Hp 50g Graphing Calculator Bedienungsanleitung

HPg graphing calculatoruser’s guideHEdition 1HP part number F2229AA-90006

Page TOC-6The PROOT function ,5-21The PTAYL function ,5-21The QUOT and REMAINDER functions ,5-21The EPSX0 function and the CAS variable EPS ,5-22The P

Page 2-40The Object input field, the first input field in the form, is highlighted by default. This input field can hold the contents of a new variab

Page 2-41To move into the MANS directory, press the corresponding soft menu key (Ain this case), and ` if in algebraic mode. The directory tree will

Page 2-42Use the down arrow key (˜) to select the option 2. MEMORY… , or justpress 2. Then, press @@OK@@. This will produce the following pull-downm

Page 2-43Press the @@OK@ soft menu key to activate the command, to create the sub-directory:Moving among subdirectoriesTo move down the directory tree

Page 2-44The ‘S2’ string in this form is the name of the sub-directory that is being deleted. The soft menu keys provide the following options:@YES@

Page 2-45Use the down arrow key (˜) to select the option 2. MEMORY… Then,press @@OK@@. This will produce the following pull-down menu:Use the down a

Page 2-46Press @@OK@@, to get:Then, press )@@S3@@ to enter ‘S3’ as the argument to PGDIR. Press ` to delete the sub-directory:Command PGDIR in RPN mo

Page 2-47Using the PURGE command from the TOOL menuThe TOOL menu is available by pressing the I key (Algebraic and RPN modes shown): The PURGE comm

Page 2-48Using the FILES menuWe will use the FILES menu to enter the variable A. We assume that we are in the sub-directory {HOME M NS INTRO}. To ge

Page 2-49To enter variable A (see table above), we first enter its contents, namely, the number 12.5, and then its name, A, as follows: 12.5@@OK@@ ~

Page TOC-7Variable EQ ,6-26The SOLVR sub-menu ,6-26The DIFFE sub-menu ,6-29The POLY sub-menu ,6-29The SYS sub-menu ,6-30The TVM sub-menu ,6-30Chapter

Page 2-50Using the STO commandA simpler way to create a variable is by using the STO command (i.e., the Kkey). We provide examples in both the Algeb

Page 2-51z1: 3+5*„¥ K~„z1` (if needed,accept change to Complex mode)p1: ‚å‚é~„r³„ì*~„rQ2™™™ K~„p1`.The screen, at this point, will look as follows:Yo

Page 2-52z1: ³3+5*„¥ ³~„z1 K(ifneeded, accept change to Complex mode)p1: ‚å‚é~„r³„ì*~„rQ2™™™ ³ ~„p1™` K.The screen, at this point, will look as follow

Page 2-53Pressing the soft menu key corresponding to p1 will provide an error message (try L @@@p1@@ `):Note: By pressing @@@p1@@ ` we are trying

Page 2-54At this point, the screen looks like this:To see the contents of A, use: L @@@A@@@.To run program p1 with r = 5, use: L5 @@@p1@@@.Notice that

Page 2-55Notice that this time the contents of program p1 are listed in the screen. To see the remaining variables in this directory, press L:Listing

Page 2-56followed by the variable’s soft menu key. For example, in RPN, if we want to change the contents of variable z1 to ‘a+b⋅i ’, use:³~„a+~„b*„¥

Page 2-57Use the up arrow key — to select the sub-directory MANS and press @@OK@@. If you now press „§, the screen will show the contents of sub-dire

Page 2-58Next, use the delete key three times, to remove the last three lines in the display: ƒ ƒ ƒ. At this point, the stack is ready to execute th

Page 2-59Copying two or more variables using the stack in RPN modeThe following is an exercise to demonstrate how to copy two or more variables using

Page TOC-8List size ,8-10Extracting and inserting elements in a list ,8-10Element position in the list ,8-11HEAD and TAIL functions ,8-11The SEQ funct

Page 2-60The screen now shows the new ordering of the variables:RPN modeIn RPN mode, the list of re-ordered variables is listed in the stack before ap

Page 2-61Notice that variable A12 is no longer there. If you now press „§, the screen will show the contents of sub-directory MANS, including variab

Page 2-62variable p1. Press I @PURGE@ J@@p1@@ `. The screen will now show variable p1 removed:You can use the PURGE command to erase more than one

Page 2-63the HIST key: UNDO results from the keystroke sequence ‚¯, while CMD results from the keystroke sequence „®. To illustrate the use of UNDO,

Page 2-64As you can see, the numbers 3, 2, and 5, used in the first calculation above, are listed in the selection box, as well as the algebraic ‘SIN

Page 2-65Example of flag setting: general solutions vs. principal valueFor example, the default value for system flag 01 is General solutions. What t

Page 2-66` (keeping a second copy in the RPN stack)³~ „t`Use the following keystroke sequence to enter the QUAD command: ‚N~q (use the up and down ar

Page 2-67CHOOSE boxes vs. Soft MENUIn some of the exercises presented in this chapter we have seen menu lists of commands displayed in the screen. Th

Page 2-68The screen shows flag 117 not set (CHOOSE boxes), as shown here:Press the @@CHK@@ soft menu key to set flag 117 to soft MENU. The screen w

Page 2-69Note: most of the examples in this user guide assume that the current setting of flag 117 is its default setting (that is, not set). If you

Page TOC-9Changing coordinate system ,9-12Application of vector operations ,9-15Resultant of forces ,9-15Angle between vectors ,9-15Moment of a force

Page 2-70• The CMDS (CoMmanDS) menu, activated within the Equation Writer, i.e., ‚O L @CMDS

Page 3-1Chapter 3Calculation with real numbersThis chapter demonstrates the use of the calculator for operations and functions related to real numbers

Page 3-22. Coordinate system specification (XYZ, R∠Z, R∠∠). The symbol ∠stands for an angular coordinate.XYZ: Cartesian or rectangular (x,y,z) R∠Z:

Page 3-3Real number calculations will be demonstrated in both the Algebraic (ALG) and Reverse Polish Notation (RPN) modes. Changing sign of a number,

Page 3-4Alternatively, 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

Page 3-5Squares and square rootsThe square function, SQ, is available through the keystroke combination: „º. When calculating in the stack in ALG mo

Page 3-6Using powers of 10 in entering dataPowers of ten, i.e., numbers of the form -4.5´10-2, etc., are entered by using the V key. For example, in

Page 3-7the inverse trigonometric functions represent angles, the answer from these functions will be given in the selected angular measure (DEG, RAD,

Page 3-8combination „´. With the default setting of CHOOSE boxes for system flag 117 (see Chapter 2), the MTH menu is shown as the following menu li

Page 3-9Hyperbolic functions and their inversesSelecting Option 4. HYPERBOLIC.. , in the MTH menu, and pressing @@OK@@,produces the hyperbolic functio

Page TOC-10Function VANDERMONDE ,10-13Function HILBERT ,10-14A program to build a matrix out of a number of lists ,10-14Lists represent columns of the

Page 3-10The result is:The operations shown above assume that you are using the default setting for system flag 117 (CHOOSE boxes). If you have change

Page 3-11For example, to calculate tanh(2.5), in the ALG mode, when using SOFT menusover CHOOSE boxes, follow this procedure:„´ Select MTH menu)@@HYP@

Page 3-12 Option 19. MATH.. returns the user to the MTH menu. The remaining functions are grouped into six different groups described below.If sy

Page 3-13The result is shown next:In RPN mode, recall that argument y is located in the second level of the stack, while argument x is located in the

Page 3-14Please notice that MOD is not a function, but rather an operator, i.e., in ALG mode, MOD should be used as y MOD x, and not as MOD(y,x). Th

Page 3-15GAMMA: The Gamma function Γ(α)PSI: N-th derivative of the digamma functionPsi: Digamma function, derivative of the ln(Gamma)The Gamma functio

Page 3-16Examples of these special functions are shown here using both the ALG and RPN modes. As an exercise, verify that GAMMA(2.3) = 1.166711…, PS

Page 3-17Selecting any of these entries will place the value selected, whether a symbol (e.g., e, i,π, MINR, or MAXR) or a value (2.71.., (0,1), 3.14.

Page 3-18 The user will recognize most of these units (some, e.g., dyne, are not used very often nowadays) from his or her physics classes: N =

Page 3-19 Available unitsThe following is a list of the units available in the UNITS menu. The unit symbol is shown first followed by the unit na

Page TOC-11Function TRAN ,11-15Additional matrix operations (The matrix OPER menu) ,11-15Function AXL ,11-16Function AXM ,11-16Function LCXM ,11-16Sol

Page 3-20SPEEDm/s (meter per second), cm/s (centimeter per second), ft/s (feet per second), kph (kilometer per hour), mph (mile per hour), knot (nauti

Page 3-21ANGLE (planar and solid angle measurements)o (sexagesimal degree), r (radian), grad (grade), arcmin (minute of arc), arcs (second of arc), sr

Page 3-22Converting to base unitsTo convert any of these units to the default units in the SI system, use the function UBASE. For example, to find ou

Page 3-23` Convert the unitsIn RPN mode, system flag 117 set to SOFT menus:1 Enter 1 (no underline)‚Û Select the UNITS menu„« @)VISC Select the VISCOS

Page 3-24Notice that the underscore is entered automatically when the RPN mode is active. The result is the following screen:As indicated earlier, if

Page 3-25Yyotta+24 ddeci-1Z zetta +21 c centi -2E exa +18 m milli -3P peta +15μ micro -6T tera +12 n nano -9Ggiga+9 p pico -12Mmega+6 f femto-15k,K ki

Page 3-26which shows as 65_(m⋅yd). To convert to units of the SI system, use function UBASE: To calculate a division, say, 3250 mi / 50 h, enter it a

Page 3-27Stack calculations in the RPN mode, do not require you to enclose the different terms in parentheses, e.g., 12_m ` 1.5_yd ` *3250_mi ` 50_h

Page 3-28UFACT(x,y): factors a unit y from unit object xUNIT(x,y): combines value of x with units of yThe UBASE function was discussed in detail in a

Page 3-29Examples of UNITUNIT(25,1_m) `UNIT(11.3,1_mph) `Physical constants in the calculatorFollowing along the treatment of units, we discuss the

Page TOC-12Function QXA ,11-53Function SYLVESTER ,11-54Function GAUSS ,11-54Linear Applications ,11-54Function IMAGE ,11-55Function ISOM ,11-55Functi

Page 3-30The soft menu keys corresponding to this CONSTANTS LIBRARY screen include the following functions:SI when selected, constants values are show

Page 3-31To see the values of the constants in the English (or Imperial) system, press the @ENGL option:If we de-select the UNITS option (press @UNITS

Page 3-32Special physical functionsMenu 117, triggered by using MENU(117) in ALG mode, or 117 ` MENU in RPN mode, produces the following menu (labels

Page 3-33ZFACTOR(xT, yP), where xT is the reduced temperature, i.e., the ratio of actual temperature to pseudo-critical temperature, and yP is the re

Page 3-34Function TINCFunction TINC(T0,ΔT) calculates T0+DT. The operation of this function is similar to that of function TDELTA in the sense that i

Page 3-35Press 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

Page 3-36The contents of the variable K are: <<  α β ‘α+β’ >>.Functions defined by more than one expressionIn this section we discuss the

Page 3-37Combined IFTE functionsTo program a more complicated function such as you can combine several levels of the IFTE function, i.e.,‘g(x) = IFTE(

Page 4-1Chapter 4Calculations with complex numbersThis chapter shows examples of calculations and application of functions to complex numbers.Definiti

Page 4-2Press @@OK@@ , twice, to return to the stack.Entering complex numbersComplex numbers in the calculator can be entered in either of the two Car

Page TOC-13Fast 3D plots ,12-34Wireframe plots ,12-36Ps-Contour plots ,12-38Y-Slice plots ,12-39Gridmap plots ,12-40Pr-Surface plots ,12-41The VPAR va

Page 4-3Notice that the last entry shows a complex number in the form x+iy. This is so because the number was entered between single quotes, which re

Page 4-4On the other hand, if the coordinate system is set to cylindrical coordinates (use CYLIN), entering a complex number (x,y), where x and y are

Page 4-5Changing sign of a complex numberChanging the sign of a complex number can be accomplished by using the \ key, e.g., -(5-3i) = -5 + 3iEntering

Page 4-6CMPLX menu through the MTH menuAssuming that system flag 117 is set to CHOOSE boxes (see Chapter 2), the CMPLX sub-menu within the MTH menu is

Page 4-7This first screen shows functions RE, IM, and CR. Notice that the last function returns a list {3. 5.} representing the real and imaginary

Page 4-8 The resulting menu include some of the functions already introduced in the previous section, namely, ARG, ABS, CONJ, IM, NEG, RE, and SIG

Page 4-9Functions from the MTH menuThe hyperbolic functions and their inverses, as well as the Gamma, PSI, and Psi functions (special functions) were

Page 4-10Function DROITE is found in the command catalog (‚N).Using EVAL(ANS(1)) simplifies the result to:

Page 5-1Chapter 5Algebraic and arithmetic operationsAn algebraic object, or simply, algebraic, is any number, variable name or algebraic expression th

Page 5-2(exponential, logarithmic, trigonometry, hyperbolic, etc.), as you would any real or complex number. To demonstrate basic operations with al

Page TOC-14The SYMBOLIC menu and graphs ,12-49The SYMB/GRAPH menu ,12-50Function DRAW3DMATRIX ,12-52Chapter 13 - Calculus Applications ,13-1The CALC

Page 5-3‚¹@@A1@@ „¸@@A2@@ The same results are obtained in RPN mode if using the following keystrokes:@@A1@@ @@A2@@ +μ @

Page 5-4We notice that, at the bottom of the screen, the line See: EXPAND FACTOR suggests links to other help facility entries, the functions EXPAND a

Page 5-5FACTOR: LNCOLLECT: LIN: PARTFRAC: SOLVE: SUBST: TEXPAND:Note: Recall that, to use these

Page 5-6Other forms of substitution in algebraic expressionsFunctions SUBST, shown above, is used to substitute a variable in an expression. A second

Page 5-7A different approach to substitution consists in defining the substitution expressions in calculator variables and placing the name of the var

Page 5-8LNCOLLECT, and TEXPAND are also contained in the ALG menu presented earlier. Functions LNP1 and EXPM were introduced in menu HYPERBOLIC, unde

Page 5-9Functions in the ARITHMETIC menuThe ARITHMETIC menu contains a number of sub-menus for specific applications in number theory (integers, polyn

Page 5-10LGCD (Greatest Common Denominator): PROPFRAC (proper fraction) SIMP2:The functions associated with the ARITHMETIC submenus:

Page 5-11FACTOR Factorizes an integer number or a polynomialFCOEF Generates fraction given roots and multiplicityFROOTS Returns roots and multiplicity

Page 5-12Applications of the ARITHMETIC menuThis section is intended to present some of the background necessary for application of the ARITHMETIC men

Page TOC-15Integration with units ,13-21Infinite series ,13-22Taylor and Maclaurin’s series ,13-23Taylor polynomial and reminder ,13-23Functions TAYLR

Page 5-13multiplying j times k in modulus n arithmetic is, in essence, the integer remainder of j⋅k/n in infinite arithmetic, if j⋅k>n. For examp

Page 5-14Notice that, whenever a result in the right-hand side of the “congruence” symbol produces a result that is larger than the modulo (in this ca

Page 5-15[SPC] entry, and then press the corresponding modular arithmetic function. For example, using a modulus of 12, try the following operations

Page 5-16operating on them. You can also convert any number into a ring number by using the function EXPANDMOD. For example,EXPANDMOD(125) ≡ 5 (mod

Page 5-17PolynomialsPolynomials are algebraic expressions consisting of one or more terms containing decreasing powers of a given variable. For examp

Page 5-18numbers (function ICHINREM). The input consists of two vectors [expression_1, modulo_1] and [expression_2, modulo_2]. The output is a vecto

Page 5-19An alternate definition of the Hermite polynomials iswhere dn/dxn = n-th derivative with respect to x. This is the definition used in the ca

Page 5-20For example, for n = 2, we will write:Check this result with your calculator:LAGRANGE([[ x1,x2],[y1,y2]]) = ‘((y1-y2)*X+(y2*x1-y1*x2))/(x1-x2

Page 5-21The PCOEF functionGiven an array containing the roots of a polynomial, the function PCOEF generates an array containing the coefficients of t

Page 5-22The EPSX0 function and the CAS variable EPSThe variable ε (epsilon) is typically used in mathematical textbooks to represent a very small num

NoticeREGISTER YOUR PRODUCT AT: www.register.hp.comTHIS MANUAL AND ANY EXAMPLES CONTAINED HEREIN AREPROVIDED “AS IS” AND ARE SUBJECT TO CHANGE WITHOUT

Page TOC-16Checking solutions in the calculator ,16-2Slope field visualization of solutions ,16-3The CALC/DIFF menu ,16-3Solution to linear and non-li

Page 5-23FractionsFractions can be expanded and factored by using functions EXPAND and FACTOR, from the ALG menu (‚×). For example:EXPAND(‘(1+X)^3/((

Page 5-24If you have the Complex mode active, the result will be: ‘2*X+(1/2/(X+i)+1/2/(X-2)+5/(X-5)+1/2/X+1/2/(X-i))’The FCOEF functionThe function FC

Page 5-25mode selected, then the results would be: [0 –2. 1 –1. – ((1+i*√3)/2) –1. – ((1–i*√3)/2) –1. 3

Page 5-26The CONVERT Menu and algebraic operationsThe CONVERT menu is activated by using „Ú key (the 6 key). This menu summarizes all conversion menu

Page 5-27BASE convert menu (Option 2)This menu is the same as the UNITS menu obtained by using ‚ã. The applications of this menu are discussed in det

Page 5-28Function NUM has the same effect as the keystroke combination ‚ï(associated with the ` key). Function NUM converts a symbolic result into

Page 5-29LIN LNCOLLECT POWEREXPAND SIMPLIFY

Page 6-1Chapter 6Solution to single equationsIn this chapter we feature those functions that the calculator provides for solving single equations of t

Page 6-2Using the RPN mode, the solution is accomplished by entering the equation in the stack, followed by the variable, before entering function ISO

Page 6-3The screen shot shown above displays two solutions. In the first one, β4-5β=125, SOLVE produces no solutions { }. In the second one, β4 - 5β

Page TOC-17Numerical solution of first-order ODE ,16-57Graphical solution of first-order ODE ,16-59Numerical solution of second-order ODE ,16-61Graphi

Page 6-4In the first case SOLVEVX could not find a solution. In the second case, SOLVEVX found a single solution, X = 2.The following screens show th

Page 6-5 The Symbolic Solver functions presented above produce solutions to rational equations (mainly, polynomial equations). If the equation to

Page 6-6Polynomial EquationsUsing the Solve poly… option in the calculator’s SOLVE environment you can: (1) find the solutions to a polynomial equatio

Page 6-7All the solutions are complex numbers: (0.432,-0.389), (0.432,0.389), (-0.766, 0.632), (-0.766, -0.632).Generating polynomial coefficients g

Page 6-8Press ˜ to trigger the line editor to see all the coefficients.Generating an algebraic expression for the polynomialYou can use the calculator

Page 6-9To expand the products, you can use the EXPAND command. The resulting expression is: 'X^4+-3*X^3+ -3*X^2+11*X-6'.A different approa

Page 6-10Example 1 – Calculating payment on a loanIf $2 million are borrowed at an annual interest rate of 6.5% to be repaid in 60 monthly payments, w

Page 6-11payments. Suppose that we use 24 periods in the first line of the amortization screen, i.e., 24 @@OK@@. Then, press @@AMOR@@. You will ge

Page 6-12˜ Skip PMT, since we will be solving for it0 @@OK@@ Enter FV = 0, the option End is highlighted@@CHOOS !—@@OK@@ Change payment option to Begi

Page 6-13™ ‚í Enter a comma³ ‚@@PYR@@ Enter name of variable PYR™ ‚í Enter a comma³ ‚@@FV@@. Enter name of variable FV` Execute PURGE commandThe

Page TOC-18Chapter 18 - Statistical Applications ,18-1Pre-programmed statistical features ,18-1Entering data ,18-1Calculating single-variable statis

Page 6-14³„¸~„x™-S„ì *~„x/3™‚Å 0™K~e~q`Press J to see the newly created EQ variable:Then, enter the SOLVE environment and select Solve equation…, by u

Page 6-15This, however, is not the only possible solution for this equation. To obtain a negative solution, for example, enter a negative number in t

Page 6-16The equation is here exx is the unit strain in the x-direction,σxx,σyy, andσzz, are the normal stresses on the particle in the directions o

Page 6-17With the ex: field highlighted, press @SOLVE@ to solve for ex:The solution can be seen from within the SOLVE EQUATION input form by pressing

Page 6-18Specific energy in an open channel is defined as the energy per unit weight measured with respect to the channel bottom. Let E = specific en

Page 6-19Θ Solve for y.The result is 0.149836.., i.e., y = 0.149836. Θ It is known, however, that there are actually two solutions available for y in

Page 6-20In the next example we will use the DARCY function for finding friction factors in pipelines. Thus, we define the function in the following

Page 6-21Example 3 – Flow in a pipeYou may want to create a separate sub-directory (PIPES) to try this example. The main equation governing flow in

Page 6-22The combined equation has primitive variables: hf, Q, L, g, D, ε, and Nu.Launch the numerical solver (‚Ï@@OK@@) to see the primitive variable

Page 6-23Example 4 – Universal gravitationNewton’s law of universal gravitation indicates that the magnitude of the attractive force between two bodie

Page TOC-19Paired sample tests ,18-41Inferences concerning one proportion ,18-41Testing the difference between two proportions ,18-42Hypothesis testin

Page 6-24Solve for F, and press to return to normal calculator display. The solution is F : 6.67259E-15_N, or F = 6.67259×10-15 N.Different ways to e

Page 6-25Type an equation, say X^2 - 125 = 0, directly on the stack, and press @@@OK@@@ . At this point the equation is ready for solution. Altern

Page 6-26The SOLVE soft menuThe SOLVE soft menu allows access to some of the numerical solver functions through the soft menu keys. To access this me

Page 6-27Example 1 - Solving the equation t2-5t = -4For example, if you store the equation ‘t^2-5*t=-4’ into EQ, and press @)SOLVR, it will activate t

Page 6-28You can also solve more than one equation by solving one equation at a time, and repeating the process until a solution is found. For exampl

Page 6-29Using units with the SOLVR sub-menuThese are some rules on the use of units with the SOLVR sub-menu: Θ Entering a guess with units for a giv

Page 6-30This function produces the coefficients [an, an-1, … , a2, a1 , a0] of a polynomial anxn + an-1xn-1 + … + a2x2 + a1x + a0, given a vector o

Page 6-31Press J to exit the SOLVR environment. Find your way back to the TVM sub-menu within the SOLVE sub-menu to try the other functions availabl

Page 7-1Chapter 7Solving multiple equationsMany problems of science and engineering require the simultaneous solutions of more than one equation. The

Page 7-2Use command SOLVE at this point (from the S.SLV menu: „Î) After about 40 seconds, maybe more, you get as result a list:{ ‘t = (x-x0)/(COS(θ0

Page TOC-20Custom menus (MENU and TMENU functions) ,20-2Menu specification and CST variable ,20-4Customizing the keyboard ,20-5The PRG/MODES/KEYS sub-

Page 7-3the contents of T1 and T2 to the stack and adding and subtracting them. Here is how to do it with the equation writer:Enter and store term T

Page 7-4Notice that the result includes a vector [ ] contained within a list { }. To remove the list symbol, use μ. Finally, to decompose the vector,

Page 7-5Example 1 - Example from the help facilityAs with all function entries in the help facility, there is an example attached to the MSLV entry as

Page 7-6discharge (m3/s or ft3/s), A is the cross-sectional area (m2 or ft2), Cu is a coefficient that depends on the system of units (Cu = 1.0 for th

Page 7-7μ@@@EQ1@@ μ @@@EQ2@@. The equations are listed in the stack as follows (small font option selected): We can see that these equations

Page 7-8 Next, we’ll enter variable EQS: LL@@EQS@ , followed by vector [y,Q]:‚í„Ô~„y‚í~q™and by the initial guesses ‚í„Ô5‚í 10.Before pressing `, t

Page 7-9The result is a list of three vectors. The first vector in the list will be the equations solved. The second vector is the list of unknowns.

Page 7-10The cosine law indicates that: a2 = b2 + c2 – 2⋅b⋅c⋅cosα,b2 = a2 + c2 – 2⋅a⋅c⋅cosβ,c2 = a2 + b2 – 2⋅a⋅b⋅cosγ.In order to solve any triangle,

Page 7-11‘SIN(α)/a = SIN(β)/b’‘SIN(α)/a = SIN(γ)/c’‘SIN(β)/b = SIN(γ)/c’‘c^2 = a^2+b^2-2*a*b*COS(γ)’‘b^2 = a^2+c^2-2*a*c*COS(β)’‘a^2 = b^2+c^2-2*b*c*C

Page 7-12Press J, if needed, to get your variables menu. Your menu should show the variables @LVARI! !@TITLE @@EQ@@ .Preparing to run the MESThe next

Page TOC-21“De-tagging” a tagged quantity ,21-33Examples of tagged output ,21-34Using a message box ,21-37Relational and logical operators ,21-43Relat

Page 7-13Let’s try a simple solution of Case I, using a = 5, b = 3, c = 5. Use the following entries:5[ a ] a:5 is listed in the top left corner o

Page 7-14Pressing „@@ALL@@ will solve for all the variables, temporarily showing theintermediate results. Press ‚@@ALL@@ to see the solutions:When

Page 7-15Programming the MES triangle solution using User RPLTo facilitate activating the MES for future solutions, we will create a program that will

Page 7-16Use a = 3, b = 4, c = 6. The solution procedure used here consists of solving for all variables at once, and then recalling the solutions to

Page 7-17Adding an INFO button to your directoryAn information button can be useful for your directory to help you remember the operation of the funct

Page 7-18An explanation of the variables follows:SOLVEP = a program that triggers the multiple equation solver for theparticular set of equations stor

Page 7-19Notice that after you enter a particular value, the calculator displays the variable and its value in the upper left corner of the display.

Page 7-20

Page 8-1Chapter 8Operations with listsLists are a type of calculator’s object that can be useful for data processing and in programming. This Chapter

Page 8-2The figure below shows the RPN stack before pressing the K key:Composing and decomposing listsComposing and decomposing lists makes sense in R

Page TOC-22Examples of program-generated plots ,22-17Drawing commands for use in programming ,22-19PICT ,22-20PDIM ,22-20LINE ,22-20TLINE ,22-20BOX ,

Page 8-3In RPN mode, the following screen shows the three lists and their names ready to be stored. To store the lists in this case you need to press

Page 8-4Subtraction, multiplication, and division of lists of numbers of the same length produce a list of the same length with term-by-term operation

Page 8-5ABS EXP and LN LOG and ANTILOG SQ and square root SIN, ASIN COS, ACOS TAN, ATAN INVERSE (1/x)

Page 8-6TANH, ATANH SIGN, MANT, XPON IP, FP FLOOR, CEIL DR, RDExamples of functions that use two argumentsThe screen sho

Page 8-7%({10,20,30},{1,2,3}) = {%(10,1),%(20,2),%(30,3)}This description of function % for list arguments shows the general pattern of evaluation of

Page 8-8 The following example shows applications of the functions RE(Real part), IM(imaginary part), ABS(magnitude), and ARG(argument) of complex

Page 8-9 This menu contains the following functions:ΔLIST : Calculate increment among consecutive elements in listΣLIST : Calculate summation

Page 8-10Manipulating elements of a listThe PRG (programming) menu includes a LIST sub-menu with a number of functions to manipulate elements of a lis

Page 8-11Functions GETI and PUTI, also available in sub-menu PRG/ ELEMENTS/, can also be used to extract and place elements in a list. These two fu

Page 8-12SEQ is useful to produce a list of values given a particular expression and is described in more detail here.The SEQ function takes as argume

Page TOC-23Chapter 24 - Calculator objects and flags ,24-1Description of calculator objects ,24-1Function TYPE ,24-2Function VTYPE ,24-2Calculator fl

Page 8-13In both cases, you can either type out the MAP command (as in the examples above) or select the command from the CAT menu.The following call

Page 8-14to replace the plus sign (+) with ADD:Next, we store the edited expression into variable @@@G@@@:Evaluating G(L1,L2) now produces the followi

Page 8-15Applications of listsThis section shows a couple of applications of lists to the calculation of statistics of a sample. By a sample we under

Page 8-163. Divide the result above by n = 10:4. Apply the INV() function to the latest result:Thus, the harmonic mean of list S is sh = 1.6348…Geomet

Page 8-17 Thus, the geometric mean of list S is sg = 1.003203…Weighted averageSuppose that the data in list S, defined above, namely:S = {1,5,3,1,

Page 8-183. Use function ΣLIST, once more, to calculate the denominator of sw:4. Use the expression ANS(2)/ANS(1) to calculate the weighted average:Th

Page 8-19The class mark data can be stored in variable S, while the frequency count can be stored in variable W, as follows:Given the list of class ma

Page 8-20To calculate this last result, we can use the following: The standard deviation of the grouped data is the square root of the variance:Ns

Page 9-1Chapter 9VectorsThis Chapter provides examples of entering and operating with vectors, both mathematical vectors of many elements, as well as

Page 9-2where θ is the angle between the two vectors. The cross product produces a vector A×B whose magnitude is |A×B| = |A||B|sin(θ), and its direct

Page TOC-24Storing objects on an SD card ,26-9Recalling an object from an SD card ,26-10Evaluating an object on an SD card ,26-10Purging an object fro

Page 9-3Storing vectors into variables Vectors can be stored into variables. The screen shots below show the vectorsu2 = [1, 2], u3 = [-3, 2, -2], v2

Page 9-4The ←WID key is used to decrease the width of the columns in thespreadsheet. Press this key a couple of times to see the column widthdecreas

Page 9-5The @+ROW@ key will add a row full of zeros at the location of the selectedcell of the spreadsheet.The @-ROW key will delete the row corres

Page 9-6Building a vector with ARRYThe function →ARRY, available in the function catalog (‚N‚é, use —˜ to locate the function), can also be used to b

Page 9-7In RPN mode, the function [→ARRY] takes the objects from stack levels n+1, n, n-1, …, down to stack levels 3 and 2, and converts them into a

Page 9-8Highlighting the entire expression and using the @EVAL@ soft menu key, we get the result: -15.To replace an element in an array use function P

Page 9-9Simple operations with vectorsTo illustrate operations with vectors we will use the vectors A, u2, u3, v2, and v3, stored in an earlier exerci

Page 9-10Absolute value functionThe absolute value function (ABS), when applied to a vector, produces the magnitude of the vector. For a vector A = [

Page 9-11Dot product Function DOT is used to calculate the dot product of two vectors of the same length. Some examples of application of function DO

Page 9-12In the RPN mode, application of function V will list the components of a vector in the stack, e.g., V(A) will produce the following output

Page TOC-25Appendix F - The Applications (APPS) menu ,F-1Appendix G - Useful shortcuts ,G-1Appendix H - The CAS help facility ,H-1Appendix I - Co

Page 9-13 When the rectangular, or Cartesian, coordinate system is selected, the top line of the display will show an XYZ field, and any 2-D or 3

Page 9-14The figure below shows the transformation of the vector from spherical to Cartesian coordinates, with x = ρ sin(φ) cos(θ), y = ρ sin (φ) cos

Page 9-15equivalent (r,θ,z) with r = ρ sin φ, θ = θ, z = ρ cos φ. For example, the following figure shows the vector entered in spherical coordinates

Page 9-16Suppose that you want to find the angle between vectors A = 3i-5j+6k, B = 2i+j-3k, you could try the following operation (angular measure set

Page 9-17Thus, M = (10i+26j+25k) m⋅N. We know that the magnitude of M is such that |M| = |r||F|sin(θ), where θ is the angle between r and F. We can

Page 9-18Next, we calculate vector P0P = r as ANS(1) – ANS(2), i.e., Finally, we take the dot product of ANS(1) and ANS(4) and make it equal to zero t

Page 9-19In this section we will showing you ways to transform: a column vector into a row vector, a row vector into a column vector, a list into a ve

Page 9-20If we now apply function OBJ once more, the list in stack level 1:, {3.}, will be decomposed as follows:Function LISTThis function is used

Page 9-213 - Use function ARRY to build the column vectorThese three steps can be put together into a UserRPL program, entered as follows (in RPN mod

Page 9-222 - Use function OBJ to decompose the list in stack level 1:3 - Press the delete key ƒ (also known as function DROP) to eliminate the number

PrefaceYou have in your hands a compact symbolic and numerical computer that will facilitate calculation and mathematical analysis of problems in a va

Page 1-1Chapter 1Getting started This chapter provides basic information about the operation of your calculator.It is designed to familiarize you with

Page 9-23This variable, @@CXR@@, can now be used to directly transform a column vector to a row vector. In RPN mode, enter the column vector, and the

Page 9-24A new variable, @@LXV@@, will be available in the soft menu labels after pressing J:Press ‚@@LXV@@ to see the program contained in the variab

Page 10-1Chapter 10!Creating and manipulating matricesThis chapter shows a number of examples aimed at creating matrices in the calculator and demonst

Page 10-2Entering matrices in the stackIn this section we present two different methods to enter matrices in the calculator stack: (1) using the Matri

Page 10-3If you have selected the textbook display option (using H@)DISP! and checking off Textbook), the matrix will look like the one shown above.

Page 10-4 or in the MATRICES/CREATE menu available through „Ø:The MTH/MATRIX/MAKE sub menu (let’s call it the MAKE menu) contains the following fu

Page 10-5As you can see from exploring these menus (MAKE and CREATE), they both have the same functions GET, GETI, PUT, PUTI, SUB, REPL, RDM, RANM, HI

Page 10-6Functions GET and PUTFunctions GET, GETI, PUT, and PUTI, operate with matrices in a similar manner as with lists or vectors, i.e., you need t

Page 10-7 Notice that the screen is prepared for a subsequent application of GETI or GET, by increasing the column index of the original reference

Page 10-8 If the argument is a real matrix, TRN simply produces the transpose of the real matrix. Try, for example, TRN(A), and compare it with T

Page 1-2b. Insert a new CR2032 lithium battery. Make sure its positive (+) side is facingup.c. Replace the plate and push it to the original place.Af

Page 10-9In RPN mode this is accomplished by using {4,3} ` 1.5 \` CON.Function IDNFunction IDN (IDeNtity matrix) creates an identity matrix given its

Page 10-10vector’s dimension, in the latter the number of rows and columns of the matrix. The following examples illustrate the use of function RDM:R

Page 10-11If using RPN mode, we assume that the matrix is in the stack and use {6} `RDM.Function RANMFunction RANM (RANdom Matrix) will generate a mat

Page 10-12In RPN mode, assuming that the original 2×3 matrix is already in the stack, use {1,2} ` {2,3} ` SUB.Function REPL Function REPL replaces or

Page 10-13In RPN mode, with the 3×3 matrix in the stack, we simply have to activate function DI G to obtain the same result as above.Function DIAG→Fu

Page 10-14For example, the following command in ALG mode for the list {1,2,3,4}:In RPN mode, enter {1,2,3,4} ` V NDERMONDE.Function HILBERTFunction HI

Page 10-15entered in the display as you perform those keystrokes. First, we present the steps necessary to produce program CRMC.Lists represent colum

Page 10-16~„n # n„´@)MATRX! @)COL! @COL! COL`Program is displayed in level 1To save the program: !³~~crmc~ KTo see the contents

Page 10-17Lists represent rows of the matrixThe previous program can be easily modified to create a matrix when the input lists will become the rows o

Page 10-18 Both approaches will show the same functions: When system flag 117 is set to SOFT menus, the COL menu is accessible through „´!)MAT

Page 1-3At the top of the display you will have two lines of information that describe thesettings of the calculator. The first line shows the chara

Page 10-19 In this result, the first column occupies the highest stack level after decomposition, and stack level 1 is occupied by the number of c

Page 10-20In RPN mode, enter the matrix first, then the vector, and the column number, before applying function COL+. The figure below shows the RPN

Page 10-21In RPN mode, function CSWP lets you swap the columns of a matrix listed in stack level 3, whose indices are listed in stack levels 1 and 2.

Page 10-22 When system flag 117 is set to SOFT menus, the ROW menu is accessible through „´!)MATRX !)@@ROW@ , or through „Ø!)@CREAT@ !)@@ROW@ . B

Page 10-23matrix does not survive decomposition, i.e., it is no longer available in the stack. Function ROW→Function ROW→ has the opposite effect of

Page 10-24 Function ROW-Function ROW- takes as argument a matrix and an integer number representing the position of a row in the matrix. The func

Page 10-25 As you can see, the rows that originally occupied positions 2 and 3 have been swapped. Function RCIFunction RCI stands for multiplying

Page 10-26In RPN mode, enter the matrix first, followed by the constant value, then by the row to be multiplied by the constant value, and finally ent

Page 11-1Chapter 11 Matrix Operations and Linear AlgebraIn Chapter 10 we introduced the concept of a matrix and presented a number of functions for en

Page 11-2Addition and subtractionConsider a pair of matrices A = [aij]m×n and B = [bij]m×n. Addition and subtraction of these two matrices is only po

Page 1-4Each group of 6 entries is called a Menu page. The current menu, known asthe TOOL menu (see below), has eight entries arranged in two pages.

Page 11-3 By combining addition and subtraction with multiplication by a scalar we can form linear combinations of matrices of the same dimensions

Page 11-4Matrix multiplicationMatrix multiplication is defined by Cm×n = Am×p⋅Bp×n, where A = [aij]m×p, B = [bij]p×n, and C = [cij]m×n. Notice that m

Page 11-5(another row vector). For the calculator to identify a row vector, you must use double brackets to enter it: Term-by-term multiplicatio

Page 11-6In algebraic mode, the keystrokes are: [enter or select the matrix] Q [enter the power] `. In RPN mode, the keystrokes are: [enter or select

Page 11-7To verify the properties of the inverse matrix, consider the following multiplications: Characterizing a matrix (The matrix NORM menu)The

Page 11-8Function ABS Function ABS calculates what is known as the Frobenius norm of a matrix. For a matrix A = [aij]m×n, the Frobenius norm of the m

Page 11-9Functions RNRM and CNRMFunction RNRM returns the Row NoRM of a matrix, while function CNRM returns the Column NoRM of a matrix. Examples,

Page 11-10Function SRAD Function SRAD determines the Spectral RADius of a matrix, defined as the largest of the absolute values of its eigenvalues.

Page 11-11Try the following exercise for matrix condition number on matrix A33. The condition number is COND(A33) , row norm, and column norm for A33

Page 11-12For example, try finding the rank for the matrix:You will find that the rank is 2. That is because the second row [2,4,6] is equal to the f

Page 1-5This CHOOSE box is labeled BASE MENU and provides a list of numberedfunctions, from 1. HEX x to 6. BR. This display will constitute the firs

Page 11-13The determinant of a matrixThe determinant of a 2x2 and or a 3x3 matrix are represented by the samearrangement of elements of the matrices,

Page 11-14Function TRACEFunction TRACE calculates the trace of square matrix, defined as the sum of the elements in its main diagonal, or .Examples:Fo

Page 11-15 Function TRANFunction TRAN returns the transpose of a real or the conjugate transpose of a complex matrix. TRAN is equivalent to TRN.

Page 11-16MAD and RSD are related to the solution of systems of linear equations and will be presented in a subsequent section in this Chapter. In th

Page 11-17The implementation of function LCXM for this case requires you to enter:2`3`‚@@P1@@ LCXM `The following figure shows the RPN stack before

Page 11-18, , Using the numerical solver for linear systemsThere are many ways to solve a system of linear equations with the calculator. One possibi

Page 11-19This 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 uniq

Page 11-20To check that the solution is correct, enter the matrix A and multiply times this solution vector (example in algebraic mode):Under-determin

Page 11-21To see the details of the solution vector, if needed, press the @EDIT! button. This will activate the Matrix Writer. Within this environm

Page 11-22Let’s store the latest result in a variable X, and the matrix into variable A, as follows:Press K~x` to store the solution vector into vari

Page 1-6If you now press ‚ã, instead of the CHOOSE box that you saw earlier,the display will now show six soft menu labels as the first page of the ST

Page 11-23can be written as the matrix equation A⋅x = b, if This system has more equations than unknowns (an over-determined system). The system does

Page 11-24Press ` to return to the numerical solver environment. To check that the solution is correct, try the following: • Press ——, to highlight

Page 11-25• If A is a square matrix and A is non-singular (i.e., it’s inverse matrix exist, or its determinant is non-zero), LSQ returns the exact sol

Page 11-26Under-determined systemConsider the system2x1 + 3x2 –5x3 = -10,x1 – 3x2 + 8x3 = 85,withThe solution using LSQ is shown next: Over-determ

Page 11-27Compare these three solutions with the ones calculated with the numerical solver.Solution with the inverse matrixThe solution to the system

Page 11-28The procedure for the case of “dividing” b by A is illustrated below for the case2x1 + 3x2 –5x3 = 13,x1 – 3x2 + 8x3 = -13,2x1 – 2x2 + 4x3 =

Page 11-29[[14,9,-2],[2,-5,2],[5,19,12]] `[[1,2,3],[3,-2,1],[4,2,-1]] `/The result of this operation is:Gaussian and Gauss-Jordan eliminationGaussian

Page 11-30To start the process of forward elimination, we divide the first equation (E1) by 2, and store it in E1, and show the three equations again

Page 11-31an expression = 0. Thus, the last set of equations is interpreted to be the following equivalent set of equations:X +2Y+3Z = 7,Y+ Z = 3,-7

Page 11-32To obtain a solution to the system matrix equation using Gaussian elimination, we first create what is known as the augmented matrix corresp

Page 1-7The TOOL menuThe soft menu keys for the menu currently displayed, known as the TOOL menu,are associated with operations related to manipulatio

Page 11-33Multiply row 2 by –1/8: 8\Y2 @RCI!Multiply row 2 by 6 add it to row 3, replacing it: 6#2#3 @RCIJ!If you were performing these operations by

Page 11-34Multiply row 3 by –1/7: 7\Y 3 @RCI!Multiply row 3 by –1, add it to row 2, replacing it: 1\ # 3#2 @RCIJ!Multiply row 3 by –3, add it to row

Page 11-35While performing pivoting in a matrix elimination procedure, you can improve the numerical solution even more by selecting as the pivot the

Page 11-36Now we are ready to start the Gauss-Jordan elimination with full pivoting. We will need to keep track of the permutation matrix by hand, so

Page 11-37Having filled up with zeros the elements of column 1 below the pivot, now we proceed to check the pivot at position (2,2). We find that the

Page 11-382 Y \#3#1@RCIJFinally, we eliminate the –1/16 from position (1,2) by using:16 Y # 2#1@RCIJWe now have an identity matrix in the portion of t

Page 11-39Then, for this particular example, in RPN mode, use:[2,-1,41] ` [[1,2,3],[2,0,3],[8,16,-1]] `/The calculator shows an augmented matrix consi

Page 11-40To see the intermediate steps in calculating and inverse, just enter the matrix Afrom above, and press Y, while keeping the step-by-step opt

Page 11-41The result (A-1)n×n = C n×n /det(A n×n), is a general result that applies to any non-singular matrix A. A general form for the elements of

Page 11-42LINSOLVE([X-2*Y+Z=-8,2*X+Y-2*Z=6,5*X-2*Y+Z=-12],[X,Y,Z])to produce the solution: [X=-1,Y=2,Z = -3].Function LINSOLVE works with symbolic ex

Page 1-89 key the TIME choose box is activated. This operation can also berepresented as ‚Ó. The TIME choose box is shown in the figure below:As indi

Page 11-43The diagonal matrix that results from a Gauss-Jordan elimination is called a row-reduced echelon form. Function RREF ( Row-Reduced Echelo

Page 11-44 The result is the augmented matrix corresponding to the system of equations:X+Y = 0X-Y =2Residual errors in linear system solutions (F

Page 11-45Eigenvalues and eigenvectorsGiven a square matrix A, we can write the eigenvalue equation A⋅x = λ⋅x,where the values of λ that satisfy the e

Page 11-46Using the variable λ to represent eigenvalues, this characteristic polynomial is to be interpreted as λ 3-2λ 2-22λ +21=0.Functio

Page 11-47of a matrix, while the corresponding eigenvalues are the components of a vector.For example, in ALG mode, the eigenvectors and eigenvalues o

Page 11-48• A list with the eigenvectors corresponding to each eigenvalue of matrix A (stack level 2)• A vector with the eigenvectors of matrix A (sta

Page 11-49Notice that the equation (x⋅I-A)⋅p(x)=m(x)⋅I is similar, in form, to the eigenvalue equation A⋅x = λ⋅x.As an example, in RPN mode, try:[[4,

Page 11-50Function LUFunction LU takes as input a square matrix A, and returns a lower-triangular matrix L, an upper triangular matrix U, and a permut

Page 11-51decomposition, while the vector s represents the main diagonal of the matrix Sused earlier.For example, in RPN mode: [[5,4,-1],[2,-3,5],[7,

Page 11-52Function QRIn RPN, function QR produces the QR factorization of a matrix An×m returning a Qn×n orthogonal matrix, a Rn×m upper trapezoidal m

Page 1-9Let’s change the minute field to 25, by pressing: 25 !!@@OK#@ . The secondsfield is now highlighted. Suppose that you want to change the

Page 11-53 This menu includes functions AXQ, CHOLESKY, GAUSS, QXA, and SYLVESTER.Function AXQ In RPN mode, function AXQ produces the quadratic for

Page 11-54such that x = P⋅y, by using Q = x⋅A⋅xT= (P⋅y)⋅A⋅ (P⋅y)T = y⋅(PT⋅A⋅P)⋅yT = y⋅D⋅yT.Function SYLVESTERFunction SYLVESTER takes as argument a sy

Page 11-55 Information on the functions listed in this menu is presented below by using the calculator’s own help facility. The figures show the

Page 11-56Function KER Function MKISOM

Page 12-1Chapter 12 GraphicsIn this chapter we introduce some of the graphics capabilities of the calculator. We will present graphics of functions i

Page 12-2These graph options are described briefly next.Function: for equations of the form y = f(x) in plane Cartesian coordinatesPolar: for equati

Page 12-3Θ Enter the PLOT environment by pressing „ñ(press them simultaneously if in RPN mode). Press @ADD to get you into the equation writer. You

Page 12-4Θ Enter the PLOT WINDOW environment by entering „ò (press them simultaneously if in RPN mode). Use a range of –4 to 4 for H-VIEW, then pr

Page 12-5Some useful PLOT operations for FUNCTION plotsIn order to discuss these PLOT options, we'll modify the function to force it to have some

Page 12-6ROOT: 1.6635... The calculator indicated, before showing the root, that it was found through SIGN REVERSAL. Press L to recover the menu.Θ P

Page 1-10Setting the dateAfter setting the time format option, the SET TIME AND DATE input form willlook as follows:To set the date, first set the dat

Page 12-7Θ Enter the PLOT environment by pressing, simultaneously if in RPN mode, „ñ. Notice that the highlighted field in the PLOT environment now

Page 12-8To return to normal calculator function, press @)PICT @CANCL.Graphics of transcendental functionsIn this section we use some of the graphics

Page 12-910 by using 1\@@@OK@@ 10@@@OK@@@. Next, press the soft key labeled @AUTOto let the calculator determine the corresponding vertical range. A

Page 12-10Graph of the exponential functionFirst, load the function exp(X), by pressing, simultaneously if in RPN mode, theleft-shift key „ and the ñ

Page 12-11The PPAR variablePress J to recover your variables menu, if needed. In your variables menu you should have a variable labeled PPAR . Pres

Page 12-12As indicated earlier, the ln(x) and exp(x) functions are inverse of each other, i.e., ln(exp(x)) = x, and exp(ln(x)) = x. This can be verif

Page 12-13Summary of FUNCTION plot operationIn this section we present information regarding the PLOT SETUP, PLOT-FUNCTION, and PLOT WINDOW screens ac

Page 12-14Θ Use @CANCL to cancel any changes to the PLOT SETUP window and return tonormal calculator display. Θ Press @@@OK@@@ to save changes to the

Page 12-15Θ Enter lower and upper limits for horizontal view (H-View), and press @AUTO,while the cursor is in one of the V-View fields, to generate t

Page 12-16„ó, simultaneously if in RPN mode: Plots the graph based on the settings stored in variable PPAR and the current functions defined in the P

For symbolic operations the calculator includes a powerful Computer AlgebraicSystem (CAS) that lets you select different modes of operation, e.g., com

Page 1-11Introducing the calculator’s keyboardThe figure below shows a diagram of the calculator’s keyboard with the numbering of its rows and columns

Page 12-17Generating a table of values for a functionThe combinations „õ(E) and „ö(F), pressed simultaneously if in RPN mode, let’s the user produce a

Page 12-18the corresponding values of f(x), listed as Y1 by default. You can use theup and down arrow keys to move about in the table. You will noti

Page 12-19We will try to plot the function f(θ) = 2(1-sin(θ)), as follows:Θ First, make sure that your calculator’s angle measure is set to radians.Θ

Page 12-20Θ Press L@CANCL to return to the PLOT WINDOW screen. Press L@@@OK@@@ toreturn to normal calculator display.In this exercise we entered the

Page 12-21The calculator has the ability of plotting one or more conic curves by selecting Conic as the function TYPE in the PLOT environment. Make s

Page 12-22Θ To see labels: @EDIT L@)LABEL @MENUΘ To recover the menu: LL@)PICTΘ To estimate the coordinates of the point of intersection, press th

Page 12-23which involve constant values x0, y0, v0, and θ0, we need to store the values of those parameters in variables. To develop this example, cr

Page 12-24Θ Press @AUTO. This will generate automatic values of the H-View and V-Viewranges based on the values of the independent variable t and the

Page 12-25parameters. The other variables contain the values of constants used in the definitions of X(t) and Y(t). You can store different values

Page 12-26Plotting the solution to simple differential equationsThe plot of a simple differential equation can be obtained by selecting DiffEq in the

Page 1-12shift key, key (9,1), and the ALPHA key, key (7,1), can be combined with someof the other keys to activate the alternative functions shown i

Page 12-27Θ Press L to recover the menu. Press L@)PICT to recover the originalgraphics menu.Θ When we observed the graph being plotted, you'll

Page 12-28Truth plotsTruth plots are used to produce two-dimensional plots of regions that satisfy a certain mathematical condition that can be either

Page 12-29Θ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUPwindow. Θ Press ˜ and type ‘(X^2/36+Y^2/9 < 1)⋅ (X^2/16+Y^2/9 >

Page 12-30[4.5,5.6,4.4],[4.9,3.8,5.5],[5.2,2.2,6.6]] `to store it in ΣDAT, use the function STOΣ (available in the function catalog, ‚N). Press VAR

Page 12-31accommodate the maximum value in column 1 of ΣDAT. Bar plots are useful when plotting categorical (i.e., non-numerical) data. Suppose that

Page 12-32Θ Press @ERASE @DRAW to draw the bar plot. Press @EDIT L @LABEL @MENU to seethe plot unencumbered by the menu and with identifying lab

Page 12-33Slope fieldsSlope fields are used to visualize the solutions to a differential equation of the form y’ = f(x,y). Basically, what is present

Page 12-34of y(x,y) = constant, for the solution of y’ = f(x,y). Thus, slope fields are useful tools for visualizing particularly difficult equations

Page 12-35Θ Press @ERASE @DRAW to draw the three-dimensional surface. The result is awireframe picture of the surface with the reference coordinate

Page 12-36Θ Press „ô, simultaneously if in RPN mode, to access the PLOT SETUPwindow. Θ Press ˜ and type ‘SIN(X^2+Y^2)’ @@@OK@@@.Θ Press @ERASE @DRA

Page 1-13Press the !!@@OK#@ soft menu key to return to normal display. Examples of selectingdifferent calculator modes are shown next.Operating M

Page 12-37Θ Press @EDIT L @LABEL @MENU to see the graph with labels and ranges. Thisparticular version of the graph is limited to the lower part

Page 12-38Try also a Wireframe plot for the surface z = f(x,y) = x2+y2Θ Press „ô, simultaneously if in RPN mode, to access the PLOT SETUPwindow. Θ P

Page 12-39Θ Press @EDIT!L @LABEL @MENU to see the graph with labels and ranges. Θ Press LL@)PICT@CANCL to return to the PLOT WINDOW environment. Θ

Page 12-40Θ Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd: variables.Θ Press L@@@OK@@@ to return to normal calculator display.Θ Pr

Page 12-41Θ Press „ô, simultaneously if in RPN mode, to access to the PLOTSETUP window. Θ Change TYPE to Gridmap.Θ Press ˜ and type ‘SIN(X+i*Y)’ @

Page 12-42For example, to produce a Pr-Surface plot for the surface x = x(X,Y) = X sin Y, y = y(X,Y) = x cos Y, z=z(X,Y)=X, use the following:Θ Press

Page 12-43Interactive drawingWhenever we produce a two-dimensional graph, we find in the graphics screen a soft menu key labeled @)EDIT. Pressing @)E

Page 12-44Next, we illustrate the use of the different drawing functions on the resulting graphics screen. They require use of the cursor and the arr

Page 12-45should have a straight angle traced by a horizontal and a vertical segments. The cursor is still active. To deactivate it, without moving

Page 12-46DELThis command is used to remove parts of the graph between two MARK positions. Move the cursor to a point in the graph, and press @MARK.

Page 1-14To enter this expression in the calculator we will first use the equation writer,‚O. Please identify the following keys in the keyboard, be

Page 12-47X,YThis command copies the coordinates of the current cursor position, in user coordinates, in the stack.Zooming in and out in the graphics

Page 12-48You can always return to the very last zoom window by using @ZLAST.BOXZZooming in and out of a given graph can be performed by using the sof

Page 12-49cursor at the center of the screen, the window gets zoomed so that the x-axis extends from –64.5 to 65.5.ZSQRZooms the graph so that the plo

Page 12-50SOLVER.. „Î (the 7 key) Ch. 6TRIGONOMETRIC.. ‚Ñ (the 8 key) Ch. 5EXP&LN.. „Ð (the 8 key) Ch. 5The SYMB/GRAPH menuThe GRAPH sub-menu

Page 12-51 TABVAL(X^2-1,{1, 3}) produces a list of {min max} values of the function in the interval {1,3}, while SIGNTAB(X^2-1) shows the sign of

Page 12-52of F. The question marks indicates uncertainty or non-definition. For example, for X<0, LN(X) is not defined, thus the X lines shows a

Page 13-1Chapter 13 Calculus ApplicationsIn this Chapter we discuss applications of the calculator’s functions to operations related to Calculus, e.g.

Page 13-2Function limThe calculator provides function lim to calculate limits of functions. This function uses as input an expression representing a

Page 13-3To calculate one-sided limits, add +0 or -0 to the value to the variable. A “+0” means limit from the right, while a “-0” means limit from t

Page 13-4in ALG mode. Recall that in RPN mode the arguments must be entered before the function is applied. The DERIV&INTEG menuThe functions

Page 1-15Change the operating mode to RPN by first pressing the H button. Select theRPN operating mode by either using the \key, or pressing the @CHOO

Page 13-5be differentiated. Thus, to calculate the derivative d(sin(r),r), use, in ALG mode: ‚¿~„r„ÜS~„r`In RPN mode, this expression must be enclose

Page 13-6To evaluate the derivative in the Equation Writer, press the up-arrow key —,four times, to select the entire expression, then, press @EVAL.

Page 13-7Derivatives of equationsYou can use the calculator to calculate derivatives of equations, i.e., expressions in which derivatives will exist i

Page 13-8Analyzing graphics of functionsIn Chapter 11 we presented some functions that are available in the graphics screen for analyzing graphics of

Page 13-9Θ Press L @PICT @CANCL $ to return to normal calculator display. Notice that the slope and tangent line that you requested are listed in

Page 13-10This result indicates that the range of the function corresponding to the domain D = { -1,5 } is R = .Function SIGNTABFunction SIGNTAB, ava

Page 13-11Θ Level 3: the function f(VX)Θ Two lists, the first one indicates the variation of the function (i.e., where it increases or decreases) in t

Page 13-12The interpretation of the variation table shown above is as follows: the function F(X) increases for X in the interval (-∞, -1), reaching a

Page 13-13 We find two critical points, one at x = 11/3 and one at x = -1. To evaluate the second derivative at each point use: The last scre

Page 13-14Anti-derivatives and integralsAn anti-derivative of a function f(x) is a function F(x) such that f(x) = dF/dx. For example, since d(x3) /dx

Page 1-163.` Enter 3 in level 15.` Enter 5 in level 1, 3 moves to y3.` Enter 3 in level 1, 5 moves to level 2, 3 to level 33.* Place 3 and multip

Page 13-15above. Their result is the so-called discrete derivative, i.e., one defined for integer numbers only. Definite integralsIn a definite inte

Page 13-16This is the general format for the definite integral when typed directly into the stack, i.e., ∫ (lower limit, upper limit, integrand, varia

Page 13-17The following example shows the evaluation of a definite integral in the Equation Writer, step-by-step: ʳʳʳʳʳNotice that the step-by-ste

Page 13-18 Techniques of integrationSeveral techniques of integration can be implemented in the calculators, as shown in the following examples.Su

Page 13-19Integration by parts and differentialsA differential of a function y = f(x), is defined as dy = f’(x) dx, where f’(x) is the derivative of f

Page 13-20Integration by partial fractionsFunction PARTFRAC, presented in Chapter 5, provides the decomposition of a fraction into partial fractions.

Page 13-21Using the calculator, we proceed as follows: Alternatively, you can evaluate the integral to infinity from the start, e.g., Integration

Page 13-22Some notes in the use of units in the limits of integrations:1 – The units of the lower limit of integration will be the ones used in the fi

Page 13-23Taylor and Maclaurin’s seriesA function f(x) can be expanded into an infinite series around a point x=x0 by using a Taylor’s series, namely,

Page 13-24where ξ is a number near x = x0. Since ξ is typically unknown, instead of an estimate of the residual, we provide an estimate of the order

Page 1-17Notice how the expression is placed in stack level 1 after pressing `.Pressing the EVAL key at this point will evaluate the numerical value o

Page 13-25increment h. The list returned as the first output object includes the following items:1 - Bi-directional limit of the function at point of

Page 14-1Chapter 14Multi-variate Calculus ApplicationsMulti-variate calculus refers to functions of two or more variables. In this Chapter we discuss

Page 14-2 .Similarly, .We will use the multi-variate functions defined earlier to calculate partial derivatives using these definitions. Here are the

Page 14-3therefore, with DERVX you can only calculate derivatives with respect to X. Some examples of first-order partial derivatives are shown next:

Page 14-4Third-, fourth-, and higher order derivatives are defined in a similar manner.To calculate higher order derivatives in the calculator, simply

Page 14-5A different version of the chain rule applies to the case in which z = f(x,y), x = x(u,v), y = y(u,v), so that z = f[x(u,v), y(u,v)]. The f

Page 14-6We find critical points at (X,Y) = (1,0), and (X,Y) = (-1,0). To calculate the discriminant, we proceed to calculate the second derivatives,

Page 14-7Applications of function HESS are easier to visualize in the RPN mode. Consider as an example the function φ(X,Y,Z) = X2 + XY + XZ, we’ll ap

Page 14-8The resulting matrix has elements a11 = ∂2φ/∂X2 = 6., a22 = ∂2φ/∂X2 = -2., and a12= a21= ∂2φ/∂X∂Y = 0. The discriminant, for this critica

Page 14-9 Jacobian of coordinate transformationConsider the coordinate transformation x = x(u,v), y = y(u,v). The Jacobian of this transformation

Page 1-18more about reals, see Chapter 2. To illustrate this and other number formats trythe following exercises:Θ Standard format:This mode is the m

Page 14-10where the region R’ in polar coordinates is R’ = {α < θ < β, f(θ) < r < g(θ)}.Double integrals in polar coordinates can be enter

Page 15-1Chapter 15Vector Analysis ApplicationsIn this Chapter we present a number of functions from the CALC menu that apply to the analysis of scala

Page 15-2At any particular point, the maximum rate of change of the function occurs in the direction of the gradient, i.e., along a unit vector u = ∇φ

Page 15-3as the matrix H = [hij] = [∂φ/∂xi∂xj], the gradient of the function with respect to the n-variables, grad f = [ ∂φ/∂x1, ∂φ/∂x2 , … ∂φ/∂xn], a

Page 15-4not have a potential function associated with it, since, ∂f/∂z ≠∂h/∂x. The calculator response in this case is shown below: DivergenceThe

Page 15-5CurlThe curl of a vector field F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, is defined by a “cross-product” of the del operator with the vector

Page 15-6As an example, in an earlier example we attempted to find a potential function for the vector field F(x,y,z) = (x+y)i + (x-y+z)j + xzk, and g

Page 15-7produces the vector potential function Φ2 = [0, ZYX-2YX, Y-(2ZX-X)], which is different from Φ1. The last command in the screen shot shows t

Page 16-1Chapter 16 Differential EquationsIn this Chapter we present examples of solving ordinary differential equations (ODE) using calculator functi

Page 16-2(H@)DISP) is not selected. Press ˜ to see the equation in the Equation Writer.An alternative notation for derivatives typed directly in the

Page 1-19Notice that the Number Format mode is set to Fix followed by a zero (0).This number indicates the number of decimals to be shown after thedec

Page 16-3EVAL(ANS(1)) `In RPN mode:‘∂t(∂t(u(t)))+ ω0^2*u(t) = 0’ ` ‘u(t)=A*SIN (ω0*t)’ `SUBST EVALThe result is ‘0=0’. F

Page 16-4 These functions are briefly described next. They will be described in more detail in later parts of this Chapter.DESOLVE: Differential

Page 16-5Both of these inputs must be given in terms of the default independent variable for the calculator’s CAS (typically ‘X’). The output from th

Page 16-6The solution, shown partially here in the Equation Writer, is:Replacing the combination of constants accompanying the exponential terms with

Page 16-72x1’(t) + x2’(t) = 0.In algebraic form, this is written as: A⋅x’(t) = 0, where . The system can be solved by using function LDEC with argum

Page 16-8Example 2 -- Solve the second-order ODE: d2y/dx2 + x (dy/dx) = exp(x).In the calculator use:‘d1d1y(x)+x*d1y(x) = EXP(x)’ ` ‘y(x)’ ` DESOLVET

Page 16-9Performing the integration by hand, we can only get it as far as:because the integral of exp(x)/x is not available in closed form.Example 3 –

Page 16-10Press J @ODETY to get the string “Linear w/ cst coeff” for the ODE type in this case.Laplace TransformsThe Laplace transform of a function f

Page 16-11Laplace transform and inverses in the calculatorThe calculator provides the functions LAP and ILAP to calculate the Laplacetransform and the

Page 16-12Example 3 – Determine the inverse Laplace transform of F(s) = sin(s). Use:‘SIN(X)’ ` ILAP. The calculator takes a few seconds to return

Page 1-20 Press the !!@@OK#@ soft menu key to complete the selection:Press the !!@@OK#@ soft menu key return to the calculator display.

Page 16-13Θ Differentiation theorem for the n-th derivative. Let f (k)o = dkf/dxk|t = 0, and fo = f(0), thenL{dnf/dtn} = sn⋅F(s) – sn-1⋅fo−…– s⋅f

Page 16-14Θ Shift theorem for a shift to the right. Let F(s) = L{f(t)}, then L{f(t-a)}=e–as⋅L{f(t)} = e–as⋅F(s).Θ Shift theorem for a shift to the le

Page 16-15Dirac’s delta function and Heaviside’s step functionIn the analysis of control systems it is customary to utilize a type of functions that r

Page 16-16You can prove that L{H(t)} = 1/s,from which it follows that L{Uo⋅H(t)} = Uo/s,where Uo is a constant. Also, L -1{1/s}=H(t),and L -1{ Uo /s

Page 16-17Applications of Laplace transform in the solution of linear ODEsAt the beginning of the section on Laplace transforms we indicated that you

Page 16-18The result is ‘H=((X+1)*h0+a)/(X^2+(k+1)*X+k)’. To find the solution to the ODE, h(t), we need to use the inverse Laplace transform

Page 16-19With Y(s) = L{y(t)}, and L{d2y/dt2} = s2⋅Y(s) - s⋅yo – y1, where yo = h(0) and y1= h’(0), the transformed equation iss2⋅Y(s) – s⋅yo – y1+ 2⋅

Page 16-20Example 3 – Consider the equation d2y/dt2+y = δ(t-3),where δ(t) is Dirac’s delta function. Using Laplace transforms, we can write:L{d2y/dt2

Page 16-21Check what the solution to the ODE would be if you use the function LDEC:‘Delta(X-3)’ ` ‘X^2+1’ ` LDEC μNotes:[1]. An alternative way to ob

Page 16-22The result is: ‘SIN(X-3)*Heaviside(X-3) + cC1*SIN(X) + cC0*COS(X)’.Please notice that the variable X in this expression actually represents

Page TOC-1Table of contentsChapter 1 - Getting started ,1-1Basic Operations ,1-1Batteries ,1-1Turning the calculator on and off ,1-2Adjusting the di

Page 1-21same fashion that we changed the Fixed number of decimals in theexample above). Press the !!@@OK#@ soft menu key return to the calculator

Page 16-23Use of the function H(X) with LDEC, LAP, or ILAP, is not allowed in the calculator. You have to use the main results provided earlier when

Page 16-24where H(t) is Heaviside’s step function. Using Laplace transforms, we can write: L{d2y/dt2+y} = L{H(t-3)}, L{d2y/dt2} + L{y(t)} = L{H(t-3)

Page 16-25Example 4 – Plot the solution to Example 3 using the same values of yo and y1used in the plot of Example 1, above. We now plot the function

Page 16-26f(t) = Uo⋅[1-(t-a)/(b-1)]⋅[H(t-a)-H(t-b)].Examples of the plots generated by these functions, for Uo = 1, a = 2, b = 3, c = 4, horizontal ra

Page 16-27The following exercises are in ALG mode, with CAS mode set to Exact. (When you produce a graph, the CAS mode will be reset to Approx. Make

Page 16-28Function FOURIERAn alternative way to define a Fourier series is by using complex numbers as follows:whereFunction FOURIER provides the coef

Page 16-29Next, we move to the CASDIR sub-directory under HOME to change the value of variable PERIOD, e.g., „ (hold) §`J @)CASDI `2 K @PERIOD `Retur

Page 16-30The fitting is somewhat acceptable for 0<t<2, although not as good as in theprevious example. A general expression for cnThe function

Page 16-31The result is cn = (i⋅n⋅π+2)/(n2⋅π2).Putting together the complex Fourier seriesHaving determined the general expressi

Page 16-32Or, in the calculator entry line as:DEFINE(‘F(X,k,c0) = c0+Σ(n=1,k,c(n)*EXP(2*i*π*n*X/T)+c(-n)*EXP(-(2*i*π*n*X/T))’),where T is the period,

Page 1-22Press the !!@@OK#@ soft menu key return to the calculator display. The numbernow is shown as:Because this number has three figures in th

Page 16-33Accept change to Approx mode if requested. The result is the value –0.40467…. The actual value of the function g(0.5) is g(0.5) = -0.25.

Page 16-34periodicity in the graph of the series. This periodicity is easy to visualize by expanding the horizontal range of the plot to (-0.5,4):Fou

Page 16-35The calculator returns an integral that cannot be evaluated numerically because it depends on the parameter n. The coefficient can still

Page 16-36Press `` to copy this result to the screen. Then, reactivate the Equation Writer to calculate the second integral defining the coefficient

Page 16-37This result is used to define the function c(n) as follows:DEFINE(‘c(n) = - (((-1)^n-1)/(n^2*π^2*(-1)^n)’)i.e.,Next, we define function F(X,

Page 16-38From the plot it is very difficult to distinguish the original function from the Fourier series approximation. Using k = 2, or 5 terms in t

Page 16-39In this case, the period T, is 4. Make sure to change the value of variable @@@T@@@to 4 (use: 4K@@@T@@ `). Function g(X) can be defined

Page 16-40The simplification of the right-hand side of c(n), above, is easier done on paper (i.e., by hand). Then, retype the expression for c(n) as

Page 16-41We can use this result as the first input to the function LDEC when used to obtain a solution to the system d2y/dX2 + 0.25y = SW(X), where S

Page 16-42 The solution is shown below:Fourier TransformsBefore presenting the concept of Fourier transforms, we’ll discuss the general definition

Page 1-23Θ Press the !!@@OK#@ soft menu key return to the calculator display. The number123.456789012, entered earlier, now is shown as:Angle Mea

Page 16-43The amplitudes An will be referred to as the spectrum of the function and will be a measure of the magnitude of the component of f(x) with f

Page 16-44andThe continuous spectrum is given byThe functions C(ω), S(ω), and A(ω) are continuous functions of a variable ω,which becomes the transfor

Page 16-45Define this expression as a function by using function DEFINE („à). Then, plot the continuous spectrum, in the range 0 < ω < 10, as:D

Page 16-46The continuous spectrum, F(ω), is calculated with the integral: This result can be rationalized by multiplying numerator and denominator by

Page 16-47Properties of the Fourier transformLinearity: If a and b are constants, and f and g functions, then F{a⋅f + b⋅g} = a F{f }+ b F{g}.Transfor

Page 16-48the number of operations using the FFT is reduced by a factor of 10000/664 ≈15.The FFT operates on the sequence {xj} by partitioning it into

Page 16-49The figure below is a box plot of the data produced. To obtain the graph, first copy the array just created, then transform it into a colum

Page 16-50Example 2 – To produce the signal given the spectrum, we modify the program GDATA to include an absolute value, so that it reads:<< 

Page 16-51Except for a large peak at t = 0, the signal is mostly noise. A smaller vertical scale (-0.5 to 0.5) shows the signal as follows:Solution t

Page 16-52where M = n/2 or (n-1)/2, whichever is an integer.Legendre’s polynomials are pre-programmed in the calculator and can be recalled by using t

Page 1-24key. If using the latter approach, use up and down arrow keys,— ˜,to select the preferred mode, and press the !!@@OK#@ soft menu key to

Page 16-53where ν is not an integer, and the function Gamma Γ(α) is defined in Chapter 3.If ν = n, an integer, the Bessel functions of the first kind

Page 16-54Yν(x) = [Jν(x) cos νπ – J−ν(x)]/sin νπ,for non-integer ν, and for n integer, with n > 0, bywhere γ is the Euler constant, defined byand h

Page 16-55The modified Bessel functions of the second kind, Kν(x) = (π/2)⋅[I-ν (x)−Iν (x)]/sin νπ,are also solutions of this ODE.You can implement fu

Page 16-56Laguerre’s equationLaguerre’s equation is the second-order, linear ODE of the form x⋅(d2y/dx2)+(1−x)⋅ (dy/dx) + n⋅y = 0. Laguerre polynomia

Page 16-57L 2(x) = 1-2x+ 0.5x2L 3(x) = 1-3x+1.5x2-0.16666…x3.Weber’s equation and Hermite polynomialsWeber’s equation is defined as d2y/dx2+(n+1/2-x2/

Page 16-58First, create the expression defining the derivative and store it into variable EQ. The figure to the left shows the ALG mode command, whil

Page 16-59@@OK@@ @INIT+—.75 @@OK@@ ™™@SOLVE (wait) @EDIT(Changes initial value of t to 0.5, and final value of t to 0.75, solve for v(0.75)= 2.066

Page 16-60Θ „ô (simultaneously, if in RPN mode) to enter PLOT environment Θ Highlight the field in front of TYPE, using the —˜keys. Then, press@CHOO

Page 16-61LL@)PICT To recover menu and return to PICT environment.@(X,Y)@ To determine coordinates of any point on the graph.Use the š™ keys to move

Page 16-62time t = 2, the input form for the differential equation solver should look as follows (notice that the Init: value for the Soln: is a vecto

Page 1-25from the positive z axis to the radial distance ρ. The Rectangular and Sphericalcoordinate systems are related by the following quantities:T

Page 16-63(Changes initial value of t to 0.75, and final value of t to 1, solve again for w(1) = [-0.469 -0.607])Repeat for t = 1.25, 1.50, 1.75, 2.0

Page 16-64Notice that the option V-Var: is set to 1, indicating that the first element in the vector solution, namely, x’, is to be plotted against th

Page 16-65Press LL @PICT @CANCL $ to return to normal calculator display. Numerical solution for stiff first-order ODEConsider the ODE: dy/dt = -100

Page 16-66Here we are trying to obtain the value of y(2) given y(0) = 1. With the Soln: Final field highlighted, press @SOLVE. You can check that a

Page 16-67Note: The option Stiff is also available for graphical solutions of differentialequations.Numerical solution to ODEs with the SOLVE/DIFF me

Page 16-68The value of the solution, yfinal, will be available in variable @@@y@@@. This function is appropriate for programming since it leaves the

Page 16-69contain only the value of ε, and the step Δx will be taken as a small default value. After running function @@RKF@@, the stack will show the

Page 16-70 These results indicate that (Δx)next = 0.34049…Function RRKSTEPThis function uses an input list similar to that of function RRK, as wel

Page 16-71These results indicate that (Δx)next = 0.00558… and that the RKF method (CURRENT = 1) should be used.Function RKFERRThis function returns th

Page 16-72The following screen shots show the RPN stack before and after application of function RSBERR: These results indicate that Δy = 4.1514…

Page 1-26_Last Stack: Keeps the contents of the last stack entry for use with the functionsUNDO and ANS (see Chapter 2).The _Beep option can be usefu

Page 17-1Chapter 17Probability ApplicationsIn this Chapter we provide examples of applications of calculator’s functions to probability distributions.

Page 17-2To simplify notation, use P(n,r) for permutations, and C(n,r) for combinations. We can calculate combinations, permutations, and factorials

Page 17-3Random number generators, in general, operate by taking a value, called the “seed” of the generator, and performing some mathematical algorit

Page 17-4function (pmf) is represented by f(x) = P[X=x], i.e., the probability that the random variable X takes the value x. The mass distribution fu

Page 17-5Poisson distributionThe probability mass function of the Poisson distribution is given by.In this expression, if the random variable X repres

Page 17-6 Continuous probability distributionsThe probability distribution for a continuous random variable, X, is characterized by a function f(x

Page 17-7,while its cdf is given by F(x) = 1 - exp(-x/β), for x>0, β >0.The beta distributionThe pdf for the gamma distribution is given byAs in