HP 49g+ User Manual

hp 49g graphing calculator user’s manual

Preface You have in your hands a compact symb olic and numerical computer that will facilitate calculation and mathematical analysis of problems in a variety of disciplines, from elementary mathem atics to advanced engine ering and science subjects. The present Guide contains examples that illustrate the use of the basic calculator functions and operations. The chapters in this user’s manual are organized by subject in orde r of difficulty: from the setting of calculator modes, to real and complex number calculations , operations with lists, vectors, and matrices, graphics, calculus applicat ions, vector analys is, differential equations, probability and statistics. For symbolic operations the calculat or includes a powerful Computer Algebraic System (CAS), which lets you select different modes of operation, e.g., complex numbers vs. real numbers, or exact (symbolic) vs. approximate (numerical) mode. The display can be adju sted to provide textbook-type expressions, which ca n be useful wh en working with matrices, vectors, fractions, summations, derivatives, and in tegrals. The high-speed gr aphics of the calculator are very convenient for producing com plex figures in very little time. Thanks to the infrared port and the US B cable available with your calculator, you can connect your calculator with other calculators or computers. The high-speed connection through infrared or USB allows the fast and efficient exchange of programs and data with ot her calculators or computers. The calculator provides flash memory ca rd ports to facilitate storage and exchange of data with other users. We hope your calculator will become a faithful companion for your school and professional applications.

Page TOC-1 Table of Contents Chapter 1 – Getting Started , 1-1 Basic Operations , 1-1 Batteries, 1-1 Turning the calculator on and off, 1-2 Adjusting the display contrast, 1-2 Contents of the calculator’s display, 1-2 Menus, 1-3 The TOOL menu, 1-3 Setting time and date, 1-4 Introducing the calculator’s keyboard , 1-4 Selecting calculator modes , 1-6 Operating mode, 1-7 Number Format and decimal dot or comma, 1-10 Standard format, 1-11 Fixed format with decimals, 1-11 Scientific format, 1-12 Engineering format, 1-13 Decimal comma vs. decimal point, 1-14 Angle Measure, 1-14 Coordinate System, 1-15 Selecting CAS settings , 1-16 Explanation of CAS settings, 1-17 Selecting Display modes ,1-17 Selecting the display font, 1-18 Selecting properties of the line editor, 1-19 Selecting properties of the Stack, 1-20 Selecting properties of the equation writer (EQW), 1- 21 References , 1-21 Chapter 2 – Introducing the calculator , 2-1 Calculator objects , 2-1 Editing expressions in the stack , 2-1 Creating arithmetic expressions, 2-1
Page TO C-2 Creating alge braic ex press io ns, 2- 4 Using th e Equation Writer (EQW) to create exp ressions , 2-5 Crea ting arith metic exp ressions, 2-5 Creating alge braic ex press io ns, 2- 8 Organiz ing data in the calcul ator , 2-9 The HOME directory, 2-9 Subdirectories, 2-9 Variable s , 2-10 Typing variabl e name s, 2 -1 0 Creating varia bles, 2-11 Alge braic mo de, 2- 11 RPN mode, 2-13 Checkin g v aria bles content s, 2-14 Alge braic mo de, 2- 14 RPN mode, 2-14 Using the ri ght-shi ft k ey f ol l owe d by so ft me nu k ey l abel s, 2-15 Listing the con tents of all va riab les in the screen , 2-15 Deleting va riables, 2-16 Using f u nctio n PURGE i n the st ack in Al gebrai c mode , 2- 16 Using functio n PURGE in the stack in RPN mode, 2-16 UNDO and C MD fu nctions , 2-17 CHOO SE b ox es vs. So f t M ENU , 2-17 Re fer ences , 2-20 Chapter 3 – Calculati ons wit h real numbers , 3-1 Ex amples of r eal nu mber calcu lations , 3-1 Using po wer o f 1 0 in e ntering data, 3 -4 Re al nu mber functio ns in the MTH menu , 3-6 Using calculator menus, 3-6 Hyperbo lic f u nctions and their i nverse s, 3- 6 Oper ations with u nits , 3-8 The UNITS menu, 3-8 Availabl e un its, 3- 10 Attaching units to numbe rs, 3-11 Unit prefixes, 3-11 Opera tions with units, 3-12
Page TO C-3 Unit conv ersion s, 3-14 Phys ical cons tants in the calcu lator , 3-14 Defining and u s ing functio ns , 3-16 Re fer ence , 3-18 Chapter 4 – Calculati ons wit h complex numbers , 4-1 Defini ti ons , 4-1 Setting the calcu l ator to C OMP LE X mo de , 4-1 Entering comple x numbe rs, 4 -2 Polar r epresen tati on of a comp lex number , 4-2 Simple oper ations with comple x nu mber s , 4-3 The CMPLX menus , 4-4 CMPLX menu thro ugh the MTH me nu, 4 -4 CMPLX menu in the keyboard, 4-5 Fu nctions applied to co mplex numbe rs , 4-6 Fu nction D R OITE : equ ation of a str aight line , 4-6 Re fer ence , 4-7 Chapter 5 – A lgebraic and ari thmet ic operat ions , 5-1 Ente ring al gebraic o bjects , 5-1 Simple oper ations with al gebr aic objects , 5-2 Functions in the ALG menu, 5-4 Oper ations with tr ans cendental functio ns , 5-6 Expansio n and f acto ring u sin g lo g-e xp f unc tions , 5 -6 E xpa nsi on a nd fac tori ng usin g t rig onom etr ic fun ct ions, 5-6 Fu nctions in the A R ITHM E TIC me nu , 5-7 Polynomi als , 5-8 The HORNE R function, 5-8 The v aria ble VX, 5-9 The PCO EF function, 5-9 The PRO OT function, 5-9 The QUOT IENT and REMAINDER fu nctions , 5-9 The PE VA L function, 5-10 Frac t io ns , 5-10 The S IMP2 function, 5-10 The PROP FRAC function, 5-11
Page TOC-4 The PARTFRAC function, 5-11 The FCOEF function, 5-11 The FROOTS function, 5-12 Step-by-step operations with polynomials and fractions , 5-12 Reference , 5-13 Chapter 6 – Solution to equations , 6-1 Symbolic solution of algebraic equations , 6-1 Function ISOL, 6-1 Function SOLVE, 6-2 Function SOLVEVX, 6-4 Function ZEROS, 6-4 Numerical solver menu , 6-5 Polynomial Equations, 6-6 Finding the solution to a polynomial equation, 6-6 Generating polynomial coefficients given the polynomial’s roots, 6-7 Generating an algebraic ex pression for the polynomial, 6-8 Financial calculations, 6-9 Solving equations with one unknown through NU M.SLV, 6-9 Function STEQ, 6-9 Solution to simultaneo us equations with MSLV , 6-10 Reference , 6-12 Chapter 7 – Operations with lists , 7-1 Creating and storing lists , 7-1 Operations with lists of numbers , 7-1 Changing sign, 7-1 Addition, subtraction, mu ltiplication, division, 7-2 Functions applied to lists, 7-3 Lists of complex numbers , 7-4 Lists of algebraic objects , 7-4 The MTH/LIST menu , 7-4 The SEQ function , 7-6 The MAP function , 7-6 Reference , 7-6
Page TO C-5 Chapter 8 – Vectors , 8-1 Ent eri n g vec t ors , 8-1 Typin g v ectors i n the st ack, 8-1 S tori ng ve ctor s int o va riables in the s tack, 8 -2 Using the matrix writer (MTR W) to enter vecto rs, 8-2 Simple oper ations with v ector s , 8-5 Changing s ign, 8 -5 Addition, su btractio n, 8- 5 Mul tipl icati on by a s cal ar, and divis ion by a scal ar, 8 -6 A bsolut e v a lue func ti on, 8-6 The MTH/VECTOR menu , 8-7 Magnitude, 8-7 Dot product, 8- 7 Cross product, 8-8 Re fer ence , 8-8 Chapter 9 – M atrices and li near algebra , 9-1 En ter ing matrice s in the stack , 9-1 Using the matrix ed itor, 9-1 Typing the matrix directly into the stack , 9-2 Oper ations with matr ices , 9-3 Addition and s ubtracti on, 9 -3 Multip lic at ion , 9-4 Mul tipl icatio n by a scal ar, 9 -4 Matrix-vector multipl ication, 9- 4 Matrix multiplication, 9-5 Term-by-te rm mul tiplicatio n, 9-5 The identity matrix, 9-6 The inver se matrix, 9-6 Char acter izing a matrix (The matrix NORM menu ) , 9-7 Function DET, 9-7 Function TRACE, 9-7 Solution of linea r systems , 9-7 Using the n umerica l solver for linear systems, 9-8
Page TOC-6 Solution with the inverse matrix, 9-10 Solution by “division” of matrices, 9-10 References , 9-10 Chapter 10 – Graphics , 10-1 Graphs options in the calculator , 10-1 Plotting an expression of th e form y = f(x) , 10-2 Generating a table of values for a function , 10-3 Fast 3D plots , 10-5 Reference , 10-8 Chapter 11 – Calculus Applications , 11-1 The CALC (Calculus) menu , 11-1 Limits and derivatives , 11-1 Function lim, 11-1 Functions DERIV and DERVX, 11-2 Anti-derivatives and integrals , 11-3 Functions INT, INTVX, RI SCH, SIGMA and SIGMAVX, 11-3 Definite integrals, 11-4 Infinite series , 11-4 Functions TAYLR, TAYLR0, and SERIES, 11-5 Reference , 11-6 Chapter 12 – Multi-variate Calculus Applications , 12-1 Partial derivatives , 12-1 Multiple integrals , 12-2 Reference , 12-2 Chapter 13 – Vector Analysis Applications , 13-1 The del operator , 13-1 Gradient , 13-1 Divergence , 13-2 Curl , 13-2 Reference , 13-2
Page TOC-7 Chapter 14 – Differential Equations , 14-1 The CALC/DIFF menu , 14-1 Solution to linear an d non-linear equations , 14-1 Function LDEC, 14-2 Function DESOLVE, 14-3 The variable ODETYPE, 14-4 Laplace Transforms , 14-5 Laplace transforms and inverses in the calculator, 14-5 Fourier series , 14-6 Function FOURIER, 14-6 Fourier series for a quadratic function, 14-6 Reference , 14-8 Chapter 15 – Probability Distributions , 15-1 The MTH/PROBABILITY.. sub-menu – part 1 , 15-1 Factorials, combinations, and permutations, 15-1 Random numbers, 15-2 The MTH/PROB menu – part 2 , 15-3 The Normal distribution, 15-3 The Student-t distribution, 15-3 The Chi-square distribution, 15-4 The F distribution, 15-4 Reference , 15-4 Chapter 16 – Statistical Applications , 16-1 Entering data , 16-1 Calculating single-variable statistics , 16-1 Obtaining frequency distributions , 16-3 Fitting data to a function y = f(x) , 16-4 Obtaining additional summary statistics , 16-6 Confidence intervals , 16-7 Hypothesis testing , 16-9 Reference , 16-11
Page TOC-8 Chapter 17 – Numbers in Different Bases , 17-1 The BASE menu , 17-1 Writing non-decimal numbers , 17-1 Reference , 17-2 Chapter 18 – Using SD cards , 18-1 Storing objects in the SD card , 18-1 Recalling an object from the SD card , 18-2 Purging an object from the SD card , 18-2 Limited Warranty – W-1 Service , W-2 Regulatory information , W-4
Page 1-1 Chapter 1 Getting started This chapter is aimed at providing basic information in the operation of your calculator. The exercises are aimed at familiarizing yourself with the basic operations and settings before actually performing a calculation. Basic Operations The following exercises are aimed at getting you acquainted with the hardware of your calculator. Batteries The calculator uses 3 AAA batteries as main power and a CR2032 lithium battery for memory backup. Before using the calculator, please install the batteries according to the following procedure. To install the main batteries a. Slide up the battery compartment cover as illustrated. b. Insert 3 new AAA batteries into the main compartment. Make sure each battery is inserted in the indicated direction. To install the backup battery a. Press down the holder. Push the plate to the shown direction and lift it.
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 place. After installing the batteries, press [ON] to turn the power on. Warning: When the low battery icon is displayed, you need to replace the batteries as soon as possible. However, avoid removing the backup battery and main batteries at the same time to avoid data lost. Turning the calculator on and off The $ key is located at the lower left corner of the keyboard. Press it once to turn your calculator on. To turn the calculator off, press the red right-shift key @ (first key in the second row from the bottom of the keyboard), followed by the $ key. Notice that the $ key has a red OFF label printed in the upper right corner as a reminder of the OFF command. Adjusting the display contrast You can adjust the display contrast by holding the $ key while pressing the or - keys. The $(hold) key combination produces a darker display The $(hold) - key combination produces a lighter display Contents of the calculator’s display Turn your calculator on once more. At the top of the display you will have two lines of information that describe the settings of the calculator. The first line shows the characters: RAD XYZ HEX R= 'X'
Page 1-3 For details on the meaning of these specifications see Chapter 2 in the calculator’s User’s Guide. The second line shows the characters { HOME } indicating that the HOME directory is the current file directory in the calculator’s memory. At the bottom of the display you will find a number of labels, namely, @EDIT @VIEW @@ RCL @@ @@STO@ ! PURGE !CLEAR associated with the six soft menu keys, F1 through F6: A A B B C C D D E E F F The six labels displayed in the lower part of the screen will change depending on which menu is displayed. But A will always be associated with the first displayed label, B with the second displayed label, and so on. Menus The six labels associated with the keys A through F form part of a menu of functions. Since the calculator has only six soft menu keys, it only display 6 labels at any point in time. However, a menu can have more than six entries. Each group of 6 entries is called a Menu page. To move to the next menu page (if available), press the L (NeXT menu) key. This key is the third key from the left in the third row of keys in the keyboard. The TOOL menu The soft menu keys for the menu currently displayed, known as the TOOL menu, are associated with operations related to manipulation of variables (see section on variables in this Chapter): @EDIT A EDIT the contents of a variable (see Chapter 2 in this Guide
Page 1-4 and Chapter 2 and Appendix L in the User’s Guide for more information on editing) @VIEW B VIEW the contents of a variable @@ RCL @@ C ReCaLl the contents of a variable @@STO@ D STOre the contents of a variable ! PURGE E PURGE a variable CLEAR F CLEAR the display or stack These six functions form the first page of the TOOL menu. This menu has actually eight entries arranged in two pages. The second page is available by pressing the L (NeXT menu) key. This key is the third key from the left in the third row of keys in the keyboard. In this case, only the first two soft menu keys have commands associated with them. These commands are: @CASCM A CASCMD: CAS CoMmanD, used to launch a command from the CAS by selecting from a list @HELP B HELP facility describing the commands available in the calculator Pressing the L key will show the original TOOL menu. Another way to recover the TOOL menu is to press the I key (third key from the left in the second row of keys from the top of the keyboard). Setting time and date See Chapter 1 in the calculator’s User’s Guide to learn how to set time and date. Introducing the calculator’s keyboard The figure below shows a diagram of the calculator’s keyboard with the numbering of its rows and columns. Each key has three, four, or five functions. The main key function correspond to the most prominent label in the key. Also, the green left-shift key, key (8,1), the red right-shift key, key (9,1), and
Page 1-5 the blue ALPHA key, key (7,1), can be combined with some of the other keys to activate the alternative functions shown in the keyboard. For example, the P key, key(4,4), has the following six functions associated with it: P Main function, to activate the SYMBolic menu „´ Left-shift function, to activate the MTH (Math) menu … N Right-shift function, to activate the CATalog function
Page 1-6 ~p ALPHA function, to enter the upper-case letter P ~„p ALPHA-Left-Shift function, to enter the lower-case letter p ~…p ALPHA-Right-Shift function, to enter the symbol π Of the six functions associated with a key only the first four are shown in the keyboard itself. The figure in next page shows these four labels for the P key. Notice that the color and the position of the labels in the key, namely, SYMB , MTH , CAT and P , indicate which is the main function (SYMB ), and which of the other three functions is associated with the left-shift „ ( MTH ), right-shift … ( CAT ), and ~ ( P ) keys. For detailed information on the calculator keyboard operation referee to Appendix B in the calculator’s User’s Guide. Selecting calculator modes This section assumes that you are now at least partially familiar with the use of choose and dialog boxes (if you are not, please refer to appendix A in the User’s Guide). Press the H button (second key from the left on the second row of keys from the top) to show the following CALCULATOR MODES input form:
Page 1-7 Press the !!@@OK#@ F soft menu key to return to normal display. Examples of selecting different calculator modes are shown next. Operating Mode The calculator offers two operating modes: the Algebraic mode, and the Reverse Polish Notation ( RPN ) mode. The default mode is the Algebraic mode (as indicated in the figure above), however, users of earlier HP calculators may be more familiar with the RPN mode. To select an operating mode, first open the CALCULATOR MODES input form by pressing the H button. The Operating Mode field will be highlighted. Select the Algebraic or RPN operating mode by either using the \ key (second from left in the fifth row from the keyboard bottom), or pressing the @CHOOSE soft menu key ( B ). If using the latter approach, use up and down arrow keys, — ˜ , to select the mode, and press the !!@@OK#@ soft menu key to complete the operation. To illustrate the difference between these two operating modes we will calculate the following expression in both modes: 5 . 2 3 0 . 23 0 . 3 0 . 3 1 0 . 5 0 . 3 e ⋅ − ⋅       To enter this expression in the calculator we will first use the equation writer , ‚O . Please identify the following keys in the keyboard, besides the numeric keypad keys: !@.#* -/R Q¸Ü‚Oš™˜—` The equation writer is a display mode in which you can build mathematical expressions using explicit mathematical notation including fractions, derivatives, integrals, roots, etc. To use the equation writer for writing the expression shown above, use the following keystrokes: ‚OR3.*!Ü5.-
Page 1-8 1./3.*3. ——————— /23.Q3™™™ !¸2.5` After pressing ` the calculator displays the expression: √ (3.*(5.-1/(3.*3.))/(23.^3 EXP(2.5)) Pressing ` again will provide the following value (accept Approx. mode on, if asked, by pressing !!@@OK#@ ): You could also type the expression directly into the display without using the equation writer, as follows: R!Ü3.*!Ü5.- 1/3.*3.™ /23.Q3 !¸2.5` to obtain the same result. Change the operating mode to RPN by first pressing the H button. Select the RPN operating mode by either using the \ key, or pressing the @CHOOSE soft menu key. Press the !!@@OK#@ F soft menu key to complete the operation. The display, for the RPN mode looks as follows: Notice that the display shows several levels of output labeled, from bottom to top, as 1, 2, 3, etc. This is referred to as the stack of the calculator. The
Page 1-9 different levels are referred to as the stack levels , i.e., stack level 1, stack level 2, etc. Basically, what RPN means is that, instea d of writing an operation such as 3 2, in the calculator by using 3 2` we write first the operands, in the proper order, and then the operator, i.e., 3`2` As you enter the operands, they occupy different stack levels. Entering 3` puts the number 3 in stack level 1. Next, entering 2` pushes the 3 upwards to occupy stack level 2. Finally, by pressing , we are telling the calculator to apply the operator, or program, to the objects occupying levels 1 and 2. The result, 5, is then placed in level 1. Let's try some other simple operations before trying the more complicated expression used earlier for the algebraic operating mode: 123/32 123`32/ 4 2 4`2Q 3 √ 27 27`R3@» Notice the position of the y and the x in the last two operations. The base in the exponential operation is y (stack level 2) while the exponent is x (stack level 1) before the key Q is pressed. Similarly, in the cubic root operation, y (stack level 2) is the quantity under the root sign, and x (stack level 1) is the root. Try the following exercise involving 3 factors: (5 3) × 2 5`3` Calculates (5 3) first. 2X Completes the calculation. Let's try now the expression proposed earlier:
Page 1-10 5 . 2 3 23 3 3 1 5 3 e ⋅ − ⋅       3` Enter 3 in level 1 5` Enter 5 in level 1, 3 moves to level 2 3` Enter 3 in level 1, 5 moves to level 2, 3 to level 3 3* Place 3 and multiply, 9 appears in level 1 Y 1/(3 × 3), last value in lev. 1; 5 in level 2; 3 in level 3 - 5 - 1/(3 × 3) , occupies level 1 now; 3 in level 2 * 3 × (5 - 1/(3× 3)), occupies level 1 now. 23` Enter 23 in level 1, 14.66666 moves to level 2. 3Q Enter 3, calculate 23 3 into level 1. 14.666 in lev. 2. / (3 × (5-1/(3× 3)))/23 3 into level 1 2.5 Enter 2.5 level 1 !¸ e 2.5 , goes into level 1, level 2 shows previous value. (3 × (5 - 1/(3 ×3)))/23 3 e 2.5 = 12.18369, into lev. 1. R √ ((3× (5 - 1/(3× 3)))/23 3 e 2.5 ) = 3.4905156, into 1. To select between the ALG vs. RPN operating mode, you can also set/clear system flag 95 through the following keystroke sequence: H @) FLAGS —„—„—„ — @@CHK@ Number Format and decimal dot or comma Changing the number format allows you to customize the way real numbers are displayed by the calculator. You will find this feature extremely useful in operations with powers of tens or to limit the number of decimals in a result. To select a number format, first open the CALCULATOR MODES input form by pressing the H button. Then, use the down arrow key, ˜ , to select the option Number format . The default value is Std , or St and ard format. In the standard format, the calculator will show floating-point numbers with the maximum precision allowed by the calcula tor (12 significant digits). To learn
Page 1-11 more about reals, see Chapter 2 in thi s Guide. To illustrate this and other number formats try the following exercises: • Standard format : This mode is the most used mode as it shows numbers in the most familiar notation. Press the !!@@OK#@ soft menu key, with the Number format set to Std , to return to the calculator display. Enter the number 123.4567890123456 (with16 significant figures). Press the ` key. The number is rounded to the maximum 12 significant figures, and is displayed as follows: • Fixed format with decimals : Press the H button. Next, use the down arrow key, ˜ , to select the option Number format . Press the @CHOOSE soft menu key ( B ), and select the option Fixed with the arrow down key ˜ . Press the right arrow key, ™ , to highlight the zero in front of the option Fix . Press the @CHOOSE soft menu key and, using the up and down arrow keys, —˜ , select, say, 3 decimals.
Page 1-12 Press the !!@@OK#@ soft menu key to complete the selection: Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as: Notice how the number is rounded, not truncated. Thus, the number 123.4567890123456, for this setting, is displayed as 123.457, and not as 123.456 because the digit after 6 is > 5): • Scientific format To set this format, start by pressing the H button. Next, use the down arrow key, ˜ , to select the option Number format . Press the @CHOOSE soft menu key ( B ), and select the option Scientific with the arrow down key ˜ . Keep the number 3 in front of the Sci . (This number can be changed in the same fashion that we changed the Fixed number of decimals in the example above). Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as:
Page 1-13 This result, 1.23E2, is the calculator’s version of powers-of-ten notation, i.e., 1.235 × 10 2 . In this, so-called, scientific notation, the number 3 in front of the Sci number format (shown earlier) represents the number of significant figures after the decimal point. Scientific notation always includes one integer figure as shown above. For this case, therefore, the number of significant figures is four. • Engineering format The engineering format is very similar to the scientific format, except that the powers of ten are multiples of three. To set this format, start by pressing the H button. Next, use the down arrow key, ˜ , to select the option Number format . Press the @CHOOSE soft menu key ( B), and select the option Engineering with the arrow down key ˜ . K e e p t h e number 3 in front of the Eng . (This number can be changed in the same fashion that we changed the Fixed number of decimals in an earlier example). Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as: Because this number has three figures in the integer part, it is shown with four significative figures and a zero power of ten, while using the Engineering format. For example, the number 0.00256, will be shown as:
Page 1-14 • Decimal comma vs. decimal point Decimal points in floating-point numbers can be replaced by commas, if the user is more familiar with such notation. To replace decimal points for commas, change the FM option in the CALCULATOR MODES input form to commas, as follows (Notice that we have changed the Number Format to Std ): • Press the H button. Next, use the down arrow key, ˜ , once, and the right arrow key, ™, twice, to highlight the option __FM, . To select commas, press the @CHECK soft menu key (i.e., the B key). The input form will look as follows: • Press the !!@@OK#@ soft menu key return to the calculator display. The number 123.456789012, entered earlier, now is shown as: Angle Measure Trigonometric functions, for example, require arguments representing plane angles. The calculator provides three different Angle Measure modes for working with angles, namely: • Degrees : There are 360 degrees (360 o ) in a complete circumference. • Radians : There are 2 π radians ( 2 π r ) in a complete circumference.
Page 1-15 • Grades : There are 400 grades (400 g ) in a complete circumference. The angle measure affects the trig functions like SIN, COS, TAN and associated functions. To change the angle measure mode, use the following procedure: • Press the H button. Next, use the down arrow key, ˜ , twice. Select the Angle Measure mode by either using the \ key (second from left in the fifth row from the keyboard bottom), or pressing the @CHOOSE soft menu key ( B ). If using the latter approach, use up and down arrow keys, — ˜ , to select the preferred mode, and press the !!@@OK#@ F soft menu key to complete the operation. For example, in the following screen, the Radians mode is selected: Coordinate System The coordinate system selection affects the way vectors and complex numbers are displayed and entered. To learn more about complex numbers and vectors, see Chapters 4 and 8, respec tively, in this Guide. There are three coordinate systems available in the calculator: Rectangular (RECT), Cylindrical (CYLIN), and Spherical (SPHERE). To change coordinate system: • Press the H button. Next, use the down arrow key, ˜ , three times. Select the Coord System mode by either using the \ key (second from left in the fifth row from the keyboard bottom), or pressing the @CHOOSE soft menu key ( B ). If using the latter approach, use up and down arrow keys, — ˜ , to select the preferred mode, and press the !!@@OK#@ F soft menu key to complete the operation. For example, in the following screen, the Polar coordinate mode is selected:
Page 1-16 Selecting CAS settings CAS stands for C omputer A lgebraic S ystem. This is the mathematical core of the calculator where the symbolic mathematical operations and functions are programmed. The CAS offers a number of settings can be adjusted according to the type of operation of interest. To see the optional CAS settings use the following: • Press the H button to activate the CALCULATOR MODES input form. • To change CAS settings press the @@ CAS@@ soft menu key. The default values of the CAS setting are shown below: • To navigate through the many options in the CAS MODES input form, use the arrow keys: š™˜— . • To select or deselect any of the settings shown above, select the underline before the option of interest, and toggle the @@CHK@@ soft menu key until the right setting is achieved. When an op tion is selected, a check mark will be shown in the underline (e.g., the Rigorous and Simp Non-Rational
Page 1-17 options above). Unselected options will show no check mark in the underline preceding the option of interest (e.g., the _Numeric, _Approx, _Complex, _Verbose, _Step/Step, _Incr Pow options above). • After having selected and unselected all the options that you want in the CAS MODES input form, press the @@@OK@@@ soft menu key. This will take you back to the CALCULATOR MODES input form. To return to normal calculator display at this point, press the @@@OK@@@ soft menu key once more. Explanation of CAS settings • Indep var : The independent variable for CAS applications. Typically, VX = ‘X’. • Modulo : For operations in modular arithmetic this variable holds the modulus or modulo of the arithmetic ring (see Chapter 5 in the calculator’s User’s Guide). • Numeric : If set, the calculator produces a numeric, or floating-point result, in calculations. • Approx : If set, Approximate mode used numerical results in calculations. If unchecked, the CAS is in Exact mode, which produces symbolic results in algebraic calculations. • Complex : If set, complex number operations are active. If unchecked the CAS is in Real mode, i.e., real number calculations are the default. See Chapter 4 for operations with complex numbers. • Verbose : If set, provides detailed information in certain CAS operations. • Step/Step : If set, provides step-by-step results for certain CAS operations. Useful to see intermediate steps in summations, derivatives, integrals, polynomial operations (e.g., synthetic division), and matrix operations. • Incr Pow : Increasing Power, means that, if set, polynomial terms are shown in increasing order of the powers of the independent variable. • Rigorous : If set, calculator does not simplify the absolute value function |X| to X. • Simp Non-Rational : If set, the calculator will try to simplify non-rational expressions as much as possible. Selecting Display modes
Page 1-18 The calculator display can be customized to your preference by selecting different display modes. To see the opti onal display settings use the following: • First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key ( D) to display the DISPLAY MODES input form. • To navigate through the many options in the DISPLAY MODES input form, use the arrow keys: š™˜— . • To select or deselect any of the settings shown above, that require a check mark, select the underline before the option of interest, and toggle the @@CHK@@ soft menu key until the right setting is achieved. When an option is selected, a check mark will be shown in the underline (e.g., the Textbook option in the Stack: line above). Unselected options will show no check mark in the underline preceding the option of interest (e.g., the _Small, _Full page, and _Indent options in the Edit: line above). • To select the Font for the display, highlight the field in front of the Font: option in the DISPLAY MODES input form, and use the @CHOOSE soft menu key ( B ). • After having selected and unselected all the options that you want in the DISPLAY MODES input form, press the @@@OK@@@ soft menu key. This will take you back to the CALCULATOR MODES input form. To return to normal calculator display at this point, press the @@@OK@@@ soft menu key once more. Selecting the display font First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key
Page 1-19 ( D ) to display the DISPLAY MODES input form. The Font: field is highlighted, and the option Ft8_0:system 8 is selected. This is the default value of the display font. Pressing the @CHOOSE soft menu key ( B), will provide a list of available system fonts, as shown below: The options available are three standard System Fonts (sizes 8, 7, and 6 ) and a Browse.. option. The latter will let you browse the calculator memory for additional fonts that you may have cre ated (see Chapter 23) or downloaded into the calculator. Practice changing the display fonts to sizes 7 and 6 . Press the OK soft menu key to effect the selection. When done with a font selection, press the @@@OK@@@ soft menu key to go back to the CALCUL ATOR MODES input form. To return to normal calculator display at this point, press the @@@OK@@@ soft menu key once more and see how the stack display chan ge to accommodate the different font. Selecting properties of the line editor First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key ( D ) to display the DISPLAY MODES input form. Press the down arrow key, ˜ , once, to get to the Edit line. This line shows three properties that can be modified. When these properties are selected (checked) the following effects are activated: _Small Changes font size to small _Full page Allows to place the cursor after the end of the line _Indent Auto indent cursor when entering a carriage return Instructions on the use of the line e ditor are presented in Chapter 2 in this guide.
Page 1-20 Selecting properties of the Stack First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key ( D ) to display the DISPLAY MODES input form. Press the down arrow key, ˜ , once, to get to the Edit line. This line shows three properties that can be modified. When these properties are selected (checked) the following effects are activated: _Small Changes font size to small. This maximizes the amount of information displayed on the screen. Note, this selection overrides the font selection for the stack display. _Textbook Displays mathematical expressions in graphical mathematical notation To illustrate these settings, either in algebraic or RPN mode, use the equation writer to type the following definite integral: ‚O…Á0™„虄¸\x™x` In Algebraic mode, the following screen shows the result of these keystrokes with neither _Small nor _Textbook are selected: With the _Small option selected only, the display looks as shown below: With the _Textbook option selected (default value), regardless of whether the _Small option is selected or not, the display shows the following result:
Page 1-21 Selecting properties of the equation writer (EQW) First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key ( D ) to display the DISPLAY MODES input form. Press the down arrow key, ˜ , three times, to get to the EQW (Equation Writer) line. This line shows two properties that can be modified. When these properties are selected (checked) the following effects are activated: _Small Changes font size to small while using the equation editor _Small Stack Disp Shows small font in the stack after using the equation editor Detailed instructions on the use of the equation editor (EQW) are presented elsewhere in this manual. For the example of the integral ∫ ∞ − 0 dX e X , presented above, selecting the _Small Stack Disp in the EQW line of the DISPLAY MODES input form produces the following display: References Additional references on the subjects covered in this Chapter can be found in Chapter 1 and Appendix C of the calculator’s User’s Guide.
Pag e 2-1 Chapt er 2 Intro duci ng the calcu lator In this chapte r we pre sent a nu mber o f bas ic ope rations of the cal cu lato r incl uding the us e o f the Equ atio n Write r and the manipu l ation o f data o bjec ts in the cal cu lato r. Stu dy the ex ample s in this chapter to get a go od gras p of the capabil ities of the cal cul ator f o r fu ture applicati ons. Calculator objec ts Some o f the mos t co mmonl y u sed o bjec ts are: real s (real numbe rs, written with a decima l point, e.g., -0.0023, 3.56), integers (integ er numb ers, writt en without a decimal point, e.g., 1232, -123212123), compl ex nu mbers (written as an or dered pair, e .g., (3 ,-2 )), list s , etc. Cal cu lato r obje cts are desc ribed in Chapters 2 and 24 in the cal cul ator ’s us er gu ide. Editin g expressions in the stac k In this sect ion we pr esent exam ples of expression ed iting d irectly int o the calc ul ator di spl ay or s tack . Creating arit hmetic expressions For this exampl e, we s el ect the Alge braic o perating mo de and s el ect a Fix fo rmat with 3 de cimals fo r the dis play. We are going to enter th e arith metic expression : 3 0 . 2 0 . 3 5 . 7 0 . 1 0 . 1 0 . 5 − ⋅ To enter t his expression use the following keystrokes: 5.*„Ü1. 1./7.5™/ „ÜR3.-2.Q3 The resulting expr ession is: 5*(1 1/7.5)/( ƒ 3-2^3). Press ` to get the expre ssio n in the display as f ol lo ws:
Pag e 2-2 Notice that, if your CAS is set to EX AC T (see App endix C in User’s G uide) and you enter your expression using int eger n umbers for integ er va lues, the result is a symbolic quantity, e.g., 5*„Ü1 1/7.5™/ „ÜR3-2Q3 Bef ore produ cing a res ul t, yo u wil l be as ke d to change to Appro ximate mode. Accept the change to get the fo ll owing res ul t (shown in Fix de cimal mode with three d ecimal places – see C hapt er 1): In th is case, when t he expression is ente red directl y into the stac k, as soo n as you press ` , the calcu l ator will attempt to calcu late a value fo r the expression. I f the expr ession is enter ed bet ween quotes, h owever, th e calcu lato r wil l re produ ce the e xpres sio n as entere d. For e xample : ³5*„Ü1 1/7.5™/ „ÜR3-2Q3` The r esult w ill b e sh own a s follows :
Pag e 2-3 To evaluate t he expr ession we can use the EV AL function , as follo ws: µ„î If the CAS is set to Exact , yo u wi ll be as ked to approve changing the CAS setting to Appro x . Once this is done, you will get the sam e result as befo re. An a lterna tive wa y to eva luate the expr ession enter ed ear lier between q uotes is by using the option …ï . We will now enter the exp ression used a bove wh en the ca lculator is set to t he RPN o perating mo de. We als o s et the CAS to Exact and the displ ay to Textb ook . The keystr okes to enter t he expr ession bet ween quotes a re th e same used earlier, i.e., …³5*„Ü1 1/7.5™/ „ÜR3-2Q3` Resu lting in the ou tput Press ` once more t o keep two copi es of the expr ession av ailab le in th e stack f or eval uation. W e fi rst evalu ate the e xpress ion u sing the fu nction EVAL , and next u sing the fu nctio n  NUM : µ .
Pag e 2-4 This expression is semi-sym bolic in th e sense tha t th ere are floati ng-p oint compone nts to the re sul t, as wel l as a √ 3. Next, we switch stack locations and evalu ate u sing f u nctio n  NUM: ™…ï . This l atter resu lt is pu rel y numerical , so that the two re sul ts in the s tack, although repr esentin g th e sam e expression, seem different . To verify th at th ey are not, we s ubtract the two val u es and e valuate this dif f erence us ing fu nctio n EVAL: -µ . The result is zero (0.). For additio nal i nfo rmation o n editi ng arithmetic expres sio ns in the displ ay o r stack, see Ch apt er 2 in th e calculator’s User’ s Guide. Creating algebraic express ions Alge braic ex press ion s incl u de not o nl y nu mbers, but al so variabl e names . As an exampl e, w e wil l enter the fo ll ow ing alge braic ex press io n: b L y R R x L 2 1 2 We s et the c alcu l ator o perating mo de to Algebraic, the CAS to Exact , and the displ ay to Textbook . To ent er this a lgebra ic expression we use the foll owing keystrokes: 2*~l*R„Ü1 ~„x/~r™/„ Ü ~r ~„y™ 2*~l/~„b Press ` to get the f oll owing resu lt:
Pag e 2-5 Entering this express ion when the cal cul ator is s et in the R PN mode is e xactly the same as this Alge braic mo de e xerc ise . For additio nal info rmation o n edit ing alge braic expre ss ions in the cal cu lato r’s displa y or stack see C hap ter 2 in t he calculator’ s User’s G uide. Using th e Equation Wr iter (E QW) to create expressions The equat ion writ er is an extremely p owerful tool that not on ly let you ent er or see an equati on, bu t als o al lo ws you to mo dif y and wo rk/appl y f unctio ns o n all or part of the equ ation. The Eq uation Wr iter is launch ed by p ressing the keystr oke combin ation ‚O (the third key in the fou rth row f r om the top in the keyboard). T he resulting scr een is th e foll owing. P ress L to see the second menu pa ge: The six so ft menu keys fo r the Equation Writer activate fu nctions EDIT, CURS, BIG, EV AL, FACTOR, S IMPLIFY, CMDS, and HELP. D etailed information on these function s is prov ided in Cha pter 3 of the calculator’ s User’s Guid e. Creating arit hmetic expressions Entering a rithmetic expressions in the Equation Writer is very similar to enterin g an arith metic exp ression in t he sta ck enclosed in q uotes. The ma in difference is th at in the E quation Writer t he expr essions prod uced ar e written in “text book” style inst ead of a lin e-entry st yle. For examp le, try the following keystrokes in t he Eq uation Wr iter screen : 5/5 2 The result is the exp ression
Pag e 2-6 The cu rso r is s hown as a le ft- faci ng key . The c urso r indicate s the c urrent editio n lo cation. For ex ample , fo r the cu rso r in the l ocati on indicate d above , type now: *„Ü5 1/3 The edit ed expression looks as follows: Suppos e that yo u want to repl ace the quantity be twee n parenthes es in the denominator (i.e., 5 1/3) with (5 π 2 /2). First, we us e the del ete key ( ƒ ) dele te the c urrent 1 /3 ex press ion, and the n we repl ace that f ractio n with π 2 /2, as foll ows: ƒƒƒ„ìQ2 When hit this point the scree n loo ks as fol lo ws: In order to insert the denominato r 2 in the exp ression, we need to hig hlight the ent ire π 2 expres sio n. We do this by press ing the right arro w k ey ( ™ ) once. At that point, we e nter the fo ll owing keystro kes: /2
Pag e 2-7 The expr ession now looks as follows: Suppos e that now you want to add the frac tion 1/ 3 to this entire e xpres sio n, i.e., you want to ent er the exp ression: 3 1 ) 2 5 ( 2 5 5 2 ⋅ π First, we need to highlight the entire fi rst term by u sing either the right arrow ( ™ ) or the u pper arrow ( — ) keys, repea tedly, until th e entir e expression is highli ghted, i.e ., s even t imes , pro duc ing: NOTE : Alte rna ti vely , from th e orig in al posi tion of th e cursor (to t he r ig ht of t he 2 in the deno minator o f π 2 /2), we can use the keystroke comb inat ion ‚— , interpreted as ( ‚ ‘ ). Once the e xpres sio n is highli ghted as sho wn above, type 1/3 to add the f ractio n 1/3. Res u lting i n:
Pag e 2-8 Creating algebraic express ions An alge braic expre ss ion i s very simil ar to an ari thmetic e xpres sio n, e xcept that Englis h and Greek le tters may be incl u ded. T he proc ess of creati ng an algebr aic expr ession, ther efore, foll ows the sam e idea a s that of creatin g an arithmetic expres sio n, exc ept that u se o f the alphabe tic ke yboard is inclu ded. To il lu strate the u se o f the Equatio n Writer to e nter an algebraic equ ation we will use the fol lowing examp le. Suppose th at we wa nt to ent er the expr ession:       ∆ ⋅ ⋅ − 3 / 1 2 3 2 θ µ λ µ y x LN e Use the following keystrokes: 2 / R3 ™™ * ~‚n „¸\ ~‚m ™™ * ‚¹ ~„x 2 * ~‚m * ~‚c ~„y ——— / ~‚t Q1/3 This results in the outp ut: In this exa mp le we used several lo wer-case Eng lish letters, e.g., x ( ~„x ), several Greek letters, e.g., λ ( ~‚n ), and even a combin ation o f Gre ek and Engl is h le tters, namel y, ∆ y ( ~‚c ~„y ). Keep in mind that to enter a lower -ca se E n gli sh lett er, you n eed to us e the co mbination: ~„ fol lo wed by the l etter you want to e nter.
Pag e 2-9 Als o, y ou can al ways c opy s peci al charac ters by us ing the CHAR S menu ( …± ) if yo u don’t want to memo rize the k eystro ke co mbination that produces it. A listing of commonly used ~‚ keystro ke combinations was listed i n an earlier sect ion. For additio nal info rmatio n on e diting, e valu ating, f acto ring, and s implif yi ng alge braic expre ss ions see Chapter 2 o f the cal cul ator ’s Us er’s M anu al. Organ izing data in th e calc ulator You can o rganize data in yo ur calc ul ator by sto ring variabl es in a direc tory tree. The b asis of the ca lculator’s dir ectory tr ee is the HOME dir ectory descri bed nex t. The HOME directory To get to the HOME directo ry, press the UPDIR fu nction ( „§ ) -- re peat as needed -- until th e {HOME} spec is sh own in t he second line of the di splay header. Al ternati vely, us e „ (hol d) § . For this exa mple, the HO ME directory contains no thing but the CASDIR. Press ing J will sh ow t he varia bles in the soft menu keys: Subdi rectori es To store your da ta in a well organ ized d irectory t ree you may wa nt to crea te su bdirecto ries under the HOME dire ctory , and more su bdirecto ries within su bdirecto ries , in a hie rarchy of dire ctori es s imilar to fo l ders in mo dern computers. Th e subdir ectories wi ll be giv en na mes th at m ay reflect th e conte nts of each s ubdire ctor y, or any arbitrary name that y ou can think o ff . For detai ls o n manipu latio n of directo ries see Chapter 2 in the calc ul ator’s User’s Guide.
Pag e 2-10 Variables Variable s are s imil ar to f ile s o n a compu ter hard dri ve. One variabl e c an store one ob ject (numer ical v alues, alg ebra ic expr essions, list s, vect ors, matrices , programs, etc). Vari ables are ref erre d to by the ir names , which can be any co mbinatio n of alphabe tic and nu merical characters , starti ng with a lette r (eithe r Englis h or Gre ek). Some no n-al phabetic characte rs, s uch as the arrow ( → ) can be u sed in a vari able name, if co mbined wi th an alphabe tical character. T hus, ‘ → A’ is a vali d variable name, bu t ‘ → ’ is not. Valid exampl es o f variable name s are: ‘A’, ‘B’, ‘a’, ‘b’, ‘ α ’, ‘ β ’, ‘A1’, ‘AB12’, ‘  A12’,’Vel’,’Z0’,’z1’, etc. A variable can no t have the same name as a f u nction o f the calcu lato r. T he reserved ca lculator variable na mes are the following: ALRMDAT, CS T, EQ , EXPR, IERR, IOPA R, MAXR, MINR, PICT, PPAR, PRTPAR, VPAR, ZPAR, der_, e, i, n1,n2, …, s1, s2, …, Σ DAT , Σ PAR, π , ∞ Variable s can be o rganize d into s ub- directo rie s (se e Chapter 2 in the calcu lato r’s Use r’s Guide). Typi ng variabl e names To name va riables, you will have to type string s of le tters at once, which m ay or may not be combined with numbers. To type string s of characters you can lo ck the alphabeti c ke yboard as fo ll ow s: ~~ lo cks the alphabe tic ke yboard i n upper c ase. W hen l ock ed in this fashion, p ressin g th e „ befo re a l etter k ey pro duces a lo wer cas e le tter, while pressing the ‚ key be fo re a l etter k ey pro duces a spec ial characte r. If the al phabetic keybo ard is al ready l oc ked in u pper cas e, to lo ck it in l ow er case, type, „~ ~~„~ lock s the al phabetic keyboard in lo wer case. When lock ed in this fashio n, pressing the „ befo re a l etter k ey pro duce s an u pper case lett er. To unloc k lower ca se, p re ss „~
Pag e 2-11 To unl ock the u pper-case locked keybo ard, press ~ Try the following exercises: ³~~math1~` ³~~m„a„t„h~` ³~~m„~at„h~` The c alc ulat or d isp lay will s how t he followin g (left -ha nd si de i s A lgeb ra ic mode, right-hand s ide is RPN mode ): Creating variables The simp lest way to create a var iable is by using the K . The follo wing examp les are used to stor e the v aria bles listed in the following tab le (Press J if needed to see v aria bles menu): Name Con tents Type α -0.25 real A12 3 × 10 5 real Q ‘r/(m r)' alge braic R [3,2,1] vector z1 3 5i co mplex p1 << → r 'π *r^2' >> program • Al gebr aic mode To store th e va lue of –0.25 into varia ble α : 0.25\ K ~‚a . AT this point, the screen will l ook as f oll ows:
Pag e 2-12 Press ` to create th e va riable. The v ar iable is now shown in the soft menu key labels: The follo wing a re the keyst rokes required t o enter th e remain ing variable s: A12: 3V5K~a12` Q: ³~„r/„Ü ~„m ~„r™™ K~q` R: „Ô3‚í2‚í1™ K~r` z1: 3 5*„¥ K~„z1` (Accept change to Compl ex mode if aske d). p1: ‚å‚é~„r³„ì* ~„rQ2™™™ K~„p1` .. The screen, at th is point, will loo k as fol lows: You will see six of the sev en v aria bles listed a t th e bottom of the screen: p1, z1, R, Q , A12, α .
Pag e 2-13 • RPN m od e (Use I\ @@OK@@ to change to R PN mode). U se the f ol lo wing keystrokes to stor e the v alue of –0.25 into va riab le α : 0.25\` ~‚a` . At this p oint, the screen will look a s follows: This e xpres sio n means that the valu e –0 .25 i s ready to be s tore d into α . Press K to create th e var iable. The v ari able is now shown in the soft menu key labels: To enter t he v alue 3 × 10 5 into A12, we can use a shorter version of the pr ocedure: 3V5³~a12` K Here is a way to enter the co ntents of Q: Q: ³~„r/„Ü ~„m ~„r™™ ³~q` K To enter t he v alue of R, we can use an ev en sh orter v ersion of th e proced ure: R: „Ô3#2#1™ ³K Notice that to s eparate the e le ments o f a ve ctor i n RPN mode we can use the spa ce key ( # ), rather than the co mma ( ‚í ) use d above in Al gebraic mode . z1: ³3 5*„¥ ³~„z1 K
Pag e 2-14 p1: ‚å‚é~„r³„ì* ~„rQ2™™™ ³ ~„p1™` K . The screen, at th is point, will loo k as fol lows: You will see six of the sev en v aria bles listed a t th e bottom of the screen: p1, z1, R, Q , A12, α . Checking variables content s The simp lest way to check a v ar iable cont ent is by pressing the soft m enu key labe l f or the variable . For e xampl e, f or the variable s l iste d above, press the foll owing keys to see th e conten ts of the v aria bles: Al gebr aic mode Type th ese keystrokes: J @@z1@@ ` @@@R@@ ` @@@Q@@@ ` . At this point, th e screen looks as follows: RPN m od e In RPN mode , yo u o nly need to press the co rrespo nding so ft me nu key label to get the co ntents of a nume rical or al gebraic vari able. For the c ase u nder conside ration, we can try peeking in to the v aria bles z1 , R , Q , A12 , α , and A , create d above , as f o ll ow s: J @@z1@@ @@@R@@ @@@Q@@ @@A12@@ @@ª@@ At this point, the scre en lo oks like this :
Pag e 2-15 Using the right-s hift k ey ‚ follo wed by soft menu ke y labels This approach f or vi ewing the conte nts o f a variabl e w ork s the s ame in bo th Alge braic and RPN mo des . Tr y the f ol l owi ng exampl es in eithe r mode : J‚ @@p1@@ ‚ @@z1@@ ‚ @@@R@@ ‚ @@@Q@@ ‚ @@A12@@ This prod uces the follo wing scr een (Alg ebraic m ode in th e left, RPN in the right) Notice that this time the conte nts of program p1 are listed in the screen . To see the rema ining va riables in t his direct ory, use: ‚@ @@ª@@ L ‚ @@@A@@ Listi ng the c ont ent s of all variab les in t he sc reen Use the keystr oke combin ation ‚˜ to li st the conte nts of all variables in the screen . For examp le: Press $ to retu rn to no rmal cal cul ator displ ay.
Pag e 2-16 Deleti ng variabl es The simplest way of deleting v ariables is by using function PURGE. This fu nctio n can be acc ess ed dire ctly by u sing the TOOLS menu ( I ), or by usin g th e FI LE S m enu „¡ @@OK@@ . Usi ng funct ion PURG E in t he st ack in Algebrai c mode Our variabl e l ist contains variabl es p1, z1, Q, R, and α . We will u se command PU RGE to de le te variabl e p1 . Press I @PURGE@ J @@p1@@ ` . The screen will now show v ari able p1 removed: You can u se the PURGE co mmand to e rase more than one variable by pl acing their nam es in a list in the arg ument of PURGE. For examp le, if now we wanted to purge v ar iables R and Q , si m ulta neous ly, w e ca n t ry th e followi ng exercise. Press : I @PURGE@ „ä³ J @@@R!@@ ™ ‚í ³ J @@@Q!@@ At this point, the s creen wil l sho w the fo ll owing command re ady to be executed: To f inis h del eting the variable s, pre ss ` . The screen will now show th e remaining v ariables: Us ing func ti on PUR GE i n t he s tac k in RPN mode Assu ming that o ur variabl e l is t contains the variabl es p1, z1, Q, R, and α . We wi ll use c omm a nd PU RG E t o d elet e v ar ia b le p1 . Press ³ @@p1@@ ` I @PURGE@ . The screen will now show v aria ble p1 remo ved:
Pag e 2-17 To dele te two variabl es s imu ltane ou sl y, s ay variabl es R and Q , f irst create a list (in RP N mode, t he element s of the list need not b e separa ted b y comm as as in Al gebraic mode ): J „ä³ @@@R!@@ ™ ³ @@@Q!@@ ` Then, p ress I @PURGE@ use to purge the v ari ables. Additional inf ormati on o n variable manipu latio n is avai labl e in Chapte r 2 o f the calculator’ s User’s G uide. UND O an d CMD func tions Functio ns UNDO and CMD are us efu l f or re covering recent co mmands, o r to revert an o peratio n if a mistak e was made. T hese fu nctio ns are asso ciate d with the HIST key: UNDO resu lts f rom the ke ystroke sequ ence ‚¯ , while CMD r esults from th e keystroke seq uence „® . CHOOS E bo xes vs. S of t MENU In some of th e exercises presen ted in this cha pter we hav e seen men u lists of comma nds d isplayed in th e screen. Th is men u lists are referred to as CHOOSE boxes . Herein we indicate th e way to chang e from CHOSE b oxes to Sof t M ENUs, and vi ce vers a, thro ugh an ex ercis e. Altho ugh no t appli ed to a s pecif ic e xampl e, the pre sent e xer cise show s the two option s f or menus in the ca lculator (CHO OS E boxes an d soft ME NUs) . In this exercise, we use the O RDE R comma nd t o reorder v ari ables in a directory, we use: „°˜ Show PROG menu list and select MEMORY
Pag e 2-18 @@OK@@ ˜˜˜˜ Show the ME MORY menu list and select DIRE CTORY @@OK@@ —— Show the DIR ECTORY me nu l is t and sel ect ORDER @@OK@@ activate the ORDER command There is an alterna tiv e way to access th ese menus as soft ME NU keys, b y settin g system flag 117 . (For inf ormati on o n Flags s ee Chapte rs 2 and 24 in the calcu lato r’s Use r’s Guide). T o se t this fl ag try the fo llo wing: H @FLAGS! ——————— The screen shows flag 117 n ot set ( CHOOSE boxes ), as s hown he re:
Pag e 2-19 Press th e @CHECK! soft menu key to set f lag 117 to soft MENU . Th e screen will refl ect that change: Press twice to retu rn to no rmal calc ulato r display. Now, we’l l try to find the OR DER command using similar k eystrok es to those used above, i.e., we start with „° . Notice that instead of a menu list, we get soft menu labels with the different options in the PROG menu, i.e., Press B to select the MEMORY soft menu ( ) @@MEM@@ ). The d isplay now sh ows: Press E to select the DIRECTORY soft menu ( ) @@DIR@@ ) The ORDE R comma nd is not shown in this screen. To find it we use the L key to find it:
Pag e 2-20 To activate the OR DER command we pres s the C ( @ORDER ) soft menu key. Ref erences For additio nal i nfo rmation o n enteri ng and manipul ating ex press ions in the displa y or in th e Eq uation Wr iter see Ch apt er 2 of the calculator ’s User’s Guide . For CAS (Co mpute r Algebrai c Syste m) settings , se e Appendi x C in the calculator ’s User’s G uide. For informa tion on Flags see, C hap ter 24 in the calcu lato r’s Us er’s Guide .
Pag e 3-1 Chapt er 3 Calculat ions w ith real numbers This c hapter demo nstrate s the u se o f the cal cul ator f or o peratio ns and fu nctio ns rel ated to re al nu mbers . The us er sho ul d be acqu ainted w ith the keyboard to identify certain functions ava ilable in the keyboard (e.g ., SIN, COS, TA N, etc.). Also, it is assumed that th e reader knows how to chang e the cal cul ato r’s o perating s yste m (Chapter 1), us e menu s and ch oo se bo xes (Chapter 1), and operate with variabl es (Chapter 2). Examples of re al number calc ulations To pe rfo rm real numbe r cal cul ations it is pref erred to have the CAS set to Rea l (as oppo se d to Co mple x ) mode . Exact mo de is the de fau lt mo de f or mo st operations. Therefore, you may wan t to start your calcul ations in this mode. Some o peratio ns with re al nu mbers are il lu strate d next: • Use the \ key for changing sign of a number. For e xample , in AL G mode , \2.5` . In RPN mode, e.g., 2.5\ . • Use the Y key to calculate the in verse of a numb er. For e xample , in AL G mode , Y2` . In RPN mode u se 4`Y . • For additi on, s ubtract ion, mu l tiplic ation, di visio n, u se the proper operatio n ke y, namel y, - * / . Exampl es in ALG mo de : 3.7 5.2 ` 6.3 - 8.5 ` 4.2 * 2.5 ` 2.3 / 4.5 ` Example s i n RPN mo de: 3.7` 5.2
Pag e 3-2 6.3` 8.5 - 4.2` 2.5 * 2.3` 4.5 / Alternati vely , in RPN mo de, y ou can se parate the o perands w ith a space ( # ) bef ore pre ss ing the o perato r key . Exampl es: 3.7#5.2 6.3#8.5 - 4.2#2.5 * 2.3#4.5 / • Parenthes es ( „Ü ) c an be us ed to gro up o peratio ns, as wel l as to enclose a rg umen ts of func tion s. In ALG m ode: „Ü5 3.2™/„Ü7- 2.2` In RPN mode , yo u do not ne ed the parenthe sis , calc ul ation i s done directl y o n the stack : 5`3.2` 7`2.2`-/ In RPN mode , typing the expre ssio n betwe en qu ote s wil l al l ow yo u to enter the expre ssio n l ike i n alge braic mode : ³„Ü5 3.2/ „Ü72.2`µ For bo th, ALG and R PN mode s, u sin g the Equ ation W riter: ‚O5 3.2™/7-2.2 The expression can b e evaluated within t he Eq uation writ er, by using ———— @EVAL@ or, ‚— @EVAL@ • The abso lu te val u e fu ncti on, AB S, is avail abl e thro ugh „Ê . Exampl e in AL G mo de:
Pag e 3-3 „Ê \2.32` Example in R PN mode : 2.32\„Ê • The s quare fu nctio n, SQ, is avail abl e throu gh „º . Exampl e in AL G mo de: „º\2.3` Example in R PN mode : 2.3\„º The sq uare root fun ct ion, √ , is avail abl e throu gh the R key . Whe n calcu lating in the s tack in ALG mode, e nter the f unctio n befo re the argument , e.g., R123.4` In RPN mode, e nter the nu mber firs t, then the f unction, e .g., 123.4R • The po we r fu nctio n, ^, is availabl e thro ugh the Q key. When calc ul ating in the stack in ALG mo de, enter the base ( y ) f ol lo wed by the Q key, and then th e exponent ( x ), e.g., 5.2Q1.25 In RPN mode, e nter the nu mber firs t, then the f unction, e .g., 5.2`1.25Q • The root funct ion, XROO T (y,x), is avail able throu gh the ke ystro ke combination ‚» . When calculating in the stack in A LG mode,
Pag e 3-4 enter the f unctio n XROOT fo ll owed by the arguments ( y,x ), s eparated by commas, e.g ., ‚»3‚í 27` In RPN mode , enter the argu ment y , first, then, x , and f inal ly the function call , e.g., 27`3‚» • Lo garithms o f bas e 10 are cal cul ated by the key stro ke co mbinatio n ‚à (functio n LOG) while its inverse fu nction (ALOG, or antilo garithm) is calcu l ated by u sing „ . In ALG m ode, the fu nction is e ntered bef ore the argument: ‚Ã2.45` „Â\2.3` In RPN mode , the argu ment is entere d befo re the fu nctio n 2.45 ‚à 2.3\ „ Usi ng powers of 10 i n ente ring data Powers of ten, i.e., numbers of the form -4.5 × 10 -2 , etc., are ent ered by using the V key. For example, in ALG mode: \4.5V\2` Or, in RPN mode: 4.5\V2\` • Nat ura l log ar it hm s a re c a lcula ted b y usin g ‚¹ (fu nction LN) while the exponen tial function (E XP) is calculated b y using „¸ . In ALG mode, the f unctio n is entere d bef ore the argu ment: ‚¹2.45` „¸\2.3` In RPN mode , the argu ment is entere d befo re the fu nctio n
Pag e 3-5 2.45` ‚¹ 2.3\` „¸ • Three trigono metric f unc tions are readil y avail able in the k eybo ard: sine ( S ), cosine ( T ), and tangent ( U ). Ar gumen ts of these fu nctio ns are angl es in ei ther degre es, radians, grade s. T he foll owing exam ples use angles in d egrees (D EG ): In ALG m ode: S30` T45` U135` In RPN mode: 30S 45T 135U • The inv erse trig onomet ric functi ons av ailab le in th e keyboar d ar e the arcsine ( „¼ ), arccosine ( „¾ ), and arctangent ( „À ). The an swer from th ese functions will be g iven in the selected ang ular measure (DE G, RA D, G RD). S ome exam ples are shown n ext: In ALG m ode: „¼0.25` „¾0.85` „À1.35` In RPN mode: 0.25„¼ 0.85„¾ 1.35„À All the f unc tion s des cribe d above, namel y, AB S, ABS, SQ, √ , ^, XROOT, LOG, A LOG, LN, E XP, SIN, COS , TAN, AS IN, ACO S, ATA N, can be combin ed with t he fundam enta l operat ions ( -*/ ) to f orm more complex expr essions. The E quati on Writer , whose opera tions is d escribed in Chapter 2, i s ideal f or bu ildi ng suc h expre ssio ns, re gardle ss o f the cal cul ator oper ation mo de.
Pag e 3-6 Real num ber f uncti ons i n the MTH me nu The MTH ( „´ ) menu inclu de a nu mber o f mathe matical fu nctio ns mo stl y applicabl e to real numbe rs. W ith the def au lt se tting of CHOOSE boxe s for system flag 117 (see Cha pter 2), the MTH men u shows the fol lowing func tion s: The f u nctions are gro upe d by the type of argu ment (1. ve cto rs, 2. matrices, 3. lis ts, 7 . probabil ity, 9. co mple x) or by the type of fu nctio n (4. hype rbol ic, 5. real, 6. base , 8. f f t). It als o co ntains an e ntry fo r the mathe matical co nstants availabl e in the cal cul ator , entry 1 0. In general , be aw are o f the nu mber and order o f the argu ments re quire d fo r each function, and keep in mind that, in A LG mode you should select f irst the function an d th en enter the ar gumen t, while in RPN m ode, you should enter the argument in the stack f irst, and then s ele ct the fu nction. Using c alculat or menus : 1. We wi ll desc ribe in de tail the us e of the 4. HYPERBOLIC.. menu in this sectio n with the inte ntion of des cribing the gene ral operation of calculato r menus . Pay clo se attentio n to the proces s fo r sel ecting diff erent o ptions. 2. To quickly select one of the num bered option s in a m enu list (or CH OOS E box), simp ly press th e number for the opt ion in t he keyboa rd. For exampl e, to sel ect o ption 4. HYPERBOLIC.. in the MTH menu, simply press 4 . Hyperboli c funct ions and t heir i nverses Sel ec ti ng Opti on 4. HYPERBOLIC.. , in the MTH menu, and press ing @@OK@@ , pr oduces t he h yp erb olic fun ct ion m en u:
Pag e 3-7 For examp le, in ALG mode, th e keystroke sequence to ca lculate, say, tanh(2.5), is the follo wing: „´4 @@OK@@ 5 @@OK@@ 2.5` In the RPN mo de, the ke ystrok es to perfo rm this cal cul ation are the fo ll owing: 2.5`„´4 @@OK@@ 5 @@OK@@ The o peratio ns s how n above as su me that yo u are us ing the de fau lt s etting f or system flag 117 ( CHOOSE boxes ). If you have changed the se tting of this fl ag (see Cha pter 2) to SOFT me nu , th e MTH m enu w ill sh ow a s follows ( left-h an d side in ALG mo de, right – hand side in RPN mo de): Pressin g L shows t he rem ai nin g op tion s: Thus, to sel ect, for example, the hyperbolic fu nctions menu, with this menu fo rmat press ) @@HYP@ , to produce:
Pag e 3-8 Finally, in order to sel ect, for example, the hyperbo lic tangent (tanh) f unctio n, simp ly press @@TANH@ . Note: To see additional opti ons i n these sof t menu s, press the L key or the „« keystroke sequence. For ex ample, to cal cul ate tanh(2.5), in the ALG mode, when using SOFT menu s over CHOOSE boxes , f ollo w this pr ocedure: „´ @@HYP@ @@TANH@ 2.5` In RPN m ode, the sam e va lue is calcul ated using : 2.5`„´ ) @@HYP@ @@TANH@ As an ex ercis e o f appl icatio ns o f hyperbo lic fu nctio ns, ve rif y the f ol l owing valu e s: sinh (2.5) = 6.05020.. sinh -1 (2.0) = 1.4436… cosh (2.5) = 6.13228.. a cosh -1 (2.0) = 1.3169… tanh (2.5) = 0.98661.. tanh -1 (0.2) = 0.2027… expm(2.0) = 6.38905…. l np1(1.0) = 0.69314…. Oper ations w ith un its Numbers in the calculator can ha ve units associated with them. Th us, it is poss ibl e to calcu l ate res ul ts i nvol ving a cons iste nt sys tem o f u nits and produce a resu lt w ith the appro priate co mbination o f u nits . The UNIT S menu The units men u is launched by the keystroke com bina tion ‚Û (assoc iated with the 6 key). With system flag 117 set to CHOOSE boxe s , the result is the f ol lo wing me nu:
Pag e 3-9 Option 1. Tools. . conta in s function s used to oper at e on unit s ( disc ussed later ). Options 3. Length.. through 17.Viscosity .. contain menu s w ith a numbe r of units fo r each o f the quantitie s des cribed. For ex ample , sel ect ing optio n 8. Force.. shows the foll owing unit s menu: The user will recognize m ost of these units (som e, e.g., dy ne, are n ot used very o fte n nowaday s) f rom his or he r physi cs cl ass es: N = newtons, dyn = dynes , gf = grams – f orce (to di stingu ish f ro m gram-mas s, o r plai nly gram, a unit of mass), kip = kilo-poundal (1000 pounds), lbf = pound- fo rce (to disti ngui sh f rom po u nd-mass ), pdl = po u ndal. To attach a uni t obje ct to a nu mber , the nu mber mu st be fo ll o wed by an unders core . Thu s, a f orce of 5 N wil l be entered as 5_N. For extensiv e opera tions with units SO FT menus pr ovid e a more con venien t way of attaching units. Chang e system flag 117 to SOFT m enus (see Chapter 1), and us e the k eystro ke combi nation ‚Û to get the f ol lowing menus. Press L to move to the next me nu page.
Pag e 3-10 Press ing on the appropriate sof t menu ke y wil l o pen the sub- menu of uni ts f or that particul ar sel ecti on. Fo r exampl e, f or the @ ) SPEED su b-menu , the fo ll o wing unit s are avai labl e: Pressin g th e soft menu key @ ) UNITS will take you back to the UNITS menu . Reca ll t ha t y ou ca n a lwa ys li st th e full m enu la b els in t he s cre en by usin g ‚˜ , e.g., for the @ ) ENRG set of un it s t he follow in g la b els wi ll be list ed : Note: Use the L key or the „« keystro ke s eque nce to navigate throu gh the menu s. Availabl e uni ts For a co mple te l ist o f avail abl e uni ts se e Chapter 3 in the cal cu lato r’s U ser’s Gui de .
Pag e 3-11 Attach ing un its to n umb er s To attach a uni t obje ct to a nu mber , the nu mber mu st be fo ll o wed by an unders core ( ‚Ý , key (8,5)). T hus , a fo rce o f 5 N w ill be ente red as 5_N. Here is the seq uence of steps to enter t his numb er in A LG mod e, system flag 117 set to CHOOSE box es: 5‚Ý ‚Û 8 @@OK@@ @@OK@@ ` Note : If you f orget th e underscore, th e result is the expr ession 5*N, wher e N here re prese nts a pos sibl e variabl e name and not New tons. To e nter this same quantity, with the cal cul ator in R PN mode, use the foll owing keystrokes: 5‚Û8 @@OK@@ @@OK@@ Notice that the undersco re is e ntered au tomatical ly when the RPN mode is active. The keystroke seq uences to enter units when the SOFT men u opt ion i s sel ect ed, in bo th ALG and R PN mode s, are i ll us trated ne xt. Fo r exampl e, i n ALG mode, to e nter the qu antity 5_N use : 5‚Ý ‚ÛL @ ) @FORCE @ @@N@@ ` The same qu antity, entere d in RPN mode use s the f oll owing ke ystrok es: 5‚ÛL @ ) @FORCE @ @@N@@ Note: You can enter a qu antity w ith units by typing the unde rline and units with the ~ keyboard, e.g ., 5‚Ý~n will produce the en try: 5_N Unit pre fixes You can enter p refixes for units accordin g to th e foll owing ta ble of prefixes fro m the SI sys tem. T he pref ix abbreviatio n is s hown f irst, fo ll owe d by its name, and by the expo nent x i n the f actor 1 0 x corresponding to each prefix:
Pag e 3-12 ____________________________________________________ Prefix Name x Prefix Name x ____________________________________________________ Y yotta 24 d deci -1 Z zetta 21 c centi -2 E exa 18 m m illi -3 P peta 15 µ micro -6 T tera 12 n nano -9 G gig a 9 p pico -12 M mega 6 f femto -15 k,K kilo 3 a atto -18 h,H hecto 2 z zepto -21 D(*) deka 1 y yocto -24 _____________________________________________________ (*) In t he SI system, th is prefix is da rather than D . Use D for deka in the calcul ator, however. To enter th ese prefixes, simp ly type the p refix using the ~ keyboard. Fo r example, to enter 123 pm (p icometer), use: 123‚Ý~„p~„m Using UBASE to convert to the defau lt u nit (1 m) resul ts in: Ope ra tio ns with units Here are so me c alcu l ation e xampl es us ing the AL G ope rating mo de. B e warned that, w hen mu lti plyi ng or di viding qu antities with u nits, yo u mu st enclosed each quantit y with its unit s between p arent heses. Thus, to enter, for examp le, the pr oduct 12m × 1.5 yd, type it to read (12_m)*(1.5_yd ) ` :
Pag e 3-13 which sh ows as 65_(m ⋅ yd). To convert to units of the SI system , use fu nction UBASE (find it using the command catalog, ‚N ): Note: Rec all that the ANS(1) variabl e is avail able thro u gh the key stro ke combination „î (asso ciated with the ` key). To calculate a division, sa y, 3250 mi / 50 h, en ter it as (3250_mi)/(50_h) ` which transfo rmed to SI u nits, with fu nction UB ASE , produ ces: Addition and su btractio n can be pe rfo rmed, i n ALG mo de, w ithou t u sing paren theses, e.g ., 5 m 3200 mm , can be en tered sim ply as 5_m 3200_mm ` . More comp licated exp ression req uire the use of par entheses, e.g ., (12_mm)*(1_cm^2)/(2_s) ` : Stack cal cul ations in the R PN mode, do no t requi re yo u to encl os e the different term s in par entheses, e.g ., 12_m ` 1.5_yd ` * 3250_mi ` 50_h ` /
Pag e 3-14 These oper ation s prod uce the following output : Unit conversions The U NITS menu contains a TOOLS su b-me nu, w hich provides the fo ll ow ing func tion s: CONVERT(x,y): convert unit obje ct x to un its of ob jec t y UBASE (x): convert unit object x to SI units UVAL(x): extract the value f rom unit o bject x UFACT(x,y): factors a unit x from u nit obje ct x  UNIT(x,y): combines v alue of x with units of y Example s o f f u nctio n CONVERT are sho wn bel o w. Exampl es of the o ther UNIT/ TOOLS f uncti ons are avail able i n Chapter 3 o f the calcu l ator’s Use r’s Man ual. For examp le, to conver t 33 watts t o btu’s use either of the following entr ies: CONVE RT(33_W,1_hp) ` CONVE RT(33_W,11_hp) ` Physi cal const ants in t he calcula to r The cal cu lato r’s phy sical cons tants are containe d in a cons tants l ibrary activated with the command CONLIB . To launch this co mmand you cou ld s im p l y t yp e i t in t h e s ta c k : ~~conlib~` , or, you can select the command CONLIB from the command catalog, as fol lows: First, l aunch the catalo g by using: ‚N~c . Next, use the up and down arrow keys —˜ to select CONLIB. Finally, press the F ( @@OK@@ ) soft menu key. Press ` , if needed. Use t he up and down a rrow keys ( —˜ ) to navigate through the l ist o f co nstants in yo ur cal cul ator.
Pag e 3-15 The soft menu keys corresponding to this CONSTA NTS LIBRA RY screen include the following functions: SI when selected, constant s va lues are shown in S I units (*) ENG L when selected, constants va lues are shown in Eng lish units (*) UNIT when selected , constan ts ar e shown wit h units a tta ched VALUE when selected, const ant s are sh own with out units  STK co pies valu e (with or witho ut u nits) to the stack QUIT exit co nstants l ibrary (*) Activated only if the VALUE option is s ele cted. This is the wa y the t op of the CO NSTA NTS LIBRARY scr een looks when th e option VALUE is se lecte d (units in the SI sys tem): To see the v alues of the consta nts in the E ng lish (or Imp eria l) system, p ress the @ENGL opt ion: If we de-select th e UNITS opti on (p ress @UNITS ) only th e va lues are shown (En glish uni ts selected in this c ase):
Pag e 3-16 To copy the valu e o f Vm to the s tack, sel ec t the variabl e name , and pres s ! ²STK , then, pr ess @QUIT@ . Fo r the calcu lato r set to the ALG, the scre en will lo ok like this: The di spl ay sho ws w hat is c all ed a tagge d val ue , Vm:359.0394 . In here, Vm, is the tag of this resul t. Any arithmetic operation with this number will ignore th e tag. Try, for example: ‚¹2*„î ` which prod uces: The sam e opera tion in RP N mode will require t he follo wing keyst rokes (after the value of Vm was extracte d fro m the constants library): 2`*‚¹ Defi ning and usi ng f uncti ons Use rs can de fine their o wn f unc tions by us ing the DEFINE co mmand availabl e thou ght the keystro ke s eque nce „à (asso ciated with the 2 key). The function must be entered in the foll owing format: Functi on_name (argume nts) = expre ssio n_co ntaining_ arguments For ex ample , we c ou ld de fine a simpl e f uncti on H(x) = ln(x 1) exp(-x) Suppos e that yo u have a nee d to eval uate this f uncti on f or a nu mber o f discre te valu es and, there fo re, y ou want to be able to pre ss a s ingle butto n
Pag e 3-17 and get the resu lt yo u want witho ut having to type the express ion in the right- hand side for each sep arat e value. In th e foll owing exam ple, we assume you hav e set your calcu lator to ALG mode. Enter th e fol lowing sequence of keystrokes: „à³~h„Ü~„x™‚Å ‚¹~„x 1™ „¸~„x` The sc reen will look li ke th is: Press th e J k ey, and y ou wil l no tice that the re is a new variabl e in y ou r so ft menu key ( @@@H@@ ). To see th e conten ts of this v ari able p ress ‚ @@@H@@ . The screen will show now: Thu s, the variable H co ntains a pro gram def ined by: <<  x ‘LN(x 1) EXP(x)’ >> This is a s imple program in the def aul t programming l angu age of the H P 48 G series, an d also incorporated in the HP 49 G series. This p rogram ming langu age is call ed U serR PL (See Chapters 2 0 and 21 in the cal cu lato r’s U ser’s Guide). The pr ogram sh own above is relatively simp le and consists of two parts, co ntained be tween the program co ntainers << >> : • Input:  x  x • Process: ‘LN(x 1) EXP(x) ‘ This i s to be interpre ted as s aying: ente r a valu e that is tempo rarily assigne d to the name x (ref erred to as a l ocal variable ), eval uate the expre ssi on
Pag e 3-18 betwee n quo tes that contai n that lo cal variabl e, and show the eval uate d expression . To activate the f unctio n in ALG mode, type the name of the f unctio n fol lo wed by the a rgumen t between par entheses, e.g ., @@@H@@@ „Ü2` . Some exampl es are sho wn be lo w: In the RPN mode , to activate the f unctio n enter the argu ment firs t, then press the so ft menu ke y corres ponding to the variable name @@@H@@@ . For example, you co uld t ry : 2` @@@H@@@ . The othe r exampl es sho wn abo ve can be entere d by u sing: 1.2` @@@H@@@ , 2/3` @@@H@@@ . Ref erence Additional inf ormatio n on o peratio ns wi th real nu mbers with the calcu lato r is containe d in Chapter 3 of the Us er’s Manu al.
Pag e 4-1 Chapt er 4 Calculations w ith complex numbers This chapter s hows exampl es of calcu l ations and applic ation o f f u nctions to compl ex numbe rs. Def ini ti on s A co mple x nu mber z is written as z = x iy , (Cart esian r epresen tati on) wher e x and y are re al nu mbers , and i is the imaginary unit define d by i 2 = - 1. The numbe r has a real par t, x = Re(z ), and an imaginary par t, y = Im(z). The pol ar represe ntatio n o f a co mplex numbe r is z = re i θ = r ⋅ cos θ i r ⋅ sin θ , where r = |z| = 2 2 y x is the magnitu de of the comp lex number z, and θ = Arg(z) = a rctan(y/x) is the argument of the com plex number z. The complex conju gate of a co mple x nu mber z = x iy = re i θ , is  z = x – iy = re - i θ . The negative of z, –z = -x-iy = - re i θ , can be thou ght of as the ref le ction of z about the origin. Setting the calc ulator to CO MPLE X mode To work with co mple x nu mbers s el ect the CAS compl ex mo de: H ) @@CAS@ 2˜˜™ @@CHK@ The CO MPLEX m ode will be selected if the C AS MOD ES screen sh ows the opt ion _Complex checked off, i.e., Press @@OK@@ , twice, to retu rn to the stack.
Pag e 4-2 Entering complex numbers Complex numb ers in th e calculator can be ent ered in eith er of the two Cart esian r epresent ation s, nam ely, x iy , or (x,y) . The r esults in th e calculat or wil l be s hown i n the or dered- pair fo rmat, i.e ., (x,y) . For example, with th e calc ul ato r in ALG mo de, the co mple x nu mber (3.5,-1.2), is ente red as: „Ü3.5‚í\1.2` A compl ex nu mber can als o be entere d in the f orm x iy . For ex ample, in ALG mo de, 3.5-1.2i is ente red as: 3.5 -1.2*‚¥` In RPN m ode, these n umbers will be ent ered using t he follo wing keyst rokes: „Ü3.5‚í1.\` (Notice that the change-sign keys troke is entere d after the number 1.2 has been e ntered, in the o ppos ite o rder as the ALG mo de ex erci se), and ³3.5 -1.2*‚¥` (Notice th e need to ent er an a postroph e before typin g th e number 3.5-1.2i in RPN mode) . To enter the u nit imag inary number type : „¥ (the I key). Polar repr esent ation of a complex number The polar r epresent ation of the comp lex number 3.5-1.2i, en tered a bov e, is obtaine d by changing the coo rdinate sy stem t o cy lind r ica l or p ola r ( usin g fu nction CYL IN). You can fi nd this fu nction in the catal og ( ‚N ). Press µ bef ore o r af ter u sing CYL IN. Changing to pol ar sho ws, and changing the angu lar meas ure to radians , produ ce s the re su lt:
Pag e 4-3 The re su lt show n above repres ents a magnitu de, 3. 7, and an angl e 0.33029…. The ang le symbol ( ∠ ) i s show n in fron t of th e an gle m eas ure. Retu rn to Cartes ian or rectangu lar coo rdinates by u sing fu nction RECT (available in the catalog, ‚N ). A complex number in polar represe ntation is written as z = r ⋅ e i θ . You can e nter this co mplex nu mber into the cal cul ator by us ing an orde red pair o f the fo rm (r, ∠θ ). The ang le symbol ( ∠ ) can be entere d as ~‚6 . For examp le, the comp lex number z = 5.2e 1.5i , can be ente red as f o ll ows (the f igure s sho w the R PN stack , bef ore and after e ntering the number): Because the coord ina te system is se t to rectangular (o r Cartesian), the calcu lato r auto maticall y converts the nu mber entere d to Cartes ian coo rdinates, i.e., x = r cos θ , y = r sin θ , resul ting, for this case, in (0.3678…, 5.18…). On the othe r hand, if the coo rdinate sy s tem is set to cylindrical coo rdinates (use CYLIN), ent ering a comp lex number (x,y) , where x an d y ar e real number s, wil l produ ce a pol ar repre sentatio n. For e xampl e, in cy lindrical coordinates, enter the n umber (3.,2.). The figure b elow shows the RPN sta ck, befo re and af ter e ntering this nu mber: Simple oper ation s with c omplex n umber s Complex numb ers can b e combin ed using t he four fundament al opera tions ( -*/ ). The resu lts fo ll o w the ru le s of al gebra with the caveat that i 2 = -1 . Opera tions with complex numb ers are sim ilar to th ose with rea l numbe rs. Fo r exampl e, with the c alcu lato r in AL G mode and the CAS set to Compl ex , try the fo llo wing operations:
Pag e 4-4 (3 5i) (6-3i) = (9,2); (5-2i) - (3 4i) = (2,-6) (3-i)·(2-4i) = (2,-14); (5-2i)/(3 4i) = (0.28,-1.04) 1/(3 4i) = (0.12, -0.16) ; -(5-3i) = -5 3i The CMPLX me nus There a re two CMP LX (CoMPLeX n umber s) men us ava ilable in the ca lculator. One is avai labl e thro ugh the M TH me nu (intro du ced in Chapte r 3) and o ne directly into the ke yboard ( ‚ß ). The t wo CMPLX m enus ar e presen ted next. CMPLX menu thro ugh the MTH menu Assuming that system flag 117 is set to CHOOSE boxes ( see Chap ter 2), the CMPL X s ub- menu within the MT H menu is acc ess ed by u sing: „´9 @@OK@@ . The fu nctio ns avail abl e are the fo ll ow ing: The f irst menu (optio ns 1 thro ugh 6) shows the f ol lo wing fu nctions : RE(z) : Real part of a co mplex numbe r IM(z) : Imaginary part o f a co mple x numbe r C → R(z) : Separates a compl ex number into its real and imaginary parts R → C(x,y): Forms the complex number (x,y) out o f re al nu mbers x and y ABS (z) : Calculates t he ma gnit ude of a complex numb er. ARG (z): Calculates t he arg ument of a complex numb er. SIG N(z) : Calculates a complex number of u nit mag nitude as z/|z|. NEG (z) : C han ges th e sign of z
Page 4-5 CONJ(z): Produces the complex conjugate of z Examples of applications of these functions are shown next. Recall that, for ALG mode, the function must precede the argument, while in RPN mode, you enter the argument first, and then select the function. Also, recall that you can get these functions as soft menu labels by changing the setting of system flag 117 (See Chapter 3). CMPLX menu in the keyboard A second CMPLX menu is accessible by using the right-shift option associated with the 1 key, i.e., ‚ß . With system flag 117 set to CHOOSE boxes, the keyboard CMPLX menu shows up as the following screens: The resulting menu include some of the functions already introduced in the previous section, namely, ARG, ABS, CONJ, IM, NEG, RE, and SIGN. It also includes function i which serves the same purpose as the keystroke combination „¥ .
Pag e 4-6 Func tions applied to c omplex number s Many o f the key board-bas ed f unc tions and MTH menu fu nctio ns def ined i n Chapter 3 for real numbers (e.g., S Q, ,LN, e x , etc. ), can be appl ied to com plex num be rs. Th e r esult i s an oth er com plex num be r, a s illus tra t ed i n t he fo ll o wing e xample s. Note: When using t rigon ometri c functions an d their inv erses with comp lex number s the arg uments are no longer a ngles. Th erefore, the a ng ular mea sure selected for the calculator has n o bearin g in t he calculation of th ese functions with co mple x argu ments. Func tion D RO ITE: e quation of a str aight lin e Functio n DROITE takes as argument two co mple x numbers , say, x 1 iy 1 and x 2 iy 2 , and returns the eq uation of the straight line, say, y = a bx, th at contains the points (x 1 ,y 1 ) and (x 2 ,y 2 ). For exam ple, the line between points A(5,- 3) and B(6 ,2) can be fo u nd as f ol lo ws (exampl e in Al gebraic mode ):
Pag e 4-7 Functio n DROITE is fo und in the co mmand catalo g ( ‚N ). Ref erence Additional inf ormatio n on co mple x nu mber ope ratio ns is pre se nted in Chapter 4 of the calculator’ s User’s G uide.
Pag e 5-1 Chapt er 5 Algebraic and arit hmetic operations An alge braic o bject, or s impl y, al gebraic , is any numbe r, variabl e name or alge braic expre ss ion that c an be ope rated u pon, manipul ated, and combine d acco rding to the ru l es o f al gebra. Ex ample s o f al gebraic obje cts are the followin g : • A number: 12.3, 15.2_m, ‘ π ’, ‘e ’, ‘i’ • A varia ble name: ‘a’, ‘ux’, ‘width’, etc. • An expression: ‘p*D^2/4’,’f*(L/D)*(V^2/(2*g ))’, • An equation: ‘p *V = n*R*T’ , ‘Q=(Cu/n)*A(y)*R(y)^(2/3)* √ So ’ Enter ing algebraic object s Alge braic obj ects can be cr eated by ty ping the o bjec t between single quote s di rec tly i nt o sta ck le vel 1 or b y usi ng th e eq uat ion wr ite r [ EQ W] . For exampl e, to enter the alge braic o bject ‘ π *D^2/4’ directly into stack level 1 use: ³„ì*~dQ2/4` An alge braic obj ect can al so be bu il t in the Equ ation W riter and then se nt to the sta ck, or op erated upon in the E quat ion Writ er itself. Th e opera tion of the Equatio n Wri ter was des cribed in Chapte r 2. As an exe rcis e, bu il d the fo ll o wing al gebraic o bjec t in the Equ atio n Writ er:
Pag e 5-2 Afte r buil ding the o bject , pres s to s how it in the s tack (AL G and RPN mode s shown b elow): Simple operations w ith algebr aic objec ts Alge braic o bjects can be adde d, su btracte d, mu lti plie d, divide d (exc ept by zero ), rais ed to a powe r, u sed as argume nts f or a varie ty of standard funct ion s (exp onen tia l, log ar ith mi c, t rig on ometr y, h yp erb olic, et c.) , a s you wou l d any real or co mpl ex nu mber. T o de mons trate basic oper ations with algebraic objects, let’s create a couple o f objects, s ay ‘ π *R^2’ and ‘g*t^2 /4’, and s tor e them in vari ables A1 and A2 (Se e Chapter 2 to l earn how to create varia bles and store values in th em). Here are t he keystrokes for sto ring vari able s A1 in ALG mo de: ³„ì*~rQ™ K ~a1 ` resul ting in: The k eystro ke s co rrespo nding to R PN mode are : ³„ì*~rQ`~a1 K After stor ing t he va ria ble A2 an d pr essing t he key, th e screen will show the variable s as fo ll o ws:
Pag e 5-3 In ALG m ode, the foll owing keystrokes will show a number of operations with the al gebraics contai ned in variabl es @@A1@@ and @@A2@@ (press J to reco ver variable menu ): @@A1@@ @@A2@@ ` @@A1@@ @@A2@@ ` @@A1@@ 8 @@A2@@ ` @@A1@@ / @@A2@@ ` ‚¹ @@A1@@ „¸v The sam e results are obta ined in RPN m ode if using the following keystrokes: @@A1@@ ` @@A2@@ ` @@A1@@ ` @@A2@@ ` - @@A1@@ ` @@A2@@ `* @@A1@@ ` @@A2@@ ` / @@A1@@ ` ‚¹ @@A2@@ `„¸
Pag e 5-4 Functions in t he ALG menu The ALG (Alg ebra ic) menu is av ai lable by using the keystroke seq uence ‚× (asso ciated with the ‚ key). With system flag 117 set to CHOOSE boxes , the AL G menu s how s the f ol lo wing f u nctions: Rather than l isting the des cription of each fu nction in this ma nual, the user is invited to lo ok up the de scription u sing the cal cul ator’s he lp f acility: I L @ ) HELP@ ` . To l ocate a particu lar f unctio n, type the firs t lette r of the fu nction. For ex ample, fo r fu nction COL LECT, w e type ~c , then use the up a nd down arrow keys, —˜ , to lo cate COLLECT with in the help window. To complete the op erat ion pr ess @@OK@@ . Here is the help screen for fu nction COLLECT: We notice that, at the botto m of the scre en, the l ine See: EXPAND FACTOR suggest s links to oth er help fa cility en tries, t he functi ons E XPAND an d FACTOR. To move d irectly to those entries, press the soft menu key @SEE1! for EXPAND, and @SEE2! for FACTOR. Pressing @SEE1! , for example, shows t he followin g infor m at ion for E XPA N D, wh ile @SEE2! shows inform ation for FACTO R:
Pag e 5-5 Copy the exampl es provided o nto y ou r stack by pres sing @ECHO! . For e xample , for the EXP AND entr y shown a bove, p ress the @ECHO! sof t menu k ey to get the foll owing exa mple copi ed to th e stack (p ress ` to execute the comm and ): Thu s, we leave fo r the u ser to ex plo re the appl icatio ns o f the fu nctio ns in the ALG (or ALG B) menu. This is a list of the commands: For exa mp le, for fun cti on S UBST, we fi nd th e followi ng CA S h elp fa ci lit y entry: Note: Recal l that, to use these, o r any other f unctio ns in the R PN mode, you must e nter the argume nt firs t, and then the f unctio n. For example, the exampl e f or T EXPAND, in RPN mo de wi ll be se t up as : ³„¸ ~x ~y` At this point, select function TEXPAND from menu ALG (or directly from the catalo g ‚N ), to compl ete the operation.
Pag e 5-6 Operation s with tr ansc enden tal func tions The cal cu lato r of fe rs a nu mber o f f uncti ons that c an be us ed to replac e expres sio ns containing l ogarith mic and e xpo nential fu nctio ns ( „Ð ), as well a s tri gon om etri c func ti ons ( ‚Ñ ). Expansi on and factori ng us ing l og-exp funct ions The „Ð produces t he follo wing m enu: Info rmation and e xampl es o n thes e co mmands are avai labl e in the hel p facil ity o f the calcu l ator. Fo r exampl e, the desc ription o f EX PLN is show n in the le ft- hand side , and the e xample fro m the hel p fac ility is sho wn to the right: Expans ion an d fact oring us in g tri gon omet ric fu nct i ons The TRI G menu , tri ggere d by u sing ‚Ñ , shows the following funct ions:
Pag e 5-7 These function s allow to simp lify expressi ons by replac ing some ca teg ory of tri gon omet ric funct ions for an oth er one. For ex am ple, t he fun cti on A COS 2S all ows to replace the fu nction arccosine (acos(x)) wit h its exp ression i n term s of arcsine (as in(x)). Descripti on o f thes e co mmands and e xample s o f thei r applic ations are availabl e in the calc ul ator’s hel p faci lity ( IL @HELP ). The user is inv ited to explo re this f acility to find inf ormatio n o n the commands in the TR IG menu. Function s in th e ARITHM ET IC menu The ARITHM ETIC menu is triggered throu gh the keys troke combinatio n „Þ (asso ciated with the 1 key). With system flag 117 set to CHOOSE boxes , „Þ shows the following men u: Out of this menu lis t, options 5 through 9 ( DIV IS, F ACT OR S, L GCD, PROPFRAC, SIMP2 ) co rrespo nd to co mmon f unc tions that apply to integer numbe rs o r to po ly nomial s. T he re maining opti ons ( 1. INTEG ER , 2. POLYNOM IAL , 3. MODULO , and 4. PE RMUTATION ) are actual ly s ub- menu s of f unctio ns that appl y to speci fic mathematical obje cts. W hen system flag 117 is set to SOFT me nu s , the ARITHMETIC menu ( „Þ ) produ ces: Foll owing, we pre sent the hel p facil ity entries f or f unctio ns FACTORS and SIMP2 in the ARITHMETIC menu :
Pag e 5-8 FACTORS: SIMP2: The function s associat ed with the A RITHMETIC submenus: IN TEG ER, POLYNOMIAL, MOD ULO, and PE RMUTATION, are th e foll owing: Additional inf ormatio n o n applicati ons of the ARIT HMET IC menu functio ns are present ed in Ch apter 5 in the ca lculator’s User’s G uide. Polynomials Pol ynomi als are alge braic ex press io ns co nsis ting of one or mo re terms contai ning decre asing po we rs of a give n variable . Fo r exampl e, ‘X^3 2*X^2-3*X 2’ is a third -order polyno mial in X, while ‘SIN(X)^2-2’ is a second-order polynomial in SIN(X). Functions COLLECT a nd EXPA ND can be u sed o n pol yno mials , as s hown e arlie r. Other appl icatio ns o f po l ynomial functions a re pr esented next: The HORNE R f unctio n The function HORNE R produc es the Horn er di vi sion, or syn thet ic d iv ision , of a pol ynomi al P(X ) by the f actor (X- a ), i.e., HORNER(P(X),a) = {Q(X), a, P(a)}, where P( X) = Q(X)( X-a) P(a ). For exa mple, HORNER(‘X^3 2*X^2-3*X 1’ ,2 ) = {‘X^2 4*X 5, 2, 11} i.e., X 3 2X 2 -3X 1 = (X 2 4X 5)(X-2) 11. Also, HORNER(‘X^6-1’ ,-5)= {’ X^5-5*X^4 25*X^3125*X^2 625*X- 3125’, -5, 15624} i.e., X 6 -1 = (X 5 -5*X 4 25X 3 -125X 2 625X-3125)(X 5) 15624.
Pag e 5-9 The variable VX A variable call ed VX exis ts in the calcu l ator’s {HOME CASDIR} dire ctory that takes, b y default, t he v alue of ‘X’. This is t he na me of the pr eferred indepe ndent variabl e f or al gebraic and calc ul u s appl icatio ns. Avoid usi ng the variable VX in you r programs o r equatio ns, so as to not get it co nfu sed with the CAS’ VX. Fo r additio nal inf ormatio n on the CAS variable see Appendix C in the calc ul ator’s Us er’s M anual . The PCOEF fu ncti on Given an array containing the ro ots of a pol yno mial, the fu nctio n PCOEF generate s an array containing the coefficients of the co rresponding pol ynomi al. The coe ff icie nts co rrespo nd to de creas ing orde r of the independe nt variabl e. Fo r exampl e: PCOE F([-2 –1 0 1 1 2]) = [1. –1. –5. 5. 4. –4. 0.], which rep resents th e polynomial X 6 -X 5 -5X 4 5X 3 4X 2 -4X. Th e PROO T fu nct i on Given and array c ontaini ng the co ef fic ients of a polyno mial , in decre asing orde r, the fu nctio n PROOT pro vides the roo ts o f the pol yno mial. Exampl e, from X 2 5X-6 =0, PROOT([1 –5 6] ) = [2. 3.]. The QUOT IENT and REMAI NDER functions The function s QUOTI ENT a nd REMA IND ER p rovid e, resp ectiv ely, the q uotient Q(X) and the remainder R (X) , res ul ting f rom divi ding two pol yno mials , P 1 (X) and P 2 (X). In other words,, they provide th e values of Q(X) and R(X) from P 1 (X)/P 2 (X) = Q(X) R(X)/P 2 (X). For e xample , QUOTIENT(‘ X^3-2*X 2’, ‘X-1’) = ‘ X^2 X-1’ REMAIND ER(‘ X^3-2*X 2’, ‘X-1’) = 1. Thus, we can write: (X 3 -2X 2)/(X-1) = X 2 X-1 1/(X-1).
Pag e 5-10 Note : you cou ld get the l atter resu lt by u sing PARTFRAC: PARTFRA C(‘(X^3-2*X 2)/(X-1)’) = ‘ X^2 X-1 1/(X-1)’. Th e PEVA L fun ct ion The fun cti ons PE V AL ( Polyn omi al E VA Lua tion ) ca n b e used to ev alua te a pol yno mial p(x) = a n ⋅ x n a n-1 ⋅ x n-1 … a 2 ⋅ x 2 a 1 ⋅ x a 0 , given an a rray of coef ficients [ a n , a n-1 , … a 2 , a 1 , a 0 ] and a valu e o f x 0 . The result is the ev aluation p(x 0 ). Functi on PEVAL is no t availabl e in the ARITHMETIC menu, it must be accessed from the function catalog ( ‚N ). Exam ple: PEV AL([1,5,6,1],5) = 281. Additional appl icatio ns of po lyno mial fu nctio ns are pre se nted in Chapter 5 in the calculator’ s User’s G uide. Frac tions Fraction s can be e xpanded and f acto red by us ing fu nctio ns EX PAND and FACTOR, from the ALG m enu (‚×). For example: EXPAND (‘(1 X)^3/((X-1 )(X 3))’) = ‘( X^3 3*X ^2 3*X 1)/(X^2 2*X-3)’ EXPAND(‘(X^2*(X Y)/ (2*X-X^2)^2’) = ‘( X Y)/(X^2-4*X 4)’ FACTOR(‘(3*X^3-2*X^2)/( X^2-5*X 6)’) = ‘X^2*(3*X-2)/((X -2)*(X-3 ))’ FACTOR(‘(X^3-9*X)/(X^2-5*X 6)’ ) = ‘X*(X 3)/(X-2)’ The S IMP2 func tion Functio n SIMP2 takes as argu ments tw o nu mbers or pol ynomial s, representing the nume rator and de nominato r of a rational fraction, and re turns the simpl ifi ed nume rator and de nominato r. For e xampl e: SIMP2(‘X^3-1’ ,’X^2-4*X 3’) = { ‘X^2 X 1’,‘ X-3’}
Pag e 5-11 The PROPFRAC functi on The f unc tion PR OPFRAC conve rts a ratio nal f ractio n into a “proper” fractio n, i.e. , an intege r part added to a f ractio nal part, if su ch decompos itio n is poss ibl e. Fo r exampl e: PROPFRAC (‘5/4’) = ‘1 1/4’ PROPFRAC (‘(x^2 1)/x^2’) = ‘1 1/x^2’ The PA RTFRAC fu ncti on The f unc tion PAR TFRAC de compo ses a rational f raction i nto the partial fracti ons that pro duce the o riginal f ractio n. For e xampl e: PARTFRA C(‘(2*X^6-14* X^5 29*X^4-37*X^3 41*X^2-16*X 5)/(X^5- 7*X^4 11*X^3-7*X^2 10*X)’) = ‘2*X (1/2/(X-2) 5/(X-5) 1/2/X X/(X^2 1))’ The FCOEF fu nction The f unctio n FCOEF is use d to obtain a ratio nal f raction, given the ro ots and pol es o f the frac tion. Note : If a r ati ona l frac tion is g iv en as F( X) = N (X)/ D( X), t he r oots of th e fraction re sul t fro m sol ving the equation N(X) = 0, while the pole s result from solving the eq uation D (X) = 0. The input f or the fu nction is a vecto r lis ting the roo ts fo ll owe d by their mul tipl icity (i. e., ho w many time s a give n roo t is re peated), and the pol es fo ll ow ed by their mu lt iplic ity repre sente d as a negative numbe r. Fo r exampl e, if we want to create a fractio n having roots 2 with multiplicity 1, 0 with mult ip lici ty 3, an d -5 with m ulti pli ci ty 2 , a nd pole s 1 wi th mult ip lic ity 2 a nd –3 wit h m ult ip lici ty 5, us e: FCOE F([2 1 0 3 –5 2 1 -2 -3 -5] ) = ‘(X—5)^2*X^3*(X-2)/9X—3)^5*(X-1)^2’ If you press µ you wi ll g et: ‘(X^6 8*X^5 5*X^4-50*X^3)/( X^7 13*X^6 61*X^5 105*X^4 - 4 5 * X^ 3- 297*X62-81*X 243)’
Pag e 5-12 The FROOT S functi on The f u nction FR OOTS obtains the ro ots and pol es of a fracti on. As an exampl e, appl ying f unc tion FR OOTS to the resu l t produ ced abo ve, wi ll resu l t in: [1 –2 –3 –5 0 3 2 1 –5 2]. Th e result shows poles f oll owed by their mul tipl icity as a negative numbe r, and ro ots fo ll ow ed by the ir mu ltipl icity as a positive number. In th is case, the poles are (1, -3) with multiplicities (2,5) respectively, and the roots are (0, 2, -5) with multiplicities (3, 1, 2), respectively. Anothe r exampl e is : FROOT (‘(X^2-5*X 6)/(X^5- X ^2)’) = [0,–2,1, –1,3,1,2, 1], i.e., poles = 0 (2), 1(1), and roots = 3(1), 2(1). If you have had C omplex mode selected, then the results would be: [0 –2 1 –1 ‘-((1 i* √ 3)/2’ –1]. Step-by- step operations w ith poly nomials an d frac tions By se tting the CAS modes to Step/s tep the cal cul ator wil l sho w simpl ificatio ns of frac tions or o peratio ns wi th pol ynomi als in a ste p-by- step f ashio n. T his is very useful to see the step s of a synth etic di vision . The exam ple of div idin g 2 2 3 5 2 3 − − − X X X X is shown in detail in Appendix C of the calcul ator’s User’s Guide. The fo ll owi ng exampl e sho ws a l engthier s ynthetic divisi on: 1 1 2 9 − − X X
Pag e 5-13 Ref erence Additional inf ormati on, de fi nitions , and ex ample s o f al gebraic and ari thmetic opera tions ar e present ed in C hap ter 5 of the ca lculator’s User’ s Guide.
Pag e 6-1 Chapt er 6 Solu tio n to e q uati ons Associated with the 7 key ther e are t wo men us of equation -solvin g fu nctions, the Symbo lic SOLVer ( „Î ), and the NUM erical SoLVer ( ‚Ï ). Following, we pr esent som e of the functions con tain ed in t hese menu s. Symbolic solution of algebraic equations Here we describe some of the fu nctions from the Sym bolic Solver menu. Activate the menu by u sing the ke ystrok e combinatio n „Î . With system flag 117 set to CHOOS E boxes, the follo wing m enu lists will be ava ilable: Functio ns ISOL and SOL VE can be u sed to so lve f or any unk nown in a pol ynomi al e quatio n. Fu nction SOL VEVX so lve s a polyno mial equation where the unk nown is the defau lt CAS variable VX (typical ly s et to ‘X ’). Finall y, funct ion ZE RO S p rov id es th e zeros, or roots, of a p olynom ial. Func tion ISOL Functio n ISOL(Equati on, variabl e) w ill produ ce the s ol uti on(s) to Equat ion by isol ating variable . Fo r exampl e, with the calcu lato r set to ALG mode, to solve for t in the eq uation a t 3 -bt = 0 we can us e the fo ll owing:
Pag e 6-2 Using t he RPN m ode, the solution is accomp lished b y enteri ng t he equat ion in the stac k, f ol lo wed by the variabl e, bef ore enteri ng fu nctio n ISOL. Right before the execution of ISOL, t he RPN st ack sh ould look as in the fig ure to the lef t. Af ter appl ying ISOL, the resu lt is sho wn in the figu re to the right: The f irs t argument i n ISOL can be an e xpres sio n, as s how n above, or an equation. For example, in ALG m ode, try: Note: To type the equal sign (=) in a n equation, use ‚Å (associ ated with the \ key). The s ame pro blem c an be so lved i n RPN mode as il lu st rated bel ow (figu res show the RPN s tack bef o re and afte r the appl ication o f fu nction ISOL): Func ti on SOLV E Functio n SOLVE has the same s yntax as f unctio n ISOL, ex cept that SOLVE can als o be u se d to so lve a set o f po ly nomial equations . The help-f acil ity entry for functio n SOLVE, with the solutio n to equation X^4 – 1 = 3 , is shown next:
Pag e 6-3 The following examp les show the use of function S OLV E in ALG and RPN mode s: The screen shot sh own a bove d ispla ys two solutions. I n th e first one, β 4 -5 β =125, SOLV E pr oduces no solu tions { }. In the second one, β 4 - 5 β = 6, SOLV E pr oduces fou r solu tions, shown in the last output line. The very last sol u tion is not vis ible becau se the resu l t occu pie s more characters than the width of the calculator ’s screen. Howev er, you can still see all t he solutions by using the down arrow key ( ˜ ), which triggers the line editor (this operatio n can be u sed to acces s any o utpu t li ne that is w ider than the c alcu lato r’s screen): The corresp ond ing RPN screen s for these tw o examp les, before an d after the applic ation o f f u nction SOL VE, are s hown ne xt:
Pag e 6-4 Func ti on SOLV EVX The functi on S OLVE VX solv es an equati on for th e default CA S va ria ble containe d in the re serve d variable name VX. By de fau lt, thi s variabl e is set to ‘X’. Ex ample s, u si ng the ALG mo de w ith VX = ‘X’, are sho wn bel o w: In the firs t case SOLVEVX cou ld no t fi nd a solu tion. In the second ca se, SOLVEVX f ou nd a s ingle so lu tio n, X = 2. The fo llowing screens sh ow the RP N sta ck for solving the t wo exam ples shown above (be fo re and after appl icati on o f SOL VEVX): Functi on ZEROS The f unctio n ZEROS finds the sol utio ns of a pol yno mial equ ation, withou t showing their mu lt iplicity. The fu nction requ ires having as inpu t the expression for the equa tion a nd t he na me of th e va riab le to solve for. Exampl es i n ALG mo de are sho wn nex t:
Pag e 6-5 To use function ZE ROS in RPN m ode, enter first th e polynomia l expression, then the var iab le to solve for, an d th en function ZERO S. Th e following screen shots sho w the R PN stack be fo re and af ter the appl icatio n of ZEROS to the two e xampl es above : The Symbo lic Solve r fu nctio ns pres ented abo ve pro duce so lu tions to ratio nal equatio ns (mainly, pol yno mial equ ations). If the equatio n to be s ol ved for has all n ume ri ca l coeffi cie nt s, a n umer ic al so lution is pos sib le th r ough th e use of the Numer ical Solver features of the ca lculator. Numerical solver menu The calculator prov ides a very powerful envir onment for the solution of sing le algebrai c or trans cende ntal e quatio ns. T o acc ess this e nvironme nt we s tart th e num er ic al so lve r ( NUM. S LV) by usin g ‚Ï . This produ ces a dro p- down menu that inclu des the f ol lo wing options:
Pag e 6-6 Fol lo wing, w e pres ent appli cations of items 3. S olv e p oly. . , 5. Solve f inance , and 1. Solve equat ion.. , in th at order. Ap pendix 1-A, in the calculator’s User’s Guide, con tain s instructions on h ow to use input forms with exam ples fo r the nu merical sol ver appl icatio ns. Item 6. M SLV ( Multiple equat ion SoLVer) w ill be pres ented l ater in this Chapter. Notes: 1. Whe never y ou so lve f or a val u e in the NU M. SLV applic ations , the val ue solved for will be placed in the sta ck. This is useful if you need to keep th at valu e avail able fo r o ther ope ratio ns. 2. There will be one or more v ar iab les created whenev er you act iva te som e of the appli cations in the NU M.SLV me nu. Polyn omial Equat i ons Using the S olv e p oly… opt ion i n t he c alcula tor ’s SOLVE environm ent you can: (1) find the sol utio ns to a po lyno mial e quation; (2) o btain the co ef ficie nts of the po lyno mial having a number of giv en roots; and, (3) obtain an algebrai c expre ssi on f or the pol ynomi al as a f u nctio n of X. Finding the so lu tions to a po lyno mial e quatio n A pol ynomi al e quatio n is an e quatio n of the f orm: a n x n a n-1 x n-1 … a 1 x a 0 = 0 . For examp le, solve the eq uation: 3s 4 2s 3 - s 1 = 0. We want t o place th e coefficients of the equat ion in a vector : [3,2,0,-1,1]. To sol ve fo r this pol ynomial equatio n using the cal cul ator, try the fol lo wing: ‚Ϙ˜ @@OK@@ Se lect S olve poly … „Ô3‚í2‚í 0 Enter v ector of coef ficients ‚í 1\‚í1 @@OK@@ @SOLVE@ S o l v e e q u a t i o n The sc ree n wi ll show th e solut ion a s follows :
Pag e 6-7 Press ` to return to st ack. The st ack will show th e foll owing results in ALG mode (the same re su lt w ou ld be show n in RPN mo de): All the solu tions are complex numbers: (0.432,-0.389), (0.432,0.389), (- 0.766, 0.632), (-0.766, -0.632). Gene rating po lynomial coefficie nts giv en the polyno mial' s r oo ts Supp ose you want t o gener ate t he polyn omial wh ose r oots ar e t he numb ers [1, 5, -2, 4]. To use the calculator fo r this purpose, foll ow these steps: ‚Ϙ˜ @@OK@@ Se lect S olve poly … ˜„Ô1‚í5 Ent er v ect or of root s ‚í2\‚í 4 @@OK@@ @SOLVE@ S o l v e f o r c o e f f i c i e n t s Press ` to retu rn to stack , the coe ff icients will be sho wn in the stack. Press ˜ to trigg er th e line editor t o see all the coefficients.
Pag e 6-8 Gener ating an alge braic e xpre ss ion for the pol ynomial You can u se the calc ul ator to gene rate an alge braic ex press ion fo r a polynomia l given the coefficients or t he root s of the polynomi al. The resulting expre ssi on wi ll be give n in terms of the def au lt CAS variabl e X . To gene rate the al gebraic expres sion u sing the coe ff icients, try the foll owing examp le. Assume that t he polynomia l coefficients are [1,5,-2,4] . Use the followin g key str okes: ‚Ϙ˜ @@OK@@ Se lect S olve poly … „Ô1‚í5 Enter vector of coeff icients ‚í2\‚í 4 @@OK@@ — @SYMB@ Generat e symbolic expr ession ` R e t u r n t o s t a c k . The expression th us generated is sh own in the stack a s: 'X^3 5*X^2-2*X 4'. To gene rate the alge braic expres sion u sing the ro ots, try the fo ll owing exampl e. As su me that the po ly nomial ro ots are [1,3 ,-2 ,1]. U se the fo l lo wing keystrokes: ‚Ϙ˜ @@OK@@ Se lect S olve poly … ˜„Ô1‚í3 En ter v ector of root s ‚í2\‚í 1 @@OK@@ ˜ @SYMB@ Gen erat e symbolic expressi on ` R e t u r n t o s t a c k . The e xpres sio n thus generate d is s hown i n the stac k as: ' (X-1)* (X-3 )*(X 2 )*(X -1) '. To expan d th e product s, you can use the E XPAND comm and . The resulting expression is: ' X^4 -3*X^3 -3 *X^2 11*X-6' .
Pag e 6-9 Financial calculati ons The calcul ations in item 5. Sol ve finance .. in t he N um eri ca l S olve r ( NUM.SLV ) are used for calcu lations of time value of money of interest in the discip line of engine ering e cono mics and othe r financ ial appl icati ons. This applic ation c an als o be s tarted by u si ng the ke ystro ke combi nation ‚Ò (associa ted wi th the 9 key ). Detail ed expl anatio ns of thes e type s of cal cul atio ns are prese nted in Chapter 6 of the cal cul ator ’s Us er’s M anual . Solvin g equ ati ons wi th on e un known t hrou gh NU M.SLV The calculator 's NUM.S LV men u provid es item 1. Solve eq ua tio n.. solve diff ere nt types of equ ations in a si ngle variable , incl udi ng non- line ar algebraic and transcende ntal e quatio ns. Fo r exampl e, l et's s ol ve the equatio n: e x - sin( π x/3) = 0. Simply e nter the ex press io n as an al gebraic o bject and s tor e it into variabl e EQ. Th e required keyst rokes in ALG mode a re the following: ³„¸~„x™-S„ì *~„x/3™‚Å 0™ K~e~q` Fu nction S TE Q Function STEQ will store its argument into va riable EQ, e.g ., in ALG mode: In R PN mode , enter the e quatio n betwe en apo stro phes and acti vate command STEQ. Thu s, fu nctio n STEQ can be u se d as a shortcu t to store an expres sio n into vari able EQ. Press J to see the newly creat ed EQ var iable:
Pag e 6-10 Then, en ter th e SOLV E en vir onment and select Solv e equat ion… , b y using: ‚Ï @@OK@@ . The correspondin g screen will be shown as: The e quatio n we s to red in variabl e EQ is alre ady l oaded i n the Eq fiel d in th e SOLVE EQUAT ION inpu t fo rm. Al so , a fiel d label ed x is provid ed. To solve the equatio n all yo u ne ed to do is highlight the f iel d in fro nt of X : by u sing ˜ , and pres s @SOLVE@ . The solution shown is X: 4.5006E-2: This, howe ver, is not the only po ssi ble s olu tion f or this e quation. T o obtain a nega tiv e solution, for examp le, enter a nega tiv e numb er in th e X: field before solving the eq uation. Tr y 3\ @@@OK@@ ˜ @SOLVE@ . T h e s o l u t i o n i s n o w X : - 3.045. Solution to simultaneous equations w ith M SLV Functi on M SLV is avail able in the ‚Ï menu. The hel p-f acility e ntry fo r fu nction MSLV is shown ne xt:
Pag e 6-11 Notice that f unctio n MSLV requ ires three arguments: 1. A vector contai ning th e equations, i.e., ‘[S IN(X) Y,X SIN(Y)=1] ’ 2. A vector contain ing the v ariab les to so lve for, i.e., ‘[X,Y]’ 3. A vector co ntaining initial val ues fo r the sol ution, i.e., the initial v alues of both X and Y are z ero f o r this ex ample . In ALG mo de, pre ss @ECHO to co py the e xample to the stack , press ` to run the exam ple. To see all the elem ents in t he solution you need t o activ at e the line edito r by pres sing the down arro w ke y ( ˜ ) : In RPN mode , the s ol uti on f or this exampl e is produ ced by u sing: Activating functio n MSLV resu lts in the fo llo wing screen. You may have notice d that, whi le pro d ucing th e solution, th e screen shows interm ediat e informat ion on th e upper left corner . Sin ce the solution p rov ided
Pag e 6-12 by MSLV is numerical, the inf ormatio n in the up per left corner sh ows the results of the iter ativ e process used to obta in a solution. The fina l solution is X = 1.8238, Y = -0.9681 . Ref erence Additional information on so lving single and multiple equations is provided in Chapters 6 and 7 o f the c alcu l ator’s Use r’s Gu ide.
Pag e 7-1 Chapt er 7 Ope ra tio ns with l ists List s are a type of cal cul ator’s obje ct that can be u se fu l f o r data proces sing. This chapte r prese nts ex amples of ope rations with l ists. To get starte d with the exampl es i n this Chapter , we u se the Approx imate mo de (See Chapte r 1). Creatin g and storin g lists To cre ate a list in ALG mo de, firs t enter the brace s key „ä , then type or enter t he elements of th e list, sepa ratin g th em with comma s ( ‚í ). The foll owing keystrokes will enter the list {1 2 3 4} a nd st ore it int o varia ble L1. „ä 1 ‚í 2. ‚í 3. ‚í 4. ™K~l1` Entering the same l ist in RPN mo de requires the following keystrokes: „ä 1. # 2. # 3. # 4. ` ³~l1`K Operation s with lists of number s To demon strat e opera tions wit h lists of numb ers ent er an d stor e the following lists in t he corresp ond ing va riab les. L2 = {-3.,2.,1.,5.} L3 = {-6.,5.,3 .,1.,0.,3.,-4.} L4 = {3.,-2.,1.,5.,3.,2.,1.} Changing si gn The s ign- change k ey ( \ ) , when appl ie d to a l ist o f numbe rs, w ill change the sig n of all elements in th e list. For exam ple: Note : Directories TRIANG a nd ME S1 resulted from a prev ious exercise. Please ign ore them while tr ying th ese examp les in your calculator.
Pag e 7-2 Addi ti on, s ubt ract i on, mul ti pli cat i on, di vis i on Mu lti plic ation and divi sio n of a li st by a s ingl e nu mber is distribu ted ac ros s the lis t, f or ex ample : Subtractio n of a si ngle numbe r fro m a li st wil l su btract the s ame nu mber f rom each elemen t in the list , for example: Addition o f a singl e nu mber to a li st pro duce s a l ist au gmente d by the numbe r, and not an additi on o f the singl e nu mber to e ach el ement i n the li st. For ex ample: Sub tra ct ion, multi plic at ion, an d d iv ision of lists of n umb ers of th e sam e len gt h produce a list of the same length with term -by-term operations. E xamp les: The div ision L4/L3 will produce an infinit y entry b ecause one of the elem ents in L3 is z ero, and an e rror mess age is retu rned.
Pag e 7-3 Note : If we had ent ered th e elements in lists L4 a nd L3 a s integer s, the infinit e symbol would be sh own when ever a di vision by zero occurs. To p roduce th e foll owing result you need t o re-enter t he lists as in teger ( remov e decima l points ) us ing Exact mo de: If the lists inv olved in the op eration hav e different leng ths, a n error m essag e (Invalid Dimens ions ) is pro duc ed. Tr y, f or ex ample , L1- L4. The plus sign ( ), when appl ied to li sts, acts a co ncatenatio n operat or, putting to gether the two l is ts, rather than adding them term- by-term. For exampl e: In order to produ ce te rm-by- term additio n of two l ists of the same le ngth, we need to us e ope rator ADD. This ope rator can be lo aded by u si ng the f uncti on catalog ( ‚N ). The scree n bel ow s how s an appl icatio n of ADD to add lists L1 and L2, term -by-term: Functi ons applied t o lis ts Real num ber funct ions fr om t he keyb oar d ( ABS , e x , LN, 10 x , LOG, S IN, x 2 , √, COS, T AN, ASIN, ACOS, AT AN, y x ) as wel l as thos e f rom the MTH/HYPERBOLIC menu (SINH, COS H, TANH, ASINH, A COSH, A TANH), and MTH /REAL me nu (%, e tc.), can be appl ied to lis ts, e.g.,
Pag e 7-4 A B S I N V E R S E ( 1 / x ) Lists of complex n umber s You can create a complex number list, say, L5 = L1 A DD i ⋅ L2 (type th e instru ctio n as indicate d here), as fo ll ow s: Functi ons s u ch as LN , EXP, SQ, etc., can als o be applie d to a l ist of compl ex numbers, e.g ., Lists of algebraic objec ts The f ol l owi ng are exampl es of li sts o f algebrai c obje cts w ith the f unc tion SIN applie d to the m (sel ect Exac t mode fo r these exampl es -- See Chapter 1) : The MTH/LIST m enu The MTH men u provid es a numb er of functions th at exclusively to lists. With system flag 117 set to CHOOS E b oxes, the MTH/LIST menu off ers the followin g func tion s:
Pag e 7-5 With system flag 117 set to SO FT menus, the MTH/LIST menu shows the followin g func tion s: The operation of the MTH/LIST menu is as follo ws: ∆ LIST : Calculate increm ent a mong consecutive elemen ts in list Σ LIST : Ca lculate summat ion of element s in the list Π LIST : Calculate p roduct of element s in th e list SORT : S orts element s in increa sing ord er REVLIS T : Reverses order of list ADD : Operato r fo r term- by-term addi tion o f tw o l ists o f the same length (exa mp les of th is oper at or wer e show n a bov e) Example s o f appl icati on o f the se fu nctio ns i n ALG mo de are s how n next: SORT and REVL IST can be co mbined to so rt a lis t in dec reasing o rder:
Pag e 7-6 The SEQ f unctio n The SEQ fu nction, availabl e thro ugh the co mmand catalo g ( ‚N ), takes as argu ments an e xpres sio n in terms of an index , the name of the inde x, and starting, e nding, and incr ement val ue s f or the i ndex, and re turns a lis t consistin g of th e eva luation of th e expression for all possible va lues of the index. The gen eral form of the function is SEQ( expression, index, st art, en d, increm ent ) For ex ample: The list p roduced corresp onds to t he v alues {1 2 , 2 2 , 3 2 , 4 2 }. The MAP functi on The M AP functio n, availabl e throu gh the co mmand catalog ( ‚N ), takes as argu ments a l is t of numbe rs and a f u nction f (X ), and pro duce s a l ist cons isting o f the applic ation o f f unc tion f or the program to the li st of nu mbers. For ex ample, the fo ll o wing call to f u nction M AP applie s the f uncti on SIN(X) to the list {1,2,3}: Ref erence For additio nal re fe rences , ex ample s, and appl icatio ns of l ists see Chapter 8 in the calculator’ s User’s G uide.
Pag e 8-1 Chapt er 8 Vect ors This Chapter pro vides e xample s o f e ntering and o peratin g with vect ors , both mathematical vec tors of many e le ments , as w el l as physic al ve cto rs o f 2 and 3 components. Enteri ng vect ors In th e calculator, v ectors a re rep resented by a sequence of number s enclosed betwee n bracke ts, and typi call y ente red as ro w vec tors . The brackets are generated in the cal cu lator by the keys troke combination „Ô , asso ciated with the * key. The follo wing a re exam ples of vectors in the calculator: [3.5, 2.2, -1.3, 5.6, 2.3] A general row vecto r [1.5,-2.2] A 2- D vec to r [3,-1,2] A 3- D vec to r ['t','t^2','SIN(t)'] A vecto r o f alge braics Typi ng vectors in t he s tack With the calc ul ator in ALG mo de, a ve cto r is ty ped into the s tack by ope ning a set o f brack ets ( „Ô ) an d typ ing the com pon ents or elem ents of th e vecto r s eparated by commas ( ‚í ). The scr een shot s below show th e enteri ng of a nume rical vecto r f ol lo we d by an al gebraic ve cto r. The fi gure to the l ef t sho ws the alge braic vec tor bef ore press ing ` . The fi gure to the right show s the c alcu lato r’s s cree n afte r entering the alge braic vect or: In RPN mo de, yo u c an enter a ve ctor in the st ack by opening a set of brackets and typi ng t he v ector comp onent s or element s separ ated by eith er comm as
Pag e 8-2 ( ‚í ) or s paces ( # ). Notice that af ter press ing ` , in either mode, the calculator shows th e vector elements sep arat ed by spaces. Stori ng ve ctors i nt o variable s i n t he s t ack Vect ors can be stored into v ar iab les. The screen shots b elow show th e vect ors u 2 = [1, 2] , u 3 = [-3, 2, -2] , v 2 = [3,-1] , v 3 = [1, -5, 2] Stor ed int o vari ables @@@u2@@ , @@@u3@@ , @@@v2@@ , and @@@v3@@ , respectively. First, in ALG mode : Then, in RPN mod e (before pr essing K , repeate dly): Note: The ap ostroph es (‘) a re not n eeded or dina rily in ent ering the n ames u2, v2, etc. in RPN mode. In this case, t hey are used to ov erwrite th e existing variable s cre ated e arlie r in ALG mo de. T hus , the apo stro phes mus t be use d if the existin g v aria bles are n ot purged prev iously. Using t he matrix writer (M TRW) to enter vectors Vecto rs can al so be ente red by u si ng the matrix w riter „² (third key in the fo urth ro w of keys f rom the to p of the keyb oard ). Thi s comm and gen erat es a s pecie s o f spre adshee t corre spo nding to ro ws and co l umns of a matrix (Deta ils on using th e matr ix writer to ent er mat rices will be pr esented in a subsequent chapter ). For a vector we are interested in filling only element s in the top row. B y def ault, the cell in the top ro w and firs t col umn is s ele cted. At the bottom o f the spreadshe et you will find the f ol lo wing sof t menu k eys: @EDIT! @VEC  ← WID @WID → @GO →  @GO ↓
Pag e 8-3 The @EDIT key is u sed to edit t he conten ts of a selected cell in the matrix writer. The @VEC@@ key , whe n sel ec ted, w il l pro duc e a vec tor, as oppos ed to a matrix of o ne row and many co lu mns. The ← WID key is used to decrease t he widt h of the columns in the sprea dsheet. Press this key a couple of times to see th e column width decrease in your matrix writer. The @WID → key is used to increa se the widt h of the columns in t he sprea dsheet. Press this key a couple of tim es to see the column wid th increase in your matrix writer. The @GO →  key, whe n select ed, autom at ica lly se lect s th e n ext cell t o the ri ght of the current cell when you p ress ` . Thi s opt ion i s sel ecte d by def aul t. This o ption, if de sired, needs to be s el ected before enterin g element s. The @GO ↓ key, w hen sele cte d, aut oma ti ca lly selec ts the ne xt cell b elow the current cell when you press ` . This option, if d esired, needs to be selected before enter ing elemen ts. Mov ing to the r ight vs . mo v ing down in the matr ix wr iter Activate the matrix write r and enter 3`5`2`` with the @GO →  key sel ected (default). Next, enter the sam e sequence of numb ers with the @GO ↓  key sel ected to see th e difference. In the first ca se you entered a vector of three elemen ts. In t he second case you enter ed a m atr ix of th ree rows and one col umn. Activate the matrix writer again by using „² , and pres s L to check ou t the seco nd s of t key me nu at the bo ttom o f the dis play. It wil l sho w the keys: @ ROW@ @-ROW @ COL@ @-COL@ @ → STK@@ @GOTO@
Pag e 8-4 The @ ROW@ key w ill add a row fu ll of ze ros at the locatio n of the selected cell of the sprea dsheet. The @-ROW key wi ll de lete t he row c orr esp ond in g t o th e sele cte d c ell of the spr eadsh eet. The @ COL@ key will add a co lu mn fu l l o f z ero s at the loc ation of the selected cell of the sprea dsheet. The @-COL@ key will delete th e colum n c orr esp ond in g t o th e selec ted cell of the spread sheet. The @ → STK@@ key will place the con tents of th e selected cell on the stack. The @GOTO@ key, when pr essed, will request tha t the user ind icate the number of the row a nd column where he or she wan ts to positi on the curso r. Pressin g L o nce mo re produ ces the l ast me nu, which co ntains o nly o ne func tion @@DEL@ (delete). The funct ion @@DEL@ will d elet e t he c ont en ts of t he selec ted ce ll an d replac e it wi th a zero . To see these keys in a ction try the following exercise: (1) Activ ate the matrix writer by using „² . M a k e s u r e t h e @VEC  and @GO →  keys are selected. (2) E nter the fo llo wing: 1`2`3` L @GOTO@ 2 @@OK@@ 1 @@OK@@ @@OK@@ 4`5`6` 7`8`9`
Pag e 8-5 (3) Mo ve the curs or u p two posi tions by u sing —— . Then p ress @-ROW . The s eco nd row w ill dis appear. (4) Press @ ROW@ . A row of three z eroe s appe ars in the s eco nd row. (5) Press @-COL@ . The firs t col umn w ill disappe ar. (6) Press @ COL@ . A col u mn of two z ero es appe ars in the firs t col u mn. (7) Press @GOTO@ 3 @@OK@@ 3 @@OK@@ @@OK@@ to move to position (3,3). (8) Press @ → STK@@ . This will place th e conten ts of cell (3,3) on the stack, altho ugh you will not be abl e to see it ye t. Press ` to retu rn to no rmal display. Th e numb er 9, element ( 3,3), and the full matrix en tered will be availabl e i n the st ack. Simple operations w ith v ec tors To ill ustrate operations with vectors we will use the vectors u2, u 3, v2, and v3, stor ed in an earlier exercise. A lso , store vector A=[-1,-2,-3,-4,-5] to be used in the following exercises. Changing sign To chan ge th e sign of a v ector use th e key \ , e.g., Addit ion, s ubt racti on Addition and su btractio n of vecto rs re qui re that the two ve cto r ope rands have the same l ength:
Pag e 8-6 Attempting to add or s ubtract ve ctors of dif fe rent le ngth produ ces an erro r messag e: Mul ti pli cati on by a sc alar, and di vis i on by a scal ar Mu lti plicati on by a s cal ar or divi sio n by a sc alar is straightf orw ard: Absolute va lue function The abs ol u te val ue fu nctio n (ABS), whe n applie d to a ve cto r, produ ce s the mag nitude of the v ector. F or examp le: ABS([1,-2,6]) , ABS(A) , ABS(u3) , will show in the screen as foll ows:
Pag e 8-7 The MTH/VECTOR menu The MTH m enu ( „´ ) contains a me nu o f f unctio ns that spec ifical ly to vec tor ob jec ts: The VE CTOR menu contains th e fol lowing functions (system flag 117 set to CHOOSE boxes ): Magnitud e The magni tude of a vecto r, as discu ss ed e arlie r, can be f o und wi th fu nctio n ABS. Thi s f uncti on is als o avai labl e f rom the ke yboard ( „Ê ). Exampl es of appli cation o f fu nctio n ABS we re sho wn abo ve. Do t pr od uct Functio n DOT (o ption 2 in CHOOSE box abo ve) is us ed to c alcu late the do t produ ct of two vecto rs of the same le ngth. Some exampl es of applic ation o f function DOT, using the v ectors A, u2, u3, v2, and v 3, stored earlier, are shown ne xt in AL G mode. Attempts to calcu l ate the do t produc t of two vectors of diff erent l ength produce an error me ssage:
Pag e 8-8 Cr oss p ro d uct Functio n CR OSS (optio n 3 in the MTH/VECTOR menu) is use d to cal cul ate the cross p rod uct of t wo 2-D v ect ors, of t wo 3-D v ector s, or of on e 2 -D a nd one 3- D v ect or. For t he p urpose of c alc ulati ng a c ross p rod uct, a 2-D v ect or of th e form [A x , A y ], is treated as the 3-D vector [A x , A y ,0]. Exa mples in A LG mode are shown ne xt fo r two 2-D and two 3- D vectors. Notice that the cross pr oduct of two 2 -D vec tor s will p rod uce a v ec tor i n t he z-d ire cti on on ly, i. e., a vector of the fo rm [0, 0, C z ]: Ex am ples of cr oss pr oduct s of one 3 -D v ec tor w ith one 2-D v ect or, or vi ce versa, are pres ented ne xt: Attempts to cal cul ate a cros s produ ct of vectors o f l ength other than 2 o r 3, produc e an error me ssage: Ref erence Addition al inf ormati on o n ope ratio ns with ve cto rs, i nclu ding appl icatio ns in the ph ysical sciences, is p resented in Cha pter 9 of the calculator’s User’ s Gui de .
P a g e 9 -1 Chapt er 9 Matrices and linear algebra This chapter s hows exampl es o f creating matric es and o peratio ns wi th matrices , incl u ding li near al gebra appl icatio ns. Enter ing matrices in the stac k In this sectio n we pres ent two dif fe rent metho ds to ente r matrices in the calcu l ator stac k: (1) u sing the Matrix Editor, and (2) typing the matri x direct ly into the s tack. Using the matr ix editor As with the case of vecto rs, discu ss ed in Chapter 9, matrice s can be entere d into the stack by us ing the matrix edit or. For exa mple, to enter the ma trix: first, s tart the matrix writer by u sing „² . Make sure that the option @GO →  is selected. Then use th e foll owing keystrokes: 2.5\` 4.2` 2`˜ššš .3` 1.9` 2.8 ` 2` .1\` .5` At this point, the M atrix Writ er screen ma y look l ike this: , 5 . 0 1 . 0 2 8 . 2 9 . 1 3 . 0 0 . 2 2 . 4 5 . 2           − −
Pag e 9-2 Press ` o nce mo re to place the matrix on the s tack . The ALG mo de stac k is show n next, befo re and af ter pres sing , once more: If you ha v e selected th e text book d ispla y op tion (usin g H @ ) DISP! and check ing off  Textbook ), the mat rix will look like the one shown abov e. Oth erwise, th e di sp lay will sh ow: The d isp lay in RPN mod e wi ll look ve ry si mi lar to t hes e. Typin g in th e ma trix dir ec tly into the stack The s ame res ul t as above c an be achie ved by ente ring the f ol l owi ng directl y into the s tack: „Ô „Ô 2.5\ ‚í 4.2 ‚í 2 ™ ‚í „Ô .3 ‚í 1.9 ‚í 2.8 ™ ‚í „Ô 2 ‚í .1\ ‚í .5 Thus , to enter a matrix dire ctly into the stack o pen a set o f brack ets ( „Ô ) and encl ose each ro w of the matrix with an additio nal s et of brack ets ( „Ô ). Commas ( ‚í . ) should sepa rat e the elemen ts of each row, as wel l as the brack ets be tween ro ws. For f utu re exercis es, l et’s s ave this matrix unde r the name A. In ALG mode use K~a . In RPN mode, use ³~a K .
P a g e 9 -3 Operation s with matrices Matrice s, l ike othe r mathematical obje cts, can be added and s ubtrac ted. They can be mu lt iplie d by a sc alar, o r amo ng themse lves . An impo rtant oper ation f or l inear al gebra appl icatio ns is the inve rse o f a matri x. Detail s o f these op eration s are p resented next. To illustrate the op eration s we will create a numb er of mat rices tha t we will store in th e foll owing v ariab les. Here are th e matr ices A22, B22, A 23, B23, A33 an d B33: In RPN m ode, the st eps to follow are: {2,2} ` RANM 'A22' `K {2,2} ` RANM 'B22' `K {2,3} ` RANM 'A23' `K {2,3} ` RANM 'B23' `K {3,2} ` RANM 'A32' `K {3,2} ` RANM 'B32' `K {3,3} ` RANM 'A33' `K {3,3} ` RANM 'B33' `K Addit ion and s ubt racti on Fou r exampl es are sho wn be lo w u sing the matrices sto red abo ve (ALG mo de).
Pag e 9-4 In RPN m ode, try the following eight examp les: A22 ` B22 ` A22 ` B22 `- A23 ` B23 ` A23 ` B23 `- A32 ` B32 ` A32 ` B32 `- A33 ` B33 ` A33 ` B33 `- Multiplication There ar e different m ultiplication opera tions th at inv olve ma trices. These a re desc ribed nex t. T he ex ample s are s how n in alge braic mo de. Mu ltiplication by a scalar Some e xample s o f mu lt ipli cation o f a matrix by a s calar are sho wn bel ow . Matr ix- ve ctor mu ltiplica tion Matrix-vecto r mul tiplicatio n is pos sible only if the nu mber o f co lu mns of the matrix is equal to the length of the vector. A couple o f examples of matrix- ve ctor mult ip lic at ion follow: Vector- matrix mul tipli cation, o n the othe r hand, is no t def ined. T his mul tipl icatio n can be pe rfo rmed, ho we ver, as a s peci al cas e o f matrix mul tipl icatio n as def ined ne xt.
P a g e 9 -5 Matr ix mu ltipl ication Matrix multiplicatio n is defined by C m × n = A m × p ⋅ B p × n . Notice that matrix multiplicati on is only possib le if the number of columns in the first op eran d is equal to the n umber of rows of th e second op eran d. The g enera l term in the produc t, c ij , is de fi ned as . , , 2 , 1 ; , , 2 , 1 , 1 n j m i fo r b a c p k kj ik ij K K = = ⋅ = ∑ = Matrix mul tiplicatio n is not co mmutative, i.e., in general , A ⋅B ≠ B ⋅A . Further more, on e of the multiplica tions m ay not ev en exist . The following screen sh ots show t he results of m ultiplica tion s of the ma tri ces th at we st ored earlier: Term -b y -t erm mul t ip li ca t i on Term-by- term mul tipli cation of two matrice s of the same dime nsions is po ssibl e through the use of f unction HADA MARD. Th e result is, of course, another matrix of the same dimensio ns. This fu nction is available throu gh Function catalog ( ‚N ), or thro ugh the MAT RICES/OPERATIONS sub-me nu ( „Ø ). Applic ations of fu nctio n HADAMARD are pres ented ne xt:
Pag e 9-6 The ide ntity matr ix The identity matrix has the property that A ⋅ I = I ⋅ A = A . T o v e ri f y t h i s pr o p e rt y we present the following examp les using th e matr ices stored ea rlier on. Use fu nction IDN (find it in the M TH/MATR IX/MAKE menu ) to gene rate the identity matrix as sho wn here: The inverse matrix The inv erse of a squar e mat rix A is the matrix A -1 such tha t A ⋅A -1 = A -1 ⋅ A = I , where I is the identity matrix of the s ame dimensio ns as A . The inver se of a matrix is obtained in the calculator by using the inverse function, INV (i.e., the Y key). Exa mp les of the inv erse of some of the ma trices st ored ea rlier are presented next: To verify th e prop erties of th e inver se mat rix, we presen t the following mult ip lica ti ons :
Page 9 -7 Char acterizing a matr ix (Th e matrix NORM menu) The matrix NORM (NOR MALIZE) men u is accessed through the k eystroke sequ ence „´ . This menu is describe d in detai l in Chapter 10 o f the calc ulat or’s User’ s Guid e. S ome of t hes e fun cti ons are d escribed n ext. Function DET Function DET calculates the determinant of a sq u are ma t rix. F or exam p le, Func ti on TRACE Function TRACE calcu lates the trac e of square matrix, d e fined as the sum of the elemen ts in its main di ago n al, or ∑ = = n i ii a tr 1 ) ( A . Exampl es : Solution of linear systems A system of n line ar eq uat i ons in m variables c an be written as a 11 ⋅ x 1 a 12 ⋅ x 2 a 13 ⋅ x 3 … a 1,m- 1 ⋅ x m-1 a 1,m ⋅ x m = b 1 , a 21 ⋅ x 1 a 22 ⋅ x 2 a 23 ⋅ x 3 … a 2,m- 1 ⋅ x m-1 a 2,m ⋅ x m = b 2 , a 31 ⋅ x 1 a 32 ⋅ x 2 a 33 ⋅ x 3 … a 3,m- 1 ⋅ x m-1 a 3,m ⋅ x m = b 3 , . . . … . . . a n-1,1 ⋅ x 1 a n-1,2 ⋅ x 2 a n-1,3 ⋅ x 3 … a n-1,m-1 ⋅ x m-1 a n-1,m ⋅ x m = b n-1 , a n1 ⋅ x 1 a n2 ⋅ x 2 a n3 ⋅ x 3 … a n,m-1 ⋅ x m-1 a n,m ⋅ x m = b n .
Page 9 -8 This system of l inear equations can be written as a matrix equ ation, A n × m ⋅ x m × 1 = b n × 1 , if we define the fo llow ing matrix and vectors : m n nm n n m m a a a a a a a a a A ×             = L M O M M L L 2 1 2 22 21 1 12 11 , 1 2 1 ×             = m m x x x x M , 1 2 1 ×             = n n b b b b M Using the numerical s olve r for linear sys tems There a r e man y ways t o solv e a s yste m of linea r equa ti o ns with th e ca lcu lator . One possibility is through the numerica l sol ver ‚Ï . From th e n umerical solv er scr een, sho wn b e low ( le ft), s el ect the op t ion 4. Solv e lin sys. ., and press @@ @OK @@ @ . The fo llowing inp ut form wi ll be provid e (ri ght): To solv e the lin ear system A ⋅x = b , enter the matrix A , in the format [[ a 11 , a 12, … ], … [….]] in the A : field. A lso, enter th e vector b i n t he B: f i el d. When the X: field is highligh ted, press @SOLVE . If a sol ution is available, the solut i on vec t or x wi ll be sh own in the X: fie ld. Th e soluti on is a lso c opied t o sta ck lev e l 1. So m e exam ples foll ow . The s ystem of line ar eq uati ons 2x 1 3x 2 –5 x 3 = 13, x 1 – 3x 2 8x 3 = -13, 2x 1 – 2x 2 4x 3 = -6, can be written as the matrix equation A ⋅x = b , i f
Page 9 -9 . 6 13 13 , , 4 2 2 8 3 1 5 3 2 3 2 1           − − =           =           − − − = b x A and x x x This s ystem has th e same numb er o f equat i ons a s of unk nowns, and wi ll b e ref erred t o as a s quare syste m. In general, there sh ould be a unique solut ion to the system. The solution will be the p oint of intersecti on of the three plane s in t he c oord ina t e sy stem ( x 1 , x 2 , x 3 ) repr esented by t he thre e e qua tions. To e nter matrix A you can act ivate the Ma trix Writ er wh ile th e A: fie ld is selec ted. The followi ng sc r een sh ows the Matrix Writer us ed for entering matrix A , as well as the inp ut form for the n umerical solver after ente ring matrix A (press ` in the Matrix Writer): Press ˜ to se lect t he B : field . Th e vecto r b ca n be entered as a row v ector with a sing l e set of bracket s, i.e., [ 13, -1 3, -6] @@ @ O K@@ @ . Afte r entering matrix A and vector b, and with the X: field highlighted, we can press @SOLVE! to attempt a s olution t o this system of equations: A so luti on wa s found as sh own next.
Page 9 -10 Solu ti on w it h t he inverse matrix The s oluti on t o the system A ⋅ x = b , wh ere A is a sq uare ma trix is x = A -1 ⋅ b . For the exa mple us ed earlier, we c a n find the s olution in the calcul ator as follows (First e nter matrix A and vector b once mo re): So lutio n b y “ d ivisio n” of ma tr ices Whil e the opera ti on of divi sion is not de f ined f or ma tric es, we ca n use th e calculator’s / key t o “divide” vect or b by matri x A to so l ve f o r x in the matrix equation A ⋅x = b . The pr oc edur e f or th e ca se o f “d ivid ing” b by A is illu strated below for the example above . The procedure is shown in the f ollowing screen s hots (t ype in matric es A and vec to r b on ce m ore): Ref ere nc es Additional inf ormatio n on creating matrices, matrix oper ations , and matrix applic ations in l inear algebra is presente d in Chapters 1 0 and 11 of the calc ulat or’s User’ s Guide.
Pag e 10-1 Chapt er 10 Graphics In this chapte r we intro duc e so me o f the graphi cs capabil itie s o f the calc ulat or. We will p resen t g rap hi cs of fun cti ons in Ca rt esian coor din at es and pol ar coo rdinates , parametri c plo ts, graphics o f c onics , bar pl ots , scatterpl ots , and f ast 3D pl ots. Graphs option s in th e calc ulator To acces s the l ist of graphic f ormats availabl e in the cal cul ator , us e the keystroke sequen ce „ô ( D ) Please notice that if you are using the RPN mo de thes e two key s mu st be pre ss ed simu l taneo us ly to activate any of the graph fu nctions. Afte r activating the 2D/3D functio n, the calcu lato r will produ ce the PLOT SETUP window, which includes the TYPE fie ld as i ll us trated bel ow. Right in front o f the T YPE fiel d you will , most l ikel y, see the option Function highlig hted. This is th e default type of grap h for the calculator. To see the lis t of avail able graph types , pres s the so ft me nu k ey l abel ed @CHOOS . T hi s w i l l produ ce a dro p down me nu wi th the f ol lo wing o ptions (us e the up- and down- arrow keys to see all the options):
Pag e 10-2 Plotting an expr ession of th e form y = f (x) As an ex ample , le t's pl ot the fu nctio n, ) 2 exp( 2 1 ) ( 2 x x f − = π • First, en ter the P LOT SE TUP env ironm ent by pr essing, „ô . Make sure that the optio n Function is sel ected as the TYPE , and that ‘X’ is sel ect ed as the independe nt variabl e ( INDEP ). P ress L @@@OK@@@ to re t u r n to no r m al ca l c u l a t o r di s p l a y . T h e P L O T SET UP w i nd o w s h o u l d loo k simil ar to this: • Enter the PLO T envi ronmen t by p ressing „ñ (pre ss them simultaneousl y if in RPN mode). Press @ADD to get you into the equation writer. You wil l be prompted to fill the right-hand side of an equation Y1(x) =  . Type the fu nction to be plo tted so that the Eq uat ion W rit er show s th e followin g: • Press ` to retu rn to the PLOT SETUP window. The expres sion ‘Y1(X) = EXP(-X^2/2)/ √ (2*π ) ’ w ill b e h ig hli g ht ed. Pr ess L @@@OK@@@ to return to normal calcu lator display. • Ent er the P LOT WIND OW en vir onment by en terin g „ò (pr ess them simultaneously if in RPN mode). Use a rang e of –4 to 4 for H-
Pag e 10-3 VIE W, then press @AUTO to generate the V-VIEW au tomatical ly . The PLOT WINDOW screen looks as f oll ows: • Plo t the graph: @ERASE @DRAW (wait till the calcul ator finishes the graphs) • To see labels: @EDIT L @LABEL @MENU • To recover the f irst graphics me nu: LL @ ) PICT • To trace t he curve: @TRACE @@X,Y@@ . Then use the r igh t- and left-arrow keys ( š™ ) to move abo ut the cu rve. The coordinate s of the po ints you trace will be shown at the bottom of th e screen. Ch eck tha t for x = 1.05 , y = 0.0131. Also, check that for x = -1.48 , y = 0.034. Here i s pict ure of the graph in traci ng mode : • To re cover the menu , and retu rn to the PL OT WINDOW enviro nment, press L @CANCL , then L @@OK@@ . Gener ating a table of values for a func tion The combinations „õ ( E ) and „ö ( F ), pres sed simul taneo u sl y if in RPN m ode, let’s t he user pr oduce a t able of v alues of functions. For example, we will produce a table of the f unction Y(X) = X/(X 10), in the range -5 < X < 5 following these instructions:
Pag e 10-4 • We will gen er at e v alue s of th e fun ct ion f(x) , d efin ed ab ov e, for v alue s of x f rom –5 to 5, in increments o f 0.5. First, we need to ens ure that the graph type is se t to FUNCTION in the P LOT SE TUP screen ( „ô , press them simultaneously, if in RPN m ode). Th e field in front of th e Type opt ion wil l be hi ghlighte d. If this fi eld i s not al ready set to FUNCTION , press th e soft key @CHOOS and sel ect the FUNCTION opti on, t hen p ress @@@OK@@@ . • Next, p ress ˜ to highlight the f iel d in fro nt of the optio n EQ, and type the function expression: ‘X/(X 10)’ • To acce pt the changes made to the PLOT SETU P scre en pres s L @@@OK@@@ . You wil l be returne d to no rmal cal cu lato r displ ay. • • The next step is to access the Tab le Set-up screen by using the keyst roke combination „õ (i.e., soft key E ) – simulta n eously if in RP N m ode . This will prod uce a screen wh ere you can select t he star ting va lue ( Star t ) and the increm ent ( Step ). Enter the fo ll owing: 5\ @@@OK@@@ 0.5 @@@OK@@@ 0.5 @@@OK@@@ (i.e., Zoom factor = 0.5). Toggle the @  @CHK sof t menu ke y unti l a che ck mark appears in fro nt of the option Sm a ll Fon t if you so desi re. Then press @@@OK@@@ . This will return you to normal calc ul ator di splay . • • To see the ta ble, pr ess „ö (i.e., soft menu key F ) – s im ulta ne ously if i n RPN mo de. T his wi ll produ ce a tabl e o f valu es of x = -5, -4.5, …, and the co rrespo nding val ues of f (x), l ist ed as Y 1 by def au lt. Y ou can u se the up and do wn arrow key s to move abo ut in the tabl e. Yo u wil l no tice that w e did not have to indi cate an endi ng valu e f or the independe nt var iab le x. Thus, the tab le continues beyon d th e maxim um va lue fo r x suggested early, na mely x = 5. Some o ptio ns avail able whil e the table is vis ibl e are @ZOOM , @@BIG@ ,and @DEFN : • The @DEFN , when selected , shows the d efiniti on of the in depen dent variable .
Pag e 10-5 • • The @@BIG@ key simpl y change s the f o nt in the tabl e f rom s mall to bi g, and vice v ersa. Try it. • • The @ZOOM key , when pre sse d, produ ces a menu with the o ptions : In , Ou t , Dec imal , I nte ger , and Trig . Tr y the following exercises: • • W ith the optio n In highlighte d, pre ss @@@OK@@@ . The table is e xpanded s o that the x-increment is no w 0.25 rath er than 0.5. Simp ly, what the calculator does is to multiply th e origina l in cremen t, 0.5, by the zoom factor, 0.5, to produce th e new increm ent of 0.25. Thus, the zoom in option is usef ul when you want more resolu tion for the values of x in you r tabl e. • • T o inc rease the res ol utio n by an additio nal f acto r of 0.5 press @ZOOM , select In once mo re, and pres s @@@OK@@@ . The x-increm ent is now 0.0125. • • To recover the p revious x-increm ent, p ress @ZOOM — @@@OK@@@ to select the optio n Un-zoom . The x-increm ent is incr eased to 0.25. • • T o recover the original x-incr ement of 0.5 you can do a n un-zoom again, o r us e the option zoom out by pres sing @ZOUT @@@OK@@@ . • • The option D ecim al i n @ZOOM produces x-in crement s of 0.10. • • The option In teg er in @ZOOM produces x-in crement s of 1. • • T he option T rig in produces increments rel ated to f ractions o f π , thus bei ng useful when pr oduc ing ta b les of tri gon omet ric funct ion s. • To retu rn to no rmal c alcu lato r displ ay pres s ` . Fast 3D plots Fast 3D plots a re used to v isualize thr ee-dim ensiona l surfaces repr esented b y equation s of the form z = f(x,y). For exam ple, if you want to v isualize z = f(x,y) = x 2 y 2 , we can use the foll owing:
Pag e 10-6 • Pr ess „ô , simu ltaneo us ly if in RPN mode , to acces s to the PLOT SETUP win dow. • Change TYPE to Fast3D. ( @CHOOS! , find Fast3D, @@OK@@ ). • Pr ess ˜ and typ e ‘X^2 Y^2’ @@@OK@@@ . • Make su re that ‘X’ is sel ected as the Indep: and ‘Y’ as the Depnd: variable s. • Pr ess L @@@OK@@@ to retu rn to no rmal cal cul ator displ ay. • Pr ess „ò , simu ltane ou sl y if in RPN mo de, to acces s the PLOT WINDOW screen. • Keep t he default plot win dow ran ges to rea d: X-Left:-1 X-Right:1 Y-Near:-1 Y-Far: 1 Z-Low: -1 Z-High: 1 Step Indep: 10 Depnd: 8 Note : T he Step Indep: and Depnd: val ue s repre sent the numbe r of grid lines to be used in the p lot. The larg er these n umber, t he slower it is to pro duce the graph, althou gh, the times u tiliz ed fo r graphic generation are relativel y fas t. For the time being we’ll kee p the defaul t valu es of 10 and 8 f o r the Step data. • Pr ess @ERASE @DRAW to draw the three- dimensi onal su rface . The re su lt i s a wirefram e picture of the surface wit h the r eference coord ina te system shown a t the lower left corner of the screen. By using t he arr ow keys ( š™—˜ ) you can chan ge th e orient ation of th e surface. The orientatio n of the ref ere nce coo rdinate sy stem will change accordingly. Try changing the surf ace o rientatio n on yo ur o wn. The fol lo wing figu res show a couple of v iew s of th e gra ph :
Pag e 10-7 • When done , pres s @EXIT . • Pr ess @CANCL to retu rn to the PL OT WINDOW environme nt. • Change the Step data to read: Step Indep: 20 Depnd: 16 • Pr ess @ERASE @DRAW to see the surface p lot. Sa mple v iews: • When done , pres s @EXIT . • Pr ess @CANCL to return to PLOT WINDOW. • Pr ess $ , or L @@@OK@@@ , to return to no rmal calcul ator display. Try al so a Fast 3D pl ot f or the surf ace z = f (x,y) = sin (x 2 y 2 ) • Pr ess „ô , simu ltane ou sl y if in RPN mo de, to acces s the PL OT SETUP window. • Pr ess ˜ and typ e ‘SIN(X^2 Y^2)’ @@@OK@@@ . • Pr ess @ERASE @DRAW to draw the sl ope f iel d plo t. Pre ss @EXIT @EDIT L @ ) LABEL @MENU to see the plot unencumb ered b y the m enu an d with identifying labels.
Pag e 10-8 • Pr ess LL @ ) PICT to leave th e ED IT env ironm ent. • Pr ess @CANCL to retu rn to the PL OT WINDOW e nvironment. Then, pre ss $ , or L @@@OK@@@ , to return to no rmal calcul ator display. Ref erence Additional inf ormati on o n graphics i s avail able in Chapters 12 and 2 2 in the calcu lato r’s Us er’s Guide .
Pag e 11-1 Chapt er 11 Calculus A pplications In this Chapter w e disc us s appli cations of the calcu lato r’s fu nctions to operatio ns relate d to Calcu lu s, e.g., li mits, derivatives, integrals , power s eries, etc. The CA LC (Calcu lus) menu Many o f the f u nctions prese nted in this Chapter are co ntained i n the calcu l ator’s CALC menu , avail abl e throu gh the keys trok e seque nce „Ö (asso ciated with the 4 key): The f irst f ou r optio ns in this me nu are ac tual ly s ub- menu s that appl y to (1) derivative s and integral s, (2) l imits and powe r serie s, (3 ) diff erenti al equatio ns, and (4) graphics . The f unc tions i n entries (1) and (2) wil l be prese nted in this Chapter . Functi ons DERVX and INTVX are discu ss ed in detail l ater. Limits and der ivativ es Diff erenti al cal cul u s deal s wi th derivative s, o r rates of change, o f fu nctio ns and their appl icatio ns in mathe matical analys is. The derivative of a functi on is def ined as a li mit of the dif fe rence of a fu nctio n as the increment in the indep enden t va ria ble tend s to zero. Limit s are used a lso to check t he continui ty of fu nctions. Functi on lim The ca lcula tor prov i des fun cti on lim to cal cul ate l imits of f unctions . This fu nctio n u se s as inpu t an expre ssi on repre senti ng a fu nctio n and the valu e
Pag e 11-2 where th e limit is to b e calculated. Funct ion li m is availabl e through the command catalog ( ‚N~„l ) or t hr ough opt ion 2 . LIMI TS & SE RIES … of the CA LC m enu (see abov e). Function lim is ente red in AL G mode as lim(f(x),x=a) to calc ul ate the lim it ) ( lim x f a x → . In RPN mode, e nter the f unction f irst, then the express ion ‘x=a’, and f inal ly fu nctio n l im. Exampl es in ALG mo de are sho wn nex t, including so me limits to infinity. The infi nity symbol is ass ociated with the 0 key, i.e.., „è . Functi ons DERIV and DERVX The f unctio n DERIV is use d to take derivatives in terms of any inde pendent variable, while the fu nction DERVX take s derivatives with respe ct to the CAS def aul t variable VX (typi call y ‘X’). Whil e f uncti on DERVX i s avail able directl y in the CAL C menu , both f u nctions are availabl e in the D ERIV.&INT EG sub- menu within the CALCL menu ( „Ö ). Functio n DERIV requ ires a fu nctio n, say f (t), and an i ndepende nt variable , say, t, whil e f uncti on DERVX re quir es o nly a fu nctio n of VX. Ex ample s are s hown next in ALG mode . Re cal l that in R PN mode the argu ments mus t be entered bef ore the fu nctio n is appl ied.
Pag e 11-3 Anti- deriv atives an d integrals An anti-deri vative of a fu nction f (x) i s a fu nctio n F(x) su ch that f (x) = dF/dx . One way to repres ent an anti- derivative is as a indef inite i ntegral , i.e., C x F dx x f = ∫ ) ( ) ( if and o nly if, f(x ) = dF/dx, and C = cons tant. Func tions INT, INTVX, RI SCH, SIGMA and SI GMA VX The c alcu l ator pro vides fu nctio ns INT , INTVX , RI SCH, SIGMA and SIGM AVX to cal cul ate anti -derivatives of fu nctio ns. Fu nctio ns INT, RISCH, and SIGM A work with f uncti ons of any variabl e, whi le f u nctio ns INTVX , and SIGMAVX utilize f unctions of the CAS varia ble VX (typically, ‘x’). Functions INT and RISC H require, th erefore, not on ly the expression for the function being integrated, but al so the indepe ndent variabl e name. Functio n INT, re qui res also a v alue of x where the a nti-d eriv ativ e will be eva luated. Funct ions IN TVX and SIGMAVX require o nly the expres sio n of the fu nction to integrate in terms of VX. Fu nction INTVX is ava ilable in the CALC m enu, the other fu nctions are availabl e i n the co mmand catal og. Some e xampl es are sho wn ne xt in AL G mode :
Pag e 11-4 Ple ase no tice that fu nctio ns SIGM AVX and SIGM A are des igned f or integrands that involve s ome s ort o f integer f unctio n like the facto rial (!) function shown above. Their result is the so-called discrete deriva tive, i.e., one de fi ned fo r intege r numbe rs o nly . Definite integr als In a defi nite inte gral o f a f unc tion, the resu l ting anti-de rivative is e valuate d at the u pper and lo wer l imit o f an i nterval (a,b) and the eval uate d valu es subtracted. S ymbolically, we write ), ( ) ( ) ( a F b F dx x f b a − = ∫ where f(x) = dF/d x. To c alcu l ate def inite integral s o f f uncti ons us ing the CAS variabl e VX (typical ly, ‘X ’), u se f unc tion PREVAL (f(x ),a,b). Fo r exampl e, Infin ite ser ies A fu nctio n f(x ) can be ex panded into an infi nite se ries arou nd a point x =x 0 by using a Taylor’s series, nam ely,
Pag e 11-5 ∑ ∞ = − ⋅ = 0 ) ( ) ( ! ) ( ) ( n n o o n x x n x f x f , where f (n) (x) represe nts the n-th derivative of f(x) with res pect to x, f (0) (x) = f(x). If the v alue x 0 = 0, the serie s is ref erred to as a Macl aurin’s s eries. Functions TAYLR, TAYLR0, a nd S ERIES Functions TAYLR, TAYLR0, a nd S ERIE S a re used to generate Ta ylor pol ynomi als, as we ll as T aylo r se ries w ith res idual s. T hes e f unctio ns are availabl e in the CALC/L IMIT S&SERIES menu des cribed e arli er in this Chapter. Functio n TAYL OR0 pe rfo rms a M aclau rin s eries expans ion, i .e., abou t X = 0 , of an expre ssio n in the de fau lt i ndepende nt variable , VX (typi call y ‘X’). The expan sion uses a 4-th or der relat ive p ower, i.e., the d ifference between the highest and l ow est po we r in the ex pansio n is 4. For ex ample , Func tion TAY LR prod uces a Taylor series ex pa nsion of a fun cti on of a ny variable x abou t a po int x = a f or the orde r k spe cif ied by the us er. T hus , the function has the format TAYLR(f(x-a),x,k). For exam ple, Functi on SERIES pro duce s a Tay lo r pol yno mial us ing as arguments the fu nctio n f(x ) to be e xpanded, a variable name al one (f o r Macl auri n’s se ries ) or an expre ssio n o f the f o rm ‘variable = val ue ’ indicating the point o f e xpansi on o f
Pag e 11-6 a Tayl or s eries , and the o rder o f the serie s to be produ ced. Fu nctio n SERIES returns two ou tput ite ms a l ist with f ou r ite ms, and an ex press ion f or h = x - a, if the second argument in the function call is ‘x=a’, i.e., an expression for the incremen t h. Th e list returned a s the first output ob ject includes the following items: 1 - Bi-direc tional l imit of the fu nction at po int of e xpansio n, i.e., ) ( lim x f a x → 2 - An eq uiva lent va lue of the function n ear x = a 3 - Ex press ion f o r the Tay lo r pol yno mial 4 - Orde r of the res idual or re mainder Beca use of t h e rela t iv ely la rg e a m ount of outpu t, this f unction is e asier to han dle in RPN m ode. For exam ple, the following screen sh ots show th e RPN stack bef ore and after u si ng fu nctio n SERIES: Drop the c ontents of st ack l evel 1 by press ing ƒ , and then ente r µ , to decomp ose the list. Th e results are a s follo ws: In the r igh t-han d side figure a bove, we a re using th e line editor to see th e series expansion in d etail. Ref erence Additional def initio ns and appl icatio ns o f c alcu lu s o peratio ns are prese nted in Chap ter 13 in t he calculator’s User’ s Guide.
Pag e 12-1 Chapt er 12 Multi-variate C alculus A pplications Mul ti-variate cal cul us re fers to f unctio ns of two o r more variable s. In this Chapter we di scu ss bas ic co ncepts of mul ti- variate cal cu lu s: parti al de rivatives and mul tipl e inte grals . Pa rtia l d eriva ti ves To qu ick ly calcu l ate partial derivative s o f mu lti -variate fu nctio ns, u se the rul es of o rdinary derivatives with respect to the variable o f intere st, while cons idering al l othe r variable s as c onstant. For ex ample , () () ) sin ( ) co s( ), co s( ) co s( y x y x y y y x x − = ∂ ∂ = ∂ ∂ , You can u se the derivative functio ns in the cal cul ator: DERVX, DERIV, ∂ , desc ribed in detail in Chapte r 11 o f thi s Gu ide, to cal cul ate partial deri vatives (DERVX uses the CAS default variable VX, typically, ‘X’). Some examples of first- order partial derivatives are s hown next. T he fu nctions use d in the firs t two e xample s are f (x,y ) = SIN(y), and g(x ,y,z ) = (x 2 y 2 ) 1/2 sin(z).
Pag e 12-2 Multiple in tegrals A physical interpre tation of the dou ble integral of a fu nction f (x,y) over a regio n R o n the x- y plane is the vol u me of the s ol id bo dy co ntained u nder the su rface f( x,y) abo ve the regio n R. T he re gion R can be de scri bed as R = {a<x<b, f(x)<y<g(x)} or as R = {c<y<d, r(y)<x<s(y)}. Thu s, th e d ouble integra l can be written as ∫∫ ∫∫ ∫∫ = = d c y s y r b a x g x f R dydx y x dydx y x dA y x ) ( ) ( ) ( ) ( ) , ( ) , ( ) , ( φ φ φ Calcu l ating a dou ble integral in the c alcu lato r is straightf orward. A dou ble integral can be bu ilt i n the Equ ation Wr iter (s ee e xample in Chapter 2 ), as show n bel ow. This do u ble i ntegral i s cal cul ated dire ctl y in the Equ atio n Writer b y selecting t he enti re expression and using function @EVAL . The resul t is 3/2. Ref erence For additio nal de tails of mu lti- variate cal cu lu s o peratio ns and their applic ations see Chapter 14 in the cal cu lato r’s U se r Guide .
Pag e 13-1 Chapt er 13 Vector A nalysis Applic ations This c hapter des cribes the u se o f f unc tions HESS, DIV, and CURL, f o r calc ul ating o perati ons of vect or anal ys is. The del operator The f ol lo wing o perato r, ref erre d to as the ‘del ’ or ‘nabl a’ o perator , is a ve ctor - based o perato r that can be appl ie d to a sc alar o r vect or f unc tion: [] [] [] [] z k y j x i ∂ ∂ ⋅ ∂ ∂ ⋅ ∂ ∂ ⋅ = ∇ When appl ied to a scal ar fu nctio n we c an obtain the gradi ent of the f unc tion, and when appl ied to a vec tor f u nction we can o btain the di vergenc e and the curl of that f uncti on. A c ombinatio n o f gradie nt and diverge nce pro duce s the Lapl acian of a sc alar f unc tion. Gradient The gr ad ient of a scal ar f uncti on φ (x,y,z ) is a ve ctor f u nctio n def ined by φ φ ∇ = grad . Functio n HESS can be us ed to obtain the gradient of a fu nctio n.. The fu nctio n takes as inpu t a f unctio n of n indepe ndent variabl es φ (x 1 , x 2 , …,x n ), and a vecto r o f the fu nctio ns [‘x 1 ’ ‘x 2 ’…’x n ’]. The function returns the Hess ian matrix of the fu nction, H = [h ij ] = [ ∂φ / ∂x i ∂ x j ], th e gra dient of the fu nction with res pect to the n- variables, gra d f = [ ∂φ / ∂x 1 ∂φ / ∂ x 2 … ∂φ / ∂ x n ], and the l ist o f variable s [‘x 1 ’, ‘x 2 ’,…,’x n ’]. This function is easier to visualize in th e RPN m ode. Consider a s an exa mple the functi on φ (X ,Y,Z) = X 2 XY XZ , we’l l apply f u nction H ESS to this scal ar f iel d in the f ol lo wing exampl e: Thus, the grad ient is [2X Y Z, X, X].
Pag e 13-2 Alternativ ely, use f unction DERIV a s fol lows: Diverg ence The d iv erg ence of a v ect or func tion , F (x,y,z) = f(x,y,z) i g (x,y,z) j h (x,y,z) k , is def ine d by taking a “do t-pro duc t” of the del o perator with the f unc tion, i.e., F divF • ∇ = . Function D IV ca n be used to ca lculate the div ergen ce of a vecto r fi el d. For e xample , f or F (X ,Y,Z) = [XY,X 2 Y 2 Z 2 ,YZ], the div ergence is calculated, in ALG mode, as follo ws: DIV( [X*Y,X^2 Y^2 Z^2,Y*Z],[X,Y,Z]) Curl The curl o f a vector field F (x,y,z) = f(x,y,z) i g(x,y,z) j h(x,y,z) k ,is def ined by a “ cross-p rod uct” of th e del op era tor w ith th e ve ctor field, i.e., F F × ∇ = curl . The curl of vector field can be calculated with function CURL. For exam ple, for the function F (X,Y,Z) = [XY,X 2 Y 2 Z 2 ,YZ], the curl is calculated as follo ws: CURL([X*Y,X^2 Y^2 Z^2,Y*Z],[X,Y,Z]) Ref erence For additio nal info rmatio n on vec tor anal ys is appl icatio ns se e Chapter 15 in the calculator’ s User’s G uide.
Pag e 14-1 Chapt er 14 Diffe re ntial E qu atio ns In th is Cha pter we present exam ples of solving ordin ary d ifferential equa tions (ODE ) using cal culator functions. A dif ferential equ ation is an equation involving deriva tives of the indepen dent v ariable. In m ost cases, we seek the dependent f unctio n that satisf ies the diff erential equ ation. The CA LC/D IFF menu The DIFFERE NTIAL EQN S.. sub-menu within the CALC ( „Ö ) menu provides functions for the solu tion of dif ferential equations. The menu is listed below with system fl ag 117 set to CHOOSE boxes: These functions are briefly described n ext. They will be described in m ore detail in l ater parts o f this Chapte r. DE SOLV E: Differential Equation SOLVE r, sol ves differential equations, when pos sibl e ILAP: Inv erse LAPlace tran sform, L -1 [F(s)] = f(t) LAP: LAPlace tra nsform, L[f(t)]=F(s) LDEC: Linear Diffe rential Equation Command Solution to lin ear an d non -linear equations An equ ation in whi ch the depende nt variable and al l its pertinent derivative s are of the first degree is referred to as a linear differential equation . Otherwise , the equatio n is said to be no n-l inear .
Pag e 14-2 Function LDEC The calcu l ator provide s f unc tion LDEC (Line ar Dif fe rential Equatio n Command) to find the ge neral so lu tio n to a line ar ODE of any o rder w ith co nstant coeff icients, whether it is homogen eous o r not. This function requires you to prov ide two p ieces of input: • the right-hand s ide o f the ODE • the characteris tic e quatio n o f the ODE Both of these in puts must b e giv en in terms of th e default ind epend ent v aria ble for the calcu lator’s CAS (typically X). The ou tput from the fu nction is the general solu tion of the ODE . The examples belo w are shown in the RPN mode : Exam ple 1 – To sol ve the homogeneous ODE d 3 y/dx 3 -4 ⋅ (d 2 y/dx 2 )-11 ⋅ (dy/dx) 30 ⋅ y = 0. Enter: 0 ` 'X^3-4*X^2-11*X 30' ` LDEC The solution is (figure put together from EQW screensh ots): where cC0, cC1, and cC2 are constants of integra tion. This result can be re- written as y = K 1 ⋅ e –3x K 2 ⋅ e 5x K 3 ⋅ e 2x . Exam ple 2 – Using the function LDE C, sol ve the non-homogeneous ODE: d 3 y/dx 3 -4 ⋅ (d 2 y/dx 2 )-11 ⋅ (dy/dx) 30 ⋅ y = x 2 . Enter:
Pag e 14-3 'X^2' ` 'X^3-4*X^2-11*X 30' ` LDEC The solution is: which can be simplified to y = K 1 ⋅ e –3x K 2 ⋅ e 5x K 3 ⋅ e 2x (450 ⋅ x 2 330 ⋅ x 241)/13500. Func ti on DESOLV E The cal cu lato r pro vides f u nctio n DESOLVE (Diffe rential Equatio n SOLVEr) to solve certain types of differential equations. The fu nction requires a s input the diff erential equ ation and the u nkno wn fu nction, and retu rns the so lu tion to the equ ation if avai labl e. Yo u can also pro vide a vector co ntaining the diff erential equatio n and the initial co nditions , ins tead o f only a dif f erential e q u a t i o n , a s i n p u t t o D E S O L V E . T h e f u n c t i o n D E S O L V E i s a v a i l a b l e i n t h e CALC/DIFF menu. Exampl es of DESOLVE appl icati ons are sho wn bel ow using RPN mode. Exam ple 1 – Sol ve the first-o rder ODE: dy/dx x 2 ⋅ y(x) = 5. In the calculator use: 'd1y(x) x^2*y(x)=5' ` 'y(x)' ` DESOLVE The so lu tio n provi ded is {‘y = (5*INT (EXP(xt^3/3),xt,x) CC0)*1/EXP(x^3/3))’ }, which simpl ifies to ( ) . ) 3 / exp( ) 3 / exp( 5 ) ( 0 3 3 C dx x x x y ⋅ ⋅ − ⋅ = ∫
Pag e 14-4 Th e variable ODETYPE You wi ll noti ce in the s of t-men u k ey label s a new variable call ed @ODETY (ODE TYPE). This variable is produced with the call to the DESOL function and holds a string showing the type o f ODE used as input for DES OLVE . Press @ODETY to obtain the string “ 1st order linear ”. Exam ple 2 – Solving a n equation with initial conditions. Solve d 2 y/dt 2 5y = 2 cos(t/2), with initial conditio ns y(0) = 1.2, y’(0) = -0.5. In the calculator, u se: [‘d1d1y(t) 5*y(t) = 2*COS (t/2)’ ‘y(0) = 6/5’ ‘d1y(0) = -1/2’] ` ‘y(t)’ ` DESOLV E Notice that the initial conditio ns were changed to their Exact expressions, ‘y(0) = 6/5’, rather th an ‘y(0)=1.2’, a nd ‘d1y(0) = -1/2’ , rather than , ‘d1y(0) = -0.5’. Changin g to these Exact expressions facilitates the solu tion. Note : To o btain fracti onal e xpress io ns f o r de cimal val ue s u se f u nction  Q (See C hap ter 5). Press µµ to simplify the resul t. The solution is: i.e., ‘y(t) = -((19*5*SIN( √ 5*t)-(148*COS( √ 5*t) 80*COS(t/2)))/190)’. Press J @ODETY to get the s tring “ Linear w/ cst coeff ” for the ODE type in thi s case.
Pag e 14-5 Laplac e Tran sforms The Laplace tran sform of a f unction f(t) produces a fu nction F(s) in the imag e do m a in t ha t c an be u ti l i z e d t o f i n d t h e s o l u t i o n o f a l i ne a r di f f e re n t i al eq u a t io n involving f(t) through algebraic meth ods. The steps involved in thi s applicati on are three : 1. Use of the Laplace transform converts the linear ODE involving f(t) into an alge braic e quatio n. 2. The unknown F(s) is solved for in the image domain through algebr aic manipul atio n. 3. An invers e L aplac e trans fo rm is u se d to conve rt the image f u nctio n f ou nd in step 2 into the solutio n to the diffe rential equation f(t). Laplace trans form and in verses i n the cal culat or The cal cul ato r provides the fu nctions LAP and ILAP to cal cu late the Lapl ace transf orm and the inve rse L aplace transfo rm, re spec tivel y, of a f unctio n f (VX), where VX is the CAS defaul t independ ent v aria ble (typ ically X). The c a l c u l a t o r r e t u r n s t h e t r a n s f o r m o r i n v e r s e t r a n s f o r m a s a f u n c t i o n o f X . T h e fu nctio ns L AP and ILAP are available under the CALC/DIFF menu. The examples are worked out in the RPN mode, b ut translating them to ALG mode is straightfo rward. Exam ple 1 – Y o u c a n g e t t h e d e f i n i t i o n o f t h e L a p l a c e t r a n s f o r m u s e t h e followin g : ‘ f(X) ’ ` LAP i n R P N m o d e , o r LAP(F(X)) in ALG mode. The calculator returns the resul t (RPN, left; ALG, right): Compa re these expressions with the one given earlier in the definition o f the Laplace tran sform, i.e., ∫ ∞ − ⋅ = = 0 , ) ( ) ( )} ( { dt e t f s F t f st L
Pag e 14-6 and yo u wil l notice that the CAS def aul t vari able X in the e quatio n writer s c r e e n r e p l a c e s t h e v a r i a b l e s i n t h i s d e f i n i t i o n . T h e r e f o r e , w h e n u s i n g t h e f u n c t i o n L A P y o u g e t b a c k a f u n c t i o n of X , w h i c h i s t h e L a p l a c e t r a n s f o r m o f f(X). Exam ple 2 – Determin e the inverse La place transform of F(s) = sin(s). Use: ‘1/(X 1)^2’ ` ILAP The calcul ator returns the resu lt: ‘X /EXP(X)’, mean ing that L -1 {1/(s 1) 2 } = x ⋅ e -x . Fouri er s eries A complex Fourier series is defined by the foll owing expression ∑ ∞ −∞ = ⋅ = n n T t in c t f ), 2 exp( ) ( π where ∫ ∞ − − −∞ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = T n n dt t T n i t f T c 0 . ,... 2 , 1 , 0 , 1 , 2 ,..., , ) 2 exp( ) ( 1 π Function FOURIER Function FOURIER provides the co efficient c n of t he co mp lex- form of th e Fourier series given the fu nction f(t) and the value o f n. The f unction FOURIER requires you to store the value of the period (T) of a T-periodic function into th e C A S va ri a bl e P ER I O D be f o re c al l in g th e fu nc t i o n. Th e f u n c ti o n FO U R I ER i s availabl e in t he DERI V su b-me nu withi n the CALC menu ( „Ö ). Fouri er s eri es for a qu adrati c fu nct i on Determ ine the coefficients c 0 , c 1 , and c 2 for the f unction g(t) = ( t-1) 2 (t-1), with period T = 2.
Pag e 14-7 Using the ca lcul ator in ALG mode, first we define functio ns f(t) and g(t): Next, we move to the CASD IR sub-directory u nder HOME to chang e the value of varia ble PERIOD, e.g., „ (hold) §`J @ ) CASDI ` 2 K @PERIOD ` Retu rn to the s ub- direct ory where you de fine d fu nctio ns f and g, and cal cu late the coefficients. Set CAS to Complex mode (see chapter 2) b efore trying the exercises. Fu nction COLLECT is available in the ALG menu ( ‚× ).
Pag e 14-8 Thus, c 0 = 1/3, c 1 = ( π⋅ i 2)/ π 2 , c 2 = ( π⋅ i 1)/(2 π 2 ). The Fourier series with thr ee elements will be written as g(t) ≈ Re[(1/3) ( π⋅ i 2)/ π 2 ⋅ exp(i ⋅π ⋅ t) ( π⋅ i 1)/(2 π 2 ) ⋅ exp(2 ⋅i ⋅π ⋅ t)]. Ref erence For additional definit ions , appl icatio ns, and exe rcise s on so lving dif fe rential equati ons , u sing L aplace transf o rm, and Fo uri er s erie s and transforms , as well as numeric al and graphical methods, see Chapter 16 in the c alcu lato r’s User’s Guide.
Pag e 15-1 Chapt er 15 Probability D ist ributions In thi s Chapter w e provide exampl es of applic ations of the pre- def ined probabil ity dis tributi ons in the calc ul ator. The M TH /PR OBA BILIT Y.. sub-men u - part 1 The MTH/PROBA BILITY.. sub-m enu is accessible th rough th e keystroke sequence „´ . With system flag 117 set to CHOOS E boxes, the fo ll ow ing fu nctio ns are availabl e in the PROBAB ILITY .. me nu: In this se ctio n we dis cus s fu nctio ns COM B, PERM , ! (f actorial ), and RAND. Factori als, combi nati ons, and permu tat ions The f acto rial of an intege r n is def ine d as: n! = n ⋅ (n-1) ⋅ (n-2)…3 ⋅ 2⋅ 1. By definition, 0! = 1. Factori als are us ed in the calcu l ation o f the numbe r of permutatio ns and combinati ons o f o bject s. Fo r exampl e, the number of permu tations of r obje cts fr om a set of n d ist inc t ob ject s is ) /( ! ) 1 )...( 1 )( 1 ( r n n r n n n n r n P − = − − − = Also , the nu mber o f co mbinations of n objects taken r at a time is )! ( ! ! ! ) 1 )...( 2 )( 1 ( r n r n r r n n n n r n − = − − − =        
Pag e 15-2 We can cal cu late combinati ons , permu tations , and f actorial s w ith functio ns COMB, PERM, and ! from the MTH/PROBABILITY.. sub-menu. The operation of those fun ct ions i s desc rib ed n ext : • COMB(n ,r): Ca lculates the n umber of com bina tions of n items taken r at a time • PERM( n,r): Calculates the number of permuta tion s of n items t aken r at a time • n!: Factorial of a posi tive integer. For a non-intege r, x! returns Γ (x 1), wh ere Γ (x) is the Gam ma funct ion (see C hap ter 3). Th e facto rial symbo l (!) can be entere d also as the key strok e co mbination ~‚2 . Example of appl icatio ns of thes e f unc tions are sho wn ne xt: Random numbers The cal cul ator pro vides a rando m numbe r g enerato r that re turns a u nifo rml y distribu ted rando m real nu mber be tween 0 and 1. T o ge nerate a rando m number , use function RA ND from th e MTH/PROBA BILITY sub-m enu. The follo wing screen shows a numb er of ran dom n umbers p rod uced using RA ND . (Note: Th e rand om n umbers in your calculator will differ from t hese). Additional detail s o n random nu mbers in the c alcu lato r are pro vided in Chap ter 17 of the User ’s Guide. S pecifically, the use of function RD Z, to re- start lists of rand om n umbers is p resented in det ail in Ch apt er 17 of th e User’s Gui de .
Pag e 15-3 The M TH /PR OB men u - par t 2 In this se ctio n we dis cus s f ou r conti nuo us probabil ity dis tributio ns that are commonl y us ed fo r probl ems re lated to statistical infe rence: the normal distributio n, the Student’s t dis tributio n, the Chi-squ are ( χ 2 ) distribu tion, and the F-dis tributi on. T he fu nctio ns provide d by the cal cul ator to evalu ate probabil ities fo r these distribu tions are NDIST, UT PN, UTPT , UT PC, and UTPF. These function s are con tain ed in t he MTH/PROB ABILITY m enu intr oduced earlier in t his cha pter. To see t hese functions a ctiv ate th e MTH menu: „´ and sel ect the PROBAB ILITY o ption: The N ormal dis t ribu ti on Functio ns NDIST and UTPN rel ate to the Normal distribu tion with mean µ , and variance σ 2 . To c alcu late the val ue of probabil ity de nsity f u nction, or pdf , of the f (x) f or the normal distribu tion, use fu nctio n NDIST( µ ,σ 2 ,x). Fo r exampl e, che ck that for a normal distribution, NDIS T(1.0,0.5,2.0) = 0.20755374. This function is use fu l to pl ot the No rmal di stribu tion pdf ). The cal cul ator al so provides fu nction U TPN that cal cul ates the u pper-tail normal distribu tio n, i.e. , UTPN( µ ,σ 2 , x) = P(X>x) = 1 - P(X<x), where P() repres ents a probabi lity . For ex ample, check that fo r a normal distribu tio n, with µ = 1.0, σ 2 = 0.5, UTPN(1.0,0.5,0.75) = 0.638163. The S tude nt-t d istrib utio n The S tudent -t, or sim ply, the t -, distr ibution ha s one pa ram eter ν , known a s the deg rees of freedom of the d istri bution . The calc ulator p rovid es for va lues of the uppe r-tail (cumu l ative) distribu tio n fu nction fo r the t-dis tributio n, fu nction
Pag e 15-4 UTPT, giv en t he pa ram eter ν an d th e value of t, i.e., UTPT( ν ,t) = P (T>t) = 1- P(T<t). For example, UTPT(5,2.5) = 2.7245…E -2. The C hi-sq uar e d istri butio n The Ch i-square ( χ 2 ) distribu tio n has one parameter ν , known a s the d egrees of freedom. Th e calculator p rov ides for va lues of the u pper-t ail (cumulativ e) distributio n fu nction f or the χ 2 -distribution u sing [UTPC] given the value of x and the paramete r ν . The de finitio n of this fu nction is, therefo re, UTPC( ν ,x) = P(X>x) = 1 - P(X<x). For exam ple, UTPC(5, 2.5) = 0.776495… The F di s tri but ion The F dis tribu tion has two parameters ν N = numer ator d egr ees of freedom, and ν D = d enomin ator degr ees of freedom. Th e calculator p rov ides for valu es o f the uppe r-tail (cumu lative) distribu tion f unc tion f or the F di stribu tion, function UTP F, g iven the p ara met ers ν N and ν D, a nd the v alue of F. The definitio n of this fu nction is, theref ore, UTPF( ν N,ν D, F) = P( ℑ >F) = 1 - P( ℑ <F). For example, to calculate UTPF(10,5, 2.5) = 0.1618347… Ref erence For additio nal pro babil ity dist ributio ns and pro babili ty applic ations , ref er to Chap ter 17 in t he calculator’s User’ s Guide.
Pag e 16-1 Chapt er 16 Stat istic al Applicat ions The cal cu lato r provide s the f ol l owing pre -pro grammed statistical feature s accessi ble thro ugh the k eystro ke co mbination ‚Ù (the 5 key): Enter ing data Applic ations numbe r 1, 2 , and 4 f rom the lis t above requ ire that the data be availabl e as c ol umns of the matri x Σ DAT. This can be accomplished by entering the data in col umns us ing the matrix writer, „² , and then u si ng func tion s S TO Σ to sto re the matrix into Σ DAT . For example, enter the fo llowing data using th e Matr ix Writer (see Ch apt ers 8 or 9 i n this gu ide), and sto re the data i nto Σ DAT : 2.1 1.2 3.1 4.5 2.3 1.1 2.3 1.5 1.6 2.2 1.2 2.5. The screen m ay look like this: Notice the variabl e @£DAT listed in the soft menu keys. Calculatin g single-v ariable statistic s After ente ring the colu mn vector into Σ DAT, p ress ‚Ù @@@OK@@ to sel ect 1. Single-var.. The follow ing i np ut form wi ll be p rov ided:
Pag e 16-2 The f orm l ist s the data in Σ DAT, shows that col umn 1 is selected (ther e is only one column in t he current Σ DAT). Move abou t the f orm with the arrow keys, and press the @  CHK@ sof t menu ke y to sel ect tho se measu res (Me an, Standard Deviatio n, Variance , To tal nu mber o f data points, Maximu m and Minimu m valu es) that y ou want as o utpu t o f this program. When ready, press @@@OK@@ . The s el ecte d val u es wi ll be l iste d, appropriate ly label ed, in the s creen o f you r calcu lato r. Fo r exampl e: Sample vs . po pul atio n The pre -pro grammed fu nctio ns f or si ngle -variabl e stati stics us ed abo ve can be applie d to a f inite popu latio n by s ele cting the Type: Population in the SINGLE-VARIABLE STATISTICS screen. The m ain d ifference is in th e v alues of the variance and standard de viation w hich are calcu l ated u sing n in the denomi nator of the variance , rather than (n- 1). Fo r the exampl e abo ve, u se now the @CHOOSE sof t menu ke y to sel ect po pu latio n as Type: and re-cal cu late measu re s:
Pag e 16-3 Obtain ing fr eque nc y distr ibutions The appl icatio n 2. Frequencies.. i n t h e S T A T m e n u c a n b e u s e d t o o b t a i n fre quenc y distri butio ns f or a s et of data. T he data mus t be pre sent in the form of a colum n v ect or st ored i n v ar iab le Σ DAT. To get started, press ‚Ù˜ @@@OK@@@ . The resulting input form contains the follo wing fields: Σ DAT : the matrix containing the data o f intere st. Col : the column of Σ DAT that is unde r scru tiny. X-Min : the minimu m cl ass bo undary to be us ed in the fre quenc y distribu tio n (defau l t = -6 .5). Bin Count : the num ber of classes used in th e frequency d istrib ution (default = 13). Bin Width : the uniform width of each class in t he frequency distribu tio n (defau l t = 1). Given a s et o f n data valu es: {x 1 , x 2 , …, x n } lis ted in no partic ul ar order, o ne can grou p the data into a nu mber of cl ass es , or bins by cou nting the freq uen cy or nu mber o f val u es c orres ponding to each clas s. The applicati on 2. Frequencies.. in the STAT menu will perf orm this f reque ncy count, and will kee p track o f those valu es that may be be lo w the minimum and above the maximu m clas s bou ndarie s (i.e ., the o utl iers ). As an exa mple, generate a relatively large dat a set, say 200 points, b y using the comman d RANM({200,1}), and storing the result into variab le Σ DAT, by usin g fun ct ion S TO Σ (see exampl e abo ve). Ne xt, o btain si ngle -variable info rmation using: ‚Ù @@@OK@@@ . The results are:
Pag e 16-4 This i nfo rmation indi cates that o ur data range s f rom - 9 to 9 . To produ ce a fre que ncy dist ributi on we wil l u se the interval (-8 ,8) divi ding it into 8 bins o f width 2 ea ch. • Sel e ct the pro gr am 2. Frequencies.. by u sing ‚Ù˜ @@@OK@@@ . The data is al ready l oade d in Σ DA T, and the op tion Col s hould hold th e v alue 1 since w e have onl y one col umn in Σ DAT . • Change X- Min to -8, Bin Co unt to 8, and Bin Width to 2, then pres s @@@OK@@@ . Usi ng the RPN mo de, the res ul ts are sho wn in the s tack as a co lu mn vecto r in stack le vel 2, and a ro w ve cto r of two co mpone nts in s tack le vel 1. T he vector in stac k level 1 is the num ber of outliers outside of the in terv al where the fre quency cou nt was perf ormed. Fo r this case , I get the values [ 14. 8.] indicating that there are, in the Σ DAT ve ctor , 14 valu es smal le r than -8 and 8 lar ge r t ha n 8 . • Press ƒ to dr op t he v ector of outlier s from th e stac k. The r ema in ing resu lt i s the f requ ency co unt o f data. The bins for this frequency distribution will be: -8 to -6, -6 to -4, …, 4 to 6, and 6 to 8, i.e., 8 of them, with the freq uencies in th e column vector in the stack, namely (f or this case): 23, 22, 22, 17, 26, 15, 20, 33. This mean s that th ere are 23 va lues in the bin [ -8,-6], 22 in [-6,-4], 22 in [-4,- 2], 17 in [-2,0], 26 in [0,2], 15 in [ 2,4], 20 in [4,6] , and 33 in [6,8]. You can als o che ck that adding al l these valu es plu s the ou tlie rs, 1 4 and 8, s how above, you will get the total number of el ements in the sa mple, namely, 200. Fitting data to a func tion y = f(x) The pr ogr am 3. Fit data.. , avail able as optio n nu mber 3 i n the STAT menu , can be u sed to fit l inear, lo garithmic, e xpone ntial, and powe r functio ns to
Pag e 16-5 dat a sets ( x,y), st ored i n columns of th e Σ DAT matrix. For this application, you ne ed to have at l east two co lu mns in you r Σ DAT vari able . For ex ample, to f it a li near rel ations hip to the data s hown i n the table bel ow: x y 0 0.5 1 2.3 2 3.6 3 6.7 4 7.2 5 11 • First, enter the two co lu mns of data into variable Σ DAT by using the matrix writer , and fu nctio n STO Σ . • To access the program 3. Fit data.. , use the follo wing keyst rokes: ‚Ù˜˜ @@@OK@@@ The inpu t fo rm will show the current Σ DAT , already lo aded. If nee ded, change you r se t up s creen to the fo ll owi ng param eters for a linear fitting: • To o btain the data f itting press @@OK@@ . The out put from th is p rog ra m, show n be lo w f or ou r particu lar data s et, co nsis ts of the f ol lo wing three lines in RPN mode: 3: '0.195238095238 2.00857242857*X' 2: Correlation: 0.983781424465 1: Covariance: 7.03
Pag e 16-6 Level 3 shows the form of the eq uation . Level 2 shows t he sam ple corr elation coefficient, and level 1 shows the cova ria nce of x-y. For d efinition s of these para meter s see Chap ter 18 in the User’s G uide. For additio nal inf ormati on o n the data-f it f eature of the cal cu lato r see Chapter 18 in the User’ s Guide. Obtain ing addition al summary statistic s The appl icatio n 4. Summary stats.. i n t h e S TA T m e n u c a n b e u s e fu l i n s o m e calcu l ations f or s ample s tatistic s. T o get s tarted, pre ss ‚Ù once mor e, move to the f ou rth optio n using the do wn-arrow k ey ˜ , and pres s @@@OK@@@ . The re sul ting inpu t fo rm contains the fo ll ow ing fie lds : Σ DAT: the matrix containing the data of inte rest. X-Col, Y-Col: these optio ns appl y o nly w hen yo u have more than two col umns in the matrix Σ D A T . B y d e f a u l t , t h e x c o l u m n i s colum n 1, and th e y colum n i s colum n 2. I f you h av e on ly one co lu mn, then the o nly s etting that makes sense is to have X-Col: 1 . _ Σ X _ Σ Y… : summa ry sta tist ics th at you can c hoose as resul ts of t his program by che cki ng the appropri ate f iel d us ing [  CHK] when th at field is selected. Many of th ese summa ry st atisti cs are used t o calculate sta tisti cs of tw o variable s (x,y) that may be rel ated by a f unctio n y = f (x). There fo re, thi s program can be tho u ght of f as a companio n to program 3. Fit data.. As an ex ample, fo r the x- y data cu rrentl y in Σ DAT, obtain al l the su mmary statist ics. • To access the summary st ats… op ti on, use: ‚Ù˜˜˜ @@@OK@@@ • Sele ct the co l umn nu mbers corre sponding to the x - and y- data, i.e., X-Col : 1, and Y- Col : 2. • Using t he @  CHK@ key select a ll th e opt ions for outp uts, i .e., _ Σ X, _Σ Y, etc.
Pag e 16-7 • Press @@@OK@@@ to obtain the f ol lo wing resul ts: Confiden ce in ter vals The appl icatio n 6. Conf Inter val can be acce ss ed by u si ng ‚Ù— @@@OK@@@ . The appl icatio n of f ers the fo ll ow ing optio ns: These options are to be inte rpreted as f ol lo ws: 1. Z-INT: 1 µ .: Singl e sampl e co nfi dence i nterval fo r the po pul ation me an, µ , wi th kn own pop ulati on v ar ia nce , or for lar ge sa mp les wit h un known popu latio n variance . 2. Z-INT: µ1−µ2 .: Confidence in terva l for the difference of th e population means, µ 1 - µ 2 , with e ither kno wn popu latio n variances, or f or l arge sample s wi th unk nown po pu latio n variances . 3. Z-INT: 1 p.: Single sampl e co nfidenc e interval for the propo rtion, p, for lar ge s am ple s wit h un know n p opula ti on v a ria n ce.
Pag e 16-8 4. Z-INT: p 1− p2 .: C onfid enc e int erv al for th e differ enc e of two p ropor ti ons, p 1 -p 2 , for la rg e sa m ples wit h un know n p opula ti on v a ria nc es. 5. T-INT: 1 µ . : Single sample co nfide nce inte rval f or the popu latio n mean, µ , for sma ll sa mp les wi th unkn own pop ulat ion v ar ia nce . 6. T-INT: µ1−µ2 .: Confidence interv al fo r the d ifference of the p opulation means, µ 1 - µ 2 , for small samp les with unkn own pop ulation v ar ianc es. Exampl e 1 – Determ ine th e centered confidence int erva l for the mean of a popu latio n if a s ample of 60 e leme nts indic ate that the me an valu e o f the samp le is  x = 23 .2, and i ts s tandard deviatio n is s = 5.2. U se α = 0.05. The confidence lev el is C = 1- α = 0.95. Sele ct cas e 1 f rom the menu sho wn above by pre ssi ng @@@OK@@@ . Enter the values required in the input fo rm as shown: Press @HELP to obta in a screen explain ing t he mea ning of th e confidence interval in terms o f rando m number s gene rated by a cal cu lato r. To scro l l down th e resulting screen use th e d own-arr ow key ˜ . Press @@@OK@@@ when done with the help screen. Th is will retu rn you to the screen shown above. To calculate the confid ence interv al, pr ess @@@OK@@@ . The result shown i n the calculator is: Press @GRAPH to se e a graphical dis play o f the conf idenc e inte rval inf ormati on:
Pag e 16-9 The graph s hows the s tandard normal distribu tio n pdf (pro babil ity dens ity fu nction), the l ocatio n of the critical po ints ± z α/2 , the mean val u e (23 .2) and the corresponding interva l limits (21.88424 and 24.51576). Press @TEXT to return t o the pr evious results screen , and /or pr ess @@@OK@@@ to exit the confidence in ter va l en v iron m ent . Th e re sults w ill b e list ed in th e ca lcula tor ’s d isp la y. Additional ex ample s of co nfide nce inte rval c alcu lati ons are pres ented in Chap ter 18 in t he calculator’s User’ s Guide. Hypothesis testing A hypothe sis is a de claratio n made abo u t a popu latio n (f or instance , with respe ct to its me an). Acceptanc e of the hypo thes is is based o n a statis tical test o n a sample taken f ro m the popu l ation. T he co nsequ ent acti on and decis ion -maki ng are cal le d hypothe sis tes ting. The cal cu lato r provide s hypo thesi s tes ting procedu res under applic ation 6. Conf Interval can be acce sse d by u sing ‚Ù—— @@@OK@@@ . As with the calcul ation of co nfidence intervals, discu ssed earlier, this p rogram offers t he followin g 6 opt ion s: Thes e opti ons are interprete d as in the conf idence interval applic ations :
Pag e 16-10 1. Z-Test: 1 µ .: S ingle samp le hypothesis testing for the population mean , µ , wit h know n p opula tion v ari an ce, or for la rg e sam ples w ith unkn own popu latio n variance . 2. Z-Test: µ1−µ2 .: Hypothes is tes ting for the diff erence of the populatio n means, µ 1 - µ 2 , with e ither kno wn popu latio n variances, or f or l arge sample s wi th unk nown po pu latio n variances . 3. Z -Te st: 1 p.: Single sample hypo thesis te sting fo r the proportio n, p, fo r lar ge s am ple s wit h un know n p opula ti on v a ria n ce. 4. Z-Test: p 1− p 2 .: Hypothes is testing fo r the diff erence of two propo rtions, p 1 -p 2 , for la rg e sa m ples wit h un know n p opula ti on v a ria nc es. 5. T-Test: 1 µ .: Single sample hypothesis testin g for the population mean, µ , for sm all s am ples wit h un know n p opula ti on v a ria n ce. 6. T-Test: µ1−µ2 .: Hypothe sis te sting fo r the diff erence of the popu latio n means, µ 1 - µ 2 , for small samp les with unkn own pop ulation v ar ianc es. Try the following exercise: Exampl e 1 – For µ 0 = 150, σ = 10,  x = 158, n = 50, for α = 0.05, test the hypothe sis H 0 : µ = µ 0 , against the alte rnative hypothe sis , H 1 : µ ≠ µ 0 . Press ‚Ù—— @@@OK@@@ to acces s the co nfidence interval f eature in the calculator. Press @@@OK@@@ to select o ption 1. Z-Test: 1 µ . Enter the f ol l owing data and pre ss @@@OK@@@ : You are the n aske d to se le ct the al ternative hypo thes is:
Pag e 16-11 Sel e ct µ ≠ 150 . Then, pr ess @@@OK@@@ . The result is: Then, we rej ect H 0 : µ = 150 , against H 1 : µ ≠ 150 . The test z v alue is z 0 = 5.656854. The P-v alue is 1.54 × 10 -8 . The critical values of ± z α /2 = ± 1.959964, corresponding to critical  x rang e of {147.2 152.8}. This info rmatio n can be o bserve d graphical ly by pressing the soft- menu key @GRAPH : Ref erence Additional material s o n stati stical analy sis, incl uding de fini tions of concepts, and advanc ed stati stical appli cations , are avail abl e in Chapte r 18 in the User’s Guide.
Pag e 17-1 Chapt er 17 Num be r s in Di ffer e nt Base s Bes ides ou r decimal (base 10, di gits = 0 -9) number sys tem, yo u can work with a binary s yst em (base 2, digits = 0, 1), an o ctal sys tem (base 8, digi ts = 0-7 ), o r a hexade cimal sys tem (bas e 16 , digits =0- 9,A- F), among o thers . The same way th at the decim al integer 321 mea ns 3x10 2 2x10 1 1x10 0 , the number 100110, in bin ary notation, mean s 1x2 5 0x2 4 0x2 3 1x2 2 1x2 1 0x2 0 = 32 0 0 4 2 0 = 38. The BASE m enu The BAS E m enu is accessible th rough ‚ã (the 3 key). With system flag 117 set to CHOOSE b oxes (see Chapt er 1 in this guide), the following entrie s are avail abl e: With system flag 117 set to SO FT menus, the BAS E m enu shows the followin g : This f igure sho ws that the L OGIC, BIT, and BYT E entries within the BASE me nu are themse lve s su b-me nus . Thes e menu s are disc us sed in de tail in Chapter 19 of the calculator’ s User’s G uide. Wri ting non-d eci ma l num bers Numbers in non- decimal s yste ms, ref erred to as binary intege rs , are written preced ed by t he # symb ol ( „â ) in the calcul ator. To se lect the current
Pag e 17-2 base to be u sed f or bi nary intege rs, cho os e eithe r HEX(ade cimal) , DEC(imal), OCT(al ), or B IN(ary) in the B ASE menu. For example, if @HEX  ! is sele cted, binary in tegers will be a hexadecima l numbers, e.g., #53, #A5B, etc. As different systems are selected, the n umbers will be a utomatica lly converted to the new cu rrent bas e. To w rite a nu mber in a particu lar s yste m, start the number with # and e nd with eithe r h (hexade cimal ), d (deci mal), o (octal ), o r b (binary), exampl es: H E X D E C O C T B I N Ref erence For additio nal de tails on nu mbers fro m dif fere nt bases se e Chapter 1 9 in the calcu lato r’s Us er’s Guide .
Page 18-1 Chapter 18 Using SD cards The calculator provides a memory card port where you can insert an SD flash card for backing up calculator objects, or for downloading objects from other sources. The SD card in the calculator will appear as port number 3. Accessing an object from the SD card is performed similarly as if the object were located in ports 0, 1, or 2. However, Port 3 will not appear in the menu when using the LIB function ( ‚á ). The SD files can only be managed using the Filer, or File Manager ( „¡). When starting the Filer, the Tree view will show: 0: IRAM 1: ERAM 2: FLASH 3: SD HOME |-sub-directories When you enter in the SD tree, all objects will appear as backup objects. Therefore, it is not possible to tell what type a given objects by just looking at its name in the Filer. Long names are supported, however all names longer than 62 characters will be ignored. THIS IS IMPORTANT, names longer than 62 characters can’t be used with the Filer and will simply be ignored. As an alternative to using the File Manager operations, you can use functions STO and RCL to store and recall objects from the SD card, as shown below. Storing objects in the SD card You can only store an object at the root of the SD, i.e., no sub-directory tree can be build into Port 3 (This feature may be enhanced in a future flash ROM upgrade). To store an object, use function STO as follows: • In algebraic mode: Enter object, press K , type the name of the stored object using port 3 (e.g., :3:VAR1 ), press ` . • In RPN mode:
Page 18-2 Enter object, type the name of the stored object using port 3 (e.g., :3:VAR1 ), press K . Recalling an object from the SD card To recall an object from the SD card onto the screen, use function RCL, as follows: • In algebraic mode: Press „© , type the name of the stored object using port 3 (e.g., :3:VAR1 ), press ` . • In RPN mode: Type the name of the stored object using port 3 (e.g., :3:VAR1), press „© . With the RCL command, it is possible to recall variables by specifying a path in the command, e.g., in RPN mode: :3: {path} ` RCL. The path, like in a DOS drive, is a series of directory names that locate the position of the variable within a directory tree. However, some variables stored within a backup object cannot be recalled by specifying a path. In this case, the full backup object (e.g., a directory) will have to be recalled, and the individual variables then accessed in the screen. Purging an object from the SD card To purge an object from the SD card onto the screen, use function PURGE, as follows: • In algebraic mode: Press I @PURGE , type the name of the stored object using port 3 (e.g., :3:VAR1 ), press ` . • In RPN mode: Type the name of the stored object using port 3 (e.g., :3:VAR1), press I @PURGE .
Page W-1 Limited Warranty hp 49g graphing calculator; Warranty period: 12 months 1. HP warrants to you, the end- user customer, that HP hardware, accessories and supplies will be fr ee from defects in materials and workmanship after the date of pu rchase, for the period specified ab ov e . I f H P r ecei ve s no t ic e of su ch d ef ec ts d ur in g t h e wa rr a nt y p er i od , HP will, at its option, either repair or replace products which prove to be defective. Replacement products may be either new or like-new. 2. HP warrants to you that HP software will not fail to execute its programming instructions after the date of purchase, for the period specified above, due to defect s in material and workmanship when properly installed and used. If HP receives notice of such defects during the warranty period, HP wi ll replace softwar e media which does not execute its progra mming instructions due to such defects. 3. HP does not w ar r ant that the oper ation o f HP pr oducts w i ll be uninter r upted or er r or f r ee . If HP is una ble , w ithin a r ea sonable tim e , to r epair or r eplace an y produc t to a condition as w arr a nted , y ou w ill be entitled to a r ef und of the pur c hase pr i ce upon pr om pt r eturn o f the pr oduct w ith pr oof o f purc has e . 4. HP products may contain remanufa ctured parts equivalent to new in performance or may have been subject to incidental use. 5. Warranty does not apply to defects resulting from (a) improper or inadequate maintenance or calibration, (b) software, interfacing, parts or supplies not supplied by HP, (c) un authorized modification or misuse, (d) operation outside of the publishe d environmental specifications for the product, or (e) improper site preparation or maintenance. 6. HP MAKES NO OTHER EXPRESS WARRANTY OR CONDITION WHETHER WRITTEN OR ORAL. TO THE EXTENT ALLOWED BY LOCAL LAW, ANY IMPLIED WARRANTY OR CONDITION OF MERCHANTABILITY, SATISFACTORY QUALITY, OR FITNESS FOR A PARTICULAR PURPOSE IS LIMITED TO THE DURATION OF THE EXPRESS WARRANTY SET FORTH ABOVE. Some countries, states or provinces do not allow limitations on the duration of an implied warranty, so the above limitation or exclusion might not apply to you. This warranty gives you specific le ga l rights and you might also have other rights that vary from country to country, state to state, or province to province.
Page W-2 7. TO THE EXTENT ALLOWED BY LOCAL LAW, THE REMEDIES IN THIS WARRANTY STATEMENT ARE YOUR SOLE AND EXCLUSIVE REMEDIES. EXCEPT AS INDICATED ABOVE, IN NO EVENT WILL HP O R I T S S U P P L I E R S B E L I A B L E F O R L O S S O F D A T A O R F O R D I R E C T , SPECIAL, INCIDENTAL, CONSEQUENT IAL (INCLUDING LOST PROFIT OR DATA), OR OTHER DAMAGE, WHETHER BASED IN CONTRACT, TORT, OR OTHERWISE. Some count rie s, States or provinces do not allow the exclusion or limitation of incidental or consequential damages, so the above limitation or ex clusion may not apply to you. 8. The onl y w arr anties for HP pr oduc ts and servi ces ar e s et forth in the expr ess war r anty statements accompan y ing such pr oducts and servi ces. HP shall not be lia ble fo r techni cal or edito ri al er r ors or omis sio ns contained her ein . FOR CONSUMER TRANSACTIONS IN AUSTRALIA AND NEW ZEALAND: THE WARRANTY TERMS CONTAINED IN THIS STATEMENT, EXCEPT TO THE EXTENT LAWFULLY PERMITTED, DO NOT EXCLUDE, RESTRICT OR MODIFY AND ARE IN ADDITION TO THE MANDATORY STATUTORY RIGHTS APPLICABLE TO THE SALE OF THIS PRODUCT TO YOU. Service Europe Country : Telephone numbers Austria 43-1-360277120 3 Belgium 32-2-7126219 Denmark 45-8-2332844 Eastern Europe countries 420-5-41422523 Finland 35-89640009 France 33-1-49939006 Germany 49-69-95307103 Greece 420-5-41422523 Holland 31-2-06545301 Italy 39-0422-30 3069 Norway 47-63849309 Portugal 351-213-180020 Spain 34-917-820111 Sweden 46-851992065
Page W-3 Switzerland 41-1-4395358 (German) 41-22-8278780 (Fr ench) 39-0422-303069 (Ital ian) Turkey 420-5-4142 2523 UK 44-207-4580161 Czech Republic 420-5-4142 2523 South Africa 27-11-541 9573 Luxembourg 32-2-7126219 Other European countries 420-5-4142 2523 Asia Pacific Country : Telephone nu mbers Australia 61-3-9841-5211 Singapore 61-3-9841-5211 L.America Country : Telephone numbers Argentina 0-810-555-5520 Brazil Sao Paulo 3747-7799; ROTC 0-800-157751 Mexico Mx City 52 58-9922 ; ROTC 01-800-472-6684 Venezuela 0800-4746-8368 Chile 800-360999 Columbia 9-800-114726 Peru 0-800-10111 Central America & Caribbean 1-800-711-2884 Guatemala 1-800-999-5105 Puerto Rico 1-877-232-0589 Costa Rica 0-800-011-0524 N.America Country : Telephone numbers U.S. 1800-HP INVENT Canada (905) 206-4663 or 800- HP INVENT ROTC = Rest of the country
Page W-4 R R e e g g u u l l a a t t o o r r y y i i n n f f o o r r m m a a t t i i o o n n This section contains info r mation that s hows how the hp 49g graphin g calculator complies with regulations in certain regions. Any modifications to the calculator not expressly approved by Hewlett-Packard could void the authority to operate the 49g in these region s. USA This calculator generates, uses, and can radiate radio frequency energy and may interfere with radio and television re ception. The calculator complies with the limi ts for a Cl ass B digital device, pursuant to Part 15 of the FCC Rules. These limits are designed to provide reasonable protec tion against harmful interference in a residentia l installation. However, there is no guarantee that in terference will not occur in a particular installation. In the unlikely event that ther e is interference to radio or television reception(which can be determined by tu rning the calculator off and on), the user is encouraged to try to correct the interference by one or more of the following measures:  R eori ent or relocate the r ecei v ing antenna.  R elocate the calc ulator , with r esp ect to the r e cei v er . Connections to Pe ripheral Devices To maintain compliance with FCC rule s and regulations, use only the cable accessories provided. Canada This Class B digital apparatus comp lies with Canadian ICES-003. Cet appareil numerique de la classe B est conforme a la norme NMB-003 d u Canada. Japan この装置は、情報処理装置等電波障 害自主規制協議会 (VCCI ) の基準 に基づく第二情報技術装置です。こ の装置は、家庭環境で使用 することを目 的としていますが、この装置がラジ オやテレビジョン受信機に 近接して使用 されると、受信障害を引き起こすこ とがあります。 取扱説明書に従って正しい取り扱い をしてください。