatible, and that the equation is dimensionally homogeneous.
  • Page 223 At this point the equation is ready for solution. Alternatively, you can activate the equation writer after pressing @EDIT to enter your equation. Press ` to return to the numerical solver screen. Another way to enter an equation into the EQ variable is to select a variable already existing in your directory to be entered into EQ.
  • Page 224: The Solve Soft Menu

    The SOLVE soft menu The SOLVE soft menu allows access to some of the numerical solver functions through the soft menu keys. To access this menu use in RPN mode: 74 MENU, or in ALG mode: MENU(74). Alternatively, you can use ‚(hold) 7 to activate the SOLVE soft menu.
  • Page 225: The Solvr Sub-menu

    The SOLVR sub-menu The SOLVR sub-menu activates the soft-menu solver for the equation currently stored in EQ. Some examples are shown next: Example 1 - Solving the equation t -5t = -4 For example, if you store the equation ‘t^2-5*t=-4’ into EQ, and press @) S OLVR, it will activate the following menu: This result indicates that you can solve for a value of t for the equation listed at the top of the display.
  • Page 226 As variables Q, a, and b, get assigned numerical values, the assignments are listed in the upper left corner of the display. At this point we can solve for t, by using „[ t ]. The result is t: 2. Pressing @EXPR= shows the results: Example 3 - Solving two simultaneous equations, one at a time You can also solve more than one equation by solving one equation at a time, and repeating the process until a solution is found.
  • Page 227: The Diffe Sub-menu

    After solving the two equations, one at a time, we notice that, up to the third decimal, X is converging to a value of 7.500, while Y is converging to a value o 0.799. Using units with the SOLVR sub-menu These are some rules on the use of units with the SOLVR sub-menu: •...
  • Page 228: The Sys Sub-menu

    Function PROOT This function is used to find the roots of a polynomial given a vector containing the polynomial coefficients in decreasing order of the powers of the independent variable. In other words, if the polynomial is a + … + a x + a , the vector of coefficients should be entered as [a , …...
  • Page 229 The SOLVR sub-menu The SOLVR sub-menu in the TVM sub-menu will launch the solver for solving TVM problems. For example, pressing @) S OLVR, at this point, will trigger the following screen: As an exercise, try using the values n = 10, I%YR = 5.6, PV = 10000, and FV = 0, and enter „[ PMT ] to find PMT = -1021.08….
  • Page 230 Function BEG If selected, the TMV calculations use payments at the beginning of each period. If deselected, the TMV calculations use payments at the end of each period. Page 6-33...
  • Page 231: Chapter 7 - Solving Multiple Equations

    Chapter 7 Solving multiple equations Many problems of science and engineering require the simultaneous solutions of more than one equation. The calculator provides several procedures for solving multiple equations as presented below. Please notice that no discussion of solving systems of linear equations is presented in this chapter. Linear systems solutions will be discussed in detail in subsequent chapters on matrices and linear algebra.
  • Page 232: Example 2 - Stresses In A Thick Wall Cylinder

    At this point, we need only press K twice to store these variables. Exact To solve, first change CAS mode to , then, list the contents of A2 and A1, in that order: @@@A2@@@ @@@A1@@@ . Use command SOLVE at this point (from the S.SLV menu: „Î) After about 40 seconds, maybe more, you get as result a list: { ‘t = (x-x0)/(COS(θ0)*v0)’...
  • Page 233 Notice that the right-hand sides of the two equations differ only in the sign between the two terms. Therefore, to write these equations in the calculator, I suggest you type the first term and store in a variable T1, then the second term, and store it in T2.
  • Page 234: Example 3 - System Of Polynomial Equations

    To solve for P and P , use the command SOLVE from the S.SLV menu („Î), it may take the calculator a minute to produce the result: {[‘Pi=-(((σθ-σr)*r^2-(σθ+σr)*a^2)/(2*a^2))’ ‘Po=-(((σθ-σr)*r^2-(σθ+σr)*b^2)/(2*b^2))’ ] } , i.e., Notice that the result includes a vector [ ] contained within a list { }. To remove the list symbol, use µ.
  • Page 235: Example 1 - Example From The Help Facility

    Example 1 – Example from the help facility As with all function entries in the help facility, there is an example attached to the MSLV entry as shown above. Notice that function MSLV requires three arguments: 1. A vector containing the equations, i.e., ‘[SIN(X)+Y,X+SIN(Y)=1]’ 2.
  • Page 236: Example 2 - Entrance From A Lake Into An Open Channel

    Example 2 - Entrance from a lake into an open channel This particular problem in open channel flow requires the simultaneous solution of two equations, the equation of energy: , and Manning’s equation: . In these equations, H represents the energy head (m, or ft) available for a flow at the entrance to a channel, y is the flow depth (m or ft), V = Q/A is the flow velocity (m/s or ft/s), Q is the volumetric discharge (m /s or ft...
  • Page 237 To see the original equations, EQ1 and EQ2, in terms of the primitive variables listed above, we can use function EVAL applied to each of the equations, i.e., µ@@@EQ1@@ µ @@@EQ2@@. The equations are listed in the stack as follows (small font option selected): We can see that these equations are indeed given in terms of the primitive variables b, m, y, g, S , n, Cu, Q, and H...
  • Page 238 Now, we are ready to solve the equation. First, we need to put the two equations together into a vector. We can do this by actually storing the vector into a variable that we will call EQS (EQuationS): As initial values for the variables y and Q we will use y = 5 (equal to the value of H , which is the maximum value that y can take) and Q = 10 (this is a guess).
  • Page 239 Press @@OK@@ and allow the solution to proceed. An intermediate solution step may look like this: The vector at the top representing the current value of [y,Q] as the solution progresses, and the value .358822986286 representing the criteria for convergence of the numerical method used in the solution. If the system is well posed, this value will diminish until reaching a value close to zero.
  • Page 240: Using The Multiple Equation Solver (mes)

    Using the Multiple Equation Solver (MES) The multiple equation solver is an environment where you can solve a system of multiple equations by solving for one unknown from one equation at a time. It is not really a solver to simultaneous solutions, rather, it is a one-by-one solver of a number of related equations.
  • Page 241 cosine law, and sum of interior angles of a triangle, to solve for the other three variables. If the three sides are known, the area of the triangle can be calculated with ,where s is known as the Heron’s formula semi-perimeter of the triangle, i.e., Triangle solution using the Multiple Equation Solver (MES) The Multiple Equation Solver (MES) is a feature that can be used to solve two...
  • Page 242 ‘a^2 = b^2+c^2-2*b*c*COS(α)’ ‘α+β+γ = 180’ ‘s = (a+b+c)/2’ ‘A = √ (s*(s-a)*(s-b)*(s-c))’ Then, enter the number 9, and create a list of equations by using: function LIST (use the command catalog ‚N). Store this list in the variable EQ. The variable EQ contains the list of equations that will be scanned by the MES when trying to solve for the unknowns.
  • Page 243 Preparing to run the MES The next step is to activate the MES and try one sample solution. Before we do that, however, we want to set the angular units to DEGrees, if they are not already set to that, by typing ~~deg`. Next, we want to keep in the stack the contents of TITLE and LVARI, by using: !@TITLE @LVARI! We will use the following MES functions...
  • Page 244 5[ a ] a:5 is listed in the top left corner of the display. 3[ b ] b:3 is listed in the top left corner of the display. 5[ c ] c:5 is listed in the top left corner of the display. To solve for the angles use: „[ α...
  • Page 245 When done, press $ to return to the MES environment. Press J to exit the MES environment and return to the normal calculator display. Organizing the variables in the sub directory Your variable menu will now contain the variables (press L to see the second set of variables): Variables corresponding to all the variables in the equations in EQ have been created.
  • Page 246 Programming the MES triangle solution using User RPL To facilitate activating the MES for future solutions, we will create a program that will load the MES with a single keystroke. The program should look like this: << DEG MINIT TITLE LVARI MITM MSOLVR >>, and can be typed in by using: Opens the program symbol ‚å...
  • Page 247 Example 2 - Any type of triangle Use a = 3, b = 4, c = 6. The solution procedure used here consists of solving for all variables at once, and then recalling the solutions to the stack: J @TRISO To clear up data and re-start MES 3[ a ] 4 [ b ] 6[ c ] To enter data To move to the next variables menu.
  • Page 248: Application 2 - Velocity And Acceleration In Polar Coordinates

    carry over information from the previous solution that may wreck havoc with your current calculations. ο ο ο α( β( γ( 6.9837 20.299 84.771 8.6933 14.26 22.616 130.38 23.309 21.92 52.97 37.03 115.5 17.5 13.2 41.92 29.6 328.81 10.27 3.26 16.66 10.5 31.79 50.78 97.44 210.71...
  • Page 249 ________________________________________________________________ Program or value Store into variable: SOLVEP << PEQ STEQ MINIT NAME LIST MITM MSOLVR >> NAME "vel. & acc. polar coord." { r rD rDD θD θDD vr vθ v ar aθ a } LIST { 'vr = rD' 'vθ = r*θD' 'v = √(vr^2 + vθ^2)' 'ar = rDD −...
  • Page 250 Start the multiple equation solver by pressing J@SOLVE. The calculator produces a screen labeled , "vel. & acc. polar coord.", that looks as follows: To enter the values of the known variables, just type the value and press the button corresponding to the variable to be entered. Use the following keystrokes: 2.5 [ r ] 0.5 [ rD ] 1.5 \ [ rDD ] 2.3 [ θD ] 6.5 \ [ θDD ].
  • Page 251 To use a new set of values press, either @EXIT @@ALL@ LL, or J @SOLVE. Let's try another example using r = 2.5, vr = rD = -0.5, rDD = 1.5, v = 3.0, a = 25.0. Find, θD, θDD, vθ, ar, and aθ. You should get the following results: Page 7-21...
  • Page 252: Chapter 8 - Operations With Lists

    Chapter 8 Operations with lists Lists are a type of calculator’s object that can be useful for data processing and in programming. This Chapter presents examples of operations with lists. Definitions A list, within the context of the calculator, is a series of objects enclosed between braces and separated by spaces (#), in the RPN mode, or commas (‚í), in both modes.
  • Page 253: Composing And Decomposing Lists

    „ä 1 # 2 # 3 # 4 ` ~l1`™K The figure below shows the RPN stack before pressing the K key: Composing and decomposing lists Composing and decomposing lists makes sense in RPN mode only. Under such operating mode, decomposing a list is achieved by using function OBJ .
  • Page 254: Operations With Lists Of Numbers

    Operations with lists of numbers To demonstrate operations with lists of numbers, we will create a couple of other lists, besides list L1 created above: L2={-3,2,1,5}, L3={-6,5,3,1,0,3,-4}, L4={3,-2,1,5,3,2,1}. In ALG mode, the screen will look like this after entering lists L2, L3, L4: In RPN mode, the following screen shows the three lists and their names ready to be stored.
  • Page 255 Addition of a single number to a list produces a list augmented by the number, and not an addition of the single number to each element in the list. For example: Subtraction, multiplication, and division of lists of numbers of the same length produce a list of the same length with term-by-term operations.
  • Page 256: Real Number Functions From The Keyboard

    Real number functions from the keyboard , √, Real number functions from the keyboard (ABS, e , LN, 10 , LOG, SIN, x COS, TAN, ASIN, ACOS, ATAN, y ) can be used on lists. Here are some examples: EXP and LN LOG and ANTILOG SQ and square root SIN, ASIN...
  • Page 257: Examples Of Functions That Use Two Arguments

    SINH, ASINH COSH, ACOSH TANH, ATANH SIGN, MANT, XPON IP, FP FLOOR, CEIL D R, R D Examples of functions that use two arguments The screen shots below show applications of the function % to list arguments. Function % requires two arguments. The first two examples show cases in which only one of the two arguments is a list.
  • Page 258: Lists Of Complex Numbers

    %({10, 20, 30},1) = {%(10,1),%(20,1),%(30,1)}, while %(5,{10,20,30}) = {%(5,10),%(5,20),%(5,30)} In the following example, both arguments of function % are lists of the same size. In this case, a term-by-term distribution of the arguments is performed, i.e., %({10,20,30},{1,2,3}) = {%(10,1),%(20,2),%(30,3)} This description of function % for list arguments shows the general pattern of evaluation of any function with two arguments when one or both arguments are lists.
  • Page 259: Lists Of Algebraic Objects

    The following example shows applications of the functions RE(Real part), IM(imaginary part), ABS(magnitude), and ARG(argument) of complex numbers. The results are lists of real numbers: Lists of algebraic objects The following are examples of lists of algebraic objects with the function SIN applied to them: The MTH/LIST menu The MTH menu provides a number of functions that exclusively to lists.
  • Page 260 Next, with system flag 117 set to SOFT menus: This menu contains the following functions: ∆LIST : Calculate increment among consecutive elements in list ΣLIST : Calculate summation of elements in the list ΠLIST : Calculate product of elements in the list SORT : Sorts elements in increasing order REVLIST...
  • Page 261: Manipulating Elements Of A List

    Manipulating elements of a list The PRG (programming) menu includes a LIST sub-menu with a number of functions to manipulate elements of a list. With system flag 117 set to CHOOSE boxes: Item 1. ELEMENTS.. contains the following functions that can be used for the manipulation of elements in lists: List size Function SIZE, from the PRG/LIST/ELEMENTS sub-menu, can be used to...
  • Page 262: Element Position In The List

    Functions GETI and PUTI, also available in sub-menu PRG/ ELEMENTS/, can also be used to extract and place elements in a list. These two functions, however, are useful mainly in programming. Function GETI uses the same arguments as GET and returns the list, the element location plus one, and the element at the location requested.
  • Page 263: The Map Function

    SEQ is useful to produce a list of values given a particular expression and is described in more detail here. The SEQ function takes as arguments an expression in terms of an index, the name of the index, and starting, ending, and increment values for the index, and returns a list consisting of the evaluation of the expression for all possible values of the index.
  • Page 264: Defining Functions That Use Lists

    Defining functions that use lists In Chapter 3 we introduced the use of the DEFINE function ( „à) to create functions of real numbers with one or more arguments. A function defined with DEF can also be used with list arguments, except that, any function incorporating an addition must use the ADD operator rather than the plus sign (+).
  • Page 265: Applications Of Lists

    Next, we store the edited expression into variable @@@G@@@: Evaluating G(L1,L2) now produces the following result: As an alternative, you can define the function with ADD rather than the plus sign (+), from the start, i.e., use DEFINE('G(X,Y)=(X ADD 3)*Y') : You can also define the function as G(X,Y) = (X--3)*Y.
  • Page 266: Harmonic Mean Of A List

    and that we store it into a variable called S (The screen shot below shows this action in ALG mode, however, the procedure in RPN mode is very similar. Just keep in mind that in RPN mode you place the arguments of functions in the stack before activating the function): Harmonic mean of a list This is a small enough sample that we can count on the screen the number of...
  • Page 267: Geometric Mean Of A List

    3. Divide the result above by n = 10: 4. Apply the INV() function to the latest result: Thus, the harmonic mean of list S is s = 1.6348… Geometric mean of a list The geometric mean of a sample is defined as ∏...
  • Page 268: Weighted Average

    Weighted average Suppose that the data in list S, defined above, namely: S = {1,5,3,1,2,1,3,4,2,1} is affected by the weights, W = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} }, we notice that the k -th If we define the weight list as W = {w ,…,w element in list W, above, can be defined by w = k.
  • Page 269: Statistics Of Grouped Data

    3. Use function ΣLIST, once more, to calculate the denominator of s 4. Use the expression ANS(2)/ANS(1) to calculate the weighted average: Thus, the weighted average of list S with weights in list W is s = 2.2. Note: ANS(1) refers to the most recent result (55), while ANS(2) refers to the previous to last result (121).
  • Page 270 Given the list of class marks S = {s , …, s }, and the list of frequency counts W = {w , …, w }, the weighted average of the data in S with weights W represents the mean value of the grouped data, that we call s, in this context: ∑...
  • Page 271 To calculate this last result, we can use the following: The standard deviation of the grouped data is the square root of the variance: Page 8-20...
  • Page 272: Chapter 9 - Vectors

    Chapter 9 Vectors This Chapter provides examples of entering and operating with vectors, both mathematical vectors of many elements, as well as physical vectors of 2 and 3 components. Definitions From a mathematical point of view, a vector is an array of 2 or more elements arranged into a row or a column.
  • Page 273: Entering Vectors

    There are two definitions of products of physical vectors, a scalar or internal product (the dot product) and a vector or external product (the cross product). The dot product produces a scalar value defined as A B = |A||B|cos( ), where is the angle between the two vectors.
  • Page 274: Storing Vectors Into Variables

    In RPN mode, you can enter a vector in the stack by opening a set of brackets and typing the vector components or elements separated by either commas (‚í) or spaces (#). Notice that after pressing ` , in either mode, the calculator shows the vector elements separated by spaces.
  • Page 275 Vectors vs. matrices To see the @VEC@ key in action, try the following exercises: With @VEC and @GO (1) Launch the Matrix Writer („²). selected, → enter 3`5`2``. This produces [3. 5. 2.]. (In RPN mode, you can use the following keystroke sequence to produce the same result: 3#5#2``).
  • Page 276 Activate the Matrix Writer again by using „², and press L to check out the second soft key menu at the bottom of the display. It will show the keys: @+ROW@ @-ROW @+COL@ @-COL@ STK@@ @GOTO@ @→ The @+ROW@ key will add a row full of zeros at the location of the selected cell of the spreadsheet.
  • Page 277: Building A Vector With Arry

    (5) Press @-COL@. The first column will disappear. (6) Press @+COL@. A row of two zeroes appears in the first row. (7) Press @GOTO@ 3@@OK@@ 3@@OK@@ @@OK@@ to move to position (3,3). STK@@. (8) Press This will place the contents of cell (3,3) on the stack, @→...
  • Page 278: Identifying, Extracting, And Inserting Vector Elements

    The following screen shots show the RPN stack before and after applying function ARRY: In RPN mode, the function [ ARRY] takes the objects from stack levels n+1, n, → n-1, …, down to stack levels 3 and 2, and converts them into a vector of n elements.
  • Page 279 More complicated expressions involving elements of A can also be written. For example, using the Equation Writer (‚O), we can write the following summation of the elements of A: Highlighting the entire expression and using the @EVAL@ soft menu key, we get the result: -15.
  • Page 280: Simple Operations With Vectors

    Note: This approach for changing the value of an array element is not allowed in ALG mode, if you try to store 4.5 into A(3) in this mode you get the following error message: Invalid Syntax. To find the length of a vector you can use the function SIZE, available through the command catalog (N) or through the PRG/LIST/ELEMENTS sub-menu.
  • Page 281: Absolute Value Function

    Attempting to add or subtract vectors of different length produces an error message (Invalid Dimension), e.g., v2+v3, u2+u3, A+v3, etc. Multiplication by a scalar, and division by a scalar Multiplication by a scalar or division by a scalar is straightforward: Absolute value function The absolute value function (ABS), when applied to a vector, produces the magnitude of the vector.
  • Page 282: Magnitude,

    Magnitude The magnitude of a vector, as discussed earlier, can be found with function ABS. This function is also available from the keyboard („Ê). Examples of application of function ABS were shown above. Dot product Function DOT is used to calculate the dot product of two vectors of the same length.
  • Page 283: Decomposing A Vector

    Examples of cross products of one 3-D vector with one 2-D vector, or vice versa, are presented next: Attempts to calculate a cross product of vectors of length other than 2 or 3, produce an error message (Invalid Dimension), e.g., CROSS(v3,A), etc. Decomposing a vector Function V is used to decompose a vector into its elements or components.
  • Page 284: Building A Three-dimensional Vector

    Building a three-dimensional vector Function V3 is used in the RPN mode to build a vector with the values in stack levels 1: , 2:, and 3:. The following screen shots show the stack before and after applying function Changing coordinate system Functions RECT, CYLIN, and SPHERE are used to change the current coordinate system to rectangular (Cartesian), cylindrical (polar), or spherical coordinates.
  • Page 285 „Ô5 ‚í ~‚6 25 ‚í 2.3 Before pressing `, the screen will look as in the left-hand side of the following figure. After pressing `, the screen will look as in the right-hand side of the figure (For this example, the numerical format was changed to Fix, with three decimals).
  • Page 286 The conversion from Cartesian to cylindrical coordinates is such that r = , θ = tan (y/x), and z = z. For the case shown above the transformation was such that (x,y,z) = (3.204, 2.112, 2.300), produced (r,θ,z) = (3.536,25 ,3.536).
  • Page 287: Application Of Vector Operations

    Notice that the vectors that were written in cylindrical polar coordinates have now been changed to the spherical coordinate system. The transformation is such that ρ = (r , θ = θ, and φ = tan (r/z). However, the vector that originally was set to Cartesian coordinates remains in that form.
  • Page 288: Moment Of A Force

    Thus, the result is θ = 122.891 In RPN mode use the following: [3,-5,6] ` [2,1,-3] ` DOT [3,-5,6] ` ABS [2,1,-3] ` ABS * ACOS Moment of a force The moment exerted by a force F about a point O is defined as the cross- product M = r×F, where r, also known as the arm of the force, is the position vector based at O and pointing towards the point of application of the force.
  • Page 289: Equation Of A Plane In Space

    Thus the angle between vectors r and F is θ = 41.038 . RPN mode, we can use: [3,-5,4] ` [2,5,-6] ` CROSS ABS [3,-5,4] ` ABS [2,5,-6] ` ABS * / ASIN Equation of a plane in space ) and a vector N = N k normal to a Given a point in space P plane containing point P...
  • Page 290: Row Vectors, Column Vectors, And Lists

    We can now use function EXPAND (in the ALG menu) to expand this expression: Thus, the equation of the plane through point P (2,3,-1) and having normal vector N = 4i+6j+2k, is 4x + 6y + 2z – 24 = 0. In RPN mode, use: [2,3,-1] ` ['x','y','z'] ` - [4,6,2] DOT EXPAND Row vectors, column vectors, and lists The vectors presented in this chapter are all row vectors.
  • Page 291: Function Obj

    LIST will be available in soft menu keys A, B, OBJ , ARRY, and and C. Function DROP is available by using „°@) S TACK @DROP. Following we introduce the operation of functions OBJ , LIST, ARRY, and DROP with some examples. Function OBJ This function decomposes an object into its components.
  • Page 292: Function Drop

    n+1:. For example, to create the list {1, 2, 3}, type: 1` 2` 3` 3` „°@) T YPE! ! LIST@. Function ARRY This function is used to create a vector or a matrix. In this section, we will use it to build a vector or a column vector (i.e., a matrix of n rows and 1 column). To build a regular vector we enter the elements of the vector in the stack, and in stack level 1: we enter the vector size as a list, e.g., 1` 2` 3` „ä...
  • Page 293: Transforming A Column Vector Into A Row Vector

    A new variable, @@RXC@@, will be available in the soft menu labels after pressing Press ‚@@RXC@@ to see the program contained in the variable RXC: << OBJ ARRY >> This variable, @@RXC@@, can now be used to directly transform a row vector to a column vector.
  • Page 294 3 - Press the delete key ƒ (also known as function DROP) to eliminate the number in stack level 1: 4 - Use function LIST to create a list 5 - Use function ARRY to create the row vector These five steps can be put together into a UserRPL program, entered as follows (in RPN mode, still): ‚å„°@) T YPE! @OBJ @ @OBJ @ „°@) S TACK @DROP „°@) T YPE! ! LIST@ ! ARRY@ `...
  • Page 295: Transforming A List Into A Vector

    resulting in: Transforming a list into a vector To illustrate this transformation, we’ll enter the list {1,2,3} in RPN mode. Then, follow the next exercise to transform a list into a vector: 1 - Use function OBJ to decompose the column vector 2 - Type a 1 and use function LIST to create a list in stack level 1: 3 - Use function...
  • Page 296: Transforming A Vector (or Matrix) Into A List

    After having defined variable @@LXV@@, we can use it in ALG mode to transform a list into a vector. Thus, change your calculator’s mode to ALG and try the following procedure: {1,2,3} ` J @@LXV@@ „Ü „î, resulting Transforming a vector (or matrix) into a list To transform a vector into a list, the calculator provides function AXL.
  • Page 297: Definitions

    Chapter 10 Creating and manipulating matrices This chapter shows a number of examples aimed at creating matrices in the calculator and demonstrating manipulation of matrix elements. Definitions A matrix is simply a rectangular array of objects (e.g., numbers, algebraics) having a number of rows and columns. A matrix A having n rows and m columns will have, therefore, n×m elements.
  • Page 298: Entering Matrices In The Stack

    Entering matrices in the stack In this section we present two different methods to enter matrices in the calculator stack: (1) using the Matrix Writer, and (2) typing the matrix directly into the stack. Using the Matrix Writer As with the case of vectors, discussed in Chapter 9, matrices can be entered into the stack by using the Matrix Writer.
  • Page 299: Typing The Matrix Directly Into The Stack

    If you have selected the textbook display option (using H@) D ISP! and checking Textbook ), the matrix will look like the one shown above. Otherwise, the display will show: The display in RPN mode will look very similar to these. Note: Details on the use of the matrix writer were presented in Chapter 9.
  • Page 300 or in the MATRICES/CREATE menu available through „Ø: The MTH/MATRIX/MAKE sub menu (let’s call it the MAKE menu) contains the following functions: while the MATRICES/CREATE sub-menu (let’s call it the CREATE menu) has the following functions: Page 10-4...
  • Page 301 As you can see from exploring these menus (MAKE and CREATE), they both have the same functions GET, GETI, PUT, PUTI, SUB, REPL, RDM, RANM, HILBERT, VANDERMONDE, IDN, CON, DIAG, and DIAG . The CREATE → → menu includes the COLUMN and ROW sub-menus, that are also available under the MTH/MATRIX menu.
  • Page 302: Functions Get And Put

    Functions GET and PUT Functions GET, GETI, PUT, and PUTI, operate with matrices in a similar manner as with lists or vectors, i.e., you need to provide the location of the element that you want to GET or PUT. However, while in lists and vectors only one index is required to identify an element, in matrices we need a list of two indices {row, column} to identify matrix elements.
  • Page 303: Function Size

    Notice that the screen is prepared for a subsequent application of GETI or GET, by increasing the column index of the original reference by 1, (i.e., from {2,2} to {2,3}), while showing the extracted value, namely A(2,2) = 1.9, in stack level 1.
  • Page 304: Function Trn

    Function TRN Function TRN is used to produce the transconjugate of a matrix, i.e., the transpose (TRAN) followed by its complex conjugate (CONJ). For example, the following screen shot shows the original matrix in variable A and its transpose, shown in small font display (see Chapter 1): If the argument is a real matrix, TRN simply produces the transpose of the real matrix.
  • Page 305: Function Idn

    value. Function CON generates a matrix with constant elements. For example, in ALG mode, the following command creates a 4×3 matrix whose elements are all equal to –1.5: In RPN mode this is accomplished by using {4,3} ` 1.5 \ ` CON.
  • Page 306: Function Rdm

    Function RDM Function RDM (Re-DiMensioning) is used to re-write vectors and matrices as matrices and vectors. The input to the function consists of the original vector or matrix followed by a list of a single number, if converting to a vector, or two numbers, if converting to a matrix.
  • Page 307: Function Ranm

    If using RPN mode, we assume that the matrix is in the stack and use {6} ` RDM. Note: Function RDM provides a more direct and efficient way to transform lists to arrays and vice versa, than that provided at the end of Chapter 9. Function RANM Function RANM (RANdom Matrix) will generate a matrix with random integer elements given a list with the number of rows and columns (i.e., the...
  • Page 308: Function Repl

    want to extract elements a , and a from the last result, as a 2×2 sub-matrix, in ALG mode, use: In RPN mode, assuming that the original 2×3 matrix is already in the stack, use {1,2} ` {2,3} ` SUB. Function REPL Function REPL replaces or inserts a sub-matrix into a larger one.
  • Page 309: Function Diag

    Function →DIAG Function DIAG takes the main diagonal of a square matrix of dimensions → n×n, and creates a vector of dimension n containing the elements of the main diagonal. For example, for the matrix remaining from the previous exercise, we can extract its main diagonal by using: In RPN mode, with the 3×3 matrix in the stack, we simply have to activate DIAG to obtain the same result as above.
  • Page 310: Function Vandermonde

    so the main diagonal included only the elements in positions (1,1) and (2,2). Thus, only the first two elements of the vector were required to form the main diagonal. Function VANDERMONDE Function VANDERMONDE generates the Vandermonde matrix of dimension n based on a given list of input data.
  • Page 311: A Program To Build A Matrix Out Of A Number Of Lists

    The Hilbert matrix has application in numerical curve fitting by the method of linear squares. A program to build a matrix out of a number of lists In this section we provide a couple of UserRPL programs to build a matrix out of a number of lists of objects.
  • Page 312 „° @) B RCH! @) F OR@! @NEXT NEXT „° @) B RCH! @) @ IF@ @@IF@@ ~ „n #1 „° @) T EST! @@@>@@@ > „° @) B RCH! @@IF@ @THEN THEN ~ „n #1- n 1 - „° @) B RCH! @) F OR@! @FOR@ ~ „j # ~ „j #1+...
  • Page 313: Lists Represent Rows Of The Matrix

    To use the program in ALG mode, press @CRMC followed by a set of parentheses („Ü). Within the parentheses type the lists of data representing the columns of the matrix, separated by commas, and finally, a comma, and the number of columns. The command should look like this: CRMC({1,2,3,4}, {1,4,9,16}, {1,8,27,64}, 3) The ALG screen showing the execution of program CRMC is shown below:...
  • Page 314: Manipulating Matrices By Columns

    Manipulating matrices by columns The calculator provides a menu with functions for manipulating matrices by operating in their columns. This menu is available through the MTH/MATRIX/COL.. sequence: („´) shown in the figure below with system flag 117 set to CHOOSE boxes: or through the MATRICES/CREATE/COLUMN sub-menu: Both approaches will show the same functions: When system flag 117 is set to SOFT menus, the COL menu is accessible...
  • Page 315: Function Col

    decomposed in columns. To see the full result, use the line editor (triggered by pressing ˜). In RPN mode, you need to list the matrix in the stack, and the activate function COL, i.e., @@@A@@@ COL. The figure below shows the RPN stack before and after the application of function COL.
  • Page 316: Function Col

    as columns in the resulting matrix. The following figure shows the RPN stack before and after using function COL . Function COL+ Function COL+ takes as argument a matrix, a vector with the same length as the number of rows in the matrix, and an integer number n representing the location of a column.
  • Page 317: Function Cswp

    In RPN mode, place the matrix in the stack first, then enter the number representing a column location before applying function COL-. The following figure shows the RPN stack before and after applying function COL-. Function CSWP Function CSWP (Column SWaP) takes as arguments two indices, say, i and j, (representing two distinct columns in a matrix), and a matrix, and produces a new matrix with columns i and j swapped.
  • Page 318: Function Row

    MTH/MATRIX/ROW.. sequence: („´) shown in the figure below with system flag 117 set to CHOOSE boxes: or through the MATRICES/CREATE/ROW sub-menu: Both approaches will show the same functions: When system flag 117 is set to SOFT menus, the ROW menu is accessible through „´!) M ATRX !) @ MAKE@ !) @ @ROW@ , or through „Ø!) @ CREAT@ !) @ @ROW@ .
  • Page 319: Function Row ,

    In RPN mode, you need to list the matrix in the stack, and the activate function ROW, i.e., @@@A@@@ ROW. The figure below shows the RPN stack before and after the application of function ROW. In this result, the first row occupies the highest stack level after decomposition, and stack level 1 is occupied by the number of rows of the original matrix.
  • Page 320: Function Row Function Row

    Function ROW+ Function ROW+ takes as argument a matrix, a vector with the same length as the number of rows in the matrix, and an integer number n representing the location of a row. Function ROW+ inserts the vector in row n of the matrix. For example, in ALG mode, we’ll insert the second row in matrix A with the vector [-1,-2,-3], i.e., In RPN mode, enter the matrix first, then the vector, and the row number,...
  • Page 321: Function Rswp

    Function RSWP Function RSWP (Row SWaP) takes as arguments two indices, say, i and j, (representing two distinct rows in a matrix), and a matrix, and produces a new matrix with rows i and j swapped. The following example, in ALG mode, shows an application of this function.
  • Page 322: Function Rcij

    This same exercise done in RPN mode is shown in the next figure. The left- hand side figure shows the setting up of the matrix, the factor and the row number, in stack levels 3, 2, and 1. The right-hand side figure shows the resulting matrix after function RCI is activated.
  • Page 323: Chapter 11 - Matrix Operations And Linear Algebra

    Chapter 11 Matrix Operations and Linear Algebra In Chapter 10 we introduced the concept of a matrix and presented a number of functions for entering, creating, or manipulating matrices. In this Chapter we present examples of matrix operations and applications to problems of linear algebra.
  • Page 324: Addition And Subtraction

    Addition and subtraction Consider a pair of matrices A = [a and B = [b . Addition and × × subtraction of these two matrices is only possible if they have the same number of rows and columns. The resulting matrix, C = A ± B = [c ×...
  • Page 325 By combining addition and subtraction with multiplication by a scalar we can form linear combinations of matrices of the same dimensions, e.g., In a linear combination of matrices, we can multiply a matrix by an imaginary number to obtain a matrix of complex numbers, e.g., Matrix-vector multiplication Matrix-vector multiplication is possible only if the number of columns of the matrix is equal to the length of the vector.
  • Page 326 Vector-matrix multiplication, on the other hand, is not defined. This multiplication can be performed, however, as a special case of matrix multiplication as defined next. Matrix multiplication Matrix multiplication is defined by C ⋅B , where A = [a , B = ×...
  • Page 327 The product of a vector with a matrix is possible if the vector is a row vector, i.e., a 1×m matrix, which multiplied with a matrix m×n produces a 1xn matrix (another row vector). For the calculator to identify a row vector, you must use double brackets to enter it, e.g., Term-by-term multiplication Term-by-term multiplication of two matrices of the same dimensions is possible...
  • Page 328: Characterizing A Matrix (the Matrix Norm Menu)

    The inverse matrix The inverse of a square matrix A is the matrix A such that A⋅A ⋅A = I, where I is the identity matrix of the same dimensions as A. The inverse of a matrix is obtained in the calculator by using the inverse function, INV (i.e., the Y key).
  • Page 329: Function Abs

    These functions are described next. Because many of these functions use concepts of matrix theory, such as singular values, rank, etc., we will include short descriptions of these concepts intermingled with the description of functions. Function ABS Function ABS calculates what is known as the Frobenius norm of a matrix. For a matrix A = [a , the Frobenius norm of the matrix is defined as ×...
  • Page 330: Functions Rnrm And Cnrm

    Singular value decomposition To understand the operation of Function SNRM, we need to introduce the concept of matrix decomposition. Basically, matrix decomposition involves the determination of two or more matrices that, when multiplied in a certain order (and, perhaps, with some matrix inversion or transposition thrown in), produce the original matrix.
  • Page 331: Function Srad

    Function SRAD Function SRAD determines the Spectral RADius of a matrix, defined as the largest of the absolute values of its eigenvalues. For example, Definition of eigenvalues and eigenvectors of a matrix The eigenvalues of a square matrix result from the matrix equation A⋅x = λ⋅x. The values of λ...
  • Page 332: Function Rank

    The condition number of a singular matrix is infinity. The condition number of a non-singular matrix is a measure of how close the matrix is to being singular. The larger the value of the condition number, the closer it is to singularity.
  • Page 333: Function Det

    are constant, we say that c where the values d is linearly dependent on the columns included in the summation. (Notice that the values of j include any value in the set {1, 2, …, n}, in any combination, as long as j≠k.) If the expression shown above cannot be written for any of the column vectors then we say that all the columns are linearly independent.
  • Page 334 The determinant of a matrix The determinant of a 2x2 and or a 3x3 matrix are represented by the same arrangement of elements of the matrices, but enclosed between vertical lines, i.e., A 2×2 determinant is calculated by multiplying the elements in its diagonal and adding those products accompanied by the positive or negative sign as indicated in the diagram shown below.
  • Page 335: Function Trace

    For square matrices of higher order determinants can be calculated by using smaller order determinant called cofactors. The general idea is to “expand” a determinant of a n×n matrix (also referred to as a n×n determinant) into a sum of the cofactors, which are (n-1)×(n-1) determinants, multiplied by the elements of a single row or column, with alternating positive and negative signs.
  • Page 336: Function Tran

    Function TRAN Function TRAN returns the transpose of a real or the conjugate transpose of a complex matrix. TRAN is equivalent to TRN. The operation of function TRN was presented in Chapter 10. Additional matrix operations (The matrix OPER menu) The matrix OPER (OPERATIONS) is available through the keystroke sequence „Ø...
  • Page 337: Function Axl

    Function AXL Function AXL converts an array (matrix) into a list, and vice versa. For examples, Note: the latter operation is similar to that of the program CRMR presented in Chapter 10. Function AXM Function AXM converts an array containing integer or fraction elements into its corresponding decimal, or approximate, form.
  • Page 338: Solution Of Linear Systems

    The implementation of function LCXM for this case requires you to enter: 2`3`‚@@P1@@ LCXM ` The following figure shows the RPN stack before and after applying function LCXM: In ALG mode, this example can be obtained by using: The program P1 must still have been created and stored in RPN mode. Solution of linear systems A system of n linear equations in m variables can be written as ⋅x...
  • Page 339: Using The Numerical Solver For Linear Systems

    Using the numerical solver for linear systems There are many ways to solve a system of linear equations with the calculator. One possibility is through the numerical solver ‚Ï. From the numerical solver screen, shown below (left), select the option 4. Solve lin sys.., and press @@@OK@@@.
  • Page 340 To enter matrix A you can activate the Matrix Writer while the A: field is selected. The following screen shows the Matrix Writer used for entering matrix A, as well as the input form for the numerical solver after entering matrix A (press ` in the Matrix Writer): Press ˜...
  • Page 341 Under-determined system The system of linear equations + 3x –5x = -10, – 3x + 8x = 85, can be written as the matrix equation A⋅x = b, if This system has more unknowns than equations, therefore, it is not uniquely determined.
  • Page 342 To see the details of the solution vector, if needed, press the @EDIT! button. This will activate the Matrix Writer. Within this environment, use the right- and left-arrow keys to move about the vector, e.g., Thus, the solution is x = [15.373, 2.4626, 9.6268]. To return to the numerical solver environment, press `.
  • Page 343 Let’s store the latest result in a variable X, and the matrix into variable A, as follows: Press K~x` to store the solution vector into variable X Press ƒ ƒ ƒ to clear three levels of the stack Press K~a` to store the matrix into variable A Now, let’s verify the solution by using: @@@A@@@ * @@@X@@@ `, which results in (press ˜...
  • Page 344 Such is the approach followed by the HP 49 G numerical solver.
  • Page 345: Least-square Solution (function Lsq)

    Press ` to return to the numerical solver environment. To check that the solution is correct, try the following: • Press ——, to highlight the A: field. • Press L @CALC@ `, to copy matrix A onto the stack. • Press @@@OK@@@ to return to the numerical solver environment.
  • Page 346 • If A is a square matrix and A is non-singular (i.e., it’s inverse matrix exist, or its determinant is non-zero), LSQ returns the exact solution to the linear system. • If A has less than full row rank (underdetermined system of equations), LSQ returns the solution with the minimum Euclidean length out of an infinity number of solutions.
  • Page 347 Under-determined system Consider the system + 3x –5x = -10, – 3x + 8x = 85, with The solution using LSQ is shown next: Over-determined system Consider the system + 3x = 15, – 5x = 5, = 22, with The solution using LSQ is shown next: Page 11-25...
  • Page 348: Solution With The Inverse Matrix

    Compare these three solutions with the ones calculated with the numerical solver. Solution with the inverse matrix The solution to the system A⋅x = b, where A is a square matrix is x = A ⋅ b. This results from multiplying the first equation by A , i.e., A ⋅A⋅x = A ⋅b.
  • Page 349: Solving Multiple Set Of Equations With The Same Coefficient Matrix

    previous section. The procedure for the case of “dividing” b by A is illustrated below for the case + 3x –5x = 13, – 3x + 8x = -13, – 2x + 4x = -6, The procedure is shown in the following screen shots: The same solution as found above with the inverse matrix.
  • Page 350: Gaussian And Gauss-jordan Elimination

    The sub-indices in the variable names X, Y, and Z, determine to which equation system they refer to. To solve this expanded system we use the following procedure, in RPN mode, [[14,9,-2],[2,-5,2],[5,19,12]] ` [[1,2,3],[3,-2,1],[4,2,-1]] `/ The result of this operation is: Gaussian and Gauss-Jordan elimination Gaussian elimination is a procedure by which the square matrix of coefficients belonging to a system of n linear equations in n unknowns is reduced to an...
  • Page 351 To start the process of forward elimination, we divide the first equation (E1) by 2, and store it in E1, and show the three equations again to produce: Next, we replace the second equation E2 by (equation 2 – 3×equation 1, i.e., E1-3×E2), and the third by (equation 3 –...
  • Page 352 Y+ Z = 3, -7Z = -14. The process of backward substitution in Gaussian elimination consists in finding the values of the unknowns, starting from the last equation and working upwards. Thus, we solve for Z first: Next, we substitute Z=2 into equation 2 (E2), and solve E2 for Y: Next, we substitute Z=2 and Y = 1 into E1, and solve E1 for X: The solution is, therefore, X = -1, Y = 1, Z = 2.
  • Page 353 The matrix A is the same as the original matrix A with a new row, corresponding to the elements of the vector b, added (i.e., augmented) to the right of the rightmost column of A. Once the augmented matrix is put together, we can proceed to perform row operations on it that will reduce the original A matrix into an upper-triangular matrix.
  • Page 354 If you were performing these operations by hand, you would write the following:         ≅ − − − ≅         − − − − − − ...
  • Page 355 Multiply row 3 by –3, add it to row 1, replacing it: 3\#3#1@RCIJ! Multiply row 2 by –2, add it to row 1, replacing it: 2\#2#1 @RCIJ! Writing this process by hand will result in the following steps:   ...
  • Page 356 pivoting operations. When row and column exchanges are allowed in pivoting, the procedure is known as full pivoting. When exchanging rows and columns in partial or full pivoting, it is necessary to keep track of the exchanges because the order of the unknowns in the solution is altered by those exchanges.
  • Page 357 First, we check the pivot a . We notice that the element with the largest absolute value in the first row and first column is the value of a = 8. Since we want this number to be the pivot, then we exchange rows 1 and 3, by using: 1#3L @RSWP.
  • Page 358 25/8 -25/82 Checking the pivot at position (2,2), we now find that the value of 25/8, at position (3,2), is larger than 3. Thus, we exchange rows 2 and 3 by using: 2#3 L@RSWP -1/16 1/2 41/16 25/8 -25/8 Now, we are ready to divide row 2 by the pivot 25/8, by using ³...
  • Page 359: Step-by-step Calculator Procedure For Solving Linear Systems

    Finally, we eliminate the –1/16 from position (1,2) by using: 16 Y # 2#1@RCIJ 0 1 0 0 0 1 1 0 0 We now have an identity matrix in the portion of the augmented matrix corresponding to the original coefficient matrix A, thus we can proceed to obtain the solution while accounting for the row and column exchanges coded in the permutation matrix P.
  • Page 360 Then, for this particular example, in RPN mode, use: [2,-1,41] ` [[1,2,3],[2,0,3],[8,16,-1]] `/ The calculator shows an augmented matrix consisting of the coefficients matrix A and the identity matrix I, while, at the same time, showing the next procedure to calculate: L2 = L2-2⋅L1 stands for “replace row 2 (L2) with the operation L2 –...
  • Page 361 To see the intermediate steps in calculating and inverse, just enter the matrix A from above, and press Y, while keeping the step-by-step option active in the calculator’s CAS. Use the following: [[ 1,2,3],[3,-2,1],[4,2,-1]] `Y After going through the different steps, the solution returned is: What the calculator showed was not exactly a Gauss-Jordan elimination with full pivoting, but a way to calculate the inverse of a matrix by performing a Gauss-Jordan elimination, without pivoting.
  • Page 362: Solution To Linear Systems Using Calculator Functions

    Based on the equation A = C/det(A), sketched above, the inverse matrix, , is not defined if det(A) = 0. Thus, the condition det(A) = 0 defines also a singular matrix. Solution to linear systems using calculator functions The simplest way to solve a system of linear equations, A⋅x = b, in the calculator is to enter b, enter A, and then use the division function /.
  • Page 363 to produce the solution: [X=-1,Y=2,Z = -3]. Function LINSOLVE works with symbolic expressions. Functions REF, rref, and RREF, work with the augmented matrix in a Gaussian elimination approach. Functions REF, rref, RREF The upper triangular form to which the augmented matrix is reduced during the forward elimination part of a Gaussian elimination procedure is known as an "echelon"...
  • Page 364 The diagonal matrix that results from a Gauss-Jordan elimination is called a row-reduced echelon form. Function RREF ( Row-Reduced Echelon Form) The results of this function call is to produce the row-reduced echelon form so that the matrix of coefficients is reduced to an identity matrix. The extra column in the augmented matrix will contain the solution to the system of equations.
  • Page 365: Residual Errors In Linear System Solutions (function Rsd)

    The result is the augmented matrix corresponding to the system of equations: X+Y = 0 X-Y =2 Residual errors in linear system solutions (Function RSD) Function RSD calculates the ReSiDuals or errors in the solution of the matrix equation A⋅x=b, representing a system of n linear equations in n unknowns. We can think of solving this system as solving the matrix equation: f(x) = b - A⋅x = 0.
  • Page 366: Eigenvalues And Eigenvectors

    Eigenvalues and eigenvectors Given a square matrix A, we can write the eigenvalue equation A⋅x = λ⋅x, where the values of λ that satisfy the equation are known as the eigenvalues of matrix A. For each value of λ, we can find, from the same equation, values of x that satisfy the eigenvalue equation.
  • Page 367: Function Egvl

    Using the variable λ to represent eigenvalues, this characteristic polynomial is λ to be interpreted as -2λ -22λ +21=0. Function EGVL Function EGVL (EiGenVaLues) produces the eigenvalues of a square matrix. For example, the eigenvalues of the matrix shown below are calculated in ALG mode using function EGVL: The eigenvalues λ...
  • Page 368: Function Egv

    Change mode to Approx and repeat the entry, to get the following eigenvalues: [(1.38,2.22), (1.38,-2.22), (-1.76,0)]. Function EGV Function EGV (EiGenValues and eigenvectors) produces the eigenvalues and eigenvectors of a square matrix. The eigenvectors are returned as the columns of a matrix, while the corresponding eigenvalues are the components of a vector.
  • Page 369: Function Jordan

    Function JORDAN Function JORDAN is intended to produce the diagonalization or Jordan-cycle decomposition of a matrix. In RPN mode, given a square matrix A, function JORDAN produces four outputs, namely: • The minimum polynomial of matrix A (stack level 4) •...
  • Page 370: Matrix Factorization

    In RPN mode, function MAD generate a number of properties of a square matrix, namely: • the determinant (stack level 4) • the formal inverse (stack level 3), • in stack level 2, the matrix coefficients of the polynomial p(x) defined by (x⋅I-A) ⋅p(x)=m(x)⋅I, •...
  • Page 371: Function Lu

    Function contained in this menu are: LQ, LU, QR,SCHUR, SVD, SVL. Function LU Function LU takes as input a square matrix A, and returns a lower-triangular matrix L, an upper triangular matrix U, and a permutation matrix P, in stack levels 3, 2, and 1, respectively.
  • Page 372: Function Schur

    The Singular Value Decomposition (SVD) of a rectangular matrix A consists × in determining the matrices U, S, and V, such that A ⋅S ⋅V × × × × where U and V are orthogonal matrices, and S is a diagonal matrix. The diagonal elements of S are called the singular values of A and are usually ≥...
  • Page 373: Function Lq

    1: [[-1.03 1.02 3.86 ][ 0 5.52 8.23 ][ 0 –1.82 5.52]] Function LQ The LQ function produces the LQ factorization of a matrix A returning a × lower L trapezoidal matrix, a Q orthogonal matrix, and a P × ×...
  • Page 374: The Quadf Menu

    Finally, = 2X +6XY+2XZ+7ZY The QUADF menu The HP 49 G calculator provides the QUADF menu for operations related to QUADratic Forms. The QUADF menu is accessed through „Ø. This menu includes functions AXQ, CHOLESKY, GAUSS, QXA, and SYLVESTER. Function AXQ...
  • Page 375 Function QXA Function QXA takes as arguments a quadratic form in stack level 2 and a vector of variables in stack level 1, returning the square matrix A from which the quadratic form is derived in stack level 2, and the list of variables in stack level 1.
  • Page 376: Linear Applications

    • The list of variables (stack level 1) For example, 'X^2+Y^2-Z^2+4*X*Y-16*X*Z' ` ['X','Y','Z'] ` GAUSS returns 4: [1 –0.333 20.333] 3: [[1 2 –8][0 –3 16][0 0 1]] 2: ’61/3*Z^2+ -1/3*(16*Z+-3*Y)^2+(-8*z+2*Y+X)^2‘ 1: [‘X’ ‘Y’ ‘Z’] Linear Applications The LINEAR APPLICATIONS menu is available through the „Ø. Information on the functions listed in this menu is presented below by using the calculator’s own help facility.
  • Page 377: Function Ker

    Function KER Function MKISOM Page 11-55...
  • Page 378: Chapter 12 - Graphics

    Chapter 12 Graphics In this chapter we introduce some of the graphics capabilities of the calculator. We will present graphics of functions in Cartesian coordinates and polar coordinates, parametric plots, graphics of conics, bar plots, scatterplots, and a variety of three-dimensional graphs. Graphs options in the calculator To access the list of graphic formats available in the calculator, use the keystroke sequence „ô(D) Please notice that if you are using the...
  • Page 379: Plotting An Expression Of The Form Y = F(x)

    These graph options are described briefly next. Function: for equations of the form y = f(x) in plane Cartesian coordinates Polar: for equations of the from r = f(θ) in polar coordinates in the plane Parametric: for plotting equations of the form x = x(t), y = y(t) in the plane Diff Eq: for plotting the numerical solution of a linear differential equation Conic: for plotting conic equations (circles, ellipses, hyperbolas, parabolas) Truth: for plotting inequalities in the plane...
  • Page 380 return to normal calculator display. The PLOT SET UP window should look similar to this: • Note: You will notice that a new variable, called PPAR, shows up in your soft menu key labels. This stands for Plot PARameters. To see its contents, press ‚@PPAR.
  • Page 381 << X ‘EXP(-X^2/2)/ √(2*π)‘ >>. → Press ƒ, twice, to drop the contents of the stack. • Enter the PLOT WINDOW environment by entering „ò (press them simultaneously if in RPN mode). Use a range of –4 to 4 for H- VIEW, then press @AUTO to generate the V-VIEW automatically.
  • Page 382: Some Useful Plot Operations For Function Plots

    Some useful PLOT operations for FUNCTION plots In order to discuss these PLOT options, we'll modify the function to force it to have some real roots (Since the current curve is totally contained above the x Press ‚@@@Y1@@ to list the contents of the function axis, it has no real roots.) X ‘EXP(-X^2/2)/ √(2*π) ‘...
  • Page 383 • If you move the cursor towards the right-hand side of the curve, by pressing the right-arrow key (™), and press @ROOT, the result now is ROOT: 1.6635... The calculator indicated, before showing the root, that it was found through SIGN REVERSAL. Press L to recover the menu.
  • Page 384: Saving A Graph For Future Use

    curves intercept at two points. Move the cursor near the left intercept point and press @) @ FCN! @ISECT, to get I-SECT: (-0.6834…,0.21585). Press L to recover the menu. • To leave the FCN environment, press @) P ICT (or L) P ICT). •...
  • Page 385: Graphics Of Transcendental Functions

    Move the cursor to the upper left corner of the display, by using the š and — keys. To display the figure currently in level 1 of the stack press L REPL . To return to normal calculator function, press @) P ICT @CANCL. Note: To save printing space, we will not include more graphs produced by following the instructions in this Chapter.
  • Page 386 . Type LN(X). Press ` to equation writer with the expression Y1(X)= return to the PLOT-FUNCTION window. Press L@@@OK@@@ to return to normal calculator display. The next step is to press, simultaneously if in RPN mode, the left-shift key „ and the ò(B) key to produce the PLOT WINDOW - FUNCTION window.
  • Page 387: Graph Of The Exponential Function

    Note: When you press J , your variables list will show new variables called @@@X@@ and @@Y1@@ Press ‚@@Y1@@ to see the contents of this variable. You will get the program << X ‘LN(X)’ >> , which you will → recognize as the program that may result from defining the function ‘Y1(X) = LN(X)’...
  • Page 388: The Ppar Variable

    To add labels to the graph press @EDIT L@) L ABEL. Press @MENU to remove the menu labels, and get a full view of the graph. Press LL@) P ICT! @CANCL to Press ` to return to normal return to the PLOT WINDOW – FUNCTION. calculator display.
  • Page 389: Inverse Functions And Their Graphs

    Inverse functions and their graphs Let y = f(x), if we can find a function y = g(x), such that, g(f(x)) = x, then we say that g(x) is the inverse function of f(x). Typically, the notation g(x) = f is used to denote an inverse function.
  • Page 390: Summary Of Function Plot Operation

    Press @CANCL to return to the PLOT FUNCTION – WINDOW screen. Modify the vertical and horizontal ranges to read: H-View: -8 8, V-View: -4 By selecting these ranges we ensure that the scale of the graph is kept 1 vertical to 1 horizontal. Press @ERASE @DRAW and you will get the plots of the natural logarithm, exponential, and y = x functions.
  • Page 391 Note: the soft menu keys @EDIT and @CHOOS are not available at the same time. One or the other will be selected depending on which input field is highlighted. • Press the AXES soft menu key to select or deselect the plotting of axes in the graph.
  • Page 392 • Use @MOVE° and @MOVE³ to move the selected equation one location up or down, respectively. • Use @CLEAR if you want to clear all the equations currently active in the PLOT – FUNCTION window. The calculator will verify whether or not you want to clear all the functions before erasing all of them.
  • Page 393: Plots Of Trigonometric And Hyperbolic Functions And Their Inverses,

    • Use @ERASE to erase any graph currently existing in the graphics display window. • Use @DRAW to produce the graph according to the current contents of PPAR for the equations listed in the PLOT-FUNCTION window. • Press L to activate the second menu list. •...
  • Page 394: Generating A Table Of Values For A Function

    the function Y=X when plotting simultaneously a function and its inverse to verify their ‘reflection’ about the line Y = X. H-View range V-View range Function Minimum Maximum Minimum Maximum SIN(X) -3.15 3.15 AUTO ASIN(X) -1.2 AUTO SIN & ASIN -3.2 -1.6 COS(X)
  • Page 395: The Tpar Variable

    will be highlighted. If this field is not already set to , press the FUNCTION soft key @CHOOS and select the option, then press @@@OK@@@. FUNCTION • Next, press ˜ to highlight the field in front of the option EQ, and type the function expression: ‘X/(X+10)’...
  • Page 396: Plots In Polar Coordinates

    • The @ZOOM key, when pressed, produces a menu with the options: In, Out, • Decimal, Integer, and Trig. Try the following exercises: • With the option In highlighted, press @@@OK@@@. The table is expanded so • that the x-increment is now 0.25 rather than 0.5. Simply, what the calculator does is to multiply the original increment, 0.5, by the zoom factor, 0.5, to produce the new increment of 0.25.
  • Page 397 • field. Press ³~‚t @@@OK@@@ to The cursor is now in the Indep change the independent variable to θ. • Press L@@@OK@@@ to return to normal calculator display. • Press „ò, simultaneously if in RPN mode, to access the PLOT window (in this case it will be called PLOT –POLAR window).
  • Page 398: Plotting Conic Curves

    will get the equation ‘2*(1-SIN(θ))’ highlighted. Let’s say, we want to plot also the function ‘2*(1-COS(θ))’ along with the previous equation. • Press @@ADD@! , and type 2*„Ü1- T~‚t`, to enter the new equation. • Press @ERASE @DRAW to see the two equations plotted in the same figure. The result is two intersecting cardioids.
  • Page 399 { ‘(X-1)^2+(Y-2)^2=3’ , ‘X^2/4+Y^2/3=1’ } into the variable EQ. These equations we recognize as those of a circle centered at (1,2) with radius √3, and of an ellipse centered at (0,0) with semi-axis lengths a = 2 and b = √3. •...
  • Page 400: Parametric Plots

    centered at the origin (0,0), will extend from -2 to 2 in x, and from -√3 to √3 in y. Notice that for the circle and the ellipse the region corresponding to the left and right extremes of the curves are not plotted. This is the case with all circles or ellipses plotted using as the Conic...
  • Page 401 X(t) = X0 + V0*COS(θ0)*t Y(t) = Y0 + V0*SIN(θ0)*t – 0.5*g*t^2 which will add the variables @@@Y@@@ and @@@X@@@ to the soft menu key labels. To produce the graph itself, follow these steps: • Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP window.
  • Page 402 • Press @ERASE @DRAW to draw the parametric plot. • Press @EDIT L @LABEL @MENU to see the graph with labels. The window parameters are such that you only see half of the labels in the x-axis. • Press L to recover the menu. Press L@) P ICT to recover the original graphics menu.
  • Page 403: Plotting The Solution To Simple Differential Equations

    if in RPN mode). Then, press @ERASE @DRAW. Press @CANCL to return to the PLOT, PLOT WINDOW, or PLOT SETUP screen. Press $, or L@@@OK@@@, to return to normal calculator display. Generating a table for parametric equations In an earlier example we generated a table of values (X,Y) for an expression of the form Y=f(X), i.e., a Function type of graph.
  • Page 404 of differential equations of the form Y'(T) = F(T,Y). For our case, we let Y x and T t, therefore, F(T,Y) f(t,x) = exp(-t Before plotting the solution, x(t), for t = 0 to 5, delete the variables EQ and PPAR.
  • Page 405 • Press L to recover the menu. Press L@) P ICT to recover the original graphics menu. • When we observed the graph being plotted, you'll notice that the graph is not very smooth. That is because the plotter is using a time step that is too large.
  • Page 406: Truth Plots

    Truth plots Truth plots are used to produce two-dimensional plots of regions that satisfy a certain mathematical condition that can be either true or false. For example, suppose that you want to plot the region for X^2/36 + Y^2/9 < 1, proceed as follows: •...
  • Page 407: Plotting Histograms, Bar Plots, And Scatterplots

    You can have more than one condition plotted at the same time if you multiply the conditions. For example, to plot the graph of the points for which X /9 < 1, and X /16 + Y /9 > 1, use the following: •...
  • Page 408 [[3.1,2.1,1.1],[3.6,3.2,2.2],[4.2,4.5,3.3], [4.5,5.6,4.4],[4.9,3.8,5.5],[5.2,2.2,6.6]] ` to store it in ΣDAT, use the function STOΣ (available in the function catalog, ‚N). Press VAR to recover your variables menu. A soft menu key labeled ΣDAT should be available in the stack. The figure below shows the storage of this matrix in ALG mode: To produce the graph: •...
  • Page 409: Scatter Plots

    • Press @CANCL to return to the PLOT WINDOW environment. Then, press $ , or L@@@OK@@@, to return to normal calculator display. The number of bars to be plotted determines the width of the bar. The H- and V-VIEW are set to 10, by default. We changed the V-VIEW to better accommodate the maximum value in column 1 of ΣDAT.
  • Page 410 • Press ˜˜ to highlight the field. Enter 1@@@OK@@@ 2@@@OK@@@ to Cols: select column 1 as X and column 2 as Y in the Y-vs.-X scatter plot. • Press L@@@OK@@@ to return to normal calculator display. • Press „ò, simultaneously if in RPN mode, to access the PLOT WINDOW screen.
  • Page 411: Slope Fields

    • Press LL@) P ICT to leave the EDIT environment. • Press @CANCL to return to the PLOT WINDOW environment. Then, press $ , or L@@@OK@@@, to return to normal calculator display. Slope fields Slope fields are used to visualize the solutions to a differential equation of the form y’...
  • Page 412: Fast 3d Plots

    • Press @CANCL to return to the PLOT WINDOW environment. Then, press $ , or L@@@OK@@@, to return to normal calculator display. If you could reproduce the slope field plot in paper, you can trace by hand lines that are tangent to the line segments shown in the plot. This lines constitute lines of y(x,y) = constant, for the solution of y’...
  • Page 413 • Make sure that ‘X’ is selected as the and ‘Y’ as the variables. Indep: Depnd: • Press L@@@OK@@@ to return to normal calculator display. • Press „ò, simultaneously if in RPN mode, to access the PLOT WINDOW screen. • Keep the default plot window ranges to read: X-Left:-1, X-Right:1, Y-Near:-1, Y-Far: 1, Z-Low: -1, Z-High: 1, Step Indep: 10, Depnd: 8...
  • Page 414: Wireframe Plots

    • When done, press @EXIT. • Press @CANCL to return to PLOT WINDOW. • Press $ , or L@@@OK@@@, to return to normal calculator display. Try also a Fast 3D plot for the surface z = f(x,y) = sin (x •...
  • Page 415 The coordinates XE, YE, ZE, stand for “eye coordinates,” i.e., the coordinates from which an observer sees the plot. The values shown are the default values. The Step Indep: and Depnd: values represent the number of gridlines to be used in the plot. The larger these number, the slower it is to produce the graph.
  • Page 416: Ps-contour Plots

    • Press @ERASE @DRAW to see the surface plot. This time the bulk of the plot is located towards the right –hand side of the display. • Press @CANCL to return to the PLOT WINDOW environment. • Press $ , or L@@@OK@@@, to return to normal calculator display. Try also a Wireframe plot for the surface z = f(x,y) = x •...
  • Page 417 • Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP window. • Change TYPE Ps-Contour. • Press ˜ and type ‘X^2+Y^2’ @@@OK@@@. • Make sure that ‘X’ is selected as the and ‘Y’ as the Indep: Depnd: variables.
  • Page 418: Y-slice Plots

    • Press LL@) P ICT to leave the EDIT environment. • Press @CANCL to return to the PLOT WINDOW environment. Then, press $ , or L@@@OK@@@, to return to normal calculator display. Y-Slice plots Y-Slice plots are animated plots of z-vs.-y for different values of x from the function z = f(x,y).
  • Page 419: Gridmap Plots

    • Press „ô, simultaneously if in RPN mode, to access the PLOT SETUP window. • Press ˜ and type ‘(X+Y)*SIN(Y)’ @@@OK@@@. • Press @ERASE @DRAW to produce the Y-Slice animation. • Press $ to stop the animation. • Press @CANCL to return to the PLOT WINDOW environment. Then, press $ , or L@@@OK@@@, to return to normal calculator display.
  • Page 420: Pr-surface Plots

    • Press $ , or L@@@OK@@@, to return to normal calculator display. Other functions of a complex variable worth trying for Gridmap plots are: (1) SIN((X,Y)) i.e., F(z) = sin(z) (2)(X,Y)^2 i.e., F(z) = z (3) EXP((X,Y)) i.e., F(z) = e (4) SINH((X,Y)) i.e., F(z) = sinh(z) (5) TAN((X,Y)) i.e., F(z) = tan(z)
  • Page 421: The Vpar Variable

    • Press LL@) P ICT @CANCL to return to the PLOT WINDOW environment. • Press $ , or L@@@OK@@@, to return to normal calculator display. The VPAR variable The VPAR (Volume Parameter) variable contains information regarding the “volume” used to produce a three dimensional graph. Therefore, you will see it produced whenever you create a three dimensional plot such as Fast3D, Wireframe, or Pr-Surface.
  • Page 422: Dot+ And Dot

    points, lines, circles, etc. on the graphics screen, as described below. To see how to use these functions we will try the following exercise: First, we get the graphics screen corresponding to the following instructions: • Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP window.
  • Page 423: Mark

    MARK This command allows the user to set a mark point which can be used for a number of purposes, such as: • Start of line with the LINE or TLINE command • Corner for a BOX command • Center for a CIRCLE command Using the MARK command by itself simply leaves an x in the location of the mark (Press L@MARK to see it in action).
  • Page 424: Box

    This command is used to draw a box in the graph. Move the cursor to a clear area of the graph, and press @BOX@. This highlights the cursor. Move the cursor with the arrow keys to a point away, and in a diagonal direction, from Press @BOX@ again.
  • Page 425: Erase

    ERASE The function ERASE clears the entire graphics window. This command is available in the PLOT menu, as well as in the plotting windows accessible through the soft menu keys. MENU Pressing @MENU will remove the soft key menu labels to show the graphic unencumbered by those labels.
  • Page 426: Zooming In And Out In The Graphics Display

    Zooming in and out in the graphics display Whenever you produce a two-dimensional FUNCTION graphic interactively, the first soft-menu key, labeled @) Z OOM, lets you access functions that can be used to zoom in and out in the current graphics display. The ZOOM menu includes the following functions (press Lto move to the next menu): We present each of these functions following.
  • Page 427: Boxz

    BOXZ Zooming in and out of a given graph can be performed by using the soft- menu key BOXZ. With BOXZ you select the rectangular sector (the “box”) that you want to zoom in into. Move the cursor to one of the corners of the box (using the arrow keys), and press @) Z OOM @BOXZ.
  • Page 428: Zintg

    ZINTG Zooms the graph so that the pixel units become user-define units. For example, the minimum PICT window has 131 pixels. When you use ZINTG, with the cursor at the center of the screen, the window gets zoomed so that the x-axis extends from –64.5 to 65.5.
  • Page 429: The Symb/graph Menu

    ‚× (the 4 key) ALGEBRA.. Ch. 5 „Þ (the 1 key) ARITHMETIC.. Ch. 5 „Ö (the 4 key) CALCULUS.. Ch. 13 „Î (the 7 key) SOLVER.. Ch. 6 ‚Ñ (the 8 key) TRIGONOMETRIC.. Ch. 5 „Ð (the 8 key) EXP&LN.. Ch.
  • Page 430 PLOTADD(X^2-X) is similar to „ô but adding this function to EQ: X^2 -1. Using @ERASE @DRAW produces the plot: TABVAL(X^2-1,{1, 3}) produces a list of {min max} values of the function in the interval {1,3}, while SIGNTAB(X^2-1) shows the sign of the function in the interval (-∞,+), with f(x) >...
  • Page 431: Function Draw3dmatrix

    The output is in a graphical format, showing the original function, F(X), the derivative F’(X) right after derivation and after simplification, and finally a table of variation. The table consists of two rows, labeled in the right-hand side. Thus, the top row represents values of X and the second row represents values of F.
  • Page 432: Chapter 13 - Calculus Applications

    Chapter 13 Calculus Applications In this Chapter we discuss applications of the calculator’s functions to operations related to Calculus, e.g., limits, derivatives, integrals, power series, etc. The CALC (Calculus) menu Many of the functions presented in this Chapter are contained in the calculator’s CALC menu, available through the keystroke sequence „Ö...
  • Page 433: Function Lim

    Function lim The calculator provides function lim to calculate limits of functions. This function uses as input an expression representing a function and the value Function lim is available through the where the limit is to be calculated. command catalog (‚N~„l) or through option 2. LIMITS & SERIES…...
  • Page 434: The Deriv&integ Menu

    Derivatives The derivative of a function f(x) at x = a is defined as the limit − − > Some examples of derivatives using this limit are shown in the following screen shots: Functions DERIV and DERVX The function DERIV is used to take derivatives in terms of any independent variable, while the function DERVX takes derivatives with respect to the CAS default variable VX (typically ‘X’).
  • Page 435 Out of these functions DERIV and DERVX are used for derivatives. The other functions include functions related to anti-derivatives and integrals (IBP, INTVX, PREVAL, RISCH, SIGMA, and SIGMAVX), to Fourier series (FOURIER),and to vector analysis (CURL, DIV, HESS, LAPL). Next we discuss functions DERIV and DERVX, the remaining functions are presented either later in this Chapter or in subsequent Chapters.
  • Page 436 The insert cursor ( ) will be located right at the denominator awaiting for the user to enter an independent variable, say, s: ~„s. Then, press the right-arrow key (™) to move to the placeholder between parentheses: Next, enter the function to be differentiated, say, s*ln(s): To evaluate the derivative in the Equation Writer, press the up-arrow key —, four times, to select the entire expression, then, press @EVAL.
  • Page 437 derivatives, utilizing the same symbol for both. The user must keep this distinction in mind when translating results from the calculator to paper. The chain rule The chain rule for derivatives applies to derivatives of composite functions. A general expression for the chain-rule is d{f[g(x)]}/dx = (df/dg)⋅...
  • Page 438: Implicit Derivatives

    Notice that in the expressions where the derivative sign (∂) or function DERIV was used, the equal sign is preserved in the equation, but not in the cases where function DERVX was used. In these cases, the equation was re-written with all its terms moved to the left-hand side of the equal sign.
  • Page 439 maxima) of the function, to plot the derivative, and to find the equation of the tangent line. Try the following example for the function y = tan(x). • Press „ô, simultaneously in RPN mode, to access to the PLOT SETUP window. •...
  • Page 440: Function Domain

    • Press L @PICT @CANCL $ to return to normal calculator display. Notice that the slope and tangent line that you requested are listed in the stack. Function DOMAIN Function DOMAIN, available through the command catalog (‚N), provides the domain of definition of a function as a list of numbers and specifications.
  • Page 441: Function Signtab

    This result indicates that the range of the function     corresponding to the domain D = { -1,5 } is R =   Function SIGNTAB Function SIGNTAB, available through the command catalog (‚N), provides information on the sign of a function through its domain. example, for the TAN(X) function, SIGNTAB indicates that TAN(X) is negative between –π/2 and 0, and positive between 0 and π...
  • Page 442 • Two lists, the first one indicates the variation of the function (i.e., where it increases or decreases) in terms of the independent variable VX, the second one indicates the variation of the function in terms of the dependent variable. •...
  • Page 443: Using Derivatives To Calculate Extreme Points

    The interpretation of the variation table shown above is as follows: the function F(X) increases for X in the interval (-∞, -1), reaching a maximum equal to 36 at X = -1 Then, F(X) decreases until X 11/3, reaching a minimum of Also, at X = ±∞, F(X)= 400/27 After that F(X) increases until reaching +∞...
  • Page 444: Higher-order Derivatives

    For example, to determine where the critical points of function 'X^3-4*X^2- 11*X+30' occur, we can use the following entries in ALG mode: We find two critical points, one at x = 11/3 and one at x = -1. To evaluate the second derivative at each point use: The last screen shows that f”(11/3) = 14, thus, x = 11/3 is a relative minimum.
  • Page 445 Anti-derivatives and integrals An anti-derivative of a function f(x) is a function F(x) such that f(x) = dF/dx. For example, since d(x ) /dx = 3x , an anti-derivative of f(x) = 3x is F(x) = x + C, where C is a constant. One way to represent an anti-derivative is as a ∫...
  • Page 446 Please notice that functions SIGMAVX and SIGMA are designed for integrands that involve some sort of integer function like the factorial (!) function shown above. Their result is the so-called discrete derivative, i.e., one defined for integer numbers only. Definite integrals In a definite integral of a function, the resulting anti-derivative is evaluated at the upper and lower limit of an interval (a,b) and the evaluated values ∫...
  • Page 447 At this point, you can press ` to return the integral to the stack, which will show the following entry (ALG mode shown): This is the general format for the definite integral when typed directly into the stack, i.e., ∫ (lower limit, upper limit, integrand, variable of integration) Pressing ` at this point will evaluate the integral in the stack: The integral can be evaluated also in the Equation Writer by selecting the entire expression an using the soft menu key @EVAL.
  • Page 448 Notice the application of the chain rule in the first step, leaving the derivative of the function under the integral explicitly in the numerator. In the second step, the resulting fraction is rationalized (eliminating the square root from the denominator), and simplified. The final version is shown in the third step. Each step is shown by pressing the @EVAL menu key, until reaching the point where further application of function EVAL produce no more changes in the expression.
  • Page 449: Integrating An Equation

    Integrating an equation Integrating an equation is straightforward, the calculator simply integrates both sides of the equation simultaneously, e.g., Techniques of integration Several techniques of integration can be implemented in the calculators, as shown in the following examples. Substitution or change of variables ∫...
  • Page 450 The last four steps show the progression of the solution: a square root, followed by a fraction, a second fraction, and the final result. This result can be simplified by using function @SIMP, to read: Integration by parts and differentials A differential of a function y = f(x), is defined as dy = f’(x) dx, where f’(x) is the derivative of f(x).
  • Page 451 Thus, we can use function IBP to provide the components of an integration by parts. The next step will have to be carried out separately. It is important to mention that the integral can be calculated directly by using, for example, Integration by partial fractions Function PARTFRAC, presented in Chapter 5, provides the decomposition of a fraction into partial fractions.
  • Page 452: Integration With Units

    Improper integrals These are integrals with infinite limits of integration. Typically, an improper integral is dealt with by first calculating the integral as a limit to infinity, e.g., ∞ ε ∫ ∫ ε → ∞ Using the calculator, we proceed as follows: Alternatively, you can evaluate the integral to infinity from the start, e.g., Integration with units An integral can be calculated with units incorporated into the limits of...
  • Page 453 If you enter the integral with the CAS set to Exact mode, you will be asked to change to Approx mode, however, the limits of the integral will be shown in a different format as shown here: These limits represent 1×1_mm and 0×1_mm, which is the same as 1_mm and 0_mm, as before.
  • Page 454: Infinite Series

    Infinite series ∑ − An infinite series has the form . The infinite series typically starts with indices n = 0 or n = 1. Each term in the series has a coefficient h(n) that depends on the index n. Taylor and Maclaurin’s series A function f(x) can be expanded into an infinite series around a point x=x using a Taylor’s series, namely,...
  • Page 455 ∑ ∑ i.e., The polynomial P (x) is referred to as Taylor’s polynomial. The order of the residual is estimated in terms of a small quantity h = x-x , i.e., evaluating the polynomial at a value of x very close to x .
  • Page 456 Function TAYLR produces a Taylor series expansion of a function of any variable x about a point x = a for the order k specified by the user. Thus, the function has the format TAYLR(f(x-a),x,k). For example, Function SERIES produces a Taylor polynomial using as arguments the function f(x) to be expanded, a variable name alone (for Maclaurin’s series) or an expression of the form ‘variable = value’...
  • Page 457 In the right-hand side figure above, we are using the line editor to see the series expansion in detail. Page 13-26...
  • Page 458: Chapter 14 - Multi-variate Calculus Applications

    Chapter 14 Multi-variate Calculus Applications Multi-variate calculus refers to functions of two or more variables. In this Chapter we discuss the basic concepts of multi-variate calculus including partial derivatives and multiple integrals. Multi-variate functions A function of two or more variables can be defined in the calculator by using the DEFINE function („à).
  • Page 459 → Similarly, → We will use the multi-variate functions defined earlier to calculate partial derivatives using these definitions. Here are the derivatives of f(x,y) with respect to x and y, respectively: Notice that the definition of partial derivative with respect to x, for example, requires that we keep y fixed while taking the limit as h 0.
  • Page 460: Higher-order Derivatives

    therefore, with DERVX you can only calculate derivatives with respect to X. Some examples of first-order partial derivatives are shown next: Higher-order derivatives The following second-order derivatives can be defined The last two expressions represent cross-derivatives, the partial derivatives signs in the denominator shows the order of derivation. In the left-hand side, the derivation is taking first with respect to x and then with respect to y, and in the right-hand side, the opposite is true.
  • Page 461: The Chain Rule For Partial Derivatives

    Third-, fourth-, and higher order derivatives are defined in a similar manner. To calculate higher order derivatives in the calculator, simply repeat the derivative function as many times as needed. Some examples are shown below: The chain rule for partial derivatives Consider the function z = f(x,y), such that x = x(t), y = y(t).
  • Page 462: Total Differential Of A Function Z = Z(x,y)

    Total differential of a function z = z(x,y) From the last equation, if we multiply by dt, we get the total differential of the ∂z/∂x) function z = z(x,y), i.e., dz = ⋅ dx + (∂z/∂y) ⋅ A different version of the chain rule applies to the case in which z = f(x,y), x = x(u,v), y = y(u,v), so that z = f[x(u,v), y(u,v)].
  • Page 463: Using Function Hess To Analyze Extrema

    We find critical points at (X,Y) = (1,0), and (X,Y) = (-1,0). To calculate the discriminant, we proceed to calculate the second derivatives, fXX(X,Y) = ∂ , fXY(X,Y) = ∂ f/∂X/∂Y, and fYY(X,Y) = ∂ f/∂X f/∂Y The last result indicates that the discriminant is ∆ = -12X, thus, for (X,Y) = (1,0), ∆...
  • Page 464 Applications of function HESS are easier to visualize in the RPN mode. Consider as an example the function φ(X,Y,Z) = X + XY + XZ, we’ll apply function HESS to function φ in the following example. The screen shots show the RPN stack before and after applying function HESS.
  • Page 465: Multiple Integrals

    = ∂ φ/∂X = ∂ φ/∂X The resulting matrix has elements a = 6., a = -2., = ∂ φ/∂X∂Y = 0. The discriminant, for this critical point s2(1,0) and a ⋅ is ∆ = (∂ f/∂x (∂ f/∂y )-[∂ f/∂x∂y] = (6.)(-2.) = -12.0 <...
  • Page 466: Jacobian Of Coordinate Transformation

    Jacobian of coordinate transformation Consider the coordinate transformation x = x(u,v), y = y(u,v). The Jacobian of this transformation is defined as det( When calculating an integral using such transformation, the expression to use φ φ | )] dydx dudv , where R’...
  • Page 467 β θ φ θ φ θ θ rdrd α θ where the region R’ in polar coordinates is R’ = {α < θ < β, f(θ) < r < g(θ)}. Double integrals in polar coordinates can be entered in the calculator, making sure that the Jacobian |J| = r is included in the integrand.
  • Page 468: Chapter 15 - Vector Analysis Applications

    Chapter 15 Vector Analysis Applications In this Chapter we present a number of functions from the CALC menu that apply to the analysis of scalar and vector fields. The CALC menu was presented in detail in Chapter 13. In particular, in the DERIV&INTEG menu we identified a number of functions that have applications in vector analysis, namely, CURL, DIV, HESS, LAPL.
  • Page 469: A Program To Calculate The Gradient

    At any particular point, the maximum rate of change of the function occurs in the direction of the gradient, i.e., along a unit vector u = The value of that directional derivative is equal to the magnitude of the gradient at any point D (x,y,z) = | = | The equation (x,y,z) = 0 represents a surface in space.
  • Page 470: Potential Of A Gradient,

    n independent variables (x , …,x ), and a vector of the functions [‘x ’ ‘x ’…’x ’]. Function HESS returns the Hessian matrix of the function , defined as the matrix H = [h ] = [ / x ], the gradient of the function with respect to the n-variables, grad f = [ , …...
  • Page 471: Divergence

    function (x,y,z) does not exist. In such case, function POTENTIAL returns an error message. For example, the vector field F(x,y,z) = (x+y)i + (x-y+z)j + xzk, does not have a potential function associated with it, since, f/ z h/ x. The calculator response in this case is shown below: Divergence The divergence of a vector function, F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, is defined by taking a “dot-product”...
  • Page 472: Curl

    Curl The curl of a vector field F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, is defined by a “cross-product” of the del operator with the vector field, i.e., curl The curl of vector field can be calculated with function CURL. For example, for the function F(X,Y,Z) = [XY,X ,YZ], the curl is calculated as follows: Irrotational fields and potential function In an earlier section in this chapter we introduced function POTENTIAL to...
  • Page 473: Vector Potential

    As an example, in an earlier example we attempted to find a potential function for the vector field F(x,y,z) = (x+y)i + (x-y+z)j + xzk, and got an error message back from function POTENTIAL. To verify that this is a rotational field (i.e., 0), we use function CURL on this field: On the other hand, the vector field F(x,y,z) = xi + yj + zk, is indeed...
  • Page 474 produces the vector potential function Φ = [0, ZYX-2YX, Y-(2ZX-X)], which is different from Φ . The last command in the screen shot shows that indeed F = Φ . Thus, a vector potential function is not uniquely determined. The components of the given vector field, F(x,y,z) = f(x,y,z)i+g(x,y,z)j +h(x,y,z)k, and those of the vector potential function, Φ(x,y,z) = (x,y,z)i+ (x,y,z)j+ (x,y,z)k, are related by f = / y -...
  • Page 475: Chapter 16 - Differential Equations ,

    Chapter 16 Differential Equations In this Chapter we present examples of solving ordinary differential equations (ODE) using calculator functions. A differential equation is an equation involving derivatives of the independent variable. In most cases, we seek the dependent function that satisfies the differential equation. Basic operations with differential equations In this section we present some uses of the calculator for entering, checking and visualizing the solution of ODEs.
  • Page 476: Checking Solutions In The Calculator

    ∂ ∂ The result is ‘∂ ’. This format x(u(x)))+3*u(x)* x(u(x))+u^2=1/x shows up in the screen when the _Textbook option in the display setting (H@) D ISP) is not selected. Press ˜ to see the equation in the Equation Writer. An alternative notation for derivatives typed directly in the stack is to use ‘d1’...
  • Page 477: Slope Field Visualization Of Solutions

    result by using function EVAL to verify the solution. For example, to check that u = A sin ω + ω ⋅u = 0, use the t is a solution of the equation d u/dt following: In ALG mode: SUBST(‘∂t(∂t(u(t)))+ ω0^2*u(t) = 0’,‘u(t)=A*SIN (ω0*t)’) ` EVAL(ANS(1)) ` In RPN mode: ‘∂t(∂t(u(t)))+ ω0^2*u(t) = 0’...
  • Page 478: The Calc/diff Menu

    The CALC/DIFF menu The DIFFERENTIAL EQNS.. sub-menu within the CALC („Ö) menu provides functions for the solution of differential equations. The menu is listed below with system flag 117 set to CHOOSE boxes: These functions are briefly described next. They will be described in more detail in later parts of this Chapter.
  • Page 479: Function Ldec

    Function LDEC The calculator provides function LDEC (Linear Differential Equation Command) to find the general solution to a linear ODE of any order with constant coefficients, whether it is homogeneous or not. This function requires you to provide two pieces of input: •...
  • Page 480 of constants result from factoring out the exponential terms after the Laplace transform solution is obtained. Example 2 – Using the function LDEC, solve the non-homogeneous ODE: y/dx -4⋅(d y/dx )-11⋅(dy/dx)+30⋅y = x Enter: 'X^2' ` 'X^3-4*X^2-11*X+30' ` LDEC The solution, shown partially here in the Equation Writer, is: Replacing the combination of constants accompanying the exponential terms –3x ⋅e...
  • Page 481: Function Desolve

    Allow the calculator about ten seconds to produce the result: ‘X^2 = X^2’. Example 3 - Solving a system of linear differential equations with constant coefficients. Consider the system of linear differential equations: ’(t) + 2x ’(t) = 0, ’(t) + x ’(t) = 0.
  • Page 482: The Variable Odetype

    ⋅y(x) = 5. dy/dx + x In the calculator use: 'd1y(x)+x^2*y(x)=5' ` 'y(x)' ` DESOLVE solution provided {‘y (INT(5*EXP(xt^3/3),xt,x)+cC0)*1/EXP(x^3/3))’ }, i.e., exp( exp( The variable ODETYPE You will notice in the soft-menu key labels a new variable called @ODETY (ODETYPE). This variable is produced with the call to the DESOL function and holds a string showing the type of ODE used as input for DESOLVE.
  • Page 483 Next, we can write dy/dx = (C + exp x)/x = C/x + e In the calculator, you may try to integrate: ‘d1y(x) = (C + EXP(x))/x’ ` ‘y(x)’ ` DESOLVE The result is { ‘y(x) = INT((EXP(xt)+C)/xt,xt,x)+C0’ }, i.e., Performing the integration by hand, we can only get it as far as: because the integral of exp(x)/x is not available in closed form.