1. 1 Using the TI-89 Calculator in Undergraduate Mathematics Courses LTC Troy Siemers Virginia Military Institute © by Troy Siemers, 2006

2. 2 Calculator basics Algebra Graphing Calculus Matrices and Vectors Differential Equations Statistics Assorted Topics Course Outline

3. 3 The manual that came with the calculator is your friend. Read it and bring it to every class!! Manual Organization Front cover: Short Cuts Pg 388-391 : Condensed list of all functions Pg 392-513 : Function descriptions (with examples) Pg 537 : Reserved variable names Important Information!!! Online manual http://education.ti.com/us/product/tech/89/guide/89guideus.html

4. 4 1)Current folder 2)Radians (vs degrees) 3)Exact (vs approximate, auto) 4)Graph type 5)Number of pairs in history area Entry line History Area Screen Layout (1) (2) (3) (4) (5) Menus

5. 5 2nd ESC alpha F1F4F2F3F5 HOME APPS MODE CATALOG ENTER CLEAR Important Keys

6. 6 To adjust brightness: andor When fed up : (QUIT) HOME ESC or 2nd ESC or Important Keys (cont) To clear history area : F18 CATALOG Lists all functions and syntax on their use (to scroll : or press beginning letter)

7. 7 MODE Most of the entries in mode are self explanatory. Make sure that the Angle is in radians, the Base is decimal,and the Exact/Approx is exact. F1F2F3 or + (or up/down) Some of the things in mode that we will look at later include changing the Graph type, and splitting the screen. Other parts deal with how information is displayed. Try changing the Pretty Print to OFF and see how it affects things. Basic Operations NOTE!!! You must press twice to save any changes!!!!! (This is true on many popup screens as well) ENTER

8. 8 Basic Operations (cont) F1F8 to These function keys bring up menus that depend on the screen you are currently on. Note: they can be used in conjunction with the 2nd keys to bring up different screens and menus. and

9. 9 Basic Operations (cont) 2nd and These are the 2 nd and 3 rd buttons to access the functions in orange and green on the keyboard. Important ones to keep in mind are: 2nd + 5 Math menus (–) Last answer 6 Memory access Last entry ENTER + Gives approx. numerical value

10. 10 2) If something in the history area is too big (pg 91), either press or or 2nd 1) To view all variables: (var-link) From here, one can also delete, copy, rename, etc. variables. This is also used when transmitting data between TI89s. 2nd Tips

11. 11 Tips (cont) 3) Use when entering letters. Press again to get out alpha mode. 2nd alpha 4) To use copy/paste/cut, you need to highlight the object. Hold and use the arrow keys. 5) When selecting an item from a menu, you can either scroll down/up and press enter, or you can press the number or letter next to the item.

12. 12 One of the main difficulties that people have in using a calculator to do mathematics is entering the information properly. It should appear as it does on your piece of paper. Don’t forget rules of operations, and don’t forget your PARENTHESES !!! The TI89 is a great tool to check your algebra even if it includes variable names. Let’s look at some examples. F2 (from home: ) Algebra

13. 13 Questions: 1)Type in both xy*x*y and x*y*x*y. Why are the answers different? 2)Factor the polynomial y = x 5 – 1. 3)Solve the equation x 5 – 1 = 0. 4)Find the common denominator for 1/3465 + 1/8085 5)Enter the expression:

14. 14 Answers: 1)The TI89 thinks xy is a variable name. 2) cfactor(x^5 – 1, x) gives (complex roots) 2) factor(x^5 – 1, x) gives

15. 15 Answers: (cont) 4) comDenom(1/3465 + 1/8085) gives 2/4851. 3) csolve(x^5 – 1 = 0, x) gives (complex roots) 5) Work it out by hand. The expression equals 1. The TI89 doesn’t give 1. Why? 3) solve(x^5 – 1 = 0, x) gives (real roots).

16. 16 Homework Assignment #1 1) Find the partial fraction decomposition for 2) Find all of the zeros of the function 3) Expand in terms of and

17. 17 Homework Assignment #1 (cont) 4) In the expression make the substitutions and simplify into the form What are in terms of ?

18. 18 HW Assignment #1 Solutions 1) gives 2) gives 3)gives

19. 19 Homework Assignment #1 Solutions (cont) 4) Use xx instead of x (otherwise circular definition error). Use the following sequence of steps (press enter between each): gives STO

20. 20 The TI83-TI92 make up the “graphing calculator”part of the TI calculator lineup. Displaying pictures of graphs of functions, differential equation vector fields, statistical data, sequences of points, etc. add to understanding of information. The TI89 separates itself from its predecessors with its ability to create 3D graphs as well. Graphing

21. 21 Overview: 1)Accessing graphs and related operations: with F1 to F5 (note: menus will change depending on current screen) 2) MODE : Graph type select, split screen to display function with its graph. 3) Up to 100 functions of each type (2D, 3D, etc.) can be stored simultaneously. ON 4) Halting the graphing process : press

22. 22 “y =’’ screen: ( from anywhere ) F1 Enter the functions (up to 100) - zoom possibilities (in, out, standard, trig, etc.) F2 - (de)select (functions to be graphed have check mark) F4 - drawing/shading (line, dot, animated, shade above/below) F6 F19 - Gives format possibilities (including simultaneous/sequential creation of graphs) and

23. 23 Graphing Example To graph, F3 Window: F2 x: -3 to 3, y: 0 to 1/2 (look familiar?) The button in “graph’’ mode gives the math menu. It includes the function’s: value, zeros, minimum, maximum, intersection, (numerical) derivatives and integrals, inflection points, distance, tangent line, and arc length. F5

24. 24 Find the following for y1(x): 1)The inflection point in the second quadrant. 2)The derivative at that point. 3)The tangent line at that point. 4)The length of the curve from x = –1 to 1. 5)The total area under the curve. Questions:

25. 25 1) 8: Inflection (enter range from x = -3 to x = 0). 2) 6: Derivatives, select dy/dx (enter x = -1). 3) A: Tangent (enter x = -1). 4) B: Arc (enter x = -1 to x = 1). 5) 7: (enter x= - 3 to x = 3). Answers: (from the graph screen and menu) F5 ans.: x = –1, y=.24191 ans.: dy/dx=.24197 ans.: y=.24191x+.483 ans.: Length = 2.02983 ans.: (look familiar?)

26. 26 3D graphing (In change “Graph” entry to “3D”. “y=” shows z 1, z 2,...) MODE Example: z1=(x^3*y - y^3*x)/360 to graph F3 type F1

27. 27

28. 28 3D graphing (cont) Options: F19 - type of axes, coords. Gives expanded view YXZ Gives projected views Changes style (wire frame, contour, shaded, etc.) (Note: returns to original view) 0 Rotate (hold to animate)

29. 29 Other types of graphing (In change the “Graph” entry) MODE Parametric : “y=” screen shows x 1 (t)= and y 1 (t)= Polar: “y=” screen shows r 1 = (Note: button now gives dx/dt and dy/dt) F5 (Note: button now gives dr/d  ) F5 Sequence: “y=” screen shows u 1 = (Note: Sequence can be recursive, formulaic, etc.)

30. 30 1) Graphand Where do they intersect? Find the area between the curves from x = 1 to x = 3. Find the tangent line of each at x = 1. 2) Graph and How are the two functions related? (note: these are the first few terms in the Fourier series for some common functions) Homework Assignment #2

31. 31 Homework Assignment #2 (cont) 3) Create a graph where the area under the function and above the x-axis is shaded. 4) Multipart Function: Define a “step” function as Let

32. 32 Homework Assignment #2 (cont) 4) Multipart Function: (cont) Using the definitions of u c (t) and h(t), graph the function Note: Have a look at the “Multi-statement” functions, pg. 195. Also, this is the graph of the solution to the differential equation 2y’’ + y’ + 2y = u 5 (t) – u 20 (t) which models a spring-mass system with variable mass.

33. 33 HW Assignment #2 Solutions Intersection: specify the two functions and range. (the graphs intersect at each non-zero integer) 1) Graph both functions and use the calculus menu to get the intersection, integration and tangent line commands. F5 Integration: Find each integral from 1 to 3 and subtract. Areas: 2.60269 – 1.50408 = 1.09861 (approx) Tangent line: specify the function and x=1. No solution found since y’(1) doesn’t exist for each.

34. 34 Homework Assignment #2 Solutions (cont) 2) The derivative of the first equals the second.

35. 35 Homework Assignment #2 Solutions (cont) 3) Plot the graph. Under the calculus menu select shade. It will prompt you for shading above/below the axis. A possible graph is:

36. 36 Homework Assignment #2 Solutions (cont) 4) This is most easily done by defining our function as the product of a couple others. Two of these are multipart functions defined by the “when” command. gives

37. 37 Homework Assignment #2 Solutions (cont) 4) (cont) The graph looks like: (use x: 0..40, y=(-.3)..(.8))

38. 38 F3 (from home: ) Limits Symbolic and numerical (partial) Derivatives, Integrals Sums, Products, Max/Min Arc Length, Tangent lines Taylor Polynomials Calculus

39. 39 Questions: 1)Find the derivative of y = x*cos(x). 2)Find the integral of x*ln(x). 3)Compute 4)Compute 5) Find the Taylor polynomial of order 4 centered at x=0 for y=cos(x). Now graph both the function together with this polynomial.

40. 40 Answers: 1) d(x*cos(x), x) gives cos(x) – x*sin(x) 2) gives 3) gives4)

41. 41 Answers:(cont) or (approx) 5) taylor(cos(x),x,3,4) gives

42. 42 1) Compute 2) Compute (how would you do it by hand?) 3) Compute the derivative of 4) Compute the following Homework Assignment #3 and

43. 43 Homework Assignment #3 (cont) 5) Compute 6) Using nested functions, find dy/dx for 7) In one line, using repeated derivatives, find for

44. 44 Homework Assignment #3 (cont) 8) Find the area under and above the x-axis from x = – 2  to x = . 9) Find the arc length of the curve from x = – 3 to x = 3. Note: This is the perimeter of the upper half of an ellipse. 10) Find the maximum and minimum of the function on the interval

45. 45 HW Assignment #3 Solutions 1) 2) By hand, you use l’Hopitals rule. By calculator, it’s 3)

46. 46 Homework Assignment #3 Solutions (cont) 4) (in exact mode, the integral will be returned. You must force it to approximate) (approximate mode is not exact) 5)

47. 47 Homework Assignment #3 Solutions (cont) 6) 7) 8) (or do it from the graph screen with these bounds)

48. 48 Homework Assignment #3 Solutions (cont) 9) Graph the function from x = – 3 to x = 3. Then use the Arc command on the math menu with parameters – 3 and 3. Answer = 7.93272 (approx.) 10) Graph the function for x = -4 to 1 and use the maximum/minimum commands on the math menu. Maximum: (-3.42562, 4.11046) (approx) Minimum: (-.860334, -.70137)(approx)

49. 49 Matrix and vector manipulations with the TI89 are very useful and fast. Vectors are treated as row or column matrices so the computations are the same as for matrices. As with many applications, the bulk of the time is spent in entering the information. Let’s start with an example. Solve the system of equations: (from home: ) APPS6 Matrices and Vectors

50. 50 Type : Matrix Folder : main Variable : m Rows : 3 Columns : 4 We create the coefficient matrix and use the reduced row echelon form to read off the solutions. APPS6 Data/Matrix editor. 3

51. 51 Enter coefficient matrix in spreadsheet.

52. 52 Computation: rref(m) (either type this in manually, get the function from the catalog or follow the procedure below) From the home screen: 2nd 5 Math menus 4 Matrix submenu 4 rref function Solution: x = -2.0523, y = 3.878, z = 13.356 (approx)

53. 53 Questions: 1)Find the inverse, transpose, determinant, eigenvalues, eigenvectors, and LU decomposition of the matrix 2) Find the dot and cross products of the vectors

54. 54 Questions: (cont) 3) Three armored cars, A, B, and C, are engaged in a three-way battle. Armored cat A had probability 1/3 of destroying its target, B has probability 1/2 of destroying its target, and C has probability 1/6 of destroying its target. The armored cats fire at the same time and each fires at the strongest opponent not yet destroyed. Using as states the surviving cats at any round, set up a Markov chain and answer the following questions: a) How many of the 8 states are absorbing? b) Find the expected number of rounds fired. c) Find the probability that A survives the battle.

55. 55 Questions: (cont) 4) The method of least squares (for 4 data points). Given a set of data points (x i,y i ), i=1,2,3,4, we wish to find the “line of best fit.” This line is given by y = mx + b where the m and b are found by solving the equation Find the line of best fit to the following data that describes the concentration of a certain drug in a persons body after a certain number of hours. Use the line to estimate the amount of drug present after 5 hours. HoursConc. (ppm) 22.1 41.6 61.4 81.0 A T AX = A T Y for

56. 56 Answers: 1)Use the method in the example to enter the matrix m. From the math menu (matrix submenu), or catalog, m^-1, m T, det(m), eigVl(m), eigVc(m), and LU m, m1, m2, m3 give

57. 57 Answers: 2) Use the method in the example to enter u 1, u 2 (as rows). From the math menu (matrix submenu), or catalog, 3) In order to solve this problem, we need the possible states that the system can be in after each shot is fired. Since each car can be either dead or alive after a shot, there are 2*2*2=8 states. These are: none, A, B, C, AB, AC, BC, ABC We create a transition matrix with the (i,j) entry giving the probability of moving from state i to state j in one shot. (Of course, state AB can never be achieved, but we need it for the calculations)

58. 58 Answers: 3) (cont) The transition matrix P is given below. Note: P can be blocked off into four 4X4 pieces as

59. 59 Answers: 3) (cont) From the theory of absorbing Markov chains, we define the matrix The information we need is included in the matrices T and T*S.

60. 60 Answers: 3) (cont) The sum of the entries in the last row of T gives the expected number of shots fired The last row of T*S gives the probabilities of the battle ending in the state associated with that column.

61. 61 Answers: 4) Enter the matrices A, X, and Y as before. The solution to A T AX = A T Y is X = (A T A) -1 A T Y Computations: So, our least squares line is C = –.175 t + 2.4 where C is the concentration (in ppm) and t is the time (in hours). When t = 5, C = 1.525.

62. 62 Homework Assignment #4 1) Solve the system of equations 2) Find the inverse, transpose, determinant, eigenvalues, and eigenvectors for Use 4 point decimal approximation

63. 63 Homework Assignment #4 (cont) 4) Find the (Householder) QR factorization of the matrix (use 4 point decimal approximations) 3) Find a unit vector that points in the same direction as u 1 and the dot and cross products of the vectors

64. 64 Homework Assignment #4 (cont) 5) Using the least squares example above as a guide, find the line of best fit for the data in the table below. Use the line to make a guess as to the yield of wheat if 22 inches of rain falls. Rainfall (inches) Yield of Wheat (bushels per acre) 12.962.5 7.228.7 11.352.2 18.680.6 8.841.6 10.344.5 15.971.3 13.154.4

65. 65 HW Assignment #4 Solutions 1) Enter the matrix using the matrix editor, then use the rref command. Answer: x = 683/185, y = -21/37, z = -76/185, w = -112/185 (approx: x = 3.6919, y = -.5676, z = -.4108, w = -.6054) 2) Enter the matrix and perform the calculations:

66. 66 Homework Assignment #4 Solutions (cont) 3) Input the vectors as a single row matrix and compute. 4) Input the matrix into the editor gives

67. 67 Homework Assignment #4 Solutions (cont) 5) Input Solution: The line is y = 4.42372 x +.22919. If x = 22(inches), y = 97.55(bushels/acre) (approx.)

68. 68 The TI89 can solve first and second order differential equations using the deSolve() function. It can plot the solutions of higher order differential equations by transforming an n th order differential equation into a system of n 1 st order equations. Initial conditions can be entered as well to give a specific solution (instead of a general solution). Differential Equations

69. 69 Solve: Solution: deSolve(y’ + y = x^2, x, y) : Example: Note: deSolve() is under the Calc menu, prime is2nd We can also give the vector field plot for our solution. On the y= screen, enter y1’= t^2 - y1, yi1 = {0,3} F2 - x, y min/max: -5 to 5, fldres to 20 F3 - graphs vector field and particular solutions Change the Graph setting to DIFF EQUATIONS. MODE

70. 70 This graph displays the slope field along with two solutions given by the initial contitions y(0)=0 and y(0)=3. Note: other initial conditions can be specified with 2nd F8

71. 71 Example: Solve: Since this is not 1 st or 2 nd order, the best we can do is graph the solution. This is always useful to get an idea of the behavior of the solution. First, we must transform the equation into a system of four 1 st order equations by introducing the intermediate variables y 1, y 2, y 3 as transforming the equation to the system Now we can enter the equation in the calculator.

72. 72 Example:(cont) In the y= screen, enter the system of equations with initial conditions yi1=0, yi2=1, yi3=1. A few adjustments have to be made before graphing the solution.

73. 73 Example:(cont) On the y= screen, make sure y1’ is the only one checked. Enter and set Axes=ON, Labels=ON, Solution Method=RK, Fields=FLDOFF (important!) In the y= screen, enter and set Axes=TIME 2nd F2 In the window screen enter t0=0, tmax=10, tstep=.1, tplot=0 xmin=-1, xmax=10, xscl=1, ymin=-3, ymax=3, yscl=1 ncurves=0, diftol=.001

74. 74 Example:(cont) A solution (using MA311 tools) can be found to be Now graph. F3

75. 75 Homework Assignment #5 1) Solve: 2) Solve: 3) In the solution to number 2), find the value of the constant when y=0 and x= –1. (Hint: See pg 184)

76. 76 4) Follow our example for third order differential equations to plot the solution to Homework Assignment #5 (cont) Use the window screen settings t0=0, tmax=3, tstep=.1, tplot=0, xmin=-1, xmax=2, xscl=1, ymin=-15, ymax=20, yscl=1, ncurves=0, diftol=.001 Note: the actual solution is

77. 77 HW Assignment #5 Solutions 1) gives 2) gives

78. 78 Homework Assignment #5 Solutions (cont) 3) thengives 4) Follow the example to get the graph:

79. 79 The TI89 statistical capabilities include finding one and two variable descriptive statistics (mean, median, variance, etc.), regressions (linear, quadratic, cubic, logistic, etc.), correlations, and plots of data along with their regression curves. As with any set of data, the bulk of time is spent in entering the data. Computations are very fast. 6 2nd 5 (MATH menu) (Statistics submenu) Note: from the home screen, Statistics

80. 80 Example: Input is similar to method for Matrices. APPS36 Keep as “data” and input Variable name: Example: set1 ( or - not case sensitive) alpha - now on data entry screen (“New”)

81. 81 In column c1, type 1, 4, 7, 7, 10, 39 From screen F5 Calculation Type : OneVar X : c1 ENTER gives mean,  x,  x 2, Sx, # entries, minX, quartiles, maxX

82. 82 Example: APPS36 Input data name: set2C1:1,2,3,4,5,6,10 C2:1,8,27,64,125,216,1000 Press F5 Change Calculation Type: LinRegX: C1Y:C2 Store ReGEQ to y1(x)

83. 83 Do same for QuadReg in y2(x) and CubicReg in y3(x). We now plot the data with the regression curves. The regression line is now stored in y1(x). (Change the Graph to FUNCTION to see it) MODE

84. 84 F1 - brings up the “y=” screen F3 - graphs it all F2F1 X: C1Y:C2 (note: push to see the Plots) F2 - zoom to the appropriate fit (ZoomFit) or, F2 - specify range of x and y Statistical Plots From the worksheet

85. 85 Here, the data is plotted as squares together with the linear, quadratic, and cubic regression curves.

86. 86 Homework Assignment #6 1)This is a continuation of problem 5) from assignment #4. a)Enter the rainfall vs. yield information as a data set. b)Find the linear regression between the two variables. c)Plot the regression line together with a scatter plot of the data. d)Use the value command in the Math submenu to find the yield for 22 inches of rain.

87. 87 Homework Assignment #6 (cont) 623749568952417080285445955266 435956706455627948266156624971 587774633768415260695873146084 554463472883465553725483706136 465035564361766366425065416274 456047728754674576525732557044 817254579261423057586286456328 574044553655444057286345866151 685647865270594071563462815843 466045697442554650537770495863 2) The following 150 data points are scores from a recent government achievement test. a)Find the one variable descriptive statistics. b)Create a histogram of the data using classes 10-19, 20-29, …, 90-99.

88. 88 Homework Assignment #6 (cont) 3) The following table holds the scores obtained by 44 cadets firing at a target from a kneeling position, X and from a standing position, Y. XYXYXYXY 8183817694867783 9388968186769786 76788691 908378 8683917685878689 9994908193849891 9887 8583879382 77908983818878 9294989199979093 9594 90969792 9884757696868987 918388 85848892 Create a scatter plot and describe the relationship between the scores in the two positions.

89. 89 HW Assignment #6 Solutions 1) APPS36 Create data set “s”. Input the data. F5 Change calculation type to LinReg. X : c1, Y : c2. Store RegEQ to y1(x). Gives y=4.4424 x +.2292 F2F1 X : c1, Y : c2. F2 X : 0 to 23, Y : 0 to 100. F3 Graphs it all.

90. 90 Homework Assignment #6 Solutions (cont) 1) (cont) F5 Select value and input x=22. Result y=97.55

91. 91 Homework Assignment #6 Solutions (cont) 2) Again, use the data editor to create a data set and input the values. F5 Change calculation type to OneVar. X : c1. Answers: Mean = 57.05, Standard Deviation = 15.02 Min = 14, Q1 = 46, Median = 56.5, Q3 = 67, Max = 95

92. 92 Homework Assignment #6 Solutions (cont) 2) (cont) Change Plot Type to Histogram, X : c1, Hist. Width = 10 F2 X : 0 to 100, Y : 0 to 50. F3 Graphs it all. F2F1 (Highlight Plot 2)

93. 93 Homework Assignment #6 Solutions (cont) 3) Again, use the data editor to create a data set and input the values. Change Plot Type to Scatter, X : c1, Y : c2 F2 X : 60 to 100, Y : 60 to 100. F3 Graphs it all. F2F1 (Highlight Plot 2) (Shown with y=x to indicate standing scores aren’t as good as kneeling)

94. 94 Functions, Programming, and Numeric Solver User-defined functions: Expand existing TI89 functions. Useful in evaluating the same expression with different values. Can graph or store resulting values. Numeric Solver: Provides fast solutions to expressions or equations.

95. 95 Programming: Similar syntax to common programming languages (e.g. If…EndIf, loops, etc.) Can call other programs as subroutines. Can change the TI89’s configuration inside a program (e.g. setMode command) Can prompt user for input. Can get or create Assembly-Language programs. Now, some examples…

96. 96 Example: A Millionaire in the Making Under what saving conditions can you become a millionaire? We answer this question using three methods: 1)A user-defined function. 2)A program. 3)The numeric solver.

97. 97 Assumptions and Variables: Time horizon. Variable name: t (the amount of time until $1,000,000 is achieved). Number of interest compounding periods per year. Variable name: n. The annual (nominal) interest rate expressed as a decimal. Variable name: r. We assume that the interest rate is constant over the time horizon. No inflation is assumed. Amount invested per interest period (equal per period). Variable name: P.

98. 98 Formulas: The basic formula for the future value of a one time investment P, at rate r, for t years, with n compounding periods is: To get the formula we are interested in, we use a finite sum over the entire time horizon. (next page)

99. 99 Formulas: (cont) Using a simple formula on partial sums of geometric series, we have We now use this in our function, program, and the numeric solver.

100. 100 User-defined Function: (pg. 85) We create a function called “value1” which takes the variables, P, n, r, and t, and returns the future value. There are three ways to do this (see pg 85). We use the store command here. Type: p*n/r*((1+r/n)^(t*n+1)-1) value1(p,n,r,t) STO ENTER press

101. 101 User-defined Function: (cont) To use this function, you can either type for example: value1(1000, 4, 0.08, 10) or, 2nd (var-link) and select the function from there. On the entry line of the home screen, it will place “value1(”. Input the data above and press enter.

102. 102 Program: (ch. 17) We use the program editor to enter our program which prompts the user for values of the variables and returns the value of the investment. APPS73 - opens the program editor Type: Program, Folder: main, Variable: value2

103. 103 Program: (cont) Using the key,enter the commands on the line under “Prgm”. CATALOG value2() Prgm ClrIO Disp "Enter P":Prompt p Disp "Enter n":Prompt n Disp "Enter r":Prompt r Disp "Enter t":Prompt t p*n/r*((1+r/n)^(t*n+1)-1)->val Disp "Value is" Disp val EndPrgm

104. 104 Program: (cont) To run the program, from the home screen either type value2()(no input here) or, 2nd (var-link) and select the function from there. On the entry line of the home screen, it will place “value2(”. Close the parenthesis and press enter.

105. 105 Numeric Solver: (ch. 19) At this point you may be wondering about the whole millionaire part. The trouble is that there are four variables that you can adjust to meet your goal. That’s where the numeric solver can help since you get to choose which variable to solve for. APPS9 opens the numeric solver

106. 106 Numeric Solver: (cont.) Enter the equation: 1000000=p*n/r*((1+r/n)^(t*n+1)-1) ENTER press Input values for all but one of the variables, move the cursor to the remaining variable, and press F2 Example: n = 4, r =.08, t = 10. Move cursor to p = and press F2 p = 15971 (note: this is dollars per quarter)

107. 107 1)Calculate and. What does this say about ? (Note: The exact value is unknown) Final Problem Set Instructions: The solutions to these problems MUST include details about how the calculator was used in addition to your final answers. Graph link software may be used for printouts. I can supply the software for installation on your computer.

108. 108 Final Problem Set (cont) 2a) Compute 2b) Plot (Note: This problem is related to something called Gibb’s phenomenon in signal processing)

109. 109 Final Problem Set (cont) 3a) Find the determinant of the following matrices. These special matrices are called Vandermond matrices. 3b) Find the formula for the determinant of

110. 110 Final Problem Set (cont) 4) The Hessian of a function is defined to be the determinant (Note: means ) Find the Hessian of the function

111. 111 Final Problem Set (cont) 5) The given chart represents mile run times (in seconds) by world class runners in the given year (after 1900). YearTimeYearTimeYearTimeYearTime 54239.458236.264234.168231.8 54238.060235.364234.970232.0 56238.160234.866231.370231.9 56238.562235.166232.772231.4 58234.562234.468231.472231.5 a)Find the linear, cubic, and logarithmic regression curves for the data. b) Use each curve to predict a time for the year 2002. c) The current world record of 223.1 was set in 1999 by Moroccan runner Hicham El-Guerrouj. Which of the curves in part a) best approximate this?

112. 112 Final Problem Set (cont) 6) When a tractor trailer turns into a cross street or driveway, its rear wheels follow a curve called a tractrix. The function that traces this curve is the function that is a solution to the differential equation a)Solve the differential equation. b)The solution provided by the calculator is wrong. Explain what is wrong with this solution. c)The actual solution is given below. Plot this function.

113. 113 Final Problem Set (cont) 7) A random variable X is said to have an Erlang distribution (with parameters λ and r) if the associated probability distribution function is given by The Erlang distribution is a special case of the gamma distribution and is appropriate for queuing theory applications including loss and waiting times in telephone calls. (cont next page)

114. 114 7) (cont) The mean μ and variance σ 2 formulas for any distribution are Compute the mean and standard deviation for the Erlang distribution in the case that λ=3 and r=2. Final Problem Set (cont)

115. 115 Final Problem Set (cont) 8) The center of mass of an object is where and δ(x,y,z) is the density of the object.

116. 116 Final Problem Set (cont) 8)(cont) Find the centroid of the object defined by

117. 117 Final Problem Set (cont) 9) For what values of λ does the following system of equations have a solution? For those λ, give the solution.

118. 118 Final Problem Set (cont) 1r2r2 r3r3 r4r4 r5r5 1 r2r2 r3r3 r4r4 r5r5 10) One of the roots of x 5 – 1 = 0 is 1. Find the other (complex) roots of the equation x 5 – 1 = 0. Lable the roots 1, r 2, r 3, r 4, r 5 and complete the chart below (where the entry in the i th row and j th column is r i * r j ).

119. 119 Final Problem Set Solutions 1) So,

120. 120 2) Final Problem Set Solutions (cont) Store Plot with x between –2π and 2π, y between –2 and 2

121. 121 Final Problem Set Solutions (cont) 3a) det([[1,2][2,2]]) = –2 det([[1,2,3][2,2,3][3,3,3]]) = 3 det([[1,2,3,4][2,2,3,4][3,3,3,4][4,4,4,4]]) = –4 3b) The pattern is (-1) n *n, where n is the number of rows in the matrix.

122. 122 Final Problem Set Solutions (cont) 4) Define To get the Hessian, use the command (the three dots indicate that this is entered on one line) Answer:

123. 123 Final Problem Set Solutions (cont) 5a) Store the information in the data editor as “mile”. Using the calc menu, compute the linreg, cubicreg and lnreg with x:c1, y:c2. 5b) Using the value command on the graph (or direct calc) with y = 102 (for year 2002). Best approximation

124. 124 Final Problem Set Solutions (cont) 6) gives This cannot be true since the domain of this function is empty. Look at the second term’s domain. Plot

125. 125 Final Problem Set Solutions (cont) 7)

126. 126 Final Problem Set Solutions (cont) 8) So, center of mass is (-1/2,0,5/4)

127. 127 Final Problem Set Solutions (cont) 9) giveswith So the solution is x = A, y = B, z = C if

128. 128 Final Problem Set Solutions (cont) 10) In approximate mode, gives 1 and (Note, you can’t use the system variables r 2 to r 5 ) 1rr 2 rr 3 rr 4 rr 5 11rr 2 rr 3 rr 4 rr 5 rr 2 rr 5 1rr 3 rr 4 rr 3 1rr 4 rr 5 rr 2 rr 4 rr 3 rr 5 rr 2 1 rr 5 rr 4 rr 2 1rr 3