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The present Guide contains examples that illustrate the use of the basic calculator functions and operations. The chapters in this user's manual are organized by subject in order of difficulty: from the setting of calculator modes, to real and complex number calculations, operations with lists, vectors, and matrices, graphics, calculus applications, vector analysis, differential equations, probability and statistics.
Table of Contents Chapter 1 – Getting Started 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 The TOOL menu, 1-3 Setting time and date, 1-3 Introducing the calculator’s keyboard, 1-4...
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Creating algebraic expressions, 2-4 Using the Equation Writer (EQW) to create expressions, 2-5 Creating arithmetic expressions, 2-5 Creating algebraic expressions, 2-8 Organizing data in the calculator, 2-9 The HOME directory, 2-9 Subdirectories, 2-9 Variables, 2-10 Typing variable names, 2-10 Creating variables, 2-11...
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Defining and using functions, 3-16 Reference, 3-18 Chapter 4 – Calculations with complex numbers Definitions, 4-1 Setting the calculator to COMPLEX mode, 4-1 Entering complex numbers, 4-2 Polar representation of a complex number, 4-2 Simple operations with complex numbers, 4-3...
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The PROPFRAC function, 5-11 The PARTFRAC function, 5-11 The FCOEF function, 5-11 The FROOTS function, 5-12 Step-by-step operations with polynomials and fractions, 5-12 Reference, 5-13 Chapter 6 – Solution to equations Symbolic solution of algebraic equations, 6-1 Function ISOL, 6-1 Function SOLVE, 6-2 Function SOLVEVX, 6-4 Function ZEROS, 6-4...
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The SEQ function, 7-6 The MAP function, 7-6 Reference, 7-6 Chapter 8 – Vectors , 8-1 Entering vectors, 8-1 Typing vectors in the stack, 8-1 Storing vectors into variables in the stack, 8-2 Using the matrix writer (MTRW) to enter vectors, 8-2 Simple operations with vectors, 8-5 Changing sign, 8-5 Addition, subtraction, 8-5...
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Solution by “division” of matrices, 9-10 References, 9-10 Chapter 10 – Graphics , 10-1 Graphs options in the calculator, 10-1 Plotting an expression of the form y = f(x), 10-2 Generating a table of values for a function, 10-3 Fast 3D plots, 10-5 Reference, 10-8 Chapter 11 –...
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Solution to linear and non-linear equations, 14-1 Function LDEC, 14-2 Function DESOLVE, 14-3 The variable ODETYPE, 14-4 Laplace Transforms, 14-5 Laplace transforms and inverses in the calculator, 14-5 Fourier series, 14-6 Function FOURIER, 14-6 Fourier series for a quadratic function, 14-6 Reference, 14-8 Chapter 15 –...
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Chapter 17 – Numbers in Different Bases The BASE menu, 17-1 Writing non-decimal numbers, 17-1 Reference, 17-2 Warranty – Service, W-2 Regulatory information, W-4 , 17-1 Page TOC-8...
The following exercises are aimed at getting you acquainted with the hardware of your calculator. Batteries The calculator uses 3 AAA batteries as main power and a CR2032 lithium battery for memory backup. Before using the calculator, please install the batteries according to the following procedure.
The $(hold) - key combination produces a lighter display Contents of the calculator’s display Turn your calculator on once more. At the top of the display you will have two lines of information that describe the settings of the calculator.
The six labels associated with the keys A through F form part of a menu of functions. Since the calculator has only six soft menu keys, it only display 6 labels at any point in time. However, a menu can have more than six entries.
TOOL menu is to press the I key (third key from the left in the second row of keys from the top of the keyboard). Setting time and date See Chapter 1 in the calculator’s user's guide to learn how to set time and date. Introducing the calculator’s keyboard The figure below shows a diagram of the calculator’s keyboard with the...
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the blue ALPHA key, key (7,1), can be combined with some of the other keys to activate the alternative functions shown in the keyboard. For example, the P 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 „´...
… (CAT ), and ~ (P) keys. For detailed information on the calculator keyboard operation referee to Appendix B in the calculator’s user's guide. Selecting calculator modes This section assumes that you are now at least partially familiar with the use of choose and dialog boxes (if you are not, please refer to appendix A in the user's guide).
The calculator offers two operating modes: Reverse Polish Notation (RPN) mode. 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.
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The display, for the RPN mode looks as follows: 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 Page 1-8...
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3 upwards to occupy stack level 2. Finally, by pressing +, we are telling the calculator to apply the operator, or program, + to the objects occupying levels 1 and 2. The result, 5, is then placed in level 1.
Changing the number format allows you to customize the way real numbers are displayed by the calculator. You will find this feature extremely useful in operations with powers of tens or to limit the number of decimals in a result.
This mode is the most used mode as it shows numbers in the most familiar notation. Press the !!@@OK#@ soft menu key, with the Number format set to Std, return calculator 123.4567890123456 (with16 significant figures). Press the ` key. The number is rounded to the maximum 12 significant figures, and is displayed as follows: •...
Press the !!@@OK#@ soft menu key return to the calculator display. number now is shown as: Notice how the number is rounded, not truncated. Thus, the number 123.4567890123456, for this setting, is displayed as 123.457, and not as 123.456 because the digit after 6 is > 5): •...
Fixed number of decimals in an earlier example). 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.
@CHECK soft menu key (i.e., the B key). The input form will look as follows: • Press the ! ! @@OK#@ soft menu key return to the calculator display. number 123.456789012, entered earlier, now is shown as: Angle Measure Trigonometric functions, for example, require arguments representing plane angles.
To learn more about complex numbers and vectors, see Chapters 4 and 8, respectively, in this Guide. There are three coordinate systems available in the calculator: Rectangular (RECT), Cylindrical (CYLIN), and Spherical (SPHERE). To change coordinate system: •...
To see the optional CAS settings use the following: • Press the H button to activate the CALCULATOR MODES input form. • To change CAS settings press the @@ CAS@@ soft menu key. The default values of the CAS setting are shown below: •...
After having selected and unselected all the options that you want in the CAS MODES input form, press the @@@OK@@@ soft menu key. This will take you back to the CALCULATOR MODES input form. calculator display at this point, press the @@@OK@@@ soft menu key once more. Explanation of CAS settings •...
DISPLAY MODES input form, press the @@@OK@@@ soft menu key. This will take you back to the CALCULATOR MODES input form. calculator display at this point, press the @@@OK@@@ soft menu key once more. Selecting the display font First, press the H button to activate the CALCULATOR MODES input form.
When done with a font selection, press the @@@OK@@@ soft menu key to go back to the CALCULATOR MODES input form. to normal calculator display at this point, press the @@@OK@@@ soft menu key once more and see how the stack display change to accommodate the different font.
Selecting properties of the Stack 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. Press the down arrow key, ˜, twice, to get to the Stack line.
Selecting properties of the equation writer (EQW) 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. Press the down arrow key, ˜, three times, to get to the EQW (Equation Writer) line.
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.
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– see Chapter 1): In this case, when the expression is entered directly into the stack, as soon as you press `, the calculator will attempt to calculate a value for the expression. If the expression is entered between quotes, however, the calculator will reproduce the expression as entered.
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An alternative way to evaluate the expression entered earlier between quotes is by using the option …ï. We will now enter the expression used above when the calculator is set to the RPN operating mode. We also set the CAS to Exact and the display to Textbook.
Algebraic expressions include not only numbers, but also variable names. As an example, we will enter the following algebraic expression: 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: 2*~l*R„Ü1+~„x/~r™/„...
Entering this expression when the calculator is set in the RPN mode is exactly the same as this Algebraic mode exercise. For additional information on editing algebraic expressions in the calculator’s display or stack see Chapter 2 in the calculator’s user's guide.
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The cursor is shown as a left-facing key. The cursor indicates the current edition location. For example, for the cursor in the location indicated above, type now: *„Ü5+1/3 The edited expression looks as follows: Suppose that you want to replace the quantity between parentheses in the denominator (i.e., 5+1/3) with (5+ /2).
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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: 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...
Creating algebraic expressions An algebraic expression is very similar to an arithmetic expression, except that English and Greek letters may be included. The process of creating an algebraic expression, therefore, follows the same idea as that of creating an arithmetic expression, except that use of the alphabetic keyboard is included. To illustrate the use of the Equation Writer to enter an algebraic equation we will use the following example.
Chapter 2 of the calculator’s user's guide. Organizing data in the calculator You can organize data in your calculator by storing variables in a directory tree. The basis of the calculator’s directory tree is the HOME directory described next.
‘A’, ‘B’, ‘a’, ‘b’, ‘ ’, ‘ ’, ‘A1’, ‘AB12’, ‘ A12’,’Vel’,’Z0’,’z1’, etc. A variable can not have the same name as a function of the calculator. The reserved calculator variable names are the following: ALRMDAT, CST, EQ, EXPR, IERR, IOPAR, MAXR, MINR, PICT, PPAR, PRTPAR, VPAR, ZPAR, der_, e, i, n1,n2, …, s1, s2, …, DAT, PAR, ,...
Try the following exercises: ³~~math1~` 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): Creating variables The simplest way to create a variable is by using the K . The following...
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Press ` to create the variable. The variable is now shown in the soft menu key labels: The following are the keystrokes required to enter the remaining variables: A12: 3V5K~a12` Q: ³~„r/„Ü ~„m+~„r™™ K~q` R: „Ô3‚í2‚í1™ K~r` z1: 3+5*„¥ K~„z1` (Accept change to Complex mode if asked).
RPN mode (Use I\@@OK@@ to change to RPN mode). Use the following keystrokes to store the value of –0.25 into variable : 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 .
p1: ‚å‚é~„r³„ì* ~„rQ2™™™ ³ ~„p1™` K. The screen, at this point, will look as follows: You will see six of the seven variables listed at the bottom of the screen: p1, z1, R, Q, A12, . Checking variables contents The simplest way to check a variable content is by pressing the soft menu key label for the variable.
@@@ @@ Listing the contents of all variables in the screen Use the keystroke combination ‚˜ to list the contents of all variables in the screen. For example: Press $ to return to normal calculator display. Page 2-15...
Deleting variables The simplest way of deleting variables is by using function PURGE. This function can be accessed directly by using the TOOLS menu (I), or by using the FILES menu „¡@@OK@@ . Using function PURGE in the stack in Algebraic mode Our variable list contains variables p1, z1, Q, R, and .
Soft MENUs, and vice versa, through an exercise. Although not applied to a specific example, the present exercise shows the two options for menus in the calculator (CHOOSE boxes and soft MENUs). In this exercise, we use the ORDER command to reorder variables in a directory,...
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There is an alternative way to access these menus as soft MENU keys, by setting system flag 117. (For information on Flags see Chapters 2 and 24 in the calculator’s user's guide). To set this flag try the following: H @FLAGS! ———————...
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Press the @CHECK! soft menu key to set flag 117 to soft MENU. The screen will reflect that change: 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 „°.
References For additional information on entering and manipulating expressions in the display or in the Equation Writer see Chapter 2 of the calculator’s user's guide. For CAS (Computer Algebraic System) settings, see Appendix C in the calculator’s user’s guide. For information on Flags see, Chapter 24 in the calculator’s user’s guide.
Chapter 3 Calculations with real numbers This chapter demonstrates the use of the calculator for operations and functions related to real numbers. The user should be acquainted with the keyboard to identify certain functions available in the keyboard (e.g., SIN, COS, TAN, etc.).
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6.3` 8.5 - 4.2` 2.5 * 2.3` 4.5 / Alternatively, in RPN mode, you can separate the operands with a space (#) before pressing the operator key. Examples: 3.7#5.2 + 6.3#8.5 - 4.2#2.5 * 2.3#4.5 / • Parentheses („Ü) can be used to group operations, as well as to enclose arguments of functions.
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„Ê \2.32` Example in RPN mode: 2.32\„Ê • The square function, SQ, is available through „º. Example in ALG mode: „º\2.3` Example in RPN mode: 2.3\„º The square root function, √, is available through the R key. When calculating in the stack in ALG mode, enter the function before the argument, e.g., R123.4` In RPN mode, enter the number first, then the function, e.g.,...
enter the function XROOT followed by the arguments (y,x), separated by commas, e.g., ‚»3‚í 27` In RPN mode, enter the argument y, first, then, x, and finally the function call, e.g., 27`3‚» • Logarithms of base 10 are calculated by the keystroke combination ‚Ã...
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LOG, ALOG, LN, EXP, SIN, COS, TAN, ASIN, ACOS, ATAN, can be combined with the fundamental operations (+-*/) to form more complex expressions. The Equation Writer, whose operations is described in Chapter 2, is ideal for building such expressions, regardless of the calculator operation mode. S30`...
Using calculator menus: 1. We will describe in detail the use of the 4. HYPERBOLIC.. menu in this section with the intention of describing the general operation of calculator menus. Pay close attention to the process for selecting different options.
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For example, in ALG mode, the keystroke sequence to calculate, say, tanh(2.5), is the following: „´4 @@OK@@ 5 @@OK@@ 2.5` In the RPN mode, the keystrokes to perform this calculation are the following: 2.5`„´4 @@OK@@ 5 @@OK@@ The operations shown above assume that you are using the default setting for system flag 117 (CHOOSE boxes).
(2.5) = 0.98661.. expm(2.0) = 6.38905…. 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.
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Option 1. Tools.. contains functions used to operate on units (discussed later). Options 3. Length.. through 17.Viscosity.. contain menus with a number of units for each of the quantities described. For example, selecting option 8. Force.. shows the following units menu: The user will recognize most of these units (some, e.g., dyne, are not used very often nowadays) from his or her physics classes: N = newtons, dyn = dynes, gf = grams –...
‚˜, e.g., for the @) E RG set of units the following labels will be listed: Note: Use the L key or the „«keystroke sequence to navigate through the menus. Available units For a complete list of available units see Chapter 3 in the calculator’s user's guide. Page 3-10...
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: 5‚Û8@@OK@@ @@OK@@ Notice that the underscore is entered automatically when the RPN mode is active.
_____________________________________________________ (*) In the SI system, this prefix is da rather than D. Use D for deka in the calculator, however. To enter these prefixes, simply type the prefix using the ~ keyboard. For example, to enter 123 pm (picometer), use: 123‚Ý~„p~„m...
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which shows as 65_(m⋅yd). To convert to units of the SI system, use function UBASE (find it using the command catalog, ‚N): Note: Recall that the ANS(1) variable is available through the keystroke combination „î(associated with the ` key). To calculate a division, say, 3250 mi / 50 h, enter it as (3250_mi)/(50_h) ` which transformed to SI units, with function UBASE, produces: Addition and subtraction can be performed, in ALG mode, without using...
Examples of function CONVERT are shown below. Examples of the other UNIT/TOOLS functions are available in Chapter 3 of the calculator’s user's guide. For example, to convert 33 watts to btu’s use either of the following entries:...
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The soft menu keys corresponding to this CONSTANTS LIBRARY screen include the following functions: when selected, constants values are shown in SI units (*) ENGL when selected, constants values are shown in English units (*) UNIT when selected, constants are shown with units attached VALUE when selected, constants are shown without units STK copies value (with or without units) to the stack QUIT...
To copy the value of Vm to the stack, select the variable name, and press ! STK, then, press @ UIT@. For the calculator set to the ALG, the screen will look like this: The display shows what is called a tagged value, .
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<< 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 (See Chapters 20 and 21 in the calculator’s user's guide).
2`@@@H@@@ . The other examples shown above can be entered by using: 1.2`@@@H@@@ , 2/3`@@@H@@@ . Reference Additional information on operations with real numbers with the calculator is contained in Chapter 3 of the user’s guide. Page 3-18...
θ . The negative of z, –z = -x-iy = - re z about the origin. Setting the calculator to COMPLEX mode To work with complex numbers select the CAS complex mode: H) @ @CAS@ 2˜˜™ @@CHK@ The COMPLEX mode will be selected if the CAS MODES screen shows the option _Complex checked off, i.e.,...
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:...
= r⋅e . You can enter this complex number into the calculator by using an ordered pair of the form (r, ∠θ). The angle symbol (∠) can be entered as ~‚6. For example, the complex number z = 1.5i...
1/(3+4i) = (0.12, -0.16) ; The CMPLX menus There are two CMPLX (CoMPLeX numbers) menus available in the calculator. One is available through the MTH menu (introduced in Chapter 3) and one directly into the keyboard (‚ß). The two CMPLX menus are presented next.
CONJ(z): Produces the complex conjugate of z Examples of applications of these functions are shown next. Recall that, for ALG mode, the function must precede the argument, while in RPN mode, you enter the argument first, and then select the function. Also, recall that you can get these functions as soft menu labels by changing the setting of system flag 117 (See Chapter 3).
Note: When using trigonometric functions and their inverses with complex numbers the arguments are no longer angles. Therefore, the angular measure selected for the calculator has no bearing in the calculation of these functions with complex arguments. Function DROITE: equation of a straight line...
Function DROITE is found in the command catalog (‚N). Reference Additional information on complex number operations is presented in Chapter 4 of the calculator’s user’s guide. Page 4-7...
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: •...
After building the object, press to show it in the stack (ALG and RPN modes shown below): 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.
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In ALG mode, the following keystrokes will show a number of operations with the algebraics contained in variables @@A @@ and @@A @@ (press J to recover variable menu): @@A @@ + @@A @@ ` @@A @@ + @@A @@ ` @@A @@ 8 @@A @@ ` @@A @@ / @@A @@ ` ‚¹@@A @@...
Rather than listing the description of each function in this manual, the user is invited to look up the description using the calculator’s help facility: I L @) H ELP@ ` . To locate a particular function, type the first letter of the function.
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Copy the examples provided onto your stack by pressing @ECHO!. For example, for the EXPAND entry shown above, press the @ECHO! soft menu key to get the following example copied to the stack (press ` to execute the command): Thus, we leave for the user to explore the applications of the functions in the ALG (or ALGB) menu.
Operations with transcendental functions The calculator offers a number of functions that can be used to replace expressions containing logarithmic and exponential functions („Ð), as well as trigonometric functions (‚Ñ). Expansion and factoring using log-exp functions The „Ð produces the following menu: Information and examples on these commands are available in the help facility of the calculator.
(asin(x)). Description of these commands and examples of their applications are available in the calculator’s help facility (IL@HELP). The user is invited to explore this facility to find information on the commands in the TRIG menu. Functions in the ARITHMETIC menu The ARITHMETIC menu is triggered through the keystroke combination „Þ...
The functions associated with the ARITHMETIC submenus: INTEGER, POLYNOMIAL, MODULO, and PERMUTATION, are the following: Additional information on applications of the ARITHMETIC menu functions are presented in Chapter 5 in the calculator’s user's guide. Polynomials Polynomials are algebraic expressions consisting of one or more terms containing decreasing powers of a given variable.
The variable VX A variable called VX exists in the calculator’s {HOME CASDIR} directory that takes, by default, the value of ‘X’. This is the name of the preferred independent variable for algebraic and calculus applications. Avoid using the variable VX in your programs or equations, so as to not get it confused with the CAS’...
ARITHMETIC menu, it must be accessed from the function catalog (‚N). Example: PEVAL([1,5,6,1],5) = 281. Additional applications of polynomial functions are presented in Chapter 5 in the calculator’s user's guide. Fractions Fractions can be expanded and factored by using functions EXPAND and FACTOR, from the ALG menu (‚×). For example: EXPAND(‘(1+X)^3/((X-1)(X+3))’) = ‘(X^3+3*X^2+3*X+1)/(X^2+2*X-3)’...
The PROPFRAC function The function PROPFRAC converts a rational fraction into a “proper” fraction, i.e., an integer part added to a fractional part, if such decomposition is possible. For example: 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.
This is very useful to see the steps of a synthetic division. The example of dividing is shown in detail in Appendix C of the calculator’s user's guide. The following example shows a lengthier synthetic division: −...
Reference Additional information, definitions, and examples of algebraic and arithmetic operations are presented in Chapter 5 of the calculator’s user’s guide. Page 5-13...
ZEROS provides the zeros, or roots, of a polynomial. Function ISOL Function ISOL(Equation, variable) will produce the solution(s) to Equation by isolating variable. For example, with the calculator set to ALG mode, to solve for t in the equation at -bt = 0 we can use the following:...
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.
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SOLVE produces four solutions, shown in the last output line. The very last solution is not visible because the result occupies more characters than the width of the calculator’s screen. However, you can still see all the solutions by using the down arrow key (˜), which triggers the line editor (this operation can be used to access any output line that is wider than the calculator’s...
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.
Numerical Solver features of the calculator. Numerical solver menu The calculator provides a very powerful environment for the solution of single algebraic or transcendental equations. To access this environment we start the numerical solver (NUM.SLV) by using ‚Ï. This produces a drop-...
Following, we present applications of items 3. Solve poly.., 5. Solve finance, and 1. Solve equation.., in that order. Appendix 1-A, in the calculator’s user's guide, contains instructions on how to use input forms with examples for the numerical solver applications. Item 6. MSLV (Multiple equation SoLVer) will be presented later in this Chapter.
0.766, 0.632), (-0.766, -0.632). Generating polynomial coefficients given the polynomial's roots Suppose you want to generate the polynomial whose roots are the numbers [1, 5, -2, 4]. To use the calculator for this purpose, follow these steps: ‚Ï˜˜@@OK@@ ˜„Ô1‚í5 ‚í2\‚í 4@@OK@@ @SOLVE@ Press ` to return to stack, the coefficients will be shown in the stack.
Press ˜ to trigger the line editor to see all the coefficients. Generating an algebraic expression for the polynomial You can use the calculator to generate an algebraic expression for a polynomial given the coefficients or the roots of the polynomial. The resulting expression will be given in terms of the default CAS variable X.
‚Ò (associated with the 9 key). Detailed explanations of these types of calculations are presented in Chapter 6 of the calculator’s user's guide. Solving equations with one unknown through NUM.SLV The calculator's NUM.SLV menu provides item different types of equations in a single variable, including non-linear algebraic and transcendental equations.
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In RPN mode, enter the equation between apostrophes and activate command STEQ. Thus, function STEQ can be used as a shortcut to store an expression into variable EQ. Press J to see the newly created EQ variable: Then, enter the SOLVE environment and select Solve equation…, by using: ‚Ï@@OK@@.
Solution to simultaneous equations with MSLV Function MSLV is available in the ‚Ï menu. The help-facility entry for function MSLV is shown next: Notice that function MSLV requires three arguments: 1. A vector containing the equations, i.e., ‘[SIN(X)+Y,X+SIN(Y)=1]’ 2. A vector containing the variables to solve for, i.e., ‘[X,Y]’ 3.
The final solution is X = 1.8238, Y = -0.9681. Reference Additional information on solving single and multiple equations is provided in Chapters 6 and 7 of the calculator’s user’s guide. Page 6-12...
Chapter 7 Operations with lists Lists are a type of calculator’s object that can be useful for data processing. This chapter presents examples of operations with lists. To get started with the examples in this Chapter, we use the Approximate mode (See Chapter 1).
Addition, subtraction, multiplication, division Multiplication and division of a list by a single number is distributed across the list, for example: Subtraction of a single number from a list will subtract the same number from each element in the list, for example: 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.
Note: If we had entered the elements in lists L4 and L3 as integers, the infinite symbol would be shown whenever a division by zero occurs. To produce the following result you need to re-enter the lists as integer (remove decimal points) using Exact mode: If the lists involved in the operation have different lengths, an error message (Invalid Dimensions) is produced.
INVERSE (1/x) Lists of complex numbers You can create a complex number list, say, L5 = L1 ADD i⋅L2 (type the instruction as indicated here), as follows: Functions such as LN, EXP, SQ, etc., can also be applied to a list of complex numbers, e.g., Lists of algebraic objects The following are examples of lists of algebraic objects with the function SIN...
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With system flag 117 set to SOFT menus, the MTH/LIST menu shows the following functions: The operation of the MTH/LIST menu is as follows: ∆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...
For example, the following call to function MAP applies the function SIN(X) to the list {1,2,3}: Reference For additional references, examples, and applications of lists see Chapter 8 in the calculator’s user’s guide. Page 7-6...
' ' ' Typing vectors in the stack With the calculator in ALG mode, a vector is typed into the stack by opening a set of brackets („Ô) and typing the components or elements of the vector separated by commas (‚í). The screen shots below show the entering of a numerical vector followed by an algebraic vector.
(‚í) or spaces (#). Notice that after pressing ` , in either mode, the calculator shows the vector elements separated by spaces. Storing vectors into variables in the stack Vectors can be stored into variables. The screen shots below show the vectors Stored into variables @@@ @@, @@@ @@, @@@ @@, and @@@ @@, respectively.
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The @EDIT key is used to edit the contents of a selected cell in the matrix writer. The @VEC@@ key, when selected, will produce a vector, as opposed to a matrix of one row and many columns. WID key is used to decrease the width of the columns in the ←...
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The @ ROW@ key will add a row full of zeros at the location of the selected cell of the spreadsheet. The @ ROW key will delete the row corresponding to the selected cell of the spreadsheet. The @ COL@ key will add a column full of zeros at the location of the selected cell of the spreadsheet.
(3) Move the cursor up two positions by using ——. Then press @ ROW. The second row will disappear. (4) Press @ ROW@. A row of three zeroes appears in the second row. (5) Press @ COL@. The first column will disappear. (6) Press @ COL@.
Attempting to add or subtract vectors of different length produces an error message: 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.
The MTH/VECTOR menu The MTH menu („´) contains a menu of functions that specifically to vector objects: The VECTOR menu contains the following functions (system flag 117 set to CHOOSE boxes): Magnitude The magnitude of a vector, as discussed earlier, can be found with function ABS.
Attempts to calculate a cross product of vectors of length other than 2 or 3, produce an error message: Reference Additional information on operations with vectors, including applications in the physical sciences, is presented in Chapter 9 of the calculator’s user’s guide. Page 8-8...
At this point, the Matrix Writer screen may look like this: [Note: not all lines will be visible when done with the exercises in the figures in this Chapter. The display header will cover the top lines in the calculator. ] ...
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Press ` once more to place the matrix on the stack. The ALG mode stack is shown next, before and after pressing , once more: If you have selected the textbook display option (using H@) D ISP! and checking ), the matrix will look like the one shown above. Otherwise, Textbook the display will show: The display in RPN mode will look very similar to these.
Operations with matrices Matrices, like other mathematical objects, can be added and subtracted. They can be multiplied by a scalar, or among themselves. An important operation for linear algebra applications is the inverse of a matrix. Details of these operations are presented next. To illustrate the operations we will create a number of matrices that we will store in the following variables.
In RPN mode, you can try a few more exercises: Multiplication There are different multiplication operations that involve matrices. These are described next. The examples are shown in algebraic mode. Multiplication by a scalar Some examples of multiplication of a matrix by a scalar are shown below. 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.
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 multiplication is only possible if the number of columns in the first operand is equal to the number of rows of the second operand.
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). Examples of the inverse of some of the matrices stored earlier are...
The matrix NORM (NORMALIZE) menu is accessed through the keystroke sequence „´ . This menu is described in detail in Chapter 10 of the calculator’s user's guide. Some of these functions are described next. Function DET Function DET calculates the determinant of a square matrix. For example,...
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@@@.
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− − − This system has the same number of equations as of unknowns, and will be referred to as a square system. In general, there should be a unique solution to the system.
While the operation of division is not defined for matrices, we can use the calculator’s / key to “divide” vector b by matrix A to solve for x in the matrix equation A⋅x = b. The procedure for the case of “dividing” b by A is illustrated below for the example above.
Right in front of the TYPE field you will, most likely, see the option Function highlighted. This is the default type of graph for the calculator. To see the list of available graph types, press the soft menu key labeled @CHOOS. This will...
First, enter the PLOT SETUP environment by pressing, „ô. Make sure that the option Function is selected as the TYPE, and that ‘X’ is selected as the independent variable (INDEP). return to normal calculator display. The PLOT SET UP window should look similar to this: •...
VIEW, then press @AUTO to generate the V-VIEW automatically. PLOT WINDOW screen looks as follows: • Plot the graph: @ERASE @DRAW (wait till the calculator finishes the graphs) • @EDIT L @LA EL @ME U To see labels: • To recover the first graphics menu: LL@) P ICT •...
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‘X/(X+10)’ • To accept the changes made to the PLOT SETUP screen press L @@@OK@@@. You will be returned to normal calculator display. • • The next step is to access the Table Set-up screen by using the keystroke combination„õ...
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. Thus, the zoom in option is useful when you want more resolution for the values of x in your table.
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Press ˜ and type ‘X^2+Y^2’ @@@OK@@@. • Make sure that ‘X’ is selected as the variables. • Press L@@@OK@@@ to return to normal calculator display. • Press „ò, simultaneously if in RPN mode, to access the PLOT WINDOW screen. •...
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• Press @CA CL 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 • Press „ô, simultaneously if in RPN mode, to access the PLOT SETUP window.
Press LL@) P ICT to leave the EDIT environment. • Press @CA CL to return to the PLOT WINDOW environment. Then, press $ , or L@@@OK@@@, to return to normal calculator display. Reference Additional information on graphics is available in Chapters 12 and 22 in the calculator’s user’s guide.
Chapter 11 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...
[Note: not all lines will be visible when done with the exercises in the figures in this Chapter. The display header will cover the top lines in the calculator.] The infinity symbol is associated with the 0 key, i.e.., „è.
= dF/dx, and C = constant. Functions INT, INTVX, RISCH, SIGMA and SIGMAVX The calculator provides functions INT, INTVX, RISCH, SIGMA and SIGMAVX to calculate anti-derivatives of functions. Functions INT, RISCH, and SIGMA work with functions of any variable, while functions INTVX, and SIGMAVX utilize functions of the CAS variable VX (typically, ‘x’).
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 subtracted.
Infinite series A function f(x) can be expanded into an infinite series around a point x=x using a Taylor’s series, namely, ∞ ∑ where f (x) represents the n-th derivative of f(x) with respect to x, f If the value x = 0, the series is referred to as a Maclaurin’s series.
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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’ indicating the point of expansion of a Taylor series, and the order of the series to be produced. Function SERIES returns two output items a list with four items, and an expression for h = x - a, if the second argument in the function call is ‘x=a’, i.e., an expression for the increment h.
In the right-hand side figure above, we are using the line editor to see the series expansion in detail. Reference Additional definitions and applications of calculus operations are presented in Chapter 13 in the calculator’s user’s guide. Page 11-7...
∂ • You can use the derivative functions in the calculator: DERVX, DERIV, ∂, described in detail in Chapter 11 of this Guide, to calculate partial derivatives (DERVX uses the CAS default variable VX, typically, ‘X’). Some examples of first-order partial derivatives are shown next. The...
φ dydx dydx Calculating a double integral in the calculator is straightforward. A double integral can be built in the Equation Writer (see example in Chapter 2), as shown below. This double integral is calculated directly in the Equation Writer by selecting the entire expression and using function @EVAL. The result is 3/2.
Chapter 13 Vector Analysis Applications This chapter describes the use of functions HESS, DIV, and CURL, for calculating operations of vector analysis. The del operator The following operator, referred to as the ‘del’ or ‘nabla’ operator, is a vector- based operator that can be applied to a scalar or vector function: ∂...
. The curl of vector field can be calculated with function CURL. For example, for the function F(X,Y,Z) = [XY,X calculated as follows: CURL([X*Y,X^2+Y^2+Z^2,Y*Z],[X,Y,Z]) Reference For additional information on vector analysis applications see Chapter 15 in the calculator’s user’s guide. ,YZ], the divergence is ,YZ], the curl is Page 13-2...
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.
Both of these inputs must be given in terms of the default independent variable for the calculator’s CAS (typically X). The output from the function is the general solution of the ODE. The examples below are shown in the RPN mode: Example 1 –...
–3x y = K Function DESOLVE The calculator provides function DESOLVE (Differential Equation SOLVEr) to solve certain types of differential equations. The function requires as input the differential equation and the unknown function, and returns the solution to the equation if available.
Laplace transform, respectively, of a function f(VX), where VX is the CAS default independent variable (typically X). calculator returns the transform or inverse transform as a function of X. The functions LAP and ILAP are available under the CALC/DIFF menu.
DERIV sub-menu within the CALC menu („Ö). Fourier series for a quadratic function Determine the coefficients c , and c period T = 2. Using the calculator in ALG mode, first we define functions f(t) and g(t): Therefore, when using the {1/(s+1) π +∞...
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Next, we move t o the CASDIR sub-directory under HOME to change the value of variable PERIOD, e.g. [Note: not all lines will be visible when done with the exercises in the following figures.] „ (hold) §`J @) C ASDI `2 K @PERIOD ` Return to the sub-directory where you defined functions f and g, and calculate the coefficients.
≈ Re[(1/3) + (π⋅i+2)/π ⋅exp(i⋅π⋅t)+ (π⋅i+1)/(2π )⋅exp(2⋅i⋅π⋅t)]. Reference For additional definitions, applications, and exercises on solving differential equations, using Laplace transform, and Fourier series and transforms, as well as numerical and graphical methods, see Chapter 16 in the calculator’s user’s guide. Page 14-8...
Chapter 15 Probability Distributions In this Chapter we provide examples of applications of the pre-defined probability distributions in the calculator. The MTH/PROBABILITY.. sub-menu - part 1 The MTH/PROBABILITY.. sub-menu is accessible through the keystroke sequence „´. With system flag 117 set to CHOOSE boxes, the following functions are available in the PROBABILITY..
Random numbers The calculator provides a random number generator that returns a uniformly distributed random real number between 0 and 1. number, use function RAND from the MTH/PROBABILITY sub-menu.
,x). For example, check that for a normal distribution, NDIST(1.0,0.5,2.0) = 0.20755374. This function is useful to plot the Normal distribution pdf). The calculator also provides function UTPN that calculates the upper-tail normal distribution, i.e., UTPN(µ,σ , x) = P(X>x) = 1 - P(X<x), where P() represents a probability.
The Student-t distribution The Student-t, or simply, the t-, distribution has one parameter ν, known as the degrees of freedom of the distribution. The calculator provides for values of the upper-tail (cumulative) distribution function for the t-distribution, function UTPT, given the parameter ν and the value of t, i.e., UTPT(ν,t) = P(T>t) = 1- P(T<t).
Chapter 16 Statistical Applications The calculator provides the following pre-programmed statistical features accessible through the keystroke combination ‚Ù (the 5 key): Entering data Applications number 1, 2, and 4 from the list above require that the data be available as columns of the matrix DAT.
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When ready, press @@@OK@@. The selected values will be listed, appropriately labeled, in the screen of your calculator. For example: Sample vs. population The pre-programmed functions for single-variable statistics used above can be...
Obtaining frequency distributions 2. Frequencies.. The application frequency distributions for a set of data. The data must be present in the form of a column vector stored in variable ‚Ù˜ @@@OK@@@. The resulting input form contains the following fields: the matrix containing the data of interest. the column of DAT that is under scrutiny.
This information indicates that our data ranges from -9 to 9. To produce a frequency distribution we will use the interval (-8,8) dividing it into 8 bins of width 2 each. 2. Frequencies.. Select the program data is already loaded in DAT, and the option Col should hold the value 1 since we have only one column in DAT.
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data sets (x,y), stored in columns of the DAT matrix. For this application, you need to have at least two columns in your DAT variable. For example, to fit a linear relationship to the data shown in the table below: First, enter the two columns of data into variable DAT by using the matrix writer, and function STO .
1 shows the covariance of x-y. For definitions of these parameters see Chapter 18 in the user’s guide. For additional information on the data-fit feature of the calculator see Chapter 18 in the user’s guide. Obtaining additional summary statistics Summary stats..
Press @@@OK@@@ to obtain the following results: Confidence intervals The application 6. Conf Interval can be accessed by using ‚Ù— @@@OK@@@. The application offers the following options: These options are to be interpreted as follows: 1. Z-INT: 1 .: Single sample confidence interval for the population mean, , with known population variance, or for large samples with unknown population variance.
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Press @HELP to obtain a screen explaining the meaning of the confidence interval in terms of random numbers generated by a calculator. To scroll down the resulting screen use the down-arrow key ˜. Press @@@OK@@@ when done with the help screen.
@@@OK@@@ to exit the confidence interval environment. The results will be listed in the calculator’s display. Additional examples of confidence interval calculations are presented in Chapter 18 in the calculator’s user’s guide.
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, against the alternative hypothesis, H Press ‚Ù—— @@@OK@@@ to access the confidence interval feature in the calculator. Press @@@OK@@@ to select option 1. Z-Test: 1 . Enter the following data and press @@@OK@@@: You are then asked to select the alternative hypothesis: = 0.05, test the...
. Then, press @@@OK@@@. The result is: Select Then, we reject H , against H 5.656854. The P-value is 1.54 10 1.959964, corresponding to critical x range of {147.2 152.8}. This information can be observed graphically by pressing the soft-menu key @GRAPH: Reference Additional materials on statistical analysis, including definitions of concepts,...
Writing non-decimal numbers Numbers in non-decimal systems, referred to as binary integers, are written preceded by the # symbol („â) in the calculator. To select the current base to be used for binary integers, choose either HEX(adecimal), DEC(imal), Page 17-1...
(hexadecimal), d (decimal), o (octal), or b (binary), examples: [Note: not all lines will be visible when done with the exercises in the following figures.] Reference For additional details on numbers from different bases see Chapter 19 in the calculator’s user’s guide. is selected, Page 17-2...
If HP receives notice of such defects during the warranty period, HP will, at its option, either repair or replace products which prove to be defective. Replacement products may be either new or like-new.
Regulatory information This section contains information that shows how the hp 48gII graphing calculator complies with regulations in certain regions. Any modifications to the calculator not expressly approved by Hewlett-Packard could void the authority to operate the 48gII in these regions.
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