hp 48gII graphing calculator user’s manual K
Preface You have in your hands a compact symbolic and numerical computer that will facilitate calculation and mathematical analysis of problems in a variety of disciplines, from elementary mathematics to advanced engineering and science subjects. The present Guide contains examples that illustrate the use of the basic calculator functions and operations.
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 The TOOL menu, 1-3 Setting time and date, 1-3 Introducing the calculator’s keyboard, 1-4 Selecting calculator modes, 1-6 Operating mode, 1-7 Number Format and decimal dot or comma, 1-10 Standard format, 1-11 Fixed format with decimals, 1-11 Scientific format, 1-12 Engineering format, 1-13 Decima
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 Algebraic mode, 2-11 RPN mode, 2-13 Checking variables contents, 2-14 Algebraic mode, 2-14 RPN mode, 2-14 Using the right-shift key followed by soft menu key labels, 2-15 Listing the contents
Operations with units, 3-12 Unit conversions, 3-14 Physical constants in the calculator, 3-14 Defining and using functions, 3-16 Reference, 3-18 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, 4-2 Simple operations with complex numbers, 4-3 The CMPLX menus, 4-4 CMPLX menu through the MTH menu, 4-4 CMPLX menu in the keyboard, 4-5 Functions applied to complex numbers, 4-6 Fu
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, 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 Finding the solution to a polynomial equation, 6-6 Generating polynomial coefficients given the polynomial’s
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 Multiplication by a scalar, and division by a scalar, 8-6 Absolute value function, 8-6 The MTH/VECTOR menu, 8-7 Magnitude, 8-7 Dot product, 8-7 Cross product, 8-8 Reference, 8-8 Chapter 9
Function TRACE, 9-7 Solution of linear systems, 9-7 Using the numerical solver for linear systems, 9-8 Solution with the inverse matrix, 9-10 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 – Calculus Applications, 11-1 The CALC (Calculus) menu, 11-1 Limits and derivatives, 11-1 Functi
Curl, 13-2 Reference, 13-2 Chapter 14 – Differential Equations, 14-1 The CALC/DIFF menu, 14-1 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 – Probability Distributions, 15-1 The MTH/PROBABILITY..
Chapter 17 – Numbers in Different Bases, 17-1 The BASE menu, 17-1 Writing non-decimal numbers, 17-1 Reference, 17-2 Warranty – W-1 Service, W-2 Regulatory information, W-4 Page TOC-8
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. Batteries The calculator uses 3 AAA batteries as main power and a CR2032 lithium battery for memory backup.
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. However, avoid removing the backup battery and main batteries at the same time to avoid data lost. Turning the calculator on and off The $ key is located at the lower left corner of the keyboard.
RAD XYZ HEX R= 'X' For details on the meaning of these specifications see Chapter 2 in the calculator’s user's guide. The second line shows the characters { HOME } indicating that the HOME directory is the current file directory in the calculator’s memory.
@EDIT A @VIEW @@ RCL @@ @@STO@ ! PURGE CLEAR B C D E F EDIT the contents of a variable (see Chapter 2 in this Guide and Chapter 2 and Appendix L in the user's guide for more information on editing) VIEW the contents of a variable ReCaLl the contents of a variable STOre the contents of a variable PURGE a variable CLEAR the display or stack These six functions form the first page of the TOOL menu. This menu has actually eight entries arranged in two pages.
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.
~„p ~…p ALPHA-Left-Shift function, to enter the lower-case letter p ALPHA-Right-Shift function, to enter the symbol π Of the six functions associated with a key only the first four are shown in the keyboard itself. The figure in next page shows these four labels for the P key.
Press the !!@@OK#@ F soft menu key to return to normal display. Examples of selecting different calculator modes are shown next. Operating Mode The calculator offers two operating modes: the Algebraic mode, and the 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.
1./3.*3. ————— /23.Q3™™™+!¸2.5` After pressing `the calculator displays the expression: √ (3.*(5.-1/(3.*3.))/23.^3+EXP(2.5)) Pressing `again will provide the following value (accept Approx mode on, if asked, by pressing !!@@OK#@): You could also type the expression directly into the display without using the equation writer, as follows: R!Ü3.*!Ü5.1/3.*3.™ /23.Q3+!¸2.5` to obtain the same result. Change the operating mode to RPN by first pressing the H button.
different levels are referred to as the stack levels, i.e., stack level 1, stack level 2, etc. Basically, what RPN means is that, instead of writing an operation such as 3 + 2, in the calculator by using 3+2` we write first the operands, in the proper order, and then the operator, i.e., 3`2`+ As you enter the operands, they occupy different stack levels. Entering 3`puts the number 3 in stack level 1. Next, entering 2`pushes the 3 upwards to occupy stack level 2.
3 ⋅ 5 − 23 3` 5` 3` 3* Y * 23` 3Q / 2.5 !¸ + R 3⋅3 2.5 +e 1 3 Enter 3 in level 1 Enter 5 in level 1, 3 moves to level 2 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; 3 in level 2 3× (5 - 1/(3×3)), occupies level 1 now. Enter 23 in level 1, 14.66666 moves to level 2. 3 Enter 3, calculate 23 into level 1. 14.666 in lev. 2.
more about reals, see Chapter 2 in this Guide. To illustrate this and other number formats try the following exercises: • Standard format: 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, to return to the calculator display. Enter the number 123.4567890123456 (with16 significant figures). Press the ` key.
Press the !!@@OK#@ soft menu key return to the calculator display. number now is shown as: The 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): • Scientific format To set this format, start by pressing the H button. Next, use the down arrow key, ˜, to select the option Number format.
This result, 1.23E2, is the calculator’s version of powers-of-ten notation, 2 i.e., 1.235 × 10 . In this, so-called, scientific notation, the number 3 in front of the Sci number format (shown earlier) represents the number of significant figures after the decimal point. Scientific notation always includes one integer figure as shown above. For this case, therefore, the number of significant figures is four.
• Decimal comma vs. decimal point Decimal points in floating-point numbers can be replaced by commas, if the user is more familiar with such notation. To replace decimal points for commas, change the FM option in the CALCULATOR MODES input form to commas, as follows (Notice that we have changed the Number Format to Std): • Press the H button. Next, use the down arrow key, ˜, once, and the right arrow key, ™, twice, to highlight the option __FM,. To select commas, press the @CHECK soft menu key (i.e.
• g Grades: There are 400 grades (400 ) in a complete circumference. The angle measure affects the trig functions like SIN, COS, TAN and associated functions. To change the angle measure mode, use the following procedure: • Press the H button. Next, use the down arrow key, ˜, twice. Select the Angle Measure mode by either using the \key (second from left in the fifth row from the keyboard bottom), or pressing the @CHOOSE soft menu key ( B).
Selecting CAS settings CAS stands for Computer Algebraic System. This is the mathematical core of the calculator where the symbolic mathematical operations and functions are programmed. The CAS offers a number of settings can be adjusted according to the type of operation of interest. 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.
options above). Unselected options will show no check mark in the underline preceding the option of interest (e.g., the _Numeric, _Approx, _Complex, _Verbose, _Step/Step, _Incr Pow options above). • 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. To return to normal calculator display at this point, press the @@@OK@@@ soft menu key once more.
The calculator display can be customized to your preference by selecting different display modes. To see the optional display settings use the following: • 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. • To navigate through the many options in the DISPLAY MODES input form, use the arrow keys: š™˜—.
(D) to display the DISPLAY MODES input form. The Font: field is highlighted, and the option Ft8_0:system 8 is selected. This is the default value of the display font. Pressing the @CHOOSE soft menu key (B), will provide a list of available system fonts, as shown below: The options available are three standard System Fonts (sizes 8, 7, and 6) and a Browse.. option. The latter will let you browse the calculator memory for additional fonts that you may have created or downloaded into the calculator.
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. This line shows two properties that can be modified. When these properties are selected (checked) the following effects are activated: _Small Changes font size to small.
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. This line shows two properties that can be modified.
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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. Calculator objects Some of the most commonly used objects are: reals (real numbers, written with a decimal point, e.g., -0.0023, 3.
Notice that, if your CAS is set to EXACT (see Appendix C in user's guide) 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 Before producing a result, you will be asked to change to Approximate mode.
To evaluate the expression we can use the EVAL function, as follows: µ„î If the CAS is set to Exact, 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. 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.
This expression is semi-symbolic in the sense that there are floating-point components to the result, as well as a √3. Next, we switch stack locations and evaluate using function ÆNUM: ™…ï. 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: -µ. The result is zero (0.).
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. Using the Equation Writer (EQW) to create expressions The equation writer is an extremely powerful tool that not only let you enter or see an equation, but also allows you to modify and work/apply functions on all or part of the equation.
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 2 denominator (i.e., 5+1/3) with (5+π /2).
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: 5 5 + 2 ⋅ (5 + π + 2 2 ) 1 3 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.
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.
Also, you can always copy special characters by using the CHARS menu (…±) if you don’t want to memorize the keystroke combination that produces it. A listing of commonly used ~‚keystroke combinations was listed in an earlier section. For additional information on editing, evaluating, factoring, and simplifying algebraic expressions see 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.
Variables Variables are similar to files on a computer hard drive. One variable can store one object (numerical values, algebraic expressions, lists, vectors, matrices, programs, etc). Variables are referred to by their names, which can be any combination of alphabetic and numerical characters, starting with a letter (either English or Greek). Some non-alphabetic characters, such as the arrow (→) can be used in a variable name, if combined with an alphabetical character.
To unlock the upper-case locked keyboard, press ~ 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 examples are used to store the variables listed in the following table (Press J if needed to see variables menu): • Name Contents Type α A12 Q R z1 p1 -0.
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). p1: ‚å‚é~„r³„ì* ~„rQ2™™™ K~„p1`.. 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, α.
• RPN mode (Use I\@@OK@@ to change to 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. Press K to create the variable.
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, α. Checking variables contents The simplest way to check a variable content is by pressing the soft menu key label for the variable. For example, for the variables listed above, press the following keys to see the contents of the variables: Algebraic mode Type these keystrokes: J@@ @@ ` @@@R@@ `@@@ @@@ `.
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‚@@ @@ ‚ @@ @@ ‚ @@@R@@ ‚@@@ @@ ‚ @@A @@ 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.
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 α. We will use command PURGE to delete variable p1. Press I @PURGE@ J@@ @@ `.
To delete two variables simultaneously, say variables R and Q, first create a list (in RPN mode, the elements of the list need not be separated by commas as in Algebraic mode): J „ä³ @@@R!@@ ™ ³ @@@ !@@ ` Then, press I@PURGE@ use to purge the variables. Additional information on variable manipulation is available in Chapter 2 of the calculator’s user's guide. UNDO and CMD functions Functions UNDO and CMD are useful for recovering recent commands, or to revert an operation if a mistake was made.
@@OK@@ ˜˜˜˜ Show the MEMORY menu list and select DIRECTORY @@OK@@ —— Show the DIRECTORY menu list and select ORDER @@OK@@ activate the ORDER command 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).
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 „°. 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@@).
To activate the ORDER command we press the C(@ORDER) soft menu key. 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.). Also, it is assumed that the reader knows how to change the calculator’s operating system (Chapter 1), use menus and choose boxes (Chapter 1), and operate with variables (Chapter 2).
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. In ALG mode: „Ü5+3.2™/„Ü72.2` In RPN mode, you do not need the parenthesis, calculation is done directly on the stack: 5`3.2`+7`2.
„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., 123.4R • The power function, ^, is available through the Q key.
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 (function LOG) while its inverse function (ALOG, or antilogarithm) is calculated by using „Â. In ALG mode, the function is entered before the argument: ‚Ã2.45` „Â\2.3` In RPN mode, the argument is entered before the function 2.45 ‚Ã 2.
2.45` ‚¹ 2.3\` „¸ • Three trigonometric functions are readily available in the keyboard: sine (S), cosine (T), and tangent (U). Arguments of these functions are angles in either degrees, radians, grades. The following examples use angles in degrees (DEG): In ALG mode: S30` T45` U135` In RPN mode: 30S 45T 135U • The inverse trigonometric functions available in the keyboard are the arcsine („¼), arccosine („¾), and arctangent („À).
Real number functions in the MTH menu The MTH („´) menu include a number of mathematical functions mostly applicable to real numbers. With the default setting of CHOOSE boxes for system flag 117 (see Chapter 2), the MTH menu shows the following functions: The functions are grouped by the type of argument (1. vectors, 2. matrices, 3. lists, 7. probability, 9. complex) or by the type of function (4. hyperbolic, 5. real, 6. base, 8. fft).
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).
Finally, in order to select, for example, the hyperbolic tangent (tanh) function, simply press @@TA H@. 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: „´@@H P@ @@TA H@ 2.5` In RPN mode, the same value is calculated using: 2.5`„´)@@H P@ @@TA H@ As an exercise of applications of hyperbolic functions, verify the following values: -1 sinh (2.
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.
Pressing on the appropriate soft menu key will open the sub-menu of units for that particular selection. For example, for the @)SPEED sub-menu, the following units are available: Pressing the soft menu key @)U ITS will take you back to the UNITS menu. Recall that you can always list the full menu labels in the screen by using ‚˜, 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.
Attaching units to numbers To attach a unit object to a number, the number must be followed by an underscore (‚Ý, key(8,5)). Thus, a force of 5 N will be entered as 5_N. Here is the sequence of steps to enter this number in ALG mode, system flag 117 set to CHOOSE boxes: 5‚Ý ‚Û 8@@OK@@ @@OK@@ ` Note: If you forget the underscore, the result is the expression 5*N, where N here represents a possible variable name and not Newtons.
____________________________________________________ Prefix Name x Prefix Name x ____________________________________________________ Y yotta +24 d deci -1 Z zetta +21 c centi -2 E exa +18 m milli -3 P peta +15 µ micro -6 T tera +12 n nano -9 G giga +9 p pico -12 M mega +6 f femto -15 k,K kilo +3 a atto -18 h,H hecto +2 z zepto -21 D(*) deka +1 y yocto -24 _____________________________________________________ (*) In the SI system, this prefix is da rather than D. Use D for deka in the calculator, however.
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 parentheses, e.g.
These operations produce the following output: Unit conversions The UNITS menu contains a TOOLS sub-menu, which provides the following functions: CONVERT(x,y): UBASE(x): UVAL(x): UFACT(x,y): ÆUNIT(x,y): convert unit object x to units of object y convert unit object x to SI units extract the value from unit object x factors a unit x from unit object x combines value of x with units of y Examples of function CONVERT are shown below.
The soft menu keys corresponding to this CONSTANTS LIBRARY screen include the following functions: SI ENGL UNIT VALUE ÆSTK QUIT when selected, constants values are shown in SI units (*) when selected, constants values are shown in English units (*) when selected, constants are shown with units attached when selected, constants are shown without units copies value (with or without units) to the stack exit constants library (*) Activated only if the VALUE option is selected.
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, . In here, Vm, is the tag of this result. Any arithmetic operation with this number will ignore the tag.
and get the result you want without having to type the expression in the righthand side for each separate value. In the following example, we assume you have set your calculator to ALG mode. Enter the following sequence of keystrokes: „à³~h„Ü~„x™‚Å ‚¹~„x+1™+„¸~„x` The screen will look like this: Press the J key, and you will notice that there is a new variable in your soft menu key (@@@H@@). To see the contents of this variable press ‚@@@H@@.
between quotes that contain that local variable, and show the evaluated expression. To activate the function in ALG mode, type the name of the function followed by the argument between parentheses, e.g., @@@H@@@ „Ü2`. Some examples are shown below: In the RPN mode, to activate the function enter the argument first, then press the soft menu key corresponding to the variable name @@@H@@@ . For example, you could try: 2`@@@H@@@ . The other examples shown above can be entered by using: 1.
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 written as z = x + iy, (Cartesian representation) where 2 x and y are real numbers, and i is the imaginary unit defined by i = -1. The number has a real part, x = Re(z), and an imaginary part, y = Im(z).
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. For example, in ALG mode, 3.5-1.2i is entered as: 3.5 -1.
The result shown above represents a magnitude, 3.7, and an angle 0.33029…. The angle symbol (∠) is shown in front of the angle measure. Return to Cartesian or rectangular coordinates by using function RECT (available in the catalog, ‚N). A complex number in polar i representation is written as z = r⋅e θ. 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 5.
(3+5i) + (6-3i) = (9,2); (5-2i) - (3+4i) = (2,-6) (3-i)·(2-4i) = (2,-14); (5-2i)/(3+4i) = (0.28,-1.04) 1/(3+4i) = (0.12, -0.16) ; -(5-3i) = -5 + 3i 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: not all lines will be visible when done with the exercises in the following figures.
Functions applied to complex numbers Many of the keyboard-based functions and MTH menu functions defined in x Chapter 3 for real numbers (e.g., SQ, ,LN, e , etc.), can be applied to complex numbers. The result is another complex number, as illustrated in the following examples. [Note: not all lines will be visible in the screen of your calculator when finishing with the exercises in the following figures].
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.
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: • A number: 12.3, 15.2_m, ‘π’, ‘e’, ‘i’ • A variable name: ‘a’, ‘ux’, ‘width’, etc.
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.
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 @@ „¸v The same results are obtained in RPN mode if using the following keystrokes: @@A @@ ` @@A @@ `+ @@A @@ `@@A @@ ` - @@A @@ `@@A @@ `* @@A @@ `@@A @@ ` / @@A @@ ` ‚¹ @@A @@ `„¸ Page 5-3
Functions in the ALG menu The ALG (Algebraic) menu is available by using the keystroke sequence ‚× (associated with the ‚ key). With system flag 117 set to CHOOSE boxes, the ALG menu shows the following functions: 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 @)HELP@ ` . To locate a particular function, type the first letter of the function.
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.
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). The user is invited to explore this facility to find information on the commands in the TRIG menu.
FACTORS: SIMP2: 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’ VX. For additional information on the CAS variable see Appendix C in the calculator’s user's guide.
Note: you could get the latter result by using PARTFRAC: PARTFRAC(‘(X^3-2*X+2)/(X-1)’) = ‘X^2+X-1 + 1/(X-1)’. The PEVAL function The functions PEVAL (Polynomial EVALuation) can be used to evaluate a polynomial n n-1 2 p(x) = an⋅x +an-1⋅x + …+ a2⋅x +a1⋅x+ a0, given an array of coefficients [an, an-1, … a2, a1, a0] and a value of x0. The result is the evaluation p(x0). Function PEVAL is not available in the ARITHMETIC menu, it must be accessed from the function catalog (‚N).
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.
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. In this case, the poles are (1, -3) with multiplicities (2,5) respectively, and the roots are (0, 2, -5) with multiplicities (3, 1, 2), respectively.
Reference Additional information, definitions, and examples of algebraic and arithmetic operations are presented in Chapter 5 of the calculator’s user’s guide.
Chapter 6 Solution to equations Associated with the 7 key there are two menus of equation-solving functions, the Symbolic SOLVer („Î), and the NUMerical SoLVer (‚Ï). Following, we present some of the functions contained in these menus. Symbolic solution of algebraic equations Here we describe some of the functions from the Symbolic Solver menu. Activate the menu by using the keystroke combination „Î.
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. After applying ISOL, the result is shown in the figure to the right: The first argument in ISOL can be an expression, as shown above, or an equation. For example, in ALG mode, try: Note: To type the equal sign (=) in an equation, use ‚Å (associated with the \ key).
The following examples show the use of function SOLVE in ALG and RPN modes. [Note: not all lines will be visible when done with the exercises in the following figures.] The screen shot shown above displays two solutions. In the first one, β -5β 4 =125, SOLVE produces no solutions { }. In the second one, β - 5β = 6, 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.
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. In the second case, SOLVEVX found a single solution, X = 2.
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).
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. Notes: 1. Whenever you solve for a value in the NUM.SLV applications, the value solved for will be placed in the stack.
‚í 1\‚í1@@OK@@ @SOLVE@ Solve equation The screen will show the solution as follows: Press ` to return to stack. The stack will show the following results in ALG mode (the same result would be shown in RPN mode): All the solutions are complex numbers: (0.432,-0.389), (0.432,0.389), (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].
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. To generate the algebraic expression using the coefficients, try the following example. Assume that the polynomial coefficients are [1,5,-2,4].
'(X-1)*(X-3)*(X+2)*(X-1)'. To expand the products, you can use the EXPAND command. The resulting expression is: 'X^4+-3*X^3+ -3*X^2+11*X-6'. Financial calculations The calculations in item 5. Solve finance.. in the Numerical Solver (NUM.SLV) are used for calculations of time value of money of interest in the discipline of engineering economics and other financial applications. This application can also be started by using the keystroke combination ‚Ò (associated with the 9 key).
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@@. 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. Also, a field labeled x is provided.
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. A vector containing initial values for the solution, i.e., the initial values of both X and Y are zero for this example.
You may have noticed that, while producing the solution, the screen shows intermediate information on the upper left corner. Since the solution provided by MSLV is numerical, the information in the upper left corner shows the results of the iterative process used to obtain a solution. 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.
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). Creating and storing lists To create a list in ALG mode, first enter the braces key „ä , then type or enter the elements of the list, separating them with commas (‚í).
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. Try, for example, L1-L4.
ABS 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.
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 : Sorts elements in increasing order REVLIST : Reverses order of list ADD : Operator for term-by-term addition of two lists of the same length (examples of this operator were shown above) Examples of ap
The SEQ function The SEQ function, available through the command catalog (‚N), 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. The general form of the function is SEQ(expression, index, start, end, increment) For example: 2 2 2 2 The list produced corresponds to the values {1 , 2 , 3 , 4 }.
Chapter 8 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. Entering vectors In the calculator, vectors are represented by a sequence of numbers enclosed between brackets, and typically entered as row vectors. The brackets are generated in the calculator by the keystroke combination „Ô , associated with the * key.
(‚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 u2 = , u3 = , v2 = , v3 = Stored into variables @@@ @@, @@@ @@, @@@ @@, and @@@ @@, respectively.
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. The ←WID key is used to decrease the width of the columns in the spreadsheet. Press this key a couple of times to see the column width decrease in your matrix writer. The @WID→ key is used to increase the width of the columns in the spreadsheet.
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. The @ COL@ key will delete the column corresponding to the selected cell of the spreadsheet. The @→STK@@ key will place the contents of the selected cell on the stack.
(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@. A column of two zeroes appears in the first column. (7) Press @GOTO@ 3@@OK@@ 3@@OK@@ @@OK@@ to move to position (3,3). (8) Press @→STK@@. This will place the contents of cell (3,3) on the stack, although you will not be able to see it yet. Press ` to return to normal display.
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. This function is also available from the keyboard („Ê). Examples of application of function ABS were shown above.
Cross product Function CROSS (option 3 in the MTH/VECTOR menu) is used to calculate the cross product of two 2-D vectors, of two 3-D vectors, or of one 2-D and one 3D vector. For the purpose of calculating a cross product, a 2-D vector of the form [Ax, Ay], is treated as the 3-D vector [Ax, Ay,0]. Examples in ALG mode are shown next for two 2-D and two 3-D vectors. Notice that the cross product of two 2-D vectors will produce a vector in the z-direction only, i.e.
Chapter 9 Matrices and linear algebra This chapter shows examples of creating matrices and operations with matrices, including linear algebra applications. Entering matrices in the stack In this section we present two different methods to enter matrices in the calculator stack: (1) using the Matrix Editor, and (2) typing the matrix directly into the stack. Using the matrix editor As with the case of vectors, discussed in Chapter 9, matrices can be entered into the stack by using the matrix editor.
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@)DISP! and checking off 3Textbook), 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. Typing in the matrix directly into the stack The same result as above can be achieved by entering the following directly into the stack: „Ô „Ô 2.
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: A A A A ` ` ` ` `+ `+ `+ `+ A A A A ` ` ` ` ````- 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 Cm×n = Am×p⋅Bp×n. Notice that matrix multiplication is only possible if the number of columns in the first operand is equal to the number of rows of the second operand. The general term in the product, cij, is defined as p cij = ∑ aik ⋅ bkj , for i = 1,2,K, m; j = 1,2,K, n.
The identity matrix The identity matrix has the property that A⋅I = I⋅A = A. To verify this property we present the following examples using the matrices stored earlier on. Use function IDN (find it in the MTH/MATRIX/MAKE menu) to generate the identity matrix as shown here: The inverse matrix -1 -1 -1 The inverse of a square matrix A is the matrix A such that A⋅A = A ⋅A = I, where I is the identity matrix of the same dimensions as A.
Characterizing a matrix (The matrix NORM menu) 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, Function TRACE Function TRACE calculates the trace of square matrix, defined as the sum of the elements in its main diagonal, or n tr (A) = ∑ aii .
This system of linear equations can be written as a matrix equation, An×m⋅xm×1 = bn×1, if we define the following matrix and vectors: a11 a A = 21 M an1 a12 a22 M an 2 L a1m x1 b1 b L a2 m x2 2 , x= , b= M M O M L anm n×m xm m×1 bn n×1 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.
x1 2 3 − 5 A = 1 − 3 8 , x = x 2 , and x3 2 − 2 4 13 b = − 13. − 6 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. The solution will be the point of intersection of the three planes in the coordinate system (x1, x2, x3) represented by the three equations. To enter matrix A you can activate the Matrix Writer while the A: field is selected.
Solution with the inverse matrix The solution to the system A⋅x = b, where A is a square matrix is -1 x = A ⋅ b. For the example used earlier, we can find the solution in the calculator as follows (First enter matrix A and vector b once more): Solution by “division” of matrices 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.
Chapter 10 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 fast 3D plots.
Plotting an expression of the form y = f(x) As an example, let's plot the function, f ( x) = 1 2π exp(− x2 ) 2 • 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). Press L@@@OK@@@ to return to normal calculator display. The PLOT SET UP window should look similar to this: • Enter the PLOT environment by pressing „ñ(press them simultaneously if in RPN mode).
VIEW, then press @AUTO to generate the V-VIEW automatically. PLOT WINDOW screen looks as follows: The • Plot the graph: @ERASE @DRAW (wait till the calculator finishes the graphs) • To see labels: • To recover the first graphics menu: LL@)PICT • To trace the curve: @TRACE @@ @@ . Then use the right- and left-arrow keys (š™) to move about the curve. The coordinates of the points you trace will be shown at the bottom of the screen. Check that for x = 1.05 , y = 0.0131. Also, check that for x = -1.
• We will generate values of the function f(x), defined above, for values of x from –5 to 5, in increments of 0.5. First, we need to ensure that the graph type is set to FUNCTION in the PLOT SETUP screen („ô, press them simultaneously, if in RPN mode). The field in front of the Type option will be highlighted. If this field is not already set to FUNCTION, press the soft key @CHOOS and select the FUNCTION option, then press @@@OK@@@.
• The @@ IG@ key simply changes the font in the table from small to big, and vice versa. Try it. • The @ OOM 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.
• Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP window. • Change TYPE to Fast3D. ( @CHOOS!, find Fast3D, @@OK@@). • Press ˜ and type ‘X^2+Y^2’ @@@OK@@@. • Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd: variables. • Press L@@@OK@@@ to return to normal calculator display. • Press „ò, simultaneously if in RPN mode, to access the PLOT WINDOW screen.
• When done, press @E IT. • Press @CA CL to return to the PLOT WINDOW environment. • Change the Step data to read: Step Indep: 20 • Press @ERASE @DRAW to see the surface plot. Sample views: • When done, press @E IT. • Press @CA CL to return to PLOT WINDOW. • Press $ , or L@@@OK@@@, to return to normal calculator display. Depnd: 16 2 2 Try also a Fast 3D plot for the surface z = f(x,y) = sin (x +y ) • Press „ô, simultaneously if in RPN mode, to access the PLOT SETUP window.
• Press LL@)PICT 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.
command catalog (‚N~„l) or through option 2. LIMITS & SERIES… of the CALC menu (see above). = Function lim is entered in ALG mode as to calculate the limit lim f ( x ) . In RPN mode, enter the function first, then the expression x→ a ‘x=a’, and finally function lim. Examples in ALG mode are shown next, including some limits to infinity. [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.
Anti-derivatives and integrals An anti-derivative of a function f(x) is a function F(x) such that f(x) = dF/dx. One way to represent an anti-derivative is as a indefinite integral, i.e., ∫ f ( x)dx = F ( x) + C if and only if, f(x) = 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.
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=x0 by using a Taylor’s series, namely, ∞ f ( x) = ∑ n =0 f ( n ) ( xo ) ⋅ ( x − xo ) n , n! (n) (0) where f (x) represents the n-th derivative of f(x) with respect to x, f (x) = f(x). If the value x0 = 0, the series is referred to as a Maclaurin’s series. Functions TAYLR, TAYLR0, and SERIES Functions TAYLR, TAYLR0, and SERIES are used to generate Taylor polynomials, as well as Taylor series with residuals.
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.
Chapter 12 Multi-variate Calculus Applications Multi-variate calculus refers to functions of two or more variables. In this Chapter we discuss basic concepts of multi-variate calculus: partial derivatives and multiple integrals. Partial derivatives To quickly calculate partial derivatives of multi-variate functions, use the rules of ordinary derivatives with respect to the variable of interest, while considering all other variables as constant.
Multiple integrals A physical interpretation of the double integral of a function f(x,y) over a region R on the x-y plane is the volume of the solid body contained under the surface f(x,y) above the region R. The region R can be described as R = {a
Chapter 13 Vector Analysis Applications This chapter describes the use of functions HESS, DIV, and CURL, for calculating operations of vector analysis.
Alternatively, use function DERIV as follows: 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” of the del operator with the function, i.e., divF = ∇ • F . Function DIV can be used to calculate the divergence of a 2 2 2 vector field.
Chapter 14 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. The CALC/DIFF menu The DIFFERENTIAL EQNS.. sub-menu within the CALC („Ö) menu provides functions for the solution of differential equations.
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: • • the right-hand side of the ODE the characteristic equation of the ODE Both of these inputs must be given in terms of the default independent variable for the calculator’s CAS (typically X).
'X ' ` 'X X X ' ` DE The solution is: which can be simplified to y = K1⋅e –3x + K2⋅e + K3⋅e + (450⋅x +330⋅x+241)/13500. 5x 2x 2 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.
The variable ODETYPE You will notice in the soft-menu key labels a new variable called @ODET (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. Press @ODET to obtain the string “1st order linear”. Example 2 – Solving an equation with initial conditions. Solve 2 2 d y/dt + 5y = 2 cos(t/2), with initial conditions y(0) = 1.2, y’(0) = -0.5.
Laplace Transforms The Laplace transform of a function f(t) produces a function F(s) in the image domain that can be utilized to find the solution of a linear differential equation involving f(t) through algebraic methods. The steps involved in this application are three: 1. 2. 3. Use of the Laplace transform converts the linear ODE involving f(t) into an algebraic equation. The unknown F(s) is solved for in the image domain through algebraic manipulation.
and you will notice that the CAS default variable X in the equation writer screen replaces the variable s in this definition. Therefore, when using the function LAP you get back a function of X, which is the Laplace transform of f(X). Example 2 – Determine the inverse Laplace transform of F(s) = sin(s). Use: ‘1/(X+1)^2’ ` ILAP -1 2 -x The calculator returns the result: ‘X/EXP(X)’, meaning that L {1/(s+1) } = x⋅e .
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 @)CASDI `2 K @PERIOD ` Return to the sub-directory where you defined functions f and g, and calculate the coefficients. Set CAS to Complex mode (see chapter 2) before trying the exercises. Function COLLECT is available in the ALG menu (‚×).
2 Thus, 2 c0 = 1/3, c1 = (π⋅i+2)/π , c2 = (π⋅i+1)/(2π ). The Fourier series with three elements will be written as g(t) ≈ Re[(1/3) + (π⋅i+2)/π ⋅exp(i⋅π⋅t)+ (π⋅i+1)/(2π )⋅exp(2⋅i⋅π⋅t)]. 2 2 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.
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.. menu: In this section we discuss functions COMB, PERM, ! (factorial), and RAND.
We can calculate combinations, permutations, and factorials with functions COMB, PERM, and ! from the MTH/PROBABILITY.. sub-menu. The operation of those functions is described next: • • • COMB(n,r): Calculates the number of combinations of n items taken r at a time PERM(n,r): Calculates the number of permutations of n items taken r at a time n!: Factorial of a positive integer. For a non-integer, x! returns Γ(x+1), where Γ(x) is the Gamma function (see Chapter 3).
start lists of random numbers is presented in detail in Chapter 17 of the user's guide. The MTH/PROB menu - part 2 In this section we discuss four continuous probability distributions that are commonly used for problems related to statistical inference: the normal 2 distribution, the Student’s t distribution, the Chi-square (χ ) distribution, and the F-distribution. The functions provided by the calculator to evaluate probabilities for these distributions are NDIST, UTPN, UTPT, UTPC, and UTPF.
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) = 1P(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 ΣDAT. This can be accomplished by entering the data in columns using the matrix writer, „², and then using functions STOΣ to store the matrix into ΣDAT.
The form lists the data in ΣDAT, shows that column 1 is selected (there is only one column in the current ΣDAT). Move about the form with the arrow keys, and press the @3CHK@ soft menu key to select those measures (Mean, Standard Deviation, Variance, Total number of data points, Maximum and Minimum values) that you want as output of this program. When ready, press @@@OK@@. The selected values will be listed, appropriately labeled, in the screen of your calculator. For example: Sample vs.
Obtaining frequency distributions The application 2. Frequencies.. in the STAT menu can be used to obtain frequency distributions for a set of data. The data must be present in the form of a column vector stored in variable ΣDAT. To get started, press ‚Ù˜ @@@OK@@@. The resulting input form contains the following fields: ΣDAT: Col: X-Min: Bin Count: Bin Width: the matrix containing the data of interest. the column of ΣDAT 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. • Select the program 2. Frequencies.. by using ‚Ù˜ @@@OK@@@. The data is already loaded in ΣDAT, and the option Col should hold the value 1 since we have only one column in ΣDAT. • Change X-Min to -8, Bin Count to 8, and Bin Width to 2, then press @@@OK@@@.
data sets (x,y), stored in columns of the ΣDAT matrix. For this application, you need to have at least two columns in your ΣDAT variable. For example, to fit a linear relationship to the data shown in the table below: x 0 1 2 3 4 5 y 0.5 2.3 3.6 6.7 7.2 11 • First, enter the two columns of data into variable ΣDAT by using the matrix writer, and function STOΣ. • To access the program 3. Fit data.., use the following keystrokes: ‚Ù˜˜@@@OK@@@ The input form will show the current ΣDAT, already loaded.
Level 3 shows the form of the equation. Level 2 shows the sample correlation coefficient, and level 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 The application 4. Summary stats.. in the STAT menu can be useful in some calculations for sample statistics.
• 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 µ.: Single sample confidence interval for the population mean, µ, with known population variance, or for large samples with unknown population variance. 2. Z-INT: µ1−µ2.
5. T-INT: 1 µ.: Single sample confidence interval for the population mean, µ, for small samples with unknown population variance. 6. T-INT: µ1−µ2.: Confidence interval for the difference of the population means, µ1- µ2, for small samples with unknown population variances. Example 1 – Determine the centered confidence interval for the mean of a population if a sample of 60 elements indicate that the mean value of the sample is x = 23.2, and its standard deviation is s = 5.2. Use α = 0.05.
The graph shows the standard normal distribution pdf (probability density function), the location of the critical points ±zα/2, the mean value (23.2) and the corresponding interval limits (21.88424 and 24.51576). Press @TE T to return to the previous results screen, and/or press @@@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.
1. Z-Test: 1 µ.: Single sample hypothesis testing for the population mean, µ, with known population variance, or for large samples with unknown population variance. 2. Z-Test: µ1−µ2.: Hypothesis testing for the difference of the population means, µ1- µ2, with either known population variances, or for large samples with unknown population variances. 3. Z-Test: 1 p.: Single sample hypothesis testing for the proportion, p, for large samples with unknown population variance. 4. Z-Test: p1− p2.
Select µ ≠ 150. Then, press @@@OK@@@. The result is: Then, we reject H0: µ = 150, against H1: µ ≠ 150. The test z value is z0 = -8 5.656854. The P-value is 1.54×10 . The critical values of ±zα/2 = ±1.959964, corresponding to critical x range of {147.2 152.8}.
Chapter 17 Numbers in Different Bases Besides our decimal (base 10, digits = 0-9) number system, you can work with a binary system (base 2, digits = 0,1), an octal system (base 8, digits = 0-7), or a hexadecimal system (base 16, digits=0-9,A-F), among others. The 2 1 0 same way that the decimal integer 321 means 3x10 +2x10 +1x10 , the number 100110, in binary notation, means 5 4 3 2 1 0 1x2 + 0x2 + 0x2 + 1x2 + 1x2 + 0x2 = 32+0+0+4+2+0 = 38. The BASE menu The BASE menu is accessible through ‚ã(the 3 key).
OCT(al), or BIN(ary) in the BASE menu. For example, if @HE ! is selected, binary integers will be a hexadecimal numbers, e.g., #53, #A5B, etc. As different systems are selected, the numbers will be automatically converted to the new current base. To write a number in a particular system, start the number with # and end with either h (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.
Warranty hp 48gII Graphing Calculator; Warranty period: 12 months 1. HP warrants to you, the end-user customer, that HP hardware, accessories and supplies will be free from defects in materials and workmanship after the date of purchase, for the period specified above. 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. 2.
7. TO THE EXTENT ALLOWED BY LOCAL LAW, THE REMEDIES IN THIS WARRANTY STATEMENT ARE YOUR SOLE AND EXCLUSIVE REMEDIES. EXCEPT AS INDICATED ABOVE, IN NO EVENT WILL HP OR ITS SUPPLIERS BE LIABLE FOR LOSS OF DATA OR FOR DIRECT, SPECIAL, INCIDENTAL, CONSEQUENTIAL (INCLUDING LOST PROFIT OR DATA), OR OTHER DAMAGE, WHETHER BASED IN CONTRACT, TORT, OR OTHERWISE.
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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. USA This calculator generates, uses, and can radiate radio frequency energy and may interfere with radio and television reception.
