Appendix D Calculus and the TI-86 Calculator A17 APPENDIX D Calculus and the TI-86 Calculator Functions A. Deﬁne a function, say y1, from the Home screen • Press 2nd [quit] to invoke the home screen. • Press 2nd [alpha] [y] 1 alpha [=] followed by an expression for the function, and press enter . B. Deﬁne a function from the function editor • Press graph f1 to select y(x) = from the GRAPH menu and obtain the screen for deﬁning functions; that is, the function editor. • Cursor down to a function. (To delete an existing expression, press clear . To create an additional function, cursor down to the last function and press enter .) • Type in an expression for the function. (Note: Press f1 or x-var to display x. Press f2 or 2nd [alpha] [y] to display y.) C. Select or deselect a function in the function editor Functions with highlighted equal signs are said to be selected. The graph screen displays the graphs of all selected functions, and tables contain a column for each selected function. • Press graph f1 to invoke the function editor. • Cursor down to the function to be selected or deselected. • Press f5 , that is, selct, to toggle the state of the function on and oﬀ. D. Select a style [such as Line (\), Thick (\\), or Dot ( . . . )] for the graph of a function • Move the cursor to a function in the function editor. • Press more and then press f3 repeatedly to select one of the seven styles. E. Display a function name, that is, y1, y2, y3, . . . • Press 2nd [alpha] [y] followed by the number. or • Press 2nd [catlg-vars] more f4 to invoke a list containing the function names. • Cursor down to the desired function. • Press enter to display the selected function name. F. Combine functions Suppose y1 is f (x) and y2 is g(x). • If y3 = y1 + y2, then y3 is f (x) + g(x). (Similarly for −, ×, and ÷.) • If y3 = y1(y2), then y3 is f (g(x)). Specify Window Settings A. Customize a window • Press graph f2 to invoke the window editor 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 Usually xRes = 1. Higher values speed up graphing, but with a loss of resolution. 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 graph f3 to invoke the ZOOM menu. • Press f4 , that is ZSTD, to obtain [−10, 10] by [−10, 10], xScl = yScl = 1. • Press more f2 , that is zsqr, 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 − x2 looks like the top half of a circle.) • Press more f4 , that is zdecm, to obtain [−6.3, 6.3] by [−3.1, 3.1], xScl = yScl = 1. (When trace is used with this setting, points have nice x-coordinates.) Copyright © 2010 Pearson Education, Inc. Goldstein/Schnieder/Lay/Asmar: Calculus and Its Applications 12/E 11/20/2008 Page 17 A18 Appendices • Press more f3 , that is ztrig, to obtain [−21π/8, 21π/8] 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. • [−6.3, 6.3] by [−3.7, 3.7] • [0, 12.6] by [0, 7.4] • [−3.15, 3.15] by [−1.85, 1.85] • [0, 25.2] by [0, 14.8] • [−9.45, 9.45] by [−5.56, 5.56] • [0, 63] by [0, 37] • [−12.6, 12.6] by [−7.4, 7.4] • [0, 126] by [0, 74] General principle: (xMax − xMin) should be a number of the form k · 12.6, where k is a whole number or 12 , 32 , 52 , . . . , then (yMax − yMin) should be (37/63) · (xMax − xMin). Derivative, Slopes, and Tangent Lines A. Compute f (a) from the Home screen using der1(f(x), x, a) • Press 2nd [calc] f3 to display der1(. • Enter either y1, y2, . . . or an expression for f (x). • Type in the remaining items and press enter . B. Deﬁne derivatives of the function y5 • Set y1 = der1(y5, x, x) to obtain the 1st derivative. • Set y2 = der2(y5, x, x) to obtain the 2nd derivative. • Set y3 = nDer(y2, x, x) to obtain the 3rd derivative. • Set y4 = nDer(y3, x, x) to obtain the 4th derivative. Note 1: der1, der2, and nDer are found on the menu obtained by pressing 2nd [calc]. Note 2: To speed up the display of the 3rd and 4th derivatives, set the value of xRes in the WINDOW screen to at least 5. C. Compute the slope of a graph at a point • Press graph f5 to display the graph of the function. • Press more f1 f2 to select dy/dx from the GRAPH/MATH menu. • Use the arrow keys to move to the point of the graph and then press enter . (This process usually works best with a nice window setting.) Or, type in the value of the x-coordinate of a point (any number between xMin and xMax), and press enter . D. Draw a tangent line to a graph • Press graph f5 to display the graph of the function. • Press more f1 more more f1 to select tanln from the GRAPH/MATH menu. • Move the cursor to any point on the graph or type in the ﬁrst coordinate of a point. • Press enter to draw the tangent line through the point and display the slope of the curve at that point. The slope is displayed at the bottom of the screen as dy/dx = slope. • To draw another tangent line, press graph and then repeat the previous three steps. Note: To remove all tangent lines, press graph more f2 more more f1 to select cldrw from the GRAPH/DRAW menu. Special Points on the Graph of y1 A. Find a point of intersection with the graph of y2, from the Home screen • Press 2nd [solver]. • To the right of "eqn:" enter y1 − y2 = 0 and press enter . (Note: y1 and y2 can be entered via F keys and the equal sign is entered with alpha [=].) • To the right of "x=" type in a guess for the x-coordinate of the point of intersection, and then press f5 . After a little delay, a value of x for which y1 = y2 will be displayed in place of your guess. B. Find intersection points, with graphs displayed • Press graph f5 to display the graphs of all selected functions. • Press more f1 more f3 to select isect from the GRAPH/MATH menu. • Reply to "First curve?" by using or (if necessary) to place the cursor on one of the two curves and then pressing enter . • Reply to "Second curve?" by using or (if necessary) to place the cursor on the other curve and then pressing enter . • Reply to "Guess?" by moving the cursor close to the desired point of intersection (or typing in a guess for the ﬁrst coordinate of the desired point) and pressing enter . Copyright © 2010 Pearson Education, Inc. Goldstein/Schnieder/Lay/Asmar: Calculus and Its Applications 12/E 11/20/2008 Page 18 Appendix D Calculus and the TI-86 Calculator C. Find the second coordinate of the point whose ﬁrst coordinate is a From the Home screen: • Display y1(a) and press enter . or • Press a sto x-var enter to assign the value a to the variable x. • Display y1 and press enter . From the Home screen or with the graph displayed: • Press graph more more f1 to display "Eval x=". • Type in the value of a and press enter . (The value of a must be between xMin and xMax.) • If desired, press the up-arrow key, , to move to points on graphs of other selected functions. With the graph displayed: • Press f4 ; that is, trace. D. Find the ﬁrst coordinate of a point whose second coordinate is b • Set y2 = b. • Find the point of intersection of the graphs of y1 and y2 as in part B above. E. Find an x -intercept of a graph of a function • Press graph more f1 f1 to select root from the GRAPH/MATH menu. • Select y2 and deselect all other functions. • Graph y2. • Find an x-intercept of y2, call it r, at which the graph of y2 crosses the x-axis. • The point (r, y1(r)) will be a possible relative extreme point of y1. G. Find an inﬂection point • Set y2 = der2(y1, x, x) or set y2 equal to the exact expression for the second derivative of y1. (To display der2, press 2nd [calc] f4 .) • Select y2 and deselect all other functions. • Graph y2. • Find an x-intercept of y2, call it r, at which the graph of y2 crosses the x-axis. • The point (r, y1(r)) will be a possible inﬂection point of y1. Tables • Type in the value of a and press enter . • If necessary, use or the desired graph. to place the cursor on • In response to the request for a Left Bound, move the cursor to a point whose ﬁrst coordinate is less than the desired x-intercept or type in a value less than the desired x-intercept. Then press the enter key. • In response to the request for a Right Bound, move the cursor to a point whose ﬁrst coordinate is greater than the desired x-intercept or type in a value greater than the desired x-intercept. Then press the enter key. • In response to the request for a Guess, move the cursor to a point near the desired x-intercept or type in a value close to the desired x-intercept. Then press the enter key. F. Find a relative extreme point • Set y2 = der1(y1, x, x) or set y2 equal to the exact expression for the derivative of y1. (To display der1(, press 2nd [calc] f3 .) A. Display values of f (x) for evenly spaced values of x • Press graph f1 and assign the function f (x) to y1. • Press table f2 to invoke the TABLE SETUP window. • Set TblStart = ﬁrst value of x. • Set ΔTbl = increment for values of x. • Set Indpnt to Auto. • Press f1 to display the table. Note 1: You can use or 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) to another function and also select that other function to obtain a table with columns for x and each of the selected functions. B. Display values of f (x ) for arbitrary values of x • Press graph f1 , assign the function f (x) to a function, and deselect all other functions. • Press table f2 . • Set Indpnt to Ask by moving the cursor to Ask and pressing enter . • Press f1 • Type in any value for x and press enter . • Repeat the previous step for as many values as you like. Copyright © 2010 Pearson Education, Inc. Goldstein/Schnieder/Lay/Asmar: A19 Calculus and Its Applications 12/E 11/20/2008 Page 19 A20 Appendices Riemann Sums Suppose that y1 is f (x), and c, d, and Δx are numbers, then sum(seq(y1,x,c,d,Δx)) computes f (c) + f (c + Δx) + f (c + 2Δx) + · · · + f (d). The functions sum and seq are found in the LIST/OPS menu. Compute [f (x1 ) + f (x2 ) + · · · + f (xn )] · Δx On the Home screen, evaluate sum(seq(f (x), x, x1 , xn , Δx)) ∗ Δx as follows: • Press 2nd [list] f5 more f1 ( f3 to display sum(seq(. • Enter either y1 or an expression for f (x). • Type in the remaining items and press enter . Definite Integrals and Antiderivatives A. Compute b a f (x ) dx On the Home screen, evaluate fnInt(f (x), x, a, b) as follows: • Press 2nd [calc] f5 to display fnInt(. • Enter either y1 or an expression for f (x). • Type in the remaining items and press enter . B. Shade a region under the graph of a function from x = a to x = b • Press graph more f1 f3 to select f (x) from the GRAPH/MATH menu. • If necessary, use the graph. or to move the cursor to • In response to the request for a Lower Limit, move the cursor to the left endpoint of the region (or type in the value of a) and press enter . • In response to the request for an Upper Limit, move the cursor to the right endpoint of the region (or type in the value of b) and press enter . Note: To remove the shading press graph more f2 more more f1 to select cldrw from the GRAPH/DRAW menu. C. Obtain the graph of the solution to the diﬀerential equation y = g(x), y(a) = b [That is, obtain the function f (x) that is an antiderivative of g(x) and satisﬁes the additional condition f (a) = b.] • Set y1 = g(x). • Set y2 = fnInt(y1, x, a, x) + b. (To display fnInt, press 2nd [calc] f5 .) The function y2 is an antiderivative of g(x) and can be evaluated and graphed. Note: The graphing of y2 proceeds very slowly. Graphing can be sped up by setting xRes (in the GRAPH/WIND menu) to a high value. D. Shade the region between two curves Suppose the graph of y1 lies below the graph of y2 for a ≤ x ≤ b and both functions have been selected. To shade the region between these two curves, execute the instruction Shade(y1,y2,a,b) as follows. • Press graph more f2 f1 to display Shade( from the GRAPH/DRAW menu. • Type in the remaining items and press enter . Note: To remove the shading, press graph more f2 more more f1 to execute ClrDraw from the GRAPH/DRAW menu. Functions of Several Variables A. Specify a function of several variables and its derivatives • In the y(x) = function editor, set y1 = f (x, y). (The letters x and y can be entered by pressing f1 and f2 .) • Set y3 = der1(y1, y, y). y3 will be ∂f . ∂x ∂f . ∂y • Set y4 = der2(y1, x, x). y4 will be ∂2f . ∂x2 • Set y5 = der2(y1, y, y). y5 will be ∂2f . ∂y 2 • Set y6 = nDer(y3, x, x). y6 will be ∂2f . ∂x ∂y • Set y2 = der1(y1, x, x). y2 will be 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-var . • Press b sto 2nd [alpha] [y] to assign the value b to the variable y. • Display the name of one of the functions, such as y1, y2, . . . , and press enter . Least-Squares Approximations A. Obtain the equation of the least-squares line Assume the points are (x1 , y1 ), . . . , (xn , yn ). Copyright © 2010 Pearson Education, Inc. Goldstein/Schnieder/Lay/Asmar: Calculus and Its Applications 12/E 11/20/2008 Page 20 Appendix D Calculus and the TI-86 Calculator • Press 2nd stat f2 to obtain a table for entering the data. • If necessary, clear data from columns xStat and yStat as follows. Move the cursor to xStat and press clear enter . Repeat for the yStat column. • To enter the x-coordinates of the points, move the cursor to the ﬁrst blank row the xStat column, type in the value of x1 , and press enter . Repeat with x2 , . . . , xn . • Move the cursor to the ﬁrst blank row of the yStat column and enter the values of y1 , . . . , yn . • Place a 1 in each of the ﬁrst n entries of the fStat column. (The other entries should be blank.) • Press 2nd [quit] to display the home screen. • Press 2nd stat f1 f3 enter to obtain the values of a and b where the least-squares line has equation y = a + bx. • If desired, the straight line (and the points) can be graphed with the following steps: (a) First press graph f1 and deselect all functions. (b) Press 2nd stat f4 to invoke the statistical DRAW menu. (If any graphs appear, press more f2 to clear them.) (c) Press the button for drreg (either f1 or more f1 ) to draw the least-squares line and press more f2 to draw the n points with SCAT. B. Assign the least-squares line to a function • Press graph f1 , move the cursor to the function, and press clear to erase the current expression for the function. • Press 2nd [stat] f5 more more f2 to assign the equation (known as RegEq) to the function. C. Display the points from part A • Press graph f1 and deselect all functions. • Press 2nd [stat] f3 to select STAT PLOTS menu. • Press f1 to select PLOT1. • Move the cursor to ON and press enter . • Select scat as the Type, select xStat as the Xlist Name, select yStat as the Ylist Name, and select any symbol for Mark. • Press graph f5 to display the data points. Note 1: Make sure 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] f3 f1 , move the cursor to OFF and press enter . D. Display the line and the points from part A • Press graph f1 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. Carry out Euler’s method, with a, b, y0 , and h as given in Section 10.7 • To invoke diﬀerential equation mode, press 2nd [mode], move the cursor down to the ﬁfth line, move the cursor right to DifEq, and press enter . • Press graph more f1 and select Euler from the 4th row and FldOff from the 5th row. • Press graph f1 and enter the diﬀerential equation. Use Q1 (or Q2, Q3, . . . ) instead of y1 (or y2, y3, . . . ). The letters t and Q can be entered with the keys f1 and f2 . Up to nine diﬀerential equations, with function variables Q1, Q2, . . . , can be speciﬁed. • Press graph f4 and set x = t and y = Q. • Press graph f2 to invoke the window-setting screen. • Set tMin and xMin to a, set tMax and xMax to b, and set tStep to h. (Leave the values of tPlot and EStep at their default settings of 0 and 1.) • Set the values of xScl, yMin, yMax, and yScl as you would when graphing ordinary functions. • Press graph f3 and set the initial value (denoted by QI1, QI2, etc.) to y0 . Alternately, you can simultaneously graph solutions for several diﬀerent initial values, call them v1 , v2 , . . . , by setting the initial value to {v1 , v2 , . . . }. Braces are entered via the LIST menu. • Press graph f5 to see a graph of the Euler’s method of solution of the diﬀerential equation. Note 1: To make the graph more accurate (and the graphing slower), decrease the value of tStep. Note 2: A table of values can be displayed by setting TblStart to a, ΔTbl to h, and Indpnt to Auto. Note 3: The next-to-last step shows one way to obtain a family of solutions of a diﬀerential equation. Another way is presented in item B below. Copyright © 2010 Pearson Education, Inc. Goldstein/Schnieder/Lay/Asmar: A21 Calculus and Its Applications 12/E 11/20/2008 Page 21 A22 Appendices B. Graph a family of solutions of a diﬀerential equation • To invoke diﬀerential equation mode, press 2nd [mode], move cursor down to the ﬁfth line, move the cursor right to DifEq, and press enter . • Press graph more f1 and select Euler from the 4th row and FldOff from the 5th row. • Press graph f1 and enter the diﬀerential equation. Use Q1 (or Q2, Q3, . . . ) instead of y. The letters t and Q can be entered with the keys f1 and f2 . • Press graph f2 to invoke the window-setting screen. • Set tMin and xMin to 0. • The values of tMax and xMax must be set by trial and error. Try setting them both to values like 2, 5, or 10 and see if the solutions look like those in Section 10.6. • Set tStep to a small value, such as .1 or the default (≈ .13). The smaller the value of tStep, the better the accuracy will be and the longer the time required. • Leave the values of tPlot and EStep at their default settings of 0 and 1. • Set the values of xScl, yMin, yMax, and yScl as you would when graphing ordinary functions. The value of yMin should be less than the smallest constant solution, and the value of yMax should be greater than the largest constant solution. • Press graph f3 and set the initial value to y0 . • Press graph f5 to graph the solution of the diﬀerential equation. • Press graph more f5 to explore solutions with other initial values. • Move the cursor up or down to another place on the y-axis and press enter to see the graph of the solution beginning at that point. • Repeat the process in the previous step as often as desired. Note: To erase one of the graphs drawn, select cldrw from the GRAPH/DRAW menu. C. Graph the slope ﬁeld of a diﬀerential equation • To invoke diﬀerential equation mode, press 2nd mode , move the cursor down to the ﬁfth line, move the cursor right to DifEq, and press enter. • Press graph more f1 and select SlpFld from the sixth line. • Press graph f1 and enter the diﬀerential equation. Use Q1 (or Q2, Q3, . . . ) instead of y1 (or y2, y3, . . . ). The letter t and Q can be entered with the keys F1 and F2. • Press graph f4 and set fldRes = 15. To draw more line segments in the slope ﬁeld, enter a larger value for fldRes. • Press graph f2 to invoke the window-setting screen. • Set tMin and xMin to 0. Enter all other settings as you would when graphing solutions of a diﬀerential equation. For example, set tStep to a small value, such as .15, or use the default (≈ .13). Set tMax and xMax to values like 3, 5, or 10, and see if the slope ﬁelds look like those in Section 10.1. • Press graph f3 and set an initial condition for Q1. This will add a solution curve with the speciﬁed initial condition to the slope ﬁeld. If you do not specify the initial condition, then only the slope ﬁeld will be graphed. • Press graph f5 to graph the slope ﬁeld and a solution curve (if you have speciﬁed an initial condition) or the slope ﬁeld only. • To display the slope ﬁeld with diﬀerent solution curves, press graph f3 , change the initial value for Q1, then press graph f5 . The Newton–Raphson Algorithm Perform the Newton–Raphson Algorithm • Assign the function f (x) to y1 and the function f (x) to y2. • 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-var enter . • Type in x − y1/y2 → x. (This statement calculates the value of x − y1/y2 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, y2 can be set equal to der1(y1,x,x). Copyright © 2010 Pearson Education, Inc. Goldstein/Schnieder/Lay/Asmar: Calculus and Its Applications 12/E 11/20/2008 Page 22 Appendix D Calculus and the TI-86 Calculator 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] f5 more f1 ( f3 to display sum(seq(. • Enter an expression for f (x). • Type in the remaining items and press enter . Note: About 6000 terms can be summed. Miscellaneous Items and Tips 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. D. When two menus are displayed at the same time, you can remove the top menu by pressing exit . (The remaining menu can be removed by pressing exit again.) E. To obtain the solutions of a quadratic equation, or of any equation of the form p(x) = 0, where p(x) is a polynomial of degree ≤ 30, press 2nd [poly], enter the degree of the polynomial, enter the coeﬃcients of the polynomial, and press f5 . (Some of the solutions might be complex numbers.) Copyright © 2010 Pearson Education, Inc. Goldstein/Schnieder/Lay/Asmar: A23 Calculus and Its Applications 12/E 11/20/2008 Page 23