APPENDICES. APPENDIX A Calculus and the TI-82 Calculator. Functions. Specific Window Settings

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1 APPENDICES APPENDIX A Calculus and the TI-82 Calculator Functions A. Define a function Press y= to obtain Y= edit screen. Cursor down to the function to be defined. (Press clear if the function is to be redefined.) Type in the expression for the function. (Note: Press x,t,θ to display X.) B. Select or deselect a function (Functions with highlighted equal signs are said to be selected. Pressing graph instructs the calculator to graph all selected functions and pressing 2nd [table] creates a column of the table for each selected function.) Press y= to obtain the Y= edit screen. Cursor down to the function to be selected or deselected. Move the cursor to the equal sign and press enter to toggle the state of the function on and off. C. Display a function name, that is, Y 1, Y 2, Y 3,... Press 2nd [y-vars] 1 to invoke the list of function names. Press the number of the function name, or cursor down to the function and press enter. D. Combine functions Suppose Y 1 is f(x) and Y 2 is g(x). If Y 3 =Y 1 +Y 2, then Y 3 is f(x)+g(x). (Similarly for,, and.) If Y 3 =Y 1 (Y 2 ), then Y 3 is f(g(x)). Specific Window Settings A. Customize a window Press window to open the window-setting screen, and edit the following values as desired. Xmin = the leftmost value on the x-axis Xmax = the rightmost value on the x-axis Xscl = the distance between tick marks on the x-axis Ymin = the bottom value on the y-axis Ymax = the top value on the y-axis Yscl = the distance between tick marks on the y-axis Note 1: The notation [a, b]by[c, d] stands for the window settings Xmin = a, Xmax = b, Ymin = c, Ymax = d. Note 2: The default values of Xscl and Yscl are 1. The value of Xscl should be made large (small) if the difference between Xmax and Xmin is large (small). For instance, with the window settings [0, 100] by [ 1, 1], good scale settings are Xscl = 10 and Yscl =.1. B. Use a predefined window setting Press zoom to display the list of predefined settings. Either press a number or move the cursor down to an item and press enter. Select ZStandard to obtain [ 10, 10] by [ 10, 10], Xscl = Yscl = 1. Select ZDecimal to obtain [ 4.7, 4.7] by [ 3.1, 3.1], Xscl = Yscl = 1. (When trace is used with this setting, points have nice x-coordinates.) Select ZSquare to obtain a true-aspect window. (With such a window setting, lines that should be perpendicular actually look perpendicular, and the graph of y = 1 x 2 looks like the top half of a circle.) Select ZTrig to obtain [ 2π, 2π]by[ 4, 4], Xscl = π/2, Yscl = 1, a good setting for the graphs of trigonometric functions. C. Some nice window settings (With these settings, one unit on the x-axis has the same length as one unit on the y-axis, and tracing progresses over simple values.) A1 Copyright 2010 Pearson Education, Inc.

2 A2 Appendices [ 4.7, 4.7] by [ 3.1, 3.1] [0, 9.4] by [0, 6.2] [ 2.35, 2.35] by [ 1.55, 1.55] [0, 18.8] by [0, 12.4] [ 7.05, 7.05] by [ 4.65, 4.65] [0, 47] by [0, 31] [ 9.4, 9.4] by [ 6.2, 6.2] [0, 94] by [0, 62] General principle: (Xmax Xmin) should be a number of the form k 9.4, where k is a whole number or 1 2, 3 2, 5 2,..., then (Ymax Ymin) should be (31/47) (Xmax Xmin). Derivative, Slopes, and Tangent Lines A. Compute f (a) from the Home screen using nderiv(f(x), X, a) Press math 8 to display nderiv(. Enter either Y 1, Y 2,... or an expression for f(x). B. Define derivatives of the function Y 1 Set Y 2 = nderiv(y 1, X, X) to obtain the 1st derivative. Set Y 3 = nderiv(y 2, X, X) to obtain the 2nd derivative. Note: nderiv( is obtained by pressing math 8. C. Compute the slope of a graph at a point Press 2nd [calc] 6. Use the arrow keys to move to the point of the graph. Note: This process usually works best with a nice window setting. D. Draw a tangent line to a graph Press 2nd [draw] 5 to select Tangent. Use the arrow keys to move to a point of the graph. Note 1: This process usually works best with a nice window setting. Note 2: To remove all tangent lines, press 2nd [draw] 1 to execute ClrDraw. Special Points on the Graph of Y 1 A. Find a point of intersection with the graph of Y 2, from the Home screen Evaluate solve(y 1 Y 2, X, g), where g is a guess, as follows: Press math 0 to display solve(. Enter either Y 1 Y 2 or an expression for the difference of the two functions. Type in the remaining items, where g is (hopefully) close to a value of x for which Y 1 = Y 2. Press enter. A value of x for which Y 1 = Y 2 will be displayed. B. Find intersection points, with graphs displayed Press graph to display the graphs of all selected functions. Press 2nd [calc] 5 to select intersect. Reply to "First curve?" by using or (if necessary) to move the cursor to one of the two curves and then pressing enter. Reply to "Second curve?" by using or (if necessary) to move the cursor to the other curve and then pressing enter. Reply to "Guess?" by moving the cursor near the point of intersection and pressing enter. C. Find the second coordinate of a point with first coordinate specified, call it a From the Home screen: Display Y 1 (a) and press enter. or Press a sto x,t,θ enter to assign the value a to the variable X. Display Y 1 and press enter. From the Home screen or with the graph displayed: Press 2nd [calc] 1 to select value. Type in the value of a and press enter. If desired, press the up-arrow key to move to points on graphs of other selected functions. With the graph displayed: Press trace. Move cursor with and/or until the x-coordinate of the cursor is as close as possible to a. (Note: This process usually works best if one of the nice window settings discussed above is used.) D. Find the first coordinate of a point with second coordinate specified, call it b From the Home screen: Press math 0 to display solve(. Continue typing to obtain solve(y 1 b,x,c) where c is a guess for the value of the first coordinate. (Note: The expression for Y 1 can be used in place of Y 1.)

3 Appendix A Calculus and the TI-82 Calculator A3 With the graph displayed: Set Y 2 = b. Find the point of intersection of Y 1 and Y 2 as in part B above. E. Find an x -intercept (that is, a root of Y 1 = 0) Graph Y 1. Press 2nd [calc] 2 to select root. Move the cursor to a point just to the left of a root and press enter. Move the cursor to a point just to the right of the root and press enter. Move the cursor to a point near the root and press enter. F. Find a relative extreme point Set Y 2 = nderiv(y 1, X, X) or set Y 2 equal to the exact expression for the derivative of Y 1.[To display nderiv( press math 8.] Select Y 2 and deselect all other functions. Graph Y 2. Find an x-intercept of Y 2, call it r, at which the graph of Y 2 crosses the x-axis. The point (r, Y 1 (r)) will be a possible relative extreme point of Y 1. G. Find an inflection point Set Y 2 = nderiv(y 1, X, X) and Y 3 = nderiv(y 2, X, X), or set Y 3 equal to the exact expression for the second derivative of Y 1. Select Y 3 and deselect all other functions. Graph Y 3. Find an x-intercept of Y 3, call it r, at which the graph of Y 3 crosses the x-axis. The point (r, Y 1 (r)) will be a possible inflection point of Y 1. Tables A. Display values of f (x ) for evenly spaced values of x Press y=, assign the function f(x) toy 1. Press 2nd [TblSet]. Set TblMin =first value of x. Set ΔTbl =increment for values of x. Set both Indpnt and Depend to Auto. Press 2nd [table]. Note 1: You can use the down- and up-arrow keys to look at function values for other values of x. Note 2: The table can display values of more than one function. For example, in the first step you can assign the function g(x) toy 2 and also select Y 2 to obtain a table with columns for X, Y 1, and Y 2. B. Display values of f (x ) for arbitrary values of x Press y= and assign the function f(x) toy 1. Press 2nd [TblSet]. Set Indpnt to Ask by moving the cursor to Ask and pressing enter. Leave Depend set to Auto. Press 2nd [table]. Type in any value for X and press enter. Repeat the previous step for any other values of X. Riemann Sums Suppose that Y 1 is f(x), and c, d, and Δx are numbers, then sum(seq(y 1,X,c,d,Δx)) computes f(c)+f(c +Δx)+f(c +2Δx)+ + f(d). The function sum is in the LIST/MATH menu and the function seq is in the LIST/OPS menu. A. Compute [f(x 1 )+f(x 2 )+ + f(x n )] Δx On the Home screen, evaluate sum(seq(f(x), X, x 1, x n,δx)) Δx as follows: Press 2nd [list] and move the cursor right to MATH. Press 5 ( to display sum(. Press 2nd [list] 5 to display seq(. Enter either Y 1 or an expression for f(x). Definite Integrals and Antiderivatives A. Compute b f (x ) dx a On the Home screen, evaluate fnint(f(x),x,a,b) as follows: Press math 9 to display fnint(. Enter either Y 1 or an expression for f(x).

4 A4 Appendices B. Shade a region under the graph of a function and find its area Press 2nd [calc] 7 to select f(x) dx. If necessary, use or to move the cursor to the graph. In response to the request for a Lower Limit, move the cursor to the left endpoint of the region and press enter. In response to the request for an Upper Limit, move the cursor to the right endpoint of the region and press enter. Note 1: This process often works best with a nice window setting. Note 2: To remove the shading, press 2nd [draw] 1 to execute the ClrDraw command. C. Obtain the graph of the solution to the differential equation y = g(x), y(a) =b [That is, obtain the graph of the function f(x) that is an antiderivative of g(x) and satisfies the additional condition f(a) = b.] Set Y 1 = g(x). Set Y 2 = fnint(y 1, X, a, X)+b. [Press math 9 to display fnint(.] The function Y 2 is an antiderivative of g(x) and can be evaluated and graphed. D. Shade the region between two curves Suppose the graph of Y 1 lies below the graph of Y 2 for a x b and both functions have been selected. To shade the region between these two curves execute the instructions Shade(Y 1,Y 2,1,a,b) as follows. Press 2nd [draw] 7 to display Shade(. Note 1: Replace 1 by 2 for a striped shading. Note 2: To remove the shading, press 2nd [draw] 1 enter to execute the ClrDraw command. Functions of Several Variables A. Specify a function of several variables and its derivatives In the Y= edit screen, set Y 1 = f(x, Y). (The letter Y is entered by pressing alpha [y].) Set Y 2 = nderiv(y 1, X, X). Y 2 will be f x. Set Y 3 = nderiv(y 1, Y, Y). Y 3 will be f y. Set Y 4 = nderiv(y 2, X, X). Y 4 will be 2 f x 2. Set Y 5 = nderiv(y 3, Y, Y). Y 5 will be 2 f y 2. Set Y 6 = nderiv(y 3, X, X). Y 6 will be 2 f x y. B. Evaluate one of the functions in part A at x = a and y = b On the Home screen, assign the value a to the variable X with a sto x,t,θ. Press b sto alpha [y] to assign the value b to the variable Y. Display the name of one of the functions, such as Y 1, Y 2,..., and press enter. Least-Squares Approximations A. Obtain the equation of the least-squares line Assume the points are (x 1,y 1 ),...,(x n,y n ). Press stat 1 for the EDIT screen, and obtain a table used for entering the data. If necessary, clear data from columns L 1 and L 2 as follows. Move the cursor to the top of column L 1 and press clear enter. Repeat for column L 2. To enter the x-coordinates of the points, move the cursor to the first blank row of column L 1, enter the value of x 1, and press enter. Repeat with x 2,...,x n. Move the cursor to the first blank row of column L 2, and enter the values of y 1,...,y n. Press stat for the CALC menu, and press 5 to place LinReg(ax + b) on the Home screen. Press enter to obtain the slope and y-intercept of the least-squares line. B. Assign the least-squares line to a function Press y=, move the cursor to the function, and press clear to erase the current expression. Press vars 5 to select the Statistics variables. Press to select the EQ menu, and press 7 for RegEQ (Regression Equation), which then assigns the least-squares line equation to the function. C. Display the points from part A Press y= and deselect all functions. Press 2nd [stat plot] enter to select Plot1, press enter to turn Plot1 ON. Select the first plot from the four icons for the types of plots. This icon corresponds to a scatter plot. Select L 1 from Xlist and select L 2 from Ylist.

5 Press graph to display the data points. Note 1: Press zoom 9 to ensure that the current window setting is large enough to display the points. Note 2: When you finish using the point-plotting feature, turn it off. Press 2nd [stat plot] enter to select Plot1. Then press to move to OFF and press enter. D. Display the line and the points from part A Press y= and deselect all functions except for the function containing the equation of the least-squares line. Carry out all but the first step of part C. The Differential Equation y' =g(t, y) A. Obtain a table for the approximation given by Euler s method, with a, b, y 0, h, andn as given in Section 10.7 Note: We will use N to represent the value of n in order to avoid confusion with the variable n in the sequence Y = edit screen. With the calculator in sequence mode, the values for t 0,t 1,t 2,... are stored as the sequence values u(0),u(1),u(2),..., and the values for y 0,y 1,y 2,... are stored as the sequence values v(0),v(1),v(2),... To invoke sequence mode, press mode, move the cursor down to the fourth line, move the cursor right to Seq, and press enter. Press y= to obtain the sequence Y= edit screen. Set U n =U n 1 + h. [In Seq mode, pressing 2nd [U n 1 ], the second function of 7, generates U n 1.] Set V n =V n 1 + g(u n 1, V n 1 ) h. [In Seq mode, pressing 2nd [V n 1 ] (the second function of 8) generates V n 1. To form g(u n 1, V n 1 ) h, in the function g(t, y) replace t with U n 1 and replace y by V n 1, and then multiply it by the value of h.] Press window and set UnStart to a, VnStart to y 0, and nstart to 0. Press 2nd [tblset], and set TblMin=0, ΔTbl=1, and the other items to Auto. Press 2nd [table]. The successive values of t and y are found in the U n and V n columns, respectively. Note: After completing using Euler s method, reset the calculator to function mode by pressing mode, moving to Func in the fourth line, and pressing enter. Appendix A Calculus and the TI-82 Calculator The Newton Raphson Algorithm Perform the Newton Raphson Algorithm Assign the function f(x) toy 1 and the function f (x) toy 2. Press 2nd [quit] to invoke the Home screen. Type in the initial approximation. Assign the value of the approximation to the variable X. This is accomplished with the keystrokes sto x,t,θ enter. Type in X Y 1 /Y 2 X. (This statement calculates the value of X Y 1 /Y 2 and assigns it to X.) Press enter to display the value of this new approximation. Each time enter is pressed, another approximation is displayed. Note: In the first step, Y 2 can be set equal to nderiv(y 1,X,X). In this case, the successive approximations will differ slightly from those obtained with Y 2 equal to f (x). Sum a Finite Series Compute the sum f(m) +f(m +1)+ + f(n) On the Home screen, evaluate sum(seq(f(x),x,m,n,1)) as follows: Press 2nd [list] and move the cursor right to MATH. Press 5 ( to display sum(. Press 2nd [list] 5 to display seq(. Enter an expression for f(x). Note: At most 99 terms can be summed. Miscellaneous Items and Tips A5 A. From the Home screen, if you plan to reuse a recently entered line with some minor changes, press 2nd [entry] until the previous line appears. You can then make alterations to the line and press enter to execute the line. B. If you plan to use trace to examine the values of various points on a graph, set Ymin to a value that is lower than is actually necessary for the graph. Then, the values of x and y will not obliterate the graph while you trace. C. To clear the Home screen, press clear twice.