HP 50g graphing calculator user’s manual H Edition 1 HP part number F2229AA-90001
FrontPageQS49_E.backup.fm Page 2 Friday, February 24, 2006 4:54 PM Notice REGISTER YOUR PRODUCT AT: www.register.hp.com THIS MANUAL AND ANY EXAMPLES CONTAINED HEREIN ARE PROVIDED “AS IS” AND ARE SUBJECT TO CHANGE WITHOUT NOTICE. HEWLETT-PACKARD COMPANY MAKES NO WARRANTY OF ANY KIND WITH REGARD TO THIS MANUAL, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY, NON-INFRINGEMENT AND FITNESS FOR A PARTICULAR PURPOSE. HEWLETT-PACKARD CO.
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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. This manual contains examples that illustrate the use of the basic calculator functions and operations.
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SG49A.book Page 8 Friday, September 16, 2005 1:31 PM Reference, 15-4 Chapter 16 - Statistical Applications Entering data, 16-1 Calculating single-variable statistics, 16-2 Sample vs.
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM Chapter 1 Getting started This chapter provides basic information about the operation of your calculator. It is designed to familiarize you with the basic operations and settings before you perform a calculation. Basic Operations Batteries The calculator uses 4 AAA (LR03) 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.
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 $ 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.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM 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 first line shows the characters: RAD XYZ HEX R= 'X' For details on the meaning of these symbols see Chapter 2 in the calculator’s user’s guide.
@@RCL@ C ReCaLl the contents of a variable @@STO@ D STOre the contents of a variable !PURGE E PURGE a variable @CLEAR F 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 second page is available by pressing the L (NeXT menu) key. This key is the third key from the left in the third row of keys in the keyboard. In this case, only the first two soft menu keys have commands associated with them.
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~…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. Notice that the color and the position of the labels in the key, namely, SYMB, MTH, CAT and P, indicate which is the main function (SYMB), and which of the other three functions is associated with the leftshift „(MTH), right-shift …(CAT ), and ~ (P) keys.
SG49A.book Page 7 Friday, September 16, 2005 1:31 PM 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. The Operating Mode field will be highlighted.
SG49A.book Page 8 Friday, September 16, 2005 1:31 PM 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. Select the RPN operating mode by either using the \ key, or pressing the @CHOOS soft menu key. Press the @@OK#@ soft menu key to complete the operation.
SG49A.book Page 9 Friday, September 16, 2005 1:31 PM Let's try some other simple operations before trying the more complicated expression used earlier for the algebraic operating mode: 123/32 123`32/ 42 3 4`2Q √(√27) 27R3@» Note the position of the y and x in the last two operations. The base in the exponential operation is y (stack level 2) while the exponent is x (stack level 1) before the key Q is pressed.
SG49A.book Page 10 Friday, September 16, 2005 1:31 PM + (3 × (5 - 1/(3 × 3)))/233 + e2.5 = 12.18369, into lev. 1. R √((3 × (5 - 1/(3×3)))/233 + e2.5) = 3.4905156, into 1. To select between the ALG vs. RPN operating mode, you can also set/ clear system flag 95 through the following keystroke sequence: H @FLAGS! 9˜˜˜˜ ` Number Format and decimal dot or comma Changing the number format allows you to customize the way real numbers are displayed by the calculator.
SG49A.book Page 11 Friday, September 16, 2005 1:31 PM Press the right arrow key, ™, to highlight the zero in front of the option Fix. Press the @CHOOS soft menu key and, using the up and down arrow keys, —˜, select, say, 3 decimals. Press the !!@@OK#@ soft menu key to complete the selection: Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as: Notice how the number is rounded, not truncated. Thus, the number 123.
SG49A.book Page 12 Friday, September 16, 2005 1:31 PM Keep the number 3 in front of the Sci. (This number can be changed in the same fashion that we changed the Fixed number of decimals in the example above). Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as: This result, 1.23E2, is the calculator’s version of powers-of-ten notation, i.e., 1.235 × 102.
SG49A.book Page 13 Friday, September 16, 2005 1:31 PM Press the !!@@OK#@ soft menu key return to the calculator display. The 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. For example, the number 0.00256, will be shown as: Decimal comma vs.
SG49A.book Page 14 Friday, September 16, 2005 1:31 PM Angle Measure Trigonometric functions, for example, require arguments representing plane angles. The calculator provides three different Angle Measure modes for working with angles, namely: • Degrees: There are 360 degrees (360°) in a complete circumference. • Radians: There are 2π radians (2π r) in a complete circumference. • Grades: There are 400 grades (400 g) in a complete circumference.
SG49A.book Page 15 Friday, September 16, 2005 1:31 PM soft menu key to complete the operation. For example, in the following screen, the Polar coordinate mode is selected: 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.
SG49A.book Page 16 Friday, September 16, 2005 1:31 PM Non-Rational 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.
SG49A.book Page 17 Friday, September 16, 2005 1:31 PM Selecting Display modes 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 to display the DISPLAY MODES input form.
SG49A.book Page 18 Friday, September 16, 2005 1:31 PM Selecting the display font First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key 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.
SG49A.book Page 19 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 20 Friday, September 16, 2005 1:31 PM 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 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.
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM 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, the display to Textbook, and the number format to Standard.
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM Creating algebraic expressions Algebraic expressions include not only numbers, but also variable names. As an example, we will enter the following algebraic expression: x R +2L R+ y b 2L 1 + We set the calculator operating mode to Algebraic, the CAS to Exact, and the display to Textbook.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM 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 Equation Writer is launched by pressing the keystroke combination ‚O (the third key in the fourth row from the top in the keyboard). The resulting screen is the following.
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM Suppose that you want to replace the quantity between parentheses in the denominator (i.e., 5+1/3) with (5+π2/2). First, we use the delete key (ƒ) delete the current 1/3 expression, and then we replace that fraction with π2/2, as follows: ƒƒƒ„ìQ2 When hit this point the screen looks as follows: In order to insert the denominator 2 in the expression, we need to highlight the entire π2 expression. We do this by pressing the right arrow key (™) once.
SG49A.book Page 7 Friday, September 16, 2005 1:31 PM 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 2 in the denominator of π2/2), we can use the keystroke combination ‚—, interpreted as (‚ ‘ ).
SG49A.book Page 8 Friday, September 16, 2005 1:31 PM ~„y———/~‚tQ1/3 This results in the output: In this example we used several lower-case English letters, e.g., x (~„x), several Greek letters, e.g., λ(~‚n), and even a ∆y combination of Greek and English letters, namely, (~‚c~„y). Keep in mind that to enter a lower-case English letter, you need to use the combination: ~„ followed by the letter you want to enter.
SG49A.book Page 9 Friday, September 16, 2005 1:31 PM Subdirectories To store your data in a well organized directory tree you may want to create subdirectories under the HOME directory, and more subdirectories within subdirectories, in a hierarchy of directories similar to folders in modern computers. The subdirectories will be given names that may reflect the contents of each subdirectory, or any arbitrary name that you can think off.
SG49A.book Page 10 Friday, September 16, 2005 1:31 PM To unlock the upper-case locked keyboard, press ~. Try the following exercises: ~~math` ~~m„a„t„h` ~~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 α Contents -0.
The following are the keystrokes for entering the remaining variables: A12: 3V5K~a12` Q: ~„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, a. RPN mode (Use H\@@OK@@ to change to RPN mode). Use the following keystrokes to store the value of –0.25 into variable α: .25\`³ ~‚a`.
SG49A.book Page 12 Friday, September 16, 2005 1:31 PM To enter the value 3×105 into A12, we can use a shorter version of the procedure: 3V5³~a12`K Here is a way to enter the contents of Q: Q: ~„r/„Ü ~„m+~„r™™³~q`K To enter the value of R, we can use an even shorter version of the procedure: R: „Ô3#2#1™ ³~rK Notice that to separate the elements of a vector in RPN mode we can use the space key (#), rather than the comma (‚í) used above in Algebraic mode. z1: ³3+5*„¥³~„z1K p1: ‚å‚é~„r³„ì* ~„rQ2™™™³~„p1™`K.
SG49A.book Page 13 Friday, September 16, 2005 1:31 PM 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@@z1@@ ` @@@R@@ `@@@Q@@@ `.
SG49A.book Page 14 Friday, September 16, 2005 1:31 PM 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. To see the remaining variables in this directory, press L. 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.
SG49A.book Page 15 Friday, September 16, 2005 1:31 PM You can use the PURGE command to erase more than one variable by placing their names in a list in the argument of PURGE. For example, if now we wanted to purge variables R and Q, simultaneously, we can try the following exercise. Press : I @PURGE@ „ä³J @@@R!@@ ™‚í³J @@@Q!@@ At this point, the screen will show the following command ready to be executed: To finish deleting the variables, press `.
SG49A.book Page 16 Friday, September 16, 2005 1:31 PM UNDO and CMD functions Functions UNDO and CMD are useful for recovering recent commands, or to revert an operation if a mistake was made. These functions are associated with the HIST key: UNDO results from the keystroke sequence ‚¯, while CMD results from the keystroke sequence „®. CHOOSE boxes vs. Soft MENU In some of the exercises presented in this chapter we have seen menu lists of commands displayed in the screen.
SG49A.book @@OK@@ Page 17 Friday, September 16, 2005 1:31 PM 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). To set this flag try the following: H@FLAGS! ——————— The screen shows flag 117 not set (CHOOSE boxes), as shown here: Press the soft menu key to set flag 117 to soft MENU.
SG49A.book Page 18 Friday, September 16, 2005 1:31 PM Press B to select the MEMORY soft menu ()@@MEM@@). The display now shows: Press E to select the DIRECTORY soft menu ()@@DIR@@) The ORDER command is not shown in this screen. To find it we use the L key to find it: To activate the ORDER command we press the C(@ORDER) soft menu key. NOTE: most of the examples in this user manual assume that the current setting of flag 117 is its default setting (that is, not set).
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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.).
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM 6.3#8.54.2#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`2.2-/ In RPN mode, typing the expression between single quotes will allow you to enter the expression like in algebraic mode: ³„Ü5+3.2™/ „Ü7-2.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM • The power function, ^, is available through the Q key. When calculating in the stack in ALG mode, enter the base (y) followed by the Q key, and then the exponent (x), e.g., 5.2Q1.25` In RPN mode, enter the number first, then the function, e.g., 5.2`1.25Q • The root function, XROOT(y,x), is available through the keystroke combination ‚».
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM 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).
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM 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 th 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).
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM 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).
SG49A.book Page 7 Friday, September 16, 2005 1:31 PM Finally, in order to select, for example, the hyperbolic tangent (tanh) function, simply press @@TANH@. 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: „´@@HYP@ @@TANH@ 2.5` In RPN mode, the same value is calculated using: 2.
SG49A.book Page 8 Friday, September 16, 2005 1:31 PM Option 1. Tools.. contains functions used to operate on units (discussed later). Options 2. 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.
SG49A.book Page 9 Friday, September 16, 2005 1:31 PM 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 @)UNITS 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.
SG49A.book Page 10 Friday, September 16, 2005 1:31 PM 5‚Û8@@OK@@ @@OK@@ Notice that the underscore is entered automatically when the RPN mode is active. The keystroke sequences to enter units when the SOFT menu option is selected, in both ALG and RPN modes, are illustrated next.
SG49A.book Page 11 Friday, September 16, 2005 1:31 PM 123‚Ý~„p~„m Using UBASE (type the name) to convert to the default unit (1 m) results in: Operations with units Here are some calculation examples using the ALG operating mode. Be warned that, when multiplying or dividing quantities with units, you must enclosed each quantity with its units between parentheses. Thus, to enter, for example, the product 12.5m × 5.2 yd, type it to read (12.5_m)*(5.2_yd) `: which shows as 65_(m⋅yd).
SG49A.book Page 12 Friday, September 16, 2005 1:31 PM Addition and subtraction can be performed, in ALG mode, without using parentheses, e.g., 5 m + 3200 mm, can be entered simply as 5_m + 3200_mm `. More complicated expression require the use of parentheses, e.g., (12_mm)*(1_cm^2)/(2_s) `: Stack calculations in the RPN mode do not require you to enclose the different terms in parentheses, e.g., 12 @@@m@@@ `1.
SG49A.book Page 13 Friday, September 16, 2005 1:31 PM Physical constants in the calculator The calculator’s physical constants are contained in a constants library activated with the command CONLIB. To launch this command you could simply type it in the stack: ~~conlib`, or, you can select the command CONLIB from the command catalog, as follows: First, launch the catalog by using: ‚N~c. Next, use the up and down arrow keys —˜ to select CONLIB. Finally, press @@OK@@. Press `, if needed.
SG49A.book Page 14 Friday, September 16, 2005 1:31 PM If we de-select the UNITS option (press @UNITS ) only the values are shown (English units selected in this case): To copy the value of Vm to the stack, select the variable name, and press @²STK, then, press @QUIT@. For the calculator set to the ALG, the screen will look like this: The display shows what is called a tagged value, Vm:359.0394. In here, Vm, is the tag of this result. Any arithmetic operation with this number will ignore the tag.
Ch03_RealNumbersQS.fm Page 15 Friday, February 24, 2006 6:19 PM Defining and using functions Users can define their own functions by using the DEFINE command available thought the keystroke sequence „à (associated with the 2 key).
Ch03_RealNumbersQS.fm Page 16 Friday, February 24, 2006 6:19 PM relatively simple and consists of two parts, contained between the program containers • Input: x x • Process: ‘LN(x+1) + EXP(x) ‘ This is to be interpreted as saying: enter a value that is temporarily assigned to the name x (referred to as a local variable), evaluate the expression between quotes that contain that local variable, and show the evaluated expression.
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM Chapter 4 Calculations with complex numbers This chapter shows examples of calculations and application of functions to complex numbers. Definitions A complex number z is a number z = x + iy, where x and y are real numbers, and i is the imaginary unit defined by i² = –1. The complex number x + iy has a real part, x = Re(z), and an imaginary part, y = Im(z).
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM Press @@OK@@ , twice, to return to the stack. Entering complex numbers Complex numbers in the calculator can be entered in either of the two Cartesian representations, namely, x+iy, or (x,y). The results in the calculator will be shown in the ordered-pair format, i.e., (x,y). For example, with the calculator in ALG mode, the complex number (3.5, -1.2), is entered as: „Ü3.5‚í\1.2` A complex number can also be entered in the form x+iy.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM Polar representation of a complex number The polar representation of the complex number 3.5-1.2i, entered above, is obtained by changing the coordinate system to cylindrical or polar (using function CYLIN). You can find this function in the catalog (‚N). You can also change the coordinate to polar using H.
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM Simple operations with complex numbers Complex numbers can be combined using the four fundamental operations (+-*/). The results follow the rules of algebra with the caveat that i2= -1. Operations with complex numbers are similar to those with real numbers. For example, with the calculator in ALG mode and the CAS set to Complex, try the following operations: (3+5i) + (6-3i) = (9,2); (5-2i) - (3+4i) = (2,-6) (3-i)·(2-4i) = (2,-14); (5-2i)/(3+4i) = (0.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM The first menu (options 1 through 6) shows the following functions: Examples of applications of these functions are shown next in RECT RE(z) Real part of a complex number IM(z) Imaginary part of a complex number C→R(z) Separates a complex number into its real and imaginary parts R→C(x,y) Forms the complex number (x,y) out of real numbers x and y ABS(z) Calculates the magnitude of a complex number.
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM CMPLX menu in keyboard A second CMPLX menu is accessible by using the right-shift option associated with the 1 key, i.e., ‚ß. With system flag 117 set to CHOOSE boxes, the keyboard CMPLX menu shows up as the following screens: The resulting menu include some of the functions already introduced in the previous section, namely, ARG, ABS, CONJ, IM, NEG, RE, and SIGN.
SG49A.book Page 7 Friday, September 16, 2005 1:31 PM 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.
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SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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: 12.3, 15.2_m, ‘π’, ‘e’, ‘i’ • A number: • A variable name: ‘a’, ‘ux’, ‘width’, etc.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 3 Friday, September 16, 2005 @@A1@@ *@@A2@@ ` ‚¹@@A1@@ 1:31 PM @@A1@@ / @@A2@@ ` „¸@@A2@@ The same results are obtained in RPN mode if using the following keystrokes: @@A1@@ @@A2@@ +µ @@A1@@ @@A2@@ *µ @@A1@@ ‚ ¹µ @@A1@@ @@A2@@ -µ @@A1@@ @@A2@@ /µ @@A2@@ „ ¸µ Functions in the ALG menu The ALG (Algebraic) menu is available by using the keystroke sequence ‚× (associated with the 4 key).
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM To complete the operation press @@OK@@. Here is the help screen for function COLLECT: We notice that, at the bottom of the screen, the line See: EXPAND FACTOR suggests links to other help facility entries, the functions EXPAND and FACTOR. To move directly to those entries, press the soft menu key @SEE1! for EXPAND, and @SEE2! for FACTOR.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM For example, for function SUBST, we find the following CAS help facility entry: NOTE: Recall that, to use these, or any other functions in the RPN mode, you must enter the argument first, and then the function. For example, the example for TEXPAND, in RPN mode will be set up as: ³„¸+~x+~y` At this point, select function TEXPAND from menu ALG (or directly from the catalog ‚N), to complete the operation.
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM Information and examples on these commands are available in the help facility of the calculator.
Functions in the ARITHMETIC menu The ARITHMETIC menu is triggered through the keystroke combination „Þ (associated with the 1 key). With system flag 117 set to CHOOSE boxes, „Þ shows the following menu: Out of this menu list, options 5 through 9 (DIVIS, FACTORS, LGCD, PROPFRAC, SIMP2) correspond to common functions that apply to integer numbers or to polynomials. The remaining options ( 1. INTEGER, 2. POLYNOMIAL, 3. MODULO, and 4.
SG49A.book Page 8 Friday, September 16, 2005 1:31 PM Polynomials Polynomials are algebraic expressions consisting of one or more terms containing decreasing powers of a given variable. For example, ‘X^3+2*X^2-3*X+2’ is a third-order polynomial in X, while ‘SIN(X)^2-2’ is a second-order polynomial in SIN(X). Functions COLLECT and EXPAND, shown earlier, can be used on polynomials.
SG49A.book Page 9 Friday, September 16, 2005 1:31 PM The PROOT function Given an array containing the coefficients of a polynomial, in decreasing order, the function PROOT provides the roots of the polynomial. Example, from X2+5X+6 =0, PROOT([1, –5, 6]) = [2. 3.]. The QUOT and REMAINDER functions The functions QUOT and REMAINDER provide, respectively, the quotient Q(X) and the remainder R(X), resulting from dividing two polynomials, P1(X) and P2(X).
SG49A.book Page 10 Friday, September 16, 2005 1:31 PM FACTOR(‘(X^3-9*X)/(X^2-5*X+6)’ )=‘X*(X+3)/(X-2)’ The SIMP2 function Function SIMP2, in the ARITHMETIC menu, takes as arguments two numbers or polynomials, representing the numerator and denominator of a rational fraction, and returns the simplified numerator and denominator. For example: SIMP2(‘X^3-1’,’X^2-4*X+3’) = {‘X^2+X+1’,‘X-3’} The PROPFRAC function The function PROPFRAC converts a rational fraction into a “proper” fraction, i.e.
SG49A.book Page 11 Friday, September 16, 2005 1:31 PM FCOEF([2,1,0,3,–5,2,1,–2,–3,–5])=‘(X--5)^2*X^3*(X-2)/(X-+3)^5*(X-1)^2’ If you press µ„î` (or, simply µ, in RPN mode) you will get: ‘(X^6+8*X^5+5*X^4-50*X^3)/(X^7+13*X^6+61*X^5+105*X^445*X^3-297*X62-81*X+243)’ The FROOTS function The function FROOTS, in the ARITHMETIC/POLYNOMIAL menu, 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.].
SG49A.book Page 12 Friday, September 16, 2005 1:31 PM Reference Additional information, definitions, and examples of algebraic and arithmetic operations are presented in Chapter 5 of the calculator’s user’s guide.
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM 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).
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM The following examples show the use of function SOLVE in ALG and RPN modes (Use Complex mode in the CAS): The screen shot shown above displays two solutions. In the first one, β4 -5β =125, SOLVE produces no solutions { }. In the second one, β4 - 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.
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM screen shots show the RPN stack before and after the application of ZEROS to the two examples above (Use Complex mode in the CAS): The Symbolic Solver functions presented above produce solutions to rational equations (mainly, polynomial equations). If the equation to be solved for has all numerical coefficients, a numerical solution is possible through the use of the Numerical Solver features of the calculator.
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM with examples for the numerical solver applications. Item 6. MSLV (Multiple equation SoLVer) will be presented later in page 6-10. Notes: 1. Whenever you solve for a value in the NUM.SLV applications, the value solved for will be placed in the stack. This is useful if you need to keep that value available for other operations. 2. There will be one or more variables created whenever you activate some of the applications in the NUM.SLV menu.
SG49A.book Page 7 Friday, September 16, 2005 1:31 PM 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].
SG49A.book Page 8 Friday, September 16, 2005 1:31 PM 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].
SG49A.book Page 9 Friday, September 16, 2005 1:31 PM Solving equations with one unknown through NUM.SLV The calculator's NUM.SLV menu provides item 1. Solve equation.. solve different types of equations in a single variable, including non-linear algebraic and transcendental equations. For example, let's solve the equation: ex-sin(πx/3) = 0. Simply enter the expression as an algebraic object and store it into variable EQ.
SG49A.book Page 10 Friday, September 16, 2005 1:31 PM 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. To solve the equation all you need to do is highlight the field in front of X: by using ˜, and press @SOLVE@. The solution shown is X: 4.5006E-2: This, however, is not the only possible solution for this equation.
SG49A.book Page 11 Friday, September 16, 2005 1:31 PM In ALG mode, press @ECHO to copy the example to the stack, press ` to run the example. To see all the elements in the solution you need to activate the line editor by pressing the down arrow key (˜): In RPN mode, the solution for this example is produced by using: Activating function MSLV results in the following screen. You may have noticed that, while producing the solution, the screen shows intermediate information on the upper left corner.
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SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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).
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM The division L4/L3 will produce an infinity entry because one of the elements in L3 is zero, and an error message is returned. 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.
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM Functions applied to lists Real number functions from the keyboard (ABS, ex, LN, 10x, LOG, SIN, x2, √, COS, TAN, ASIN, ACOS, ATAN, yx) as well as those from the MTH/ HYPERBOLIC menu (SINH, COSH, TANH, ASINH, ACOSH, ATANH), and MTH/REAL menu (%, etc.), can be applied to lists, e.g., ABS INVERSE (1/x) Lists of complex numbers You can create a complex number list, say L1 ADD i*L2. In RPN mode, you could enter this as L1 i L2 ADD *.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM Lists of algebraic objects The following are examples of lists of algebraic objects with the function SIN applied to them (select Exact mode for these examples -- See Chapter 1): The MTH/LIST menu The MTH menu provides a number of functions that exclusively to lists.
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM Examples of application of these functions in ALG mode are shown next: SORT and REVLIST can be combined to sort a list in decreasing order: If you are working in RPN mode, enter the list onto the stack and then select the operation you want. For example, to calculate the increment between consecutive elements in list L3, press: l3`!´˜˜#OK# #OK# This places L3 onto the stack and then selects the ∆LIST operation from the MTH menu.
SG49A.book Page 7 Friday, September 16, 2005 1:31 PM 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.
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SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM Storing vectors into variables in the stack Vectors can be stored into variables. The screen shots below show the vectors u2 = [1, 2], u3 = [-3, 2, -2], v2 = [3,-1], v3 = [1, -5, 2] Stored into variables @@@u2@@, @@@u3@@, @@@v2@@, and @@@v3@@, respectively. First, in ALG mode: Then, in RPN mode (before pressing K, repeatedly): NOTE: The apostrophes (‘) are not needed ordinarily in entering the names u2, v2, etc. in RPN mode.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM Using the Matrix Writer (MTRW) to enter vectors Vectors can also be entered by using the Matrix Writer „² (third key in the fourth row of keys from the top of the keyboard). This command generates a species of spreadsheet corresponding to rows and columns of a matrix (Details on using the Matrix Writer to enter matrices will be presented in Chapter 9). For a vector we are interested in filling only elements in the top row.
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM @+ROW@ @-ROW @+COL@ @-COL@ @GOTO@ 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.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM Simple operations with vectors To illustrate operations with vectors we will use the vectors u2, u3, v2, and v3, stored in an earlier exercise. Also, store vector A=[-1,-2,-3,-4,-5] to be used in the following exercises. Changing sign To change the sign of a vector use the key \, e.g.
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 7 Friday, September 16, 2005 1:31 PM Magnitude The magnitude of a vector, as discussed earlier, can be found with function ABS. This function is also available from the keyboard („Ê). Examples of application of function ABS were shown above. Dot product Function DOT (option 2 in CHOOSE box above) is used to calculate the dot product of two vectors of the same length.
SG49A.book Page 8 Friday, September 16, 2005 1:31 PM Examples of cross products of one 3-D vector with one 2-D vector, or vice versa, are presented next: Attempts to calculate a cross product of vectors of length other than 2 or 3, produce an error message: 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.
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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 Writer, and (2) typing the matrix directly into the stack.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM If you have selected the textbook display option (using H@)DISP! and checking off Textbook), the matrix will look like the one shown above. Otherwise, the display will show: The display in RPN mode will look very similar to these. 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 „Ô2.5\‚í4.2‚í2™ ‚í „Ô.3‚í1.9‚í2.8™ ‚í „Ô2‚í.1\‚í.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM Operations with matrices Matrices, like other mathematical objects, can be added and subtracted. They can be multiplied by a scalar, or among themselves, and raised to a real power. 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.
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM Addition and subtraction Four examples are shown below using the matrices stored above (ALG mode). In RPN mode, try the following eight examples: A22 ` B22`+ A22 ` B22`- A23 ` B23`+ A23 ` B23`- A32 ` B32`+ A32 ` B32`- A33 ` B33`+ A33 ` B33`- Multiplication There are a number of multiplication operations that involve matrices. These are described next. The examples are shown in algebraic mode.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM 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. A couple of examples of matrixvector multiplication follow: 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.
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM Term-by-term multiplication Term-by-term multiplication of two matrices of the same dimensions is possible through the use of function HADAMARD. The result is, of course, another matrix of the same dimensions. This function is available through Function catalog (‚N), or through the MATRICES/OPERATIONS submenu („Ø).
SG49A.book Page 7 Friday, September 16, 2005 1:31 PM 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 The inverse of a square matrix A is the matrix A-1 such that A⋅A-1 = A-1⋅A = I, where I is the identity matrix of the same dimensions as A.
SG49A.book Page 8 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 9 Friday, September 16, 2005 1:31 PM Solution of linear systems A system of n linear equations in m variables can be written as a11⋅x1 + a12⋅x2 + a13⋅x3 + …+ a1,m-1⋅x m-1 + a1,m⋅x m = b 1, a21⋅x1 + a22⋅x2 + a23⋅x3 + …+ a2,m-1⋅x m-1 + a2,m⋅x m = b2, a31⋅x1 + a32⋅x2 + a33⋅x3 + …+ a3,m-1⋅x . . . … m-1 . + a3,m⋅x m . = b 3, . an-1,1⋅x1 + an-1,2⋅x2 + an-1,3⋅x3 + …+ an-1,m-1⋅x m-1 + an-1,m⋅x m = bn-1, an1⋅x1 + an2⋅x2 + an3⋅x3 + …+ an,m-1⋅x m-1 + an,m⋅x m = bn.
SG49A.book Page 10 Friday, September 16, 2005 1:31 PM 2x1 + 3x2 –5x3 = 13, x1 – 3x2 + 8x3 = -13, 2x1 – 2x2 + 4x3 = -6, can be written as the matrix equation A⋅x = b, if ⎡ 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.
SG49A.book Page 11 Friday, September 16, 2005 1:31 PM A solution was found as shown next. Solution with the inverse matrix The solution to the system A⋅x = b, where A is a square matrix is x = A-1⋅ b.
SG49A.book Page 12 Friday, September 16, 2005 1:31 PM References Additional information on creating matrices, matrix operations, and matrix applications in linear algebra is presented in Chapters 10 and 11 of the calculator’s user’s guide.
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM • Press ` to return to the PLOT - FUNCTION window. The expression ‘Y1(X) = EXP(-X^2/2)/√(2*π)’ will be highlighted. Press L@@@OK@@@ to return to normal calculator display. • Enter the PLOT WINDOW environment by entering „ò (press them simultaneously if in RPN mode). Use a range of –4 to 4 for HVIEW, then press @AUTO to generate the V-VIEW automatically.
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM Generating a table of values for a function The combinations „õ(E) and „ö(F), pressed simultaneously if in RPN mode, let’s the user produce a table of values of functions. For example, we will produce a table of the function Y(X) = X/ (X+10), in the range -5 < X < 5 following these instructions: • We will generate values of the function f(x), defined above, for values of x from –5 to 5, in increments of 0.5.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM • 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. • To increase the resolution by an additional factor of 0.
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM • Keep the default plot window ranges to read: X-Left:-1 X-Right:1 Y-Near:-1 Y-Far: 1 Z-Low: -1 Z-High: 1 Step Indep: 10 Depnd: 8 NOTE: The Step Indep: and Depnd: values represent the number of gridlines to be used in the plot. The larger these number, the slower it is to produce the graph, although, the times utilized for graphic generation are relatively fast. For the time being we’ll keep the default values of 10 and 8 for the Step data.
SG49A.book Page 7 Friday, September 16, 2005 1:31 PM • When done, press @EXIT. • Press @CANCL to return to PLOT WINDOW. • Press $, or L@@@OK@@@, to return to normal calculator display. Try also a Fast 3D plot for the surface z = f(x,y) = sin (x2+y2) • Press „ô, simultaneously if in RPN mode, to access the PLOT SETUP window. • Press ˜ and type ‘SIN(X^2+Y^2)’ @@@OK@@@. • Press @ERASE @DRAW to draw the slope field plot.
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SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM Function lim is entered in ALG mode as lim(f(x),x=a) to calculate f ( x) . the limit lim x→ a In RPN mode, enter the function first, then the expression ‘x=a’, and finally function lim. Examples in ALG mode are shown next, including some limits to infinity, and one-sided limits. The infinity symbol is associated with the 0 key, i.e.., „è. To calculate one-sided limits, add +0 or -0 to the value to the variable.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM Functions DERIV and DERVX The function DERIV is used to take derivatives in terms of any independent variable, while the function DERVX takes derivatives with respect to the CAS default variable VX (typically ‘X’). While function DERVX is available directly in the CALC menu, both functions are available in the DERIV.&INTEG sub-menu within the CALCL menu ( „Ö).
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM 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! , where f(n)(x) represents the n-th derivative of f(x) with respect to x, f(0)(x) = f(x). If the value x0 = 0, the series is referred to as a Maclaurin’s series.
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM 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. The list returned as the first output object includes the following items: 1.
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM To define the functions f(x,y) and g(x,y,z), in ALG mode, use: DEF(f(x,y)=x*COS(y)) ` DEF(g(x,y,z)=√(x^2+y^2)*SIN(z) ` To type the derivative symbol use ‚¿. The derivative ∂ ( f ( x, y )) , ∂x for example, will be entered as ∂x(f(x,y)) ` in ALG mode in the screen.
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM Chapter 13 Vector Analysis Applications This chapter describes the use of functions HESS, DIV, and CURL, for calculating operations of vector analysis.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM 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 vector field.
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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..
Ch14_DifferentialEquationsQS.fm Page 2 Friday, March 17, 2006 6:23 PM • 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). The output from the function is the general solution of the ODE. The examples below are shown in the RPN mode: Example 1 – To solve the homogeneous ODE d3y/dx3-4⋅(d2y/dx2)-11⋅(dy/dx)+30⋅y = 0.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM 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. You can also provide a vector containing the differential equation and the initial conditions, instead of only a differential equation, as input to DESOLVE.
SG49A.book Page 4 Friday, September 16, 2005 ‘d1y(0) = -0.5’. solution. 1:31 PM Changing to these Exact expressions facilitates the NOTE: To obtain fractional expressions for decimal values use function Q (See Chapter 5). Press µµ to simplify the result. Use ˜ @EDIT to see this result: i.e., ‘y(t) = -((19*√5*SIN(√5*t)-(148*COS(√5*t)+80*COS(t/2)))/190)’. Press ``J@ODETY to get the string “Linear w/ cst coeff” for the ODE type in this case.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM Compare these expressions with the one given earlier in the definition of the Laplace transform, i.e., L{f (t )}= F ( s ) = ∫ f (t ) ⋅ e − st dt , ∞ 0 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).
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM Fourier series for a quadratic function Determine the coefficients c0, c1, and c2 for the function g(t) = (t-1)2+(t-1), with period T = 2. Using the calculator in ALG mode, first we define functions f(t) and g(t): Next, we move to the CASDIR sub-directory under HOME to change the value of variable PERIOD, e.g., „(hold) §`J@)CASDI`2K@PERIOD ` Return to the sub-directory where you defined functions f and g, and calculate the coefficients.
SG49A.book Thus, Page 7 Friday, September 16, 2005 1:31 PM c0 = 1/3, c1 = (π⋅i+2)/π2, c2 = (π⋅i+1)/(2π2). The Fourier series with three elements will be written as g(t) ≈ Re[(1/3) + (π⋅i+2)/π2⋅exp(i⋅π⋅t)+ (π⋅i+1)/(2π2)⋅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.
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SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM • 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). The factorial symbol (!) can be entered also as the keystroke combination ~‚2.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM 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 distribution, the Student’s t distribution, the Chi-square (χ2) distribution, and the F-distribution. The functions provided by the calculator to evaluate probabilities for these distributions are NDIST, UTPN, UTPT, UTPC, and UTPF.
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM The Chi-square distribution The Chi-square (χ2) distribution has one parameter ν, known as the degrees of freedom. The calculator provides for values of the upper-tail (cumulative) distribution function for the χ2-distribution using UTPC given the value of x and the parameter ν. The definition of this function is, therefore, UTPC(ν,x) = P(X>x) = 1 - P(X
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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 numbered 1, 2, and 4 in the list above require that the data be available as columns of the matrix ΣDAT.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM Calculating single-variable statistics After entering the column vector into ΣDAT, press ‚Ù @@@OK@@ to select 1. Single-var.. The following input form will be provided: The form lists the data in ΣDAT, shows that column 1 is selected (there is only one column in the current ΣDAT).
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM 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: the matrix containing the data of interest. Col: the column of ΣDAT that is under scrutiny.
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM ΣDAT, by using function STOΣ (see example above). Next, obtain singlevariable information using: ‚Ù @@@OK@@@. The results are: 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@@@.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM Fitting data to a function y = f(x) The program 3. Fit data.., available as option number 3 in the STAT menu, can be used to fit linear, logarithmic, exponential, and power functions to 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 y 0 0.5 1 2.3 2 3.6 3 6.7 4 7.
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM 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..
SG49A.book Page 7 Friday, September 16, 2005 1:31 PM • 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.
SG49A.book Page 8 Friday, September 16, 2005 1:31 PM 4. Z-INT: p1− p2.: Confidence interval for the difference of two proportions, p1-p2, for large samples with unknown population variances. 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.
SG49A.book Page 9 Friday, September 16, 2005 1:31 PM The graph shows the standard normal distribution pdf (probability density function), the location of the critical points ±zα/2, the mean value (23.3) and the corresponding interval limits (21.98424 and 24.61576). Press @TEXT 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.
SG49A.book Page 10 Friday, September 16, 2005 1:31 PM 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.: Hypothesis testing for the difference of two proportions, p1-p2, for large samples with unknown population variances. 5.
SG49A.book Page 11 Friday, September 16, 2005 1:31 PM Then, we reject H0: µ = 150, against H1: µ ≠ 150. The test z value is z0 = 5.656854. The P-value is 1.54×10 -8. The critical values of ±zα/2 = ±1.959964, corresponding to critical ⎯x range of {147.2 152.8}.
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SG49A.book Page 1 Friday, September 16, 2005 1:31 PM 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 same way that the decimal integer 321 means 3x102+2x101+1x100, the number 100110, in binary notation, means 1x25 + 0x24 + 0x23 + 1x22 + 1x21 + 0x20 = 32+0+0+4+2+0 = 38.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM 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), OCT (al), or BIN (ary) in the BASE menu. For example, if is selected, binary integers will be a hexadecimal numbers, e.g., #53, #A5B, etc.
Ch18_Using SD cardQS.fm Page 1 Friday, February 24, 2006 8:39 PM Chapter 18 Using SD cards The calculator has a memory card slot into which you can insert an SD flash card for backing up calculator objects, or for downloading objects from other sources. The SD card in the calculator will appear as port number 3. Inserting and removing an SD card The SD slot is located on the bottom edge of the calculator, just below the number keys. SD cards must be inserted facing down.
Ch18_Using SD cardQS.fm Page 2 Friday, February 24, 2006 8:39 PM 4. When the formatting is finished, the HP 50g displays the message "FORMAT FINISHED. PRESS ANY KEY TO EXIT". To exit the system menu, hold down the ‡ key, press and release the C key and then release the ‡ key. The SD card is now ready for use. It will have been formatted in FAT32 format. Accessing objects on an SD card Accessing an object on the SD card is similar to when an object is located in ports 0, 1, or 2.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM Note that if the name of the object you intend to store on an SD card is longer than eight characters, it will appear in 8.3 DOS format in port 3 in the Filer once it is stored on the card. Recalling an object from the SD card To recall an object from the SD card onto the screen, use function RCL, as follows: • In algebraic mode: Press „©, type the name of the stored object using port 3 (e.g., :3:VAR1), press `.
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM Purging all objects on the SD card (by reformatting) You can purge all objects from the SD card by reformatting it. When an SD card is inserted, @FORMA appears an additional menu item in File Manager. Selecting this option reformats the entire card, a process which also deletes every object on the card. Specifying a directory on an SD card You can store, recall, evaluate and purge objects that are in directories on an SD card.
SG49A.book Page 1 Friday, September 16, 2005 1:31 PM Chapter 19 Equation Library The Equation Library is a collection of equations and commands that enable you to solve simple science and engineering problems. The library consists of more than 300 equations grouped into 15 technical subjects containing more than 100 problem titles. Each problem title contains one or more equations that help you solve that type of problem.
SG49A.book Page 2 Step 4: Friday, September 16, 2005 1:31 PM View the five equations in the Projectile Motion set. All five are used interchangeably in order to solve for missing variables (see the next example). #EQN# #NXEQ# #NXEQ# #NXEQ# #NXEQ# Step 5: Examine the variables used by the equation set. #VARS# and —as ˜ needed Now use this equation set to answer the questions in the following example.
SG49A.book Page 3 Friday, September 16, 2005 1:31 PM 0 *!!!!!!X0!!!!!+ 0 *!!!!!!Y0!!!!!+ 50 *!!!!!!Ô0!!!!!+ L65*!!!!!!R!!!!!+ Step 3: Solve for the velocity, v0. (You solve for a variable by pressing ! and then the variable’s menu key.) !*!!!!!!V0!!!!!+ Step 4: Recall the range, R, divide by 2 to get the halfway distance, and enter that as the x-coordinate. Notice that pressing the right-shifted version of a variable’s menu key causes the calculator to recall its value to the stack.
SG49A.book Page 4 Friday, September 16, 2005 1:31 PM Reference For additional details on the Equation Library, see Chapter 27 in the calculator’s user’s guide.
WarrantyQS49_E.fm Page 1 Friday, February 24, 2006 8:25 PM Limited Warranty HP 50g 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.
SG49A.book Page 2 Friday, September 16, 2005 1:31 PM 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. Some countries, States or provinces do not allow the exclusion or limitation of incidental or consequential damages, so the above limitation or exclusion may not apply to you. 8.
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SG49A.book Page 4 Friday, September 16, 2005 L.America 1:31 PM Country : Telephone numbers Argentina Brazil 0-810-555-5520 Sao Paulo 3747-7799; ROTC 0-800-157751 Mx City 5258-9922; ROTC 01-800-472-6684 0800-4746-8368 800-360999 9-800-114726 0-800-10111 1-800-711-2884 Mexico Venezuela Chile Columbia Peru Central America & Caribbean Guatemala Puerto Rico Costa Rica N.America 1-800-999-5105 1-877-232-0589 0-800-011-0524 Country : Telephone numbers U.S.
SG49A.book Page 5 Friday, September 16, 2005 1:31 PM Regulatory information Federal Communications Commission Notice This equipment has been tested and found to comply with the limits for a Class B digital device, pursuant to Part 15 of the FCC Rules. These limits are designed to provide reasonable protection against harmful interference in a residential installation.
SG49A.book Page 6 Friday, September 16, 2005 1:31 PM Or, call 1-800-474-6836 For questions regarding this FCC declaration, contact: Hewlett-Packard Company P. O. Box 692000, Mail Stop 510101 Houston, Texas 77269-2000 Or, call 1-281-514-3333 To identify this product, refer to the part, series, or model number found on the product. Canadian Notice This Class B digital apparatus meets all requirements of the Canadian Interference-Causing Equipment Regulations.
SG49A.book Page 7 Friday, September 16, 2005 1:31 PM Japanese Notice こ の装置は、 情報処理装置等電波障害自主規制協議会 （VCCI） の基準に基づ く ク ラ ス B 情報技術装置です。 こ の装置は、 家庭環境で使用する こ と を目的 と し ていますが、 こ の装 置がラ ジオやテ レ ビ ジ ョ ン受信機に近接 し て使用 さ れる と 、 受信障害を引き起 こ す こ と が あ り ます。 取扱説明書に従 っ て正 し い取 り 扱い を し て く だ さ い。 Korean Notice Disposal of Waste Equipment by Users in Private Household in the European Union This symbol on the product or on its packaging indicates that this product must not be disposed of with your other household waste.
