Hp 48gII User Manual

hp 48gII graphing calculator user’s guide H Edition 4 HP part number F2226-90020

Page TOC-6 UNITS convert menu, 5-27 BASE convert menu, 5-28 TRIGONOMETRIC convert menu, 5-28 MATRICES convert menu, 5-28 REWRITE

Page 2-42 Command CRDIR in RPN mode To use the CRDIR in RPN mode you need to have the name of the directory already available in the stack before ac

Page 2-43 key to list the contents of the directory in the screen. Select the sub-directory (or variable) that you want to delete. Press L@PURGE.

Page 2-44 Use the down arrow key (˜) to select the option 2. MEMORY… Then, press @@OK@@. This will produce the following pull-down menu: Use the

Page 2-45 Press @@OK@@, to get: Then, press )@@S3@@ to enter ‘S3’ as the argument to PGDIR. Press ` to delete the sub-directory: Command PGDI

Page 2-46 Using the PURGE command from the TOOL menu The TOOL menu is available by pressing the I key (Algebraic and RPN modes shown): The

Page 2-47 sub-directory {HOME MANS INTRO}, created in an earlier example, we want to store the following variables with the values shown: Name Cont

Page 2-48 To enter variable A (see table above), we first enter its contents, namely, the number 12.5, and then its name, A, as follows: 12.5 @@OK@@

Page 2-49 Name Contents Type α -0.25 real A12 3×105 real Q ‘r/(m+r)' algebraic R [3,2,1] vector z1 3+5i complex p1 << → r 'π*r^2&ap

Page 2-50 You will see six of the seven variables listed at the bottom of the screen: p1, z1, R, Q, A12, α. • RPN mode Use the following keystr

Page 2-51 Checking variables contents As an exercise on peeking into the contents of variables we will use the seven variables entered in the exerci

Page TOC-7 Application 2 - Velocity and acceleration in polar coordinates, 7-18 Chapter 8 - Operations with Lists, 8-1 Definitions, 8-1 Cr

Page 2-52 Note: By pressing @@@p1@@ ` we are trying to activate (run) the p1 program. However, this program expects a numerical input. Try the f

Page 2-53 Using the right-shift key ‚ followed by soft menu key labels This approach for viewing the contents of a variable works the same in both

Page 2-54 Check the new contents of variable A12 by using ‚@@A12@@ . Using the RPN operating mode: ³~‚b/2` ³@@A12@@ ` K or, in a simplified way

Page 2-55 variables p1, z1, R, Q, A12, α, and A. Suppose that we want to copy variable A and place a copy in sub-directory {HOME MANS}. Also, we w

Page 2-56 Using the history in Algebraic mode Here is a way to use the history (stack) to copy a variable from one directory to another with the cal

Page 2-57 ‚@@ @Q@@ K@@@Q@@ ` „§` ƒ ƒ ƒ` ƒ ƒ ƒ ƒ ` To verify the contents of the variables, use ‚@@ @R@ and ‚@@ @Q. This procedure can be generali

Page 2-58 Next, we’ll list the new order of the variables by using their names typed between quotes: „ä ³)@INTRO ™‚í³@@@@A@@@ ™‚í³@@@z1@@™‚í³@@@Q@@

Page 2-59 Notice that variable A12 is no longer there. If you now press „§, the screen will show the contents of sub-directory MANS, including va

Page 2-60 Using function PURGE in the stack in Algebraic mode We start again at subdirectory {HOME MANS INTRO} containing now only variables p1, z1

Page 2-61 UNDO and CMD functions Functions UNDO and CMD are useful for recovering recent commands, or to revert an operation if a mistake was made.

Page TOC-8 Using the Matrix Writer (MTWR) to enter vectors, 9-3 Building a vector with ARRY, 9-6 Identifying, extracting, and inserting v

Page 2-62 Pressing „® produces the following selection box: As you can see, the numbers 3, 2, and 5, used in the first calculation above, are list

Page 2-63 Example of flag setting: general solutions vs. principal value For example, the default value for system flag 01 is General solutions. Wh

Page 2-64 ‚O~ „t Q2™+5*~ „t+6——‚Å0` ` (keeping a second copy in the RPN stack) ³~ „t` Use the following keystroke sequence to enter the QUAD comma

Page 2-65 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.

Page 2-66 Press twice to return to normal calculator display. Now, we’ll try to find the ORDER command using similar keystrokes to those used ab

Page 2-67 • The HELP menu, activated with I L @HELP • The CMDS (CoMmanDS) menu, activated within the Equation Writer, i.e., ‚O L @CMDS

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

Page 3-2 2. Coordinate system specification (XYZ, R∠Z, R∠∠). The symbol ∠ stands for an angular coordinate. XYZ: Cartesian or rectangular (x,y,z

Page 3-3 Changing sign of a number, variable, or expression Use the \ key. In ALG mode, you can press \ before entering the number, e.g., \2.5`.

Page 3-4 4.2#2.5 * 2.3#4.5 / Using parentheses Parentheses can be used to group operations, as well as to enclose arguments of functions. The paren

Page TOC-9 Functions GET and PUT, 10-6 Functions GETI and PUTI, 10-6 Function SIZE, 10-7 Function TRN, 10-8 Function CON, 10-8 Funct

Page 3-5 In RPN mode, enter the number first, then the function, e.g., 2.3\„º The square root function, √, is available through the R key. When ca

Page 3-6 Or, in RPN mode: 4.5\V2\` Natural logarithms and exponential function Natural logarithms (i.e., logarithms of base e = 2.7182818282) are ca

Page 3-7 „À1.35` In RPN mode: 0.25`„¼ 0.85`„¾ 1.35`„À All the functions described above, namely, ABS, SQ, √, ^, XROOT, LOG, ALOG, LN, EXP, SIN, CO

Page 3-8 As they are a great number of mathematic functions available in the calculator, the MTH menu is sorted by the type of object the fun

Page 3-9 example, to select option 4. HYPERBOLIC.. in the MTH menu, simply press 4. Hyperbolic functions and their inverses Selecting Option 4. HYP

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

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

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

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

Page 3-14 Please 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).

Page TOC-10 Characterizing a matrix (the matrix NORM menu), 11-6 Function ABS, 11-7 Function SNRM, 11-7 Functions RNRM and CNRM, 11-8 Fu

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

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

Page 3-17 Please notice that e is available from the keyboard as exp(1), i.e., „¸1`, in ALG mode, or 1` „¸, in RPN mode. Also, π is available dir

Page 3-18 unit of mass), kip = kilo-poundal (1000 pounds), lbf = pound-force (to distinguish from pound-mass), pdl = poundal. To attach a unit obje

Page 3-19 LENGTH m (meter), cm (centimeter), mm (millimeter), yd (yard), ft (feet), in (inch), Mpc (Mega parsec), pc (parsec), lyr (light-year), au

Page 3-20 ENERGY J (joule), erg (erg), Kcal (kilocalorie), Cal (calorie), Btu (International table btu), ft×lbf (foot-pound), therm (EEC therm), Me

Page 3-21 Units not listed Units not listed in the Units menu, but available in the calculator include: gmol (gram-mole), lbmol (pound-mole), rpm

Page 3-22 This results in the following screen (i.e., 1 poise = 0.1 kg/(m⋅s)): In RPN mode, system flag 117 set to CHOOSE boxes: 1 Enter 1 (n

Page 3-23 Here is the sequence of steps to enter this number in ALG mode, system flag 117 set to CHOOSE boxes: 5‚Ý Enter number and underscore

Page 3-24 ‚Û Access the UNITS menu L @)@FORCE Select units of force @ @@N@@ Select Newtons (N) ` Enter quantity with units in the s

Page TOC-11 Function QR, 11-51 Matrix Quadratic Forms, 11-51 The QUADF menu, 11-52 Linear Applications, 11-54 Function IMAGE, 11-54 F

Page 3-25 To enter these prefixes, simply type the prefix using the ~ keyboard. For example, to enter 123 pm (1 picometer), use: 123‚Ý~„p~„m Usin

Page 3-26 To calculate a division, say, 3250 mi / 50 h, enter it as (3250_mi)/(50_h) `: which transformed to SI units, with function UBASE, produc

Page 3-27 Also, try the following operations: 5_m ` 3200_mm ` + 12_mm ` 1_cm^2 `* 2_s ` / These last two operations produce the following output

Page 3-28 These examples produce the same result, i.e., to convert 33 watts to btu’s CONVERT(33_W,1_hp) ` CONVERT(33_W,11_hp) ` These operations a

Page 3-29 Physical constants in the calculator Following along the treatment of units, we discuss the use of physical constants that are available

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

Page 3-31 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, t

Page 3-32 In the second page of this menu (press L) we find the following items: In this menu page, there is one function (TINC) and a number of

Page 3-33 Function SIDENS Function SIDENS(T) calculates the intrinsic density of silicon (in units of 1/cm3) as a function of temperature T (T in K)

Page 3-34 Defining and using functions Users can define their own functions by using the DEF command available thought the keystroke sequence „à (a

Page TOC-12 Y-Slice plots, 12-41 Gridmap plots, 12-42 Pr-Surface plots, 12-43 The VPAR variable, 12-44 Interactive drawing, 12-44 DO

Page 3-35 • Input:  x  x • Process: ‘LN(x+1) + EXP(x) ‘ This is to be interpreted as saying: enter a value that is temporarily assigned

Page 3-36 The calculator provides the function IFTE (IF-Then-Else) to describe such functions. The IFTE function The IFTE function is written as I

Page 3-37 Define this function by any of the means presented above, and check that g(-3) = 3, g(-1) = 0, g(1) = 0, g(3) = 9.

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

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

Page 4-3 Polar representation of a complex number The result shown above represents a Cartesian (rectangular) representation of the complex number 3

Page 4-4 Simple operations with complex numbers Complex numbers can be combined using the four fundamental operations (+-*/). The results fo

Page 4-5 Entering the unit imaginary number To enter the unit imaginary number type : „¥ Notice that the number i is entered as the ordered pa

Page 4-6 RE(z) : Real part of a complex number IM(z) : Imaginary part of a complex number C→R(z) : Takes a complex number (x,y) and separates it i

Page 4-7 Also, the result of function ARG, which represents an angle, will be given in the units of angle measure currently selected. In this examp

Page TOC-13 Function lim, 13-2 Derivatives, 13-3 Function DERIV and DERVX,13-3 The DERIV&INTEG menu, 13-3 Calculating derivatives

Page 4-8 Functions applied to complex numbers Many of the keyboard-based functions defined in Chapter 3 for real numbers, e.g., SQ, ,LN, ex, LOG, 10

Page 4-9 The following screen shows that functions EXPM and LNP1 do not apply to complex numbers. However, functions GAMMA, PSI, and Psi accept com

Page 5-1 Chapter 5 Algebraic and arithmetic operations An algebraic object, or simply, algebraic, is any number, variable name or algebraic expressi

Page 5-2 Simple operations with algebraic objects Algebraic objects can be added, subtracted, multiplied, divided (except by zero), raised to a pow

Page 5-3 @@A1@@ * @@A2@@ ` @@A1@@ / @@A2@@ ` ‚¹@@A1@@ „¸@@A2@@ The same results are obtained

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

Page 5-5 The help facility will show the following information on the commands: COLLECT: EXPAND:

Page 5-6 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 e

Page 5-7 In ALG mode, substitution of more than one variable is possible as illustrated in the following example (shown before and after pressing `

Page 5-8 hyperbolic functions in terms of trigonometric identities or in terms of exponential functions. The menus containing functions to replace

Page TOC-14 The chain rule for partial derivatives, 14-4 Total differential of a function z = z(x,y) , 14-5 Determining extrema in funct

Page 5-9 These functions allow to simplify expressions by replacing some category of trigonometric functions for another one. For example, the f

Page 5-10 of functions that apply to specific mathematical objects. This distinction between sub-menus (options 1 through 4) and plain functions

Page 5-11 IABCUV Solves au + bv = c, with a,b,c = integers IBERNOULLI n-th Bernoulli number ICHINREM Chinese reminder for integers IDIV2 Euclidea

Page 5-12 MODULO menu ADDTMOD Add two expressions modulo current modulus DIVMOD Divides 2 polynomials modulo current modulus DIV2MOD Euclidean

Page 5-13 Operations in modular arithmetic Addition in modular arithmetic of modulus n, which is a positive integer, follow the rules that if j and

Page 5-14 6 does not show the result 5 in modulus 12 arithmetic. This multiplication table is shown below: 6*0 (mod 12) 0 6*6 (mod 12) 0 6*1 (

Page 5-15 Finite arithmetic rings in the calculator All along we have defined our finite arithmetic operation so that the results are always positi

Page 5-16 ADDTMOD examples 6+5 ≡ -1 (mod 12) 6+6 ≡ 0 (mod 12) 6+7 ≡ 1 (mod 12) 11+5 ≡ 4 (mod 12) 8+10 ≡ -6 (mod 12) SUBTMOD examples 5 - 7

Page 5-17 before operating on them. You can also convert any number into a ring number by using the function EXPANDMOD. For example, EXPANDMOD(12

Page 5-18 Note: Refer to the help facility in the calculator for description and examples on other modular arithmetic. Many of these functions ar

Page TOC-15 Fourier series, 16-27 Function FOURIER, 16-28 Fourier series for a quadratic function, 16-29 Fourier series for a triangular

Page 5-19 The CHINREM function CHINREM stands for CHINese REMainder. The operation coded in this command solves a system of two congruences using t

Page 5-20 The HERMITE function The function HERMITE [HERMI] uses as argument an integer number, k, and returns the Hermite polynomial of k-th degree

Page 5-21 The LAGRANGE function The function LAGRANGE requires as input a matrix having two rows and n columns. The matrix stores data points of th

Page 5-22 The LEGENDRE function A Legendre polynomial of order n is a polynomial function that solves the differential equation 0)1(2)1(222=⋅+⋅+⋅⋅−⋅

Page 5-23 The QUOT and REMAINDER functions The functions QUOT and REMAINDER provide, respectively, the quotient Q(X) and the remainder R(X), resulti

Page 5-24 The TCHEBYCHEFF function The function TCHEBYCHEFF(n) generates the Tchebycheff (or Chebyshev) polynomial of the first kind, order n, defi

Page 5-25 PROPFRAC(‘5/4’) = ‘1+1/4’ PROPFRAC(‘(x^2+1)/x^2’) = ‘1+1/x^2’ The PARTFRAC function The function PARTFRAC decomposes a rational fractio

Page 5-26 If you press µ you will get: ‘(X^6+8*X^5+5*X^4-50*X^3)/(X^7+13*X^6+61*X^5+105*X^4-45*X^3-297*X^2-81*X+243)’ The FROOTS function The func

Page 5-27 The CONVERT Menu and algebraic operations The CONVERT menu is activated by using „Ú key (the 6 key). This menu su

Page 5-28 BASE convert menu (Option 2) This menu is the same as the UNITS menu obtained by using ‚ã. The applications of this menu are discussed in

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 W

Page TOC-16 Random numbers, 17-2 Discrete probability distributions, 17-4 Binomial distribution, 17-4 Poisson distribution, 17-5 Continuou

Page 5-29 Function NUM has the same effect as the keystroke combination ‚ï (associated with the ` key). Function NUM converts a symbolic result

Page 5-30 LIN LNCOLLECT POWEREXPAND SIMPLIFY

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

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

Page 6-3 The following examples show the use of function SOLVE in ALG and RPN modes: The screen shot shown above displays two solutions. In the f

Page 6-4 Function SOLVEVX The function SOLVEVX solves an equation for the default CAS variable contained in the reserved variable name VX. By defau

Page 6-5 To use function ZEROS in RPN mode, enter first the polynomial expression, then the variable to solve for, and then function ZEROS. The f

Page 6-6 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

Page 6-7 Press ` to return to stack. The stack will show the following results in ALG mode (the same result would be shown in RPN mode): To

Page 6-8 Press ` to return to stack, the coefficients will be shown in the stack. Press ˜ to trigger the line editor to see all the coefficient

Page TOC-17 Confidence intervals for the population mean when the population variance is known, 18-23 Confidence intervals for the

Page 6-9 To generate the algebraic expression using the roots, try the following example. Assume that the polynomial roots are [1,3,-2,1]. Use the

Page 6-10 Definitions Often, to develop projects, it is necessary to borrow money from a financial institution or from public funds. The amount of

Page 6-11 The screen now shows the value of PMT as –39,132.30, i.e., the borrower must pay the lender US $ 39,132.30 at the end of each month for th

Page 6-12 This means that at the end of 60 months the US $ 2,000,000.00 principal amount has been paid, together with US $ 347,937.79 of interest,

Page 6-13 2. The values calculated in the financial calculator environment are copied to the stack with their corresponding tag (identifying label).

Page 6-14 J „ä Prepare a list of variables to be purged @@@n@@ Enter name of variable N @I©YR@ Enter name of variable I%YR @@PV@@

Page 6-15 Press J to see the newly created EQ variable: Then, enter the SOLVE environment and select Solve equation…, by using: ‚Ï@@OK@@. The co

Page 6-16 • The user then highlights the field corresponding to the unknown for which to solve the equation, and presses @SOLVE@ • The user may fo

Page 6-17 At this point follow the instructions from Chapter 2 on how to use the Equation Writer to build an equation. The equation to enter in th

Page 6-18 The solution can be seen from within the SOLVE EQUATION input form by pressing @EDIT while the ex: field is highlighted. The resulting v

Page TOC-18 Conversion between number systems, 19-3 Wordsize, 19-4 Operations with binary integers, 19-4 The LOGIC menu, 19-5 The BIT

Page 6-19 yb1m We can type in the equation for E as shown above and use auxiliary variables for A and V, so that the resulting input form will have

Page 6-20 The result is 0.149836.., i.e., y = 0.149836. • It is known, however, that there are actually two solutions available for y in the spe

Page 6-21 written as gVDLfhf22⋅⋅= . The quantity f is known as the friction factor of the flow and it has been found to be a function of the relati

Page 6-22 Example 3 – Flow in a pipe You may want to create a separate sub-directory (PIPES) to try this example. The main equation governing

Page 6-23 Thus, the equation we are solving, after combining the different variables in the directory, is: ⋅=NuDQDDDARCYgDLQhf4/,82522πε

Page 6-24 Example 4 – Universal gravitation Newton’s law of universal gravitation indicates that the magnitude of the attractive force between two b

Page 6-25 Solve 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. Note: When u

Page 6-26 At this point the equation is ready for solution. Alternatively, you can activate the equation writer after pressing @EDIT to ente

Page 6-27 The SOLVE soft menu The SOLVE soft menu allows access to some of the numerical solver functions through the soft menu keys. To access thi

Page 6-28 The SOLVR sub-menu The SOLVR sub-menu activates the soft-menu solver for the equation currently stored in EQ. Some examples are shown nex

Page TOC-19 Programs that simulate a sequence of stack operations, 21-17 Interactive input in programs, 21-19 Prompt with an input string,

Page 6-29 As variables Q, a, and b, get assigned numerical values, the assignments are listed in the upper left corner of the display. At this point

Page 6-30 After solving the two equations, one at a time, we notice that, up to the third decimal, X is converging to a value of 7.500, while

Page 6-31 Function PROOT This function is used to find the roots of a polynomial given a vector containing the polynomial coefficients in decreasing

Page 6-32 The SOLVR sub-menu The SOLVR sub-menu in the TVM sub-menu will launch the solver for solving TVM problems. For example, pressing @)SOLV

Page 6-33 Function BEG If selected, the TMV calculations use payments at the beginning of each period. If deselected, the TMV calculations use p

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

Page 7-2 At this point, we need only press K twice to store these variables. To solve, first change CAS mode to Exact, then, list the contents of A2

Page 7-3 Notice that the right-hand sides of the two equations differ only in the sign between the two terms. Therefore, to write these equations i

Page 7-4 To solve for Pi and Po, use the command SOLVE from the S.SLV menu („Î), it may take the calculator a minute to produce the result: {[‘Pi=-(

Page 7-5 Example 1 – Example from the help facility As with all function entries in the help facility, there is an example attached to the MSLV en

Page TOC-20 Description of the PLOT menu, 22-2 Generating plots with programs, 22-14 Two-dimensional graphics, 22-14 Three-dimensional gra

Page 7-6 Example 2 - Entrance from a lake into an open channel This particular problem in open channel flow requires the simultaneous solution of t

Page 7-7 To see the original equations, EQ1 and EQ2, in terms of the primitive variables listed above, we can use function EVAL applied to

Page 7-8 Now, we are ready to solve the equation. First, we need to put the two equations together into a vector. We can do this by actually stor

Page 7-9 Press @@OK@@ and allow the solution to proceed. An intermediate solution step may look like this: The vector at the top representing the

Page 7-10 Using the Multiple Equation Solver (MES) The multiple equation solver is an environment where you can solve a system of multiple equations

Page 7-11 cosine law, and sum of interior angles of a triangle, to solve for the other three variables. If the three sides are known, the area o

Page 7-12 ‘a^2 = b^2+c^2-2*b*c*COS(α)’ ‘α+β+γ = 180’ ‘s = (a+b+c)/2’ ‘A = √ (s*(s-a)*(s-b)*(s-c))’ Then, enter the number 9, and create a list of e

Page 7-13 Preparing to run the MES The next step is to activate the MES and try one sample solution. Before we do that, however, we want to set the

Page 7-14 5[ a ] a:5 is listed in the top left corner of the display. 3[ b ] b:3 is listed in the top left corner of the display. 5[ c

Page 7-15 When done, press $ to return to the MES environment. Press J to exit the MES environment and return to the normal calculator display.

Page TOC-21 The CHARS menu, 23-2 The characters list, 23-3 Chapter 24 - Calculator objects and flags, 24-1 Description of calculator objects,

Page 7-16 Programming the MES triangle solution using User RPL To facilitate activating the MES for future solutions, we will create a program that

Page 7-17 Example 2 - Any type of triangle Use a = 3, b = 4, c = 6. The solution procedure used here consists of solving for all variables at once,

Page 7-18 carry over information from the previous solution that may wreck havoc with your current calculations. a b c α( ο) β( ο) γ( ο) A 2.5 6.9

Page 7-19 ________________________________________________________________ Program or value Store into variable:

Page 7-20 Start the multiple equation solver by pressing J@SOLVE. The calculator produces a screen labeled , "vel. & acc. polar coord.&quo

Page 7-21 To use a new set of values press, either @EXIT @@ALL@ LL, or J @SOLVE. Let's try another example using r = 2.5, vr = rD = -0.5,

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

Page 8-2 „ä 1 # 2 # 3 # 4 ` ~l1`™K The figure below shows the RPN stack before pressing the K key: Composing and decomposing lists Composing and

Page 8-3 Operations with lists of numbers To demonstrate operations with lists of numbers, we will create a couple of other lists, besides list L1 c

Page 8-4 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

Page TOC-22 Creating libraries, 26-7 Backup battery, 26-7 Appendices Appendix A - Using input forms, A-1 Appendix B - The calculator’s keyboard

Page 8-5 Real number functions from the keyboard Real number functions from the keyboard (ABS, ex, LN, 10x, LOG, SIN, x2, √, COS, TAN, ASIN, ACOS, A

Page 8-6 SINH, ASINH COSH, ACOSH TANH, ATANH SIGN, MANT, XPON IP, FP FL

Page 8-7 %({10, 20, 30},1) = {%(10,1),%(20,1),%(30,1)}, while %(5,{10,20,30}) = {%(5,10),%(5,20),%(5,30)} In the following example, both argument

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

Page 8-9 Next, with system flag 117 set to SOFT menus: This menu contains the following functions: ∆LIST : Calculate increment a

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

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

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

Page 8-13 Defining functions that use lists In Chapter 3 we introduced the use of the DEFINE function ( „à) to create functions of real numbers wi

Page 8-14 Next, we store the edited expression into variable @@@G@@@: Evaluating G(L1,L2) now produces the following result: As an alternat

Page Note-1 A note about screenshots in this guide A screenshot is a representation of the calculator screen. For example, the first time the calcu

Page 8-15 and that we store it into a variable called S (The screen shot below shows this action in ALG mode, however, the procedure in RPN mode is

Page 8-16 3. Divide the result above by n = 10: 4. Apply the INV() function to the latest result: Thus, the harmonic mean of list S is sh = 1.63

Page 8-17 Weighted average Suppose that the data in list S, defined above, namely: S = {1,5,3,1,2,1,3,4,2,1} is affected by the weights, W = {1, 2

Page 8-18 3. Use function ΣLIST, once more, to calculate the denominator of sw: 4. Use the expression ANS(2)/ANS(1) to calculate the weighted ave

Page 8-19 Given the list of class marks S = {s1, s2, …, sn }, and the list of frequency counts W = {w1, w2, …, wn }, the weighted average of the d

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

Page 9-1 Chapter 9 Vectors This Chapter provides examples of entering and operating with vectors, both mathematical vectors of many elements, as wel

Page 9-2 There are two definitions of products of physical vectors, a scalar or internal product (the dot product) and a vector or external product

Page 9-3 In RPN mode, you can enter a vector in the stack by opening a set of brackets and typing the vector components or elements separated by eit

Page 9-4 Vectors vs. matrices To see the @VEC@ key in action, try the following exercises: (1) Launch the Matrix Writer („²). With @VEC and @GO→

Page Note-2 Notice that the header lines cover the top first and a half lines of output in the calculator’s screen. Nevertheless, the lines of ou

Page 9-5 Activate the Matrix Writer again by using „², and press L to check out the second soft key menu at the bottom of the display. It will sho

Page 9-6 (5) Press @-COL@. The first column will disappear. (6) Press @+COL@. A row of two zeroes appears in the first row. (7) Press @GOTO@ 3@@O

Page 9-7 The following screen shots show the RPN stack before and after applying function ARRY: In RPN mode, the function [→ARRY] takes the

Page 9-8 More complicated expressions involving elements of A can also be written. For example, using the Equation Writer (‚O), we can writ

Page 9-9 Note: This approach for changing the value of an array element is not allowed in ALG mode, if you try to store 4.5 into A(3) in this mode y

Page 9-10 Attempting to add or subtract vectors of different length produces an error message (Invalid Dimension), e.g., v2+v3, u2+u3, A+v3, etc.

Page 9-11 Magnitude The magnitude of a vector, as discussed earlier, can be found with function ABS. This function is also available from t

Page 9-12 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 pr

Page 9-13 Building a three-dimensional vector Function V3 is used in the RPN mode to build a vector with the values in stack levels 1: , 2:, and 3:

Page 9-14 „Ô5 ‚í ~‚6 25 ‚í 2.3 Before pressing `, the screen will look as in the left-hand side of the following figure. After pressing `, the scr

Page Note-3 These simplifications of the screenshots are aimed at economizing output space in the guide. Be aware of the differences between the

Page 9-15 The conversion from Cartesian to cylindrical coordinates is such that r = (x2+y2)1/2, θ = tan-1(y/x), and z = z. For the case shown abov

Page 9-16 Notice that the vectors that were written in cylindrical polar coordinates have now been changed to the spherical coordinate system. The t

Page 9-17 Thus, the result is θ = 122.891o. In RPN mode use the following: [3,-5,6] ` [2,1,-3] ` DOT [3,-5,6] ` ABS [2,1,-3] ` ABS * /

Page 9-18 Thus the angle between vectors r and F is θ = 41.038o. RPN mode, we can use: [3,-5,4] ` [2,5,-6] ` CROSS ABS [3,-5,4] ` ABS [2,5,-6

Page 9-19 We can now use function EXPAND (in the ALG menu) to expand this expression: Thus, the equation of the plane through point P0(2,3,-1) and

Page 9-20 OBJ, ARRY, and LIST will be available in soft menu keys A, B, and C. Function DROP is available by using „°@)STACK @DROP. Following

Page 9-21 n+1:. For example, to create the list {1, 2, 3}, type: 1` 2` 3` 3` „°@)TYPE! !LIST@. Function ARRY This function is used to create

Page 9-22 A new variable, @@RXC@@, will be available in the soft menu labels after pressing J: Press ‚@@RXC@@ to see the program contained in the

Page 9-23 3 - Press the delete key ƒ (also known as function DROP) to eliminate the number in stack level 1: 4 - Use function LIST to create a li

Page 9-24 resulting in: Transforming a list into a vector To illustrate this transformation, we’ll enter the list {1,2,3} in RPN mode. Then, foll

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

Page 1-1 Chapter 1 Getting started This chapter is aimed at providing basic information in the operation of your calculator. The exercises are aim

Page 9-25 After having defined variable @@LXV@@, we can use it in ALG mode to transform a list into a vector. Thus, change your calculator’s mode

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

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

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

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 follo

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

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

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 refer

Page 10-8 Function TRN Function TRN is used to produce the transconjugate of a matrix, i.e., the transpose (TRAN) followed by its complex conjugate

Page 10-9 value. Function CON generates a matrix with constant elements. For example, in ALG mode, the following command creates a 4×3 matrix whos

Page 1-2 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 pl

Page 10-10 Function RDM Function RDM (Re-DiMensioning) is used to re-write vectors and matrices as matrices and vectors. The input to the function

Page 10-11 If using RPN mode, we assume that the matrix is in the stack and use {6} ` RDM. Note: Function RDM provides a more direct and efficien

Page 10-12 want to extract elements a12, a13, a22, and a23 from the last result, as a 2×2 sub-matrix, in ALG mode, use: In RPN mode, assuming that

Page 10-13 Function →DIAG Function →DIAG takes the main diagonal of a square matrix of dimensions n×n, and creates a vector of dimension n containin

Page 10-14 so the main diagonal included only the elements in positions (1,1) and (2,2). Thus, only the first two elements of the vector were requ

Page 10-15 The Hilbert matrix has application in numerical curve fitting by the method of linear squares. A program to build a matrix out of a num

Page 10-16 „°@)BRCH! @)FOR@! @NEXT NEXT „°@)BRCH! @)@IF@ @@IF@@ IF ~ „n #1 n 1 „°@)TEST! @@@>@@@ > „°@)BRCH! @@

Page 10-17 To use the program in ALG mode, press @CRMC followed by a set of parentheses („Ü). Within the parentheses type the lists of data represe

Page 10-18 Manipulating matrices by columns The calculator provides a menu with functions for manipulating matrices by operating in their columns.

Page 10-19 decomposed in columns. To see the full result, use the line editor (triggered by pressing ˜). In RPN mode, you need to list the m

Page 1-3 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

Page 10-20 as columns in the resulting matrix. The following figure shows the RPN stack before and after using function COL. Function COL

Page 10-21 In RPN mode, place the matrix in the stack first, then enter the number representing a column location before applying function COL-. T

Page 10-22 MTH/MATRIX/ROW.. sequence: („´) shown in the figure below with system flag 117 set to CHOOSE boxes: or through the MATRICES/CREATE/

Page 10-23 In RPN mode, you need to list the matrix in the stack, and the activate function ROW, i.e., @@@A@@@ ROW. The figure below show

Page 10-24 Function ROW+ Function ROW+ takes as argument a matrix, a vector with the same length as the number of rows in the matrix, and a

Page 10-25 Function RSWP Function RSWP (Row SWaP) takes as arguments two indices, say, i and j, (representing two distinct rows in a matrix),

Page 10-26 This same exercise done in RPN mode is shown in the next figure. The left-hand side figure shows the setting up of the matrix, the facto

Page 11-1 Chapter 11 Matrix Operations and Linear Algebra In Chapter 10 we introduced the concept of a matrix and presented a number of functions fo

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

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

Page 1-4 pressing the L (NeXT menu) key. This key is the third key from the left in the third row of keys in the keyboard. Press L once more to r

Page 11-4 Vector-matrix multiplication, on the other hand, is not defined. This multiplication can be performed, however, as a special case of mat

Page 11-5 The product of a vector with a matrix is possible if the vector is a row vector, i.e., a 1×m matrix, which multiplied with a matrix m×n pr

Page 11-6 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 s

Page 11-7 These functions are described next. Because many of these functions use concepts of matrix theory, such as singular values, rank, etc., w

Page 11-8 Singular value decomposition To understand the operation of Function SNRM, we need to introduce the concept of matrix decomposition. Basi

Page 11-9 Function SRAD Function SRAD determines the Spectral RADius of a matrix, defined as the largest of the absolute values of its eigenvalues.

Page 11-10 The condition number of a singular matrix is infinity. The condition number of a non-singular matrix is a measure of how close the matri

Page 11-11 where the values dj are constant, we say that ck is linearly dependent on the columns included in the summation. (Notice that the value

Page 11-12 The determinant of a matrix The determinant of a 2x2 and or a 3x3 matrix are represented by the same arrangement of elements of the matr

Page 11-13 For square matrices of higher order determinants can be calculated by using smaller order determinant called cofactors. The general id

Page 1-5 using the up and down arrow keys, —˜, or by pressing the number corresponding to the function in the CHOOSE box. After the function name i

Page 11-14 Function TRAN Function TRAN returns the transpose of a real or the conjugate transpose of a complex matrix. TRAN is equivalent to

Page 11-15 Function AXL Function AXL converts an array (matrix) into a list, and vice versa. For examples, Note: the latter operation is si

Page 11-16 The implementation of function LCXM for this case requires you to enter: 2`3`‚@@P1@@ LCXM ` The following figure shows the RPN stack b

Page 11-17 Using the numerical solver for linear systems There are many ways to solve a system of linear equations with the calculator. One possibi

Page 11-18 To enter matrix A you can activate the Matrix Writer while the A: field is selected. The following screen shows the Matrix Writer used f

Page 11-19 Under-determined system The system of linear equations 2x1 + 3x2 –5x3 = -10, x1 – 3x2 + 8x3 = 85, can be written as the matrix equat

Page 11-20 To see the details of the solution vector, if needed, press the @EDIT! button. This will activate the Matrix Writer. Within this env

Page 11-21 Let’s store the latest result in a variable X, and the matrix into variable A, as follows: Press K~x` to store the solution vector int

Page 11-22 can be written as the matrix equation A⋅x = b, if This system has more equations than unknowns (an over-determined system). The system

Page 11-23 Press ` to return to the numerical solver environment. To check that the solution is correct, try the following: • Press ——, to hig

Page 1-6 To navigate through the functions of this menu, press the L key to move to the next page, or „«(associated with the L key) to move to the p

Page 11-24 • 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

Page 11-25 Under-determined system Consider the system 2x1 + 3x2 –5x3 = -10, x1 – 3x2 + 8x3 = 85, with .8510,,831532321−==−−=

Page 11-26 Compare these three solutions with the ones calculated with the numerical solver. Solution with the inverse matrix The solution to the

Page 11-27 previous section. The procedure for the case of “dividing” b by A is illustrated below for the case 2x1 + 3x2 –5x3 = 13, x1 – 3x2 + 8x3

Page 11-28 The sub-indices in the variable names X, Y, and Z, determine to which equation system they refer to. To solve this expanded system we u

Page 11-29 To start the process of forward elimination, we divide the first equation (E1) by 2, and store it in E1, and show the three equation

Page 11-30 Y+ Z = 3, -7Z = -14. The process of backward substitution in Gaussian elimination consists in finding the values of the unknowns, starti

Page 11-31 −−−−=4314124123642augA The matrix Aaug is the same as the original matrix A with a new row, corresponding to the elements of t

Page 11-32 If you were performing these operations by hand, you would write the following: −−−−≅−−−−=4371241233214314124123642a

Page 11-33 Multiply row 3 by –3, add it to row 1, replacing it: 3\#3#1@RCIJ! Multiply row 2 by –2, add it to row 1, replacing it: 2\#2#1 @RCIJ

Page 1-7 @VIEW B VIEW the contents of a variable @@ RCL @@ C ReCaLl the contents of a variable @@STO@ D STOre the contents of a var

Page 11-34 pivoting operations. When row and column exchanges are allowed in pivoting, the procedure is known as full pivoting. When exchanging r

Page 11-35 First, we check the pivot a11. We notice that the element with the largest absolute value in the first row and first column is the value

Page 11-36 0 3 2 -1 1 0 0 0 25/8 0 -25/82 0 0 1 Checking the pivot at position (2,2), we now find that the value of 25/8,

Page 11-37 Finally, we eliminate the –1/16 from position (1,2) by using: 16 Y # 2#1@RCIJ 1 0 0 2 0 1 0 0 1 0 -1 0 0 1

Page 11-38 Then, for this particular example, in RPN mode, use: [2,-1,41] ` [[1,2,3],[2,0,3],[8,16,-1]] `/ The calculator shows an augmented matr

Page 11-39 To see the intermediate steps in calculating and inverse, just enter the matrix A from above, and press Y, while keeping the step-by-step

Page 11-40 Based on the equation A-1 = C/det(A), sketched above, the inverse matrix, A-1, is not defined if det(A) = 0. Thus, the condition det(A

Page 11-41 to produce the solution: [X=-1,Y=2,Z = -3]. Function LINSOLVE works with symbolic expressions. Functions REF, rref, and RREF, work wi

Page 11-42 The diagonal matrix that results from a Gauss-Jordan elimination is called a row-reduced echelon form. Function RREF ( Row-Reduced Ech

Page 11-43 The result is the augmented matrix corresponding to the system of equations: X+Y = 0 X-Y =2 Residual errors in linear system solu

Page 1-8 As indicated above, the TIME menu provides four different options, numbered 1 through 4. Of interest to us as this point is option 3. S

Page 11-44 Eigenvalues and eigenvectors Given a square matrix A, we can write the eigenvalue equation A⋅x = λ⋅x, where the values of λ that satisfy

Page 11-45 Using the variable λ to represent eigenvalues, this characteristic polynomial is to be interpreted as λ 3-2λ 2-22λ +21=0.

Page 11-46 Change mode to Approx and repeat the entry, to get the following eigenvalues: [(1.38,2.22), (1.38,-2.22), (-1.76,0)]. Function EGV Func

Page 11-47 Function JORDAN Function JORDAN is intended to produce the diagonalization or Jordan-cycle decomposition of a matrix. In RPN mode, give

Page 11-48 In RPN mode, function MAD generate a number of properties of a square matrix, namely: • the determinant (stack level 4) • the formal i

Page 11-49 Function contained in this menu are: LQ, LU, QR,SCHUR, SVD, SVL. Function LU Function LU takes as input a square matrix A, and re

Page 11-50 The Singular Value Decomposition (SVD) of a rectangular matrix Am×n consists in determining the matrices U, S, and V, such that Am×n = U

Page 11-51 1: [[-1.03 1.02 3.86 ][ 0 5.52 8.23 ][ 0 –1.82 5.52]] Function LQ The LQ function produces the LQ factorization of a matrix An×m returni

Page 11-52 []⋅−−⋅=⋅⋅ZYXZYXT153245112xAx[]−+++−+⋅=ZYXZYXZYXZYX532452 Finally, x⋅A⋅xT = 2X2+4Y2-Z2

Page 11-53 Function QXA Function QXA takes as arguments a quadratic form in stack level 2 and a vector of variables in stack level 1, returning the

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

Page 11-54 • The list of variables (stack level 1) For example, 'X^2+Y^2-Z^2+4*X*Y-16*X*Z' ` ['X','Y','Z'

Page 11-55 Function KER Function MKISOM

Page 12-1 Chapter 12 Graphics In this chapter we introduce some of the graphics capabilities of the calculator. We will present graphics of functio

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

Page 12-3 return to normal calculator display. The PLOT SET UP window should look similar to this: • Note: You will notice that a new variable

Page 12-4 << →X ‘EXP(-X^2/2)/ √(2*π)‘ >>. Press ƒ, twice, to drop the contents of the stack. • Enter the PLOT WINDOW environment by

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

Page 12-6 • If you move the cursor towards the right-hand side of the curve, by pressing the right-arrow key (™), and press @ROOT, the result now i

Page 12-7 curves intercept at two points. Move the cursor near the left intercept point and press @)@FCN! @ISECT, to get I-SECT: (-0.6834…,0.2158

Page 12-8 Move the cursor to the upper left corner of the display, by using the š and — keys. To display the figure currently in level 1 of t

Page 1-10 To set the date, first set the date format. The default format is M/D/Y (month/day/year). To modify this format, press the down arrow

Page 12-9 equation writer with the expression Y1(X)= . Type LN(X). Press ` to return to the PLOT-FUNCTION window. Press L@@@OK@@@ to return to n

Page 12-10 Note: When you press J , your variables list will show new variables called @@@X@@ and @@Y1@@ . Press ‚@@Y1@@ to see the co

Page 12-11 To add labels to the graph press @EDIT L@)LABEL. Press @MENU to remove the menu labels, and get a full view of the graph. Press LL@)P

Page 12-12 Inverse functions and their graphs Let y = f(x), if we can find a function y = g(x), such that, g(f(x)) = x, then we say that g(x) is th

Page 12-13 Press @CANCL to return to the PLOT FUNCTION – WINDOW screen. Modify the vertical and horizontal ranges to read: H-View: -8 8, V-Vi

Page 12-14 Note: the soft menu keys @EDIT and @CHOOS are not available at the same time. One or the other will be selected depending on which inpu

Page 12-15 • Use @MOVE° and @MOVE³ to move the selected equation one location up or down, respectively. • Use @CLEAR if you want to clear all t

Page 12-16 • Use @ERASE to erase any graph currently existing in the graphics display window. • Use @DRAW to produce the graph according to t

Page 12-17 the function Y=X when plotting simultaneously a function and its inverse to verify their ‘reflection’ about the line Y = X.

Page 12-18 will be highlighted. If this field is not already set to FUNCTION, press the soft key @CHOOS and select the FUNCTION option, then press

(numerical) mode. The display can be adjusted to provide textbook-type expressions, which can be useful when working with matrices, vectors, fracti

Page 1-11 The figure shows 10 rows of keys combined with 3, 5, or 6 columns. Row 1 has 6 keys, rows 2 and 3 have 3 keys each, and rows 4 throug

Page 12-19 •• The @ZOOM key, when pressed, produces a menu with the options: In, Out, Decimal, Integer, and Trig. Try the following exercises: •

Page 12-20 • The cursor is now in the Indep field. Press ³~‚t @@@OK@@@ to change the independent variable to θ. • Press L@@@OK@@@ to return to

Page 12-21 will get the equation ‘2*(1-SIN(θ))’ highlighted. Let’s say, we want to plot also the function ‘2*(1-COS(θ))’ along with the previous eq

Page 12-22 { ‘(X-1)^2+(Y-2)^2=3’ , ‘X^2/4+Y^2/3=1’ } into the variable EQ. These equations we recognize as those of a circle centered at (1,2) wi

Page 12-23 centered at the origin (0,0), will extend from -2 to 2 in x, and from -√3 to √3 in y. Notice that for the circle and the ellipse the reg

Page 12-24 X(t) = X0 + V0*COS(θ0)*t Y(t) = Y0 + V0*SIN(θ0)*t – 0.5*g*t^2 which will add the variables @@@Y@@@ and @@@X@@@ to the soft menu key lab

Page 12-25 • Press @ERASE @DRAW to draw the parametric plot. • Press @EDIT L @LABEL @MENU to see the graph with labels. The window param

Page 12-26 if in RPN mode). Then, press @ERASE @DRAW. Press @CANCL to return to the PLOT, PLOT WINDOW, or PLOT SETUP screen. Press $, or L@@@O

Page 12-27 of differential equations of the form Y'(T) = F(T,Y). For our case, we let Yx and Tt, therefore, F(T,Y)f(t,x) = exp(-t2). Bef

Page 12-28 • Press L to recover the menu. Press L@)PICT to recover the original graphics menu. • When we observed the graph being plotted, y

Page 1-12 combined with some of the other keys to activate the alternative functions shown in the keyboard. For example, the P key, key(4,4), has

Page 12-29 Truth plots Truth plots are used to produce two-dimensional plots of regions that satisfy a certain mathematical condition that can be ei

Page 12-30 You can have more than one condition plotted at the same time if you multiply the conditions. For example, to plot the graph of the poin

Page 12-31 [[3.1,2.1,1.1],[3.6,3.2,2.2],[4.2,4.5,3.3], [4.5,5.6,4.4],[4.9,3.8,5.5],[5.2,2.2,6.6]] ` to store it in ΣDAT, use the function STOΣ

Page 12-32 • Press @CANCL to return to the PLOT WINDOW environment. Then, press $ , or L@@@OK@@@, to return to normal calculator display. The

Page 12-33 • Press ˜˜ to highlight the Cols: field. Enter 1@@@OK@@@ 2@@@OK@@@ to select column 1 as X and column 2 as Y in the Y-vs.-X scatter pl

Page 12-34 • Press LL@)PICT to leave the EDIT environment. • Press @CANCL to return to the PLOT WINDOW environment. Then, press $ , or L@@@OK@

Page 12-35 • Press @CANCL to return to the PLOT WINDOW environment. Then, press $ , or L@@@OK@@@, to return to normal calculator display. If

Page 12-36 • Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd: variables. • Press L@@@OK@@@ to return to normal calculator displa

Page 12-37 • When done, press @EXIT. • Press @CANCL to return to PLOT WINDOW. • Press $ , or L@@@OK@@@, to return to normal calculator display.

Page 12-38 The coordinates XE, YE, ZE, stand for “eye coordinates,” i.e., the coordinates from which an observer sees the plot. The values shown ar

Page 1-13 Press the !!@@OK#@ F soft menu key to return to normal display. Examples of selecting different calculator modes are shown next. O

Page 12-39 • Press @ERASE @DRAW to see the surface plot. This time the bulk of the plot is located towards the right –hand side of the display.

Page 12-40 • Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP window. • Change TYPE to Ps-Contour. • Press ˜ and type

Page 12-41 • Press LL@)PICT to leave the EDIT environment. • Press @CANCL to return to the PLOT WINDOW environment. Then, press $ , or L@@@OK

Page 12-42 • Press „ô, simultaneously if in RPN mode, to access the PLOT SETUP window. • Press ˜ and type ‘(X+Y)*SIN(Y)’ @@@OK@@@. • Press @

Page 12-43 • Press $ , or L@@@OK@@@, to return to normal calculator display. Other functions of a complex variable worth trying for Gridmap plots

Page 12-44 • Press LL@)PICT @CANCL to return to the PLOT WINDOW environment. • Press $ , or L@@@OK@@@, to return to normal calculator displ

Page 12-45 points, lines, circles, etc. on the graphics screen, as described below. To see how to use these functions we will try the following exe

Page 12-46 MARK This command allows the user to set a mark point which can be used for a number of purposes, such as: • Start of line with the LIN

Page 12-47 BOX This command is used to draw a box in the graph. Move the cursor to a clear area of the graph, and press @BOX@. This highlights t

Page 12-48 ERASE The function ERASE clears the entire graphics window. This command is available in the PLOT menu, as well as in the plotting windo

Page 1-14 The equation writer is a display mode in which you can build mathematical expressions using explicit mathematical notation including fract

Page 12-49 Zooming in and out in the graphics display Whenever you produce a two-dimensional FUNCTION graphic interactively, the first soft-menu key

Page 12-50 BOXZ Zooming in and out of a given graph can be performed by using the soft-menu key BOXZ. With BOXZ you select the rectangular sector (

Page 12-51 ZINTG Zooms the graph so that the pixel units become user-define units. For example, the minimum PICT window has 131 pixels. When you u

Page 12-52 ALGEBRA.. ‚× (the 4 key) Ch. 5 ARITHMETIC.. „Þ (the 1 key) Ch. 5 CALCULUS.. „Ö (the 4 key) Ch. 13 SOLVER.. „

Page 12-53 PLOTADD(X^2-X) is similar to „ô but adding this function to EQ: X^2 -1. Using @ERASE @DRAW produces the plot: TABVAL(X^2-1,{1,

Page 12-54 The output is in a graphical format, showing the original function, F(X), the derivative F’(X) right after derivation and after simplific

Page 13-1 Chapter 13 Calculus Applications In this Chapter we discuss applications of the calculator’s functions to operations related to Calculus,

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

Page 13-3 Derivatives The derivative of a function f(x) at x = a is defined as the limit hxfhxfxfdxdfh)()(lim)('0−+==>− Some examples of d

Page 13-4 Out of these functions DERIV and DERVX are used for derivatives. The other functions include functions related to anti-derivati

Page 1-15 Notice that the display shows several levels of output labeled, from bottom to top, as 1, 2, 3, etc. This is referred to as the stack of

Page 13-5 The insert cursor () will be located right at the denominator awaiting for the user to enter an independent variable, say, s: ~„s. Then

Page 13-6 derivatives, utilizing the same symbol for both. The user must keep this distinction in mind when translating results from the calculator

Page 13-7 Notice that in the expressions where the derivative sign (∂) or function DERIV was used, the equal sign is preserved in the equation, bu

Page 13-8 maxima) of the function, to plot the derivative, and to find the equation of the tangent line. Try the following example for the functi

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

Page 13-10 This result indicates that the range of the function 11)(2+=XXf corresponding to the domain D = { -1,5 } is R = 2626,22. Function S

Page 13-11 • Two lists, the first one indicates the variation of the function (i.e., where it increases or decreases) in terms of the independent v

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

Page 13-13 For example, to determine where the critical points of function 'X^3-4*X^2-11*X+30' occur, we can use the following entries in

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

Page 1-16 3.` Enter 3 in level 1 5.` Enter 5 in level 1, 3 moves to y 3.` Enter 3 in level 1, 5 moves to level 2, 3 to level 3 3.* Place

Page 13-15 Please notice that functions SIGMAVX and SIGMA are designed for integrands that involve some sort of integer function like the factorial

Page 13-16 At this point, you can press ` to return the integral to the stack, which will show the following entry (ALG mode shown): This is the

Page 13-17 Notice the application of the chain rule in the first step, leaving the derivative of the function under the integral explicitly in the

Page 13-18 Integrating an equation Integrating an equation is straightforward, the calculator simply integrates both sides of the equation simultane

Page 13-19 The last four steps show the progression of the solution: a square root, followed by a fraction, a second fraction, and the final result

Page 13-20 Thus, we can use function IBP to provide the components of an integration by parts. The next step will have to be carried out separately

Page 13-21 Improper integrals These are integrals with infinite limits of integration. Typically, an improper integral is dealt with by first calcu

Page 13-22 If you enter the integral with the CAS set to Exact mode, you will be asked to change to Approx mode, however, the limits of the integral

Page 13-23 Infinite series An infinite series has the formnnaxnh )()(1,0−∑∞=. The infinite series typically starts with indices n

Page 13-24 ∑∑∞+==−⋅+−⋅=1)(0)()(!)()(!)()(knnoonknnoonxxnxfxxnxfxf, i.e., ).()()( xRxPxfkk+= The polynomial Pk(x)

Page 1-17 line will execute the DUP function which copies the contents of stack level 1 of the stack onto level 2 (and pushes all the other stack le

Page 13-25 Function TAYLR produces a Taylor series expansion of a function of any variable x about a point x = a for the order k specified by the us

Page 13-26 In the right-hand side figure above, we are using the line editor to see the series expansion in detail.

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

Page 14-2 hyxfyhxfxfh),(),(lim0−+=∂∂→ . Similarly, kyxfkyxfyfk),(),(lim0−+=∂∂→. We will use the multi-variate functions defined earlier to calcula

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

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

Page 14-5 Total differential of a function z = z(x,y) From the last equation, if we multiply by dt, we get the total differential of the function z

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

Page 14-7 Applications 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

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

Page 1-18 The number is rounded to the maximum 12 significant figures, and is displayed as follows: In the standard format of decimal display, inte

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

Page 14-10 ∫∫∫∫=βαθθθθφθφ)()('),(),(gfRrdrdrdAr where the region R’ in polar coordinates is R’ = {α < θ < β, f(θ) < r < g(θ)}. Dou

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

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

Page 15-3 n independent variables φ(x1, x2, …,xn), and a vector of the functions [‘x1’ ‘x2’…’xn’]. Function HESS returns the Hessian matrix of the

Page 15-4 function φ(x,y,z) does not exist. In such case, function POTENTIAL returns an error message. For example, the vector field F(x,y,z) = (

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

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

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

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

Page 1-19 • Fixed format with decimals: This mode is mainly used when working with limited precision. For example, if you are doing financial cal

Page 16-2 The result is ‘∂x(∂x(u(x)))+3*u(x)*∂x(u(x))+u^2=1/x ’. This format shows up in the screen when the _Textbook option in the display set

Page 16-3 result by using function EVAL to verify the solution. For example, to check that u = A sin ωot is a solution of the equation d2u/dt2 + ωo

Page 16-4 The CALC/DIFF menu The DIFFERENTIAL EQNS.. sub-menu within the CALC („Ö) menu provides functions for the solution of differential equation

Page 16-5 Function LDEC The calculator provides function LDEC (Linear Differential Equation Command) to find the general solution to a linear ODE o

Page 16-6 of constants result from factoring out the exponential terms after the Laplace transform solution is obtained. Example 2 – Using the fu

Page 16-7 Allow the calculator about ten seconds to produce the result: ‘X^2 = X^2’. Example 3 - Solving a system of linear differential equations

Page 16-8 dy/dx + x2⋅y(x) = 5. In the calculator use: 'd1y(x)+x^2*y(x)=5' ` 'y(x)' ` DESOLVE The solution provided is {‘y =

Page 16-9 Next, we can write dy/dx = (C + exp x)/x = C/x + ex/x. In the calculator, you may try to integrate: ‘d1y(x) = (C + EXP(x))/x’ ` ‘y(x)’ `

Page 16-10 The solution is: Press µµto simplify the result to ‘y(t) = -((19*√5*SIN(√5*t)-(148*COS(√5*t)+80*COS(t/2)))/190)’. Press J @ODETY t

Page 16-11 circuits. In most cases one is interested in the system response after time t>0, thus, the definition of the Laplace transform, given

Page 1-20 • Scientific format The scientific format is mainly used when solving problems in the physical sciences where numbers are usually rep

Page 16-12 function LAP you get back a function of X, which is the Laplace transform of f(X). Example 2 – Determine the Laplace transform of f(t)

Page 16-13 L{df/dt} = s⋅F(s) - fo. Example 1 – The velocity of a moving particle v(t) is defined as v(t) = dr/dt, where r = r(t) is the position of

Page 16-14 Now, use ‘(-X)^3*EXP(-a*X)’ ` LAP µ. The result is exactly the same. • Integration theorem. Let F(s) = L{f(t)}, then • Convolution

Page 16-15 • Laplace transform of a periodic function of period T: • Limit theorem for the initial value: Let F(s) = L{f(t)}, then • Limit th

Page 16-16 An interpretation for the integral above, paraphrased from Friedman (1990), is that the δ-function “picks out” the value of the function

Page 16-17 Another important result, known as the second shift theorem for a shift to the right, is that L -1{e–as ⋅F(s)}=f(t-a)⋅H(t-a), with F(s)

Page 16-18 Example 1 – To solve the first order equation, dh/dt + k⋅h(t) = a⋅e–t, by using Laplace transforms, we can write: L{dh/dt + k⋅h(t)} =

Page 16-19 The result is: , i.e., h(t) = a/(k-1)⋅e-t +((k-1)⋅cCo-a)/(k-1)⋅e-kt. Thus, cC0 in the results from LDEC represents the initial cond

Page 16-20 To find the solution to the ODE, y(t), we need to use the inverse Laplace transform, as follows: OBJ ƒ ƒ Isolates right-hand side of

Page 16-21 L{d2y/dt2} + L{y(t)} = L{δ(t-3)}. With ‘Delta(X-3)’ ` LAP , the calculator produces EXP(-3*X), i.e., L{δ(t-3)} = e–3s. With Y(s) = L{y

Page TOC-1 Table of contents A note about screenshots in this guide, Note-1 Chapter 1 - Getting started, 1-1 Basic Operations, 1-1 Batteries, 1

Page 1-21 Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as: Because this number has three figur

Page 16-22 ‘X/(X^2+1)’ ` ILAP Result, ‘COS(X)’, i.e., L -1{s/(s2+1)}= cos t. ‘1/(X^2+1)’ ` ILAP Result, ‘SIN(X)’, i.e., L -1{1/(s2+1)}= si

Page 16-23 Defining and using Heaviside’s step function in the calculator The previous example provided some experience with the use of Dirac’s delt

Page 16-24  Change TYPE to FUNCTION, if needed  Change EQ to ‘0.5*COS(X)-0.25*SIN(X)+SIN(X-3)*H(X-3)’.  Make sure that Indep is set to ‘X’. 

Page 16-25 OBJ ƒ ƒ Isolates right-hand side of last expression ILAP Obtains the inverse Laplace transform The result is ‘y1*SIN(X-

Page 16-26 Again, there is a new component to the motion switched at t=3, namely, the particular solution yp(t) = [1+sin(t-3)]⋅H(t-3), which changes

Page 16-27 Fourier series Fourier series are series involving sine and cosine functions typically used to expand periodic functions. A function f(

Page 16-28 Thus, the first three terms of the function are: f(t) ≈ 1/3 – (4/π2)⋅cos (π⋅t)+(2/π)⋅sin (π⋅t). A graphical compari

Page 16-29 ∫∞−−−∞=⋅⋅⋅⋅⋅−⋅=TnndttTnitfTc0.,...2,1,0,1,2,...,,)2exp()(1π Function FOURIER provides the coefficient cn of the complex-form of the Fou

Page 16-30 Thus, c0 = 1/3, c1 = (π⋅i+2)/π2, c2 = (π⋅i+1)/(2π2). The Fourier series with three elements will be wri

Page 16-31 A general expression for cn The function FOURIER can provide a general expression for the coefficient cn of the complex Fourier series e

Page 1-22 Angle Measure Trigonometric functions, for example, require arguments representing plane angles. The calculator provides three different

Page 16-32 • First, define a function c(n) representing the general term cn in the complex Fourier series. • Next, define the finite complex Fo

Page 16-33 The function @@@F@@@ can be used to generate the expression for the complex Fourier series for a finite value of k. For example,

Page 16-34 F (0.5, 4, 1/3) = (-0.167070735979,0.) F (0.5, 5, 1/3) = (-0.294394690453,0.) F (0.5, 6, 1/3) = (-0.305652599743,0.) To compare the resu

Page 16-35 Fourier series for a triangular wave Consider the function <<−<<=21,210,)(xifxxifxxg which we assume to be periodic w

Page 16-36 The calculator returns an integral that cannot be evaluated numerically because it depends on the parameter n. The coefficient can s

Page 16-37 Press `` to copy this result to the screen. Then, reactivate the Equation Writer to calculate the second integral defining the coeffic

Page 16-38 Once again, replacing einπ = (-1)n, results in This result is used to define the function c(n) as follows: DEFINE(‘c(n) = - ((

Page 16-39 The resulting graph is shown below for k = 5 (the number of elements in the series is 2k+1, i.e., 11, in this case): From the plot it

Page 16-40 In this case, the period T, is 4. Make sure to change the value of variable @@@T@@@ to 4 (use: 4 K @@@T@@ `). Function g(X) can be d

Page 16-41 The simplification of the right-hand side of c(n), above, is easier done on paper (i.e., by hand). Then, retype the expression fo

Page 1-23 The coordinate system selection affects the way vectors and complex numbers are displayed and entered. To learn more about complex numbers

Page 16-42 Fourier series applications in differential equations Suppose we want to use the periodic square wave defined in the previous example as

Page 16-43 The latter result can be defined as a function, FW(X), as follows (cutting and pasting the last result into the command): We can now

Page 16-44 integration of the form ∫⋅⋅=badttftssF .)(),()( κ The function κ(s,t) is known as the kernel of the transformation. The use of an int

Page 16-45 A plot of the values An vs. ωn is the typical representation of a discrete spectrum for a function. The discrete spectrum will show tha

Page 16-46 In the calculator, set up and evaluate the following integrals to calculate C(ω) and S(ω), respectively. CAS modes are set to Exact and

Page 16-47 ∫∞⋅⋅⋅⋅==0)sin()(2)()}({ dtttfFtf ωπωsF Inverse sine transform ∫∞−⋅⋅⋅==01)sin()()()}({ dttFtfFsωωωF Fourier cosine transform ∫∞⋅⋅⋅⋅==0)co

Page 16-48 −−⋅+⋅=+⋅=ωωωπωπωiiiiF1111211121)( +⋅−+=2211121ωωωπi which is a complex function. The absolute value of the real an

Page 16-49 convolution: For Fourier transform applications, the operation of convolution is defined as ∫⋅⋅−⋅= .)()(21))(*( ξξξπdgxfxgf The foll

Page 16-50 Introduction to Random Vibrations, Spectral & Wavelet Analysis – Third Edition,” Longman Scientific and Technical, New York. The onl

Page 16-51 in the command catalog, ‚N). Store the array into variable ΣDAT by using function STOΣ (also available through ‚N). Select Bar in the

Page 1-24 • Press the H button. Next, use the down arrow key, ˜, three times. Select the Angle Measure mode by either using the \ key (second fr

Page 16-52 Example 2 – To produce the signal given the spectrum, we modify the program GDATA to include an absolute value, so that it reads: <&l

Page 16-53 Except 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: So

Page 16-54 Legendre’s equation An equation of the form (1-x2)⋅(d2y/dx2)-2⋅x⋅ (dy/dx)+n⋅ (n+1) ⋅y = 0, where n is a real number, is known as the Leg

Page 16-55 Bessel’s equation The ordinary differential equation x2⋅(d2y/dx2) + x⋅ (dy/dx)+ (x2-ν2) ⋅y = 0, where the parameter ν is a nonnegative r

Page 16-56 If you want to obtain an expression for J0(x) with, say, 5 terms in the series, use J(x,0,5). The result is ‘1-0.25*x^3+0.015625*x^4-4.

Page 16-57 With these definitions, a general solution of Bessel’s equation for all values of ν is given by y(x) = K1⋅Jν(x)+

Page 16-58 Un(x) = sin(n⋅arccos(x))/sin(arccos(x)). You can access the function TCHEBYCHEFF through the command catalog (‚N). The first four Cheb

Page 16-59 is the m-th coefficient of the binomial expansion (x+y)n. It also represents the number of combinations of n elements taken m at a time

Page 16-60 0 HERMITE, result: 1, i.e., H0* = 1. 1 HERMITE, result: ’2*X’, i.e., H1* = 2x. 2 HERMITE, result: ’4*X^2-2’, i.e., H2* = 4x

Page 16-61 To solve, press: @SOLVE (wait) @EDIT@. The result is 0.2499 ≈ 0.25. Press @@@OK@@@. Solution presented as a table of values Suppose