SAMPLE. Computer Algebra System (TI calculators)

Transcription

1 A P P E N D I X B B1 Computer Algebra System (TI calculators) Introduction This appendix is written for use with TI calculators. Much of this material is also relevant to the use of other calculators with a computer algebra facility. The TI calculators include the TI-89, TI-89 Titanium, the TI-92 and TI-92 plus, and the Voyage. The screen dumps are from a TI-89 or TI-89 Titanium. The home screen is as shown. It is the default screen when the calculator is first turned on. It can always be reached by pressing the button HOME or 2ND QUIT. The use of the 2ND keyisvery similar to its use with the TI-83. The screen can be made lighter or darker by pressing 2ND and holding down to make it darker, or to make it lighter. At the top of the home screen a toolbar appears. Options are labelled F1 to F6. Different screens have different toolbars. To view one of the submenus, press the corresponding key from the top row of keys. F6 is obtained from pressing 2ND F1. The following steps are useful to perform if the calculator has been used previously and you are unsure of what has been stored in various memory locations. Choose F6 and then choose 1:Clear a-z by selecting and pressing ENTER. This clears the memory locations a to z. Return to the F6 menu and select 2:New Prob. NewProb appears in the entry line of the home screen. Press ENTER again. This clears the home screen and turns off any user-defined functions or stat plots. The home screen can be cleared at anytime by choosing 8:Clear Home from the F1 menu. 703

2 704 Essential Mathematical Methods 3&4CAS Executing commands in the home screen New commands are entered in the entry line. Asnew commands are entered in the entry line the old commands and answers scroll up. These can be accessed with the up and down arrow keys. The mode settings may be changed through the MODE menu. For the screen shown to the right the settings shown are: Graph...FUNCTION Angle...RADIAN Exact/Approx...AUTO mode settings entry line The 3/30 in the bottom right-hand corner of the screen informs the user that three calculations have been made. The label MAIN to the left of the screen refers to the folder being used. This option is not referred to in this chapter. To change the mode EXACT/APPROX, press MODE and then F2 to get to the second page of the MODE menu. Use the down arrow to go to AUTO, use the right arrow to see the submenu and select 2:EXACT. Press ENTER twice. Press ENTER again and note that has been recalculated in EXACT mode. When in EXACT mode you can press (GREEN mode) to obtain approximations. Errors, deleting and editing expressions and answers An error message appears if an expression is entered with incorrect format.

3 Appendix B Computer Algebra System (TI calculators) 705 B2 Press ESC to return to the entry line. Errors are deleted by using the back arrow key. The whole entry can be deleted using CLEAR. The clear button can be used to clear other expressions and the corresponding answers anywhere in the home screen. Expressions in the home screen can be reached and selected in the home screen by using or, and then ENTER. This returns the expression to the entry line and modifications can be made. The following sequence of screens illustrates how to paste from the upper section of the home screen for the purpose of editing or using a simplified expression. Note that the calculator is still in EXACT mode. Enter expression and press ENTER. Use the up arrow to select answer. Press ENTER to have answer in entry line. The expression can be modified. Using the Algebra menu (F2) In this section the algebra menu is explored. You may work through this to become acquainted with the menu. The entire menu is as shown in these two screens. It is important that the user is aware of any variables which have been assigned values. This can be done by pressing 2ND VAR-LINK.For this machine the number 5 has been assigned to the variable x. This is done in the entry line by 5 x. The variable x has this value until some action is taken. Press and to undo the assignment.

4 706 Essential Mathematical Methods 3&4CAS Operations in the Algebra menu 1:solve( This is used to solve equations, simultaneous equations and also linear inequations. There are buttons for x, y, z and t; other letters may be accessed using the ALPHA button. Youmay need to use the 2ND button to access other operators, e.g., press 2ND MATH and find and in the 8:Test sub-menu. Forexample: solve(a x + b = 0, x) results in x = b a solve(x 2 + x 1 = 0, x) results in x = ( 5 + 1) or x = 2 Note: It is necessary to use in the expression a x + b as ax without the operator is read as a new variable. solve(a b t w + t = w (ab + 1)t t, w) results in w = t + 1 solve(x 3 x 2 x + 1 = 0, x) results in x = 1orx = 1 solve(2x + 2 < 3, x) results in x < ( 2 3) 2 solve(2x + 3y = 6 and x y = 1, {x, y}) results in x = 9 5 and y = 4 5 Note: There is no requirement to use between the 2 and x. 2:factor( This command is used for factorisation. Factorisation over the rational numbers is obtained by implementing the command without the separate designation of the variable. Forexample: to factorise x 3 2x over the rational numbers, the command is factor(x 3 2x). to factorise over the real numbers, the command is factor(x 3 2x, x). Some further examples are provided here. The results are shown on the given screens. factor(a 2 b 2 ) factor(a ( 3 b 3 ) ) 2 factor x (x 1) + 1 2

5 Appendix B Computer Algebra System (TI calculators) 707 factor(2x 4 x 2 ) factor(2x 4 x 2, x) The factor command can also be used to give the prime decomposition of integers. 3:expand( This command is used in the expansion of expressions. Forexample: expand((a + b) 3 ) expand((a + 2b) 2 ) In the following, storing an expression to a memory location will also be utilised. If the name of a variable is given as the second argument, then the expansion will be given in decreasing powers of that variable. Store (x + y) 3 to memory location d and then expand without, and then with, a designated variable. The expand command can also be used to form partial fractions. Forexample: ( ) 1 expand x 2 1 ( x 3 ) + 2x + 1 expand x 2 1 Sometimes extra conditions are required for the expansion to be carried out as expected.

6 708 Essential Mathematical Methods 3&4CAS Forexample: expand((a x ) b )gives the correct result only if a is stated to be positive. expand(ln(a*b)) does not result in the expected ln(a) + ln(b). It requires the condition a > 0or b > 0tobestated. It also works if both a and b are stated to be positive. 4:zeros( This command solves an equation where one side of the equation is zero. The variable must be named. The answer is given as a list. zeros(x 2 1, x) zeros(x 2 y 2, x) zeros(x 2 y 2, y) zeros(x 2 y, y) zeros(x 2 4x + 8, x) zeros(x 2 4x + 1, x) zeros(x 2 4x + 4, x) 5:approx( Nosolutions Two solutions One solution Using this command is equivalent to choosing APPROXIMATE from the Exact/Approx submenu of the MODE menu. When in EXACT mode you can also press (GREEN mode) to obtain approximations. 6:comDenom( This gives an expression with common denominator and in simplified ratio form. The screens demonstrate the use of this command. 7:propFrac( This command changes a fractional expression to a proper fraction form.

7 Appendix B Computer Algebra System (TI calculators) 709 B3 8:nSolve( This gives numerical solutions to equations. It will return only one solution. The syntax is nsolve(equation,variable = guess). The guess is not necessary, but can make the process faster. The screen to the left below shows the solutions obtained by giving suitable guesses. The screen to the right below shows the solutions obtained by using a restriction. The positive solution is obtained with the restriction x > 0 and the negative solution with x < 0. The syntax for the latter is shown in the entry line. 9:Trig and A:Complex 9:Trig will be used in a later section of this chapter, whereas A:Complex is not used. B:Extract This contains a submenu. The first two functions of this menu are briefly discussed here. 1:getNum( This applies the common denominator function and then returns the numerator of the result. 2:getDenom( This applies the common denominator function and then returns the denominator of the result. Graphing Graphing using a TI CAS calculator is very similar to graphing with a TI graphing calculator. The screens associated with graphing are listed in green directly below the screen. Y= screen For the Y= screen press and then F1. The default Y= screen appears as shown. The cursor appears at the y1= position. Press ENTER to take the cursor to the Edit line. This can also be achieved by pressing F3 for Edit.

8 710 Essential Mathematical Methods 3&4CAS In the Edit line enter y1(x) = x 2 and press ENTER again. The result now appears in pretty print format. Note the tick to the left. This indicates that the function is selected and will be graphed when and then F3 is pressed. Menus F2, F5 and F6 contain features similar to the TI graphing calculators. These are reproduced here. The F6 menu enables the user to change the style of the graph. To change the style the cursor needs to be placed on the required function as shown. Remember, to obtain the F6 menu press 2nd F1. The user can experiment to find the use of each of the menu items. To close the menu, and return to the Y= screen, press ESC. The F5 menu enables the Y= screen to be set. The F2 menu has the same zoom features as the TI-83 with the addition of items B:Memory and C:SetFactors. B:Memory allows a window setting to be stored and recalled. From the F2 ZOOM menu choose B:Memory. The choices in this memory are: 1: ZoomPrev This returns to the viewing window displayed before the previous zoom. 2: ZoomSto This saves the current viewing window. 3: ZoomRcl This recalls the viewing window last stored with ZoomSto. The screen on the right below is the ZOOM FACTORS screen. These are altered by first using the keys and to highlight the change to be made and then by typing in the required value. The WINDOW screen The WINDOW screen is very similar to that of the TI-83. The screen to the left below shows the standard settings. By pressing F1 and 9 the GRAPH FORMATS screen can be obtained.

9 Appendix B Computer Algebra System (TI calculators) 711 B4 This is also obtainable through the Y= screen and the GRAPH screen through the same procedure. The GRAPH screen In the Y= screen enter y = 2x 4. The WINDOW screen has the settings as shown and TRACE (F3) has been activated. The MATH (F5) menu contains options. These include 9:Distance, A:Tangent, and C:Shade. Some of these will be discussed in a later section. Defining functions Expressions can be stored in memory locations. For example x 2 3x p stores the expression x 2 3x in memory location p. This can be operated on using the functions discussed previously. Make sure that memory location x has been cleared first. Functions can be defined in terms of a variable through using store or the define command, which is found by pressing F4 (to obtain F4 Other menu) and selecting 1:Define. Inthe following, two functions are defined and then operations from the Algebra and Calculus menus are applied. It is often efficient to define functions at the beginning of a problem. Functions can be defined in terms of a variable and arbitrary constants. This is shown in the screen opposite and the method of assigning values is also shown.

10 712 Essential Mathematical Methods 3&4CAS B5 Functions involving more than one variable can also be defined. Circular functions MODE The choice of radian or degree measure is made from the MODE menu. Solving equations Solve from the Algebra (F2) menu is used. Initially the solution of the equation sin x = 0.5 will be considered. The exact solution is given as x = 2k or x = 2k + 6. The parameter k on the screens. The solution when the MODE chosen is AUTO is also shown. The indicates that it is the ith parameter to be used. In the following, the syntax for finding the solutions for sin x = 0.5 for 0 x 2 is shown. In the entry line: solve(sin x = 0.5, x) x > 0 and x < 2 to give the solutions 6 and 5 6. It is also possible to give values to the parameter. For example in the is given values 1, 0 and 1. The complete entry in the entry line is x = or x = ={ 1, 0, 1} 6

11 Appendix B Computer Algebra System (TI calculators) 713 B6 Using the Calculus menu Operations in the Calculus menu (F3) Seven of the operations of the calculus menu are considered here. These are: 1:d( differentiate, 2: (integrate, 3:limit(, 6:fMin(, 7:fMax(, A:nDeriv( and B:nInt( 1:d(differentiate This operation is used to differentiate expressions. The required form is d(expression, variable). The followings screens illustrate its use. For the second derivative the form is d(expression, variable, 2). 2: (integrate This operation is used to differentiate expressions. For indefinite integrals the required form is (expression, var). If the family of antiderivatives is required enter (expression, var, c). Make sure that c has not been assigned a value. This is done by observing the VAR-LINK screen. For definite integrals the required form is (expression, var, upper, lower). The following screens illustrate its use. 3:limit( The limits considered in this section will be those that are considered in the senior years of school mathematics. Right and left limits as well as two-sided limits will be considered. The syntax is as follows. For lim limit(expression, x, a) x a For lim limit(expression, n, ) n For lim limit(expression, x,0,1) x 0 + For lim limit(expression, n,0, 1) n 0

12 714 Essential Mathematical Methods 3&4CAS The following screens demonstrate this syntax. 6:fMin( and 7:fMax( These operations return the value for which the maximum or minimum value (or the least upper bound or greatest lower bound) of a function occurs. fmax returns the value for which a local maximum occurs only if this is the actual maximum for the interval being considered. A similar statement holds for fmin. The syntax for these functions is fmin(expression, var) and fmax(expression, var). A:nDeriv( This operation will produce the numerical derivative of a function by determining f (x + h) f (x h). The derivative is then determined using the appropriate limit. h B:nInt( This returns an approximate value for a definite integral. This is illustrated in the following screens. In the screen to the left below the technique for integration discussed above is employed for comparison.

13 Appendix B Computer Algebra System (TI calculators) 715 B7 Probability Press 2ND MATH to obtain the MATH menu. Select 7: Probability to see the available functions. The first four are 1:!, 2:nPr(, 3:nCr( and 4:rand( The way of using 3:nCr( is to enter values of n and r as follows. In the home screen for 5 C 2, enter ncr(5, 2) To generate a random integer between 1 and 6 inclusive enter rand(6) For the binomial distribution, go to the Stats/List Editor (Application) and go to F5 to see the Distribution menu. Select B: Binomial Pdf. The screen appears as shown. Enter the value of p and n required. In the screen n = 6 and p = 0.7. Press ENTER again to see the probability distribution. A better way to do this is to define a binomial function which can be kept. From F4 choose 1: Define and complete as bi(n, p, x) = ncr(n, x) p x (1 p) (n x); then this function can be used for any binomial distribution problem. Also for the cumulative distribution define the function cbi(n, p, x) = (bi(n, p, x), x, 0, x). is found in the CALC menu. This will give the cumulative distribution. This is shown in the following section. For the normal distribution, go to the Stats/List editor (Application) and go to F5 to see the Distribution menu. Select 4: Normal Cdf. Enter the value of and required, for example = 6 and = 0.7. For Pr(X < 1) enter upper value = 1 and lower value =. Press ENTER again to see the probability. Forinverse choose 1: Inverse Normal after choosing 2: Inverse from the F5 menu. Enter the probability, and to obtain the result. Plotting a distribution The example for the binomial distribution is used. In the Stats/List Editor the binomial distribution values are found in the Pdf list. Enter 0 to 6 in List1. Press F2 plots and select Plot Setup. Select Plot1 and then press F1. Complete as shown. Press ENTER twice. Go to the Graph window and select 9. Zoom Data from the Zoom menu. The plot is as shown.