TI-89 Manual With Solutions

1

Using the TI-89 Calculator in

Undergraduate Mathematics Courses

LTC Troy Siemers Virginia Military Institute© by Troy Siemers, 2006

2

• Calculator basics• Algebra• Graphing • Calculus• Matrices and Vectors• Differential Equations• Statistics • Assorted Topics

Course Outline

3

The manual that came with the calculator is yourfriend. 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 manualhttp://education.ti.com/us/product/tech/89/guide/89guideus.html

4

1) Current folder2) Radians (vs degrees)3) Exact (vs approximate, auto)4) Graph type5) Number of pairs in history area

Entry line

History Area

Screen Layout

(1) (2) (3) (4) (5)

Menus

5

2nd ESC

alpha

F1 F4F2 F3 F5

HOME

APPS

MODE CATALOG

ENTER

CLEAR

Important Keys

6

To adjust brightness: and or

When fed up :(QUIT)

HOME ESCor 2nd ESCor

Important Keys (cont)

To clear history area : F1 8

CATALOG Lists all functions and syntax on their use(to scroll : or press beginning letter)

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.

F1 F2 F3or+ (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

Basic Operations (cont)F1 F8to

These function keys bring up menus that depend onthe screen you are currently on.

Note: they can be used in conjunction with the2nd

keys to bring up different screens and menus.

and

9

Basic Operations (cont)

2nd and

These are the 2nd and 3rd buttons to access the functionsin orange and green on the keyboard. Important ones tokeep in mind are:

2nd +5 Math menus

(–) Last answer

6 Memory access

Last entryENTER

+ ENTER Gives approx. numerical value

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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

Tips (cont)

3) Use when entering letters. Press again to get out alpha mode.

2nd alpha

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 eitherscroll down/up and press enter, or you can press the number or letter next to the item.

12

One of the main difficulties that people have in usinga calculator to do mathematics is entering the information properly. It should appear as it does onyour 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 evenif it includes variable names. Let’s look at someexamples.

F2(from home: )

Algebra

13

Questions:

1) Type in both xy*x*y and x*y*x*y. Why are the answers different?

2) Factor the polynomial y = x5 – 1.3) Solve the equation x5 – 1 = 0.4) Find the common denominator for 1/3465 + 1/80855) Enter the expression:

2/13/2

2/1

3/1

11

1

baccac

cbcbac

14

Answers:

1) The TI89 thinks xy is a variable name.

1

2)15(1

2)15()1( 22 xxxxx

2) cfactor(x^5 – 1, x) gives (complex roots)

ii

ii

4552

415

4552

415

4552

4)15(

4552

4)15()1(

xx

xxx

2) factor(x^5 – 1, x) gives

15

Answers: (cont)

1

4552

4145

4552

4145

4552

4)15(

4552

4)15(

xor

xorx

xorx

ii

ii

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).1x

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Homework Assignment #11) Find the partial fraction decomposition for

233313

234

5

xxxxxx

2) Find all of the zeros of the function30023064496)( 2345 xxxxxxf

3) Expand in terms of and )5cos()2sin( xx )sin(x )cos(x

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Homework Assignment #1 (cont)

4) In the expression 0FEDCBA 22 yxyxyx

make the substitutions

cossinsincos

yxyyxx

and simplify into the form

0FEDCBA 22 yxyyxx

What are

F,E,D,C,B,Ain terms of F,E,D,C,B,A ?

18

HW Assignment #1 Solutions

xxxxx

xx ,2333

13expand 234

5

1) gives

3)2(5

27)1(2

1)1(10

7)1(10 22

x

xxxxx

2) ),30023064496czeros( 2345 xxxxxx gives

ii 1,1,10,3,5x

3) )),5cos()2sin(texpand( xxx gives

xxxxxxxx 2443 cossin10cossin8cossin32)5cos()2sin(

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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):

)sin(')cos(' yx )cos(')sin(' yx

FEDCBA 22 yyxxyyyyxxxx

FF)),cos(E)sin(D(E

),sin(E)cos(DD),cos())cos(C)sin(B()(sinAC

)),cos()sin(C2)1)(cos2(B)cos()sin(A2(B

),sin())sin(C)cos(B()(AcosA

2

2

2

gives

xx yySTO STO

20

The TI83-TI92 make up the “graphing calculator”part of the TI calculator lineup.

Displaying pictures of graphs of functions, differentialequation vector fields, statistical data, sequences ofpoints, etc. add to understanding of information.

The TI89 separates itself from its predecessors with its ability to create 3D graphs as well.

Graphing

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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.

ON4) Halting the graphing process : press

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

F1 9 - Gives format possibilities (including simultaneous/sequential creation of graphs)

and

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Graphing Example

To graph, F3

2

2

21)(1

x

exy

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

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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:

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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). dxxf )(

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.: 9973.)(3

3

dxxf

(look familiar?)

26

3D graphing

(In change “Graph” entry to “3D”. “y=” shows z1, z2, ...)MODE

360

33

1xyyxz

Example:

z1=(x^3*y - y^3*x)/360

to graphF3

type F1

27

360

33

1xyyxz

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3D graphing (cont)

Options:

F1 9 - type of axes, coords.

Gives expanded view

YX Z Gives projected views

Changes style (wire frame, contour, shaded, etc.)

(Note: returns to original view)0

Rotate (hold to animate)

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Other types of graphing(In change the “Graph” entry)MODE

Parametric : “y=” screen shows x1(t)= and y1(t)=

Polar: “y=” screen shows r1=

(Note: button now gives dx/dt and dy/dt)F5

(Note: button now gives dr/d)F5

Sequence: “y=” screen shows u1=

(Note: Sequence can be recursive, formulaic, etc.)

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1) Graph xxy and

xxy

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) Graphtt

ttty14cos01299.010cos02546.0

6cos07074.02cos63662.078540.0)(

andtt

ttty14sin18186.010sin2546.0

6sin4244.02sin2732.1)(

How are the two functions related?(note: these are the first few terms in the Fourier series for some common functions)

Homework Assignment #2

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3) Create a graph where the area under the function and above the x-axis is shaded. )cos(25.1 xxy

4) Multipart Function:Define a “step” function as

ctct

tuc ,1,0

)(

Let

415sin

1515

415cos1

21)( 4/4/ teteth tt

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4) Multipart Function: (cont)Using the definitions of uc(t) and h(t), graph the function

)20()()5()()( 205 thtuthtuty

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 = u5(t) – u20(t)which models a spring-mass system with variable mass.

33


Intersection: specify the two functions and range.(the graphs intersect at each non-zero integer)

1) Graph both functions and use the calculus menuto 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


2) The derivative of the first equals the second.

35


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


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.

)1,0,5(when)(1 xxy

else,1

5,0)(1

xxygives

)1,0,20(when)(2 xxy gives

else,1

20,0)(2

xxy

)20(3)(2)5(3)(1)(4 xyxyxyxyxy

415sin

1515

415cos1

21)(3 4/4/ tetety tt

37


4) (cont) The graph looks like: (use x: 0..40, y=(-.3)..(.8))

38

F3(from home: )

Limits

Symbolic and numerical (partial) Derivatives, Integrals Sums, Products, Max/Min

Arc Length, Tangent lines

Taylor Polynomials

Calculus

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.

12

1n n

b

n

n1

2

40

Answers:

1) d(x*cos(x), x) gives cos(x) – x*sin(x)

),)ln(*( xxx42

)ln( 22 xxx

2) gives

),1,,2^( bnn 6)12()1( bbb

gives3)

),1,,2^/1( nn gives4)6

2

41

Answers:(cont)

)3cos()3()3sin(2

)3()3cos(6

)3()3sin(24

)3()3cos( 234

xxxx

or (approx)

9899.)3(1411.)3(4949.)3(02352.)3(04125. 234 xxxx

5) taylor(cos(x),x,3,4) gives

42

1) Compute

xxxx

1cos11sin3lim 2

2) Compute

(how would you do it by hand?)

xxxxxx

x cosln)1(lim

2

1

3) Compute the derivative of xy

4) Compute the following

0

2

dxe x

22 )1()1(

3 dxxx

x


and

43


5) Compute

dx

xxxxx

22

23

)22(4852

6) Using nested functions, find dy/dx for

0

)cos(21

1

x

dtt

y

7) In one line, using repeated derivatives, find

foryzx

f

3

)(tan),,( 1 zyxzyxf

44


8) Find the area under and above the x-axis from x = – 2 to x = .

)cos(25.1 xxy

9) Find the arc length of the curve

944

2xy

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 )cos(25.1 xxy

on the interval 1,4

45


1) 0,,1cos11sin3limit 2

x

xxx

2) By hand, you use l’Hopitals rule. By calculator, it’s

00,1,,cosln

)1(limit2

x

xxxxxx

3) ))floor((),)ceiling((' xdxdxxdyxy

46


4) 886227.),0,),2^(^( xxe

)2/1(tan)5ln(),2,,))1()1/(()3(( 12 xxxx

(in exact mode, the integral will be returned. You mustforce it to approximate)

(approximate mode is not exact)

5)

221)1(tan|)22ln(|

),2)^222^/()482^53^2((

212

xxxxx

xxxxxx

47


6)

2tan)1)(cos(

1)),0),cos(,),2^1/(1((xx

xxttd

7)

3222

222

1

)12)(2()1363)(63(2

)),),),((tan((

zyzyzyxxzyzyzyxx

xzyzyxddd

8) 56748.82

,0,),cos(25.12

,23,),cos(25.1

xxxxxx

(or do it from the graph screen with these bounds)

48


9) Graph the function9

442xy

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)

)cos(25.1 xxy

49

Matrix and vector manipulations with the TI89 are very useful and fast. Vectors are treated as row orcolumn matrices so the computations are the same asfor 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:

443107121825126125

zyxzyxzyx

(from home: )APPS 6Matrices and Vectors

50

Type : Matrix Folder : main Variable : m Rows : 3Columns : 4

We create the coefficient matrix and use the reducedrow echelon form to read off the solutions.

APPS 6 Data/Matrix editor. 3

51

Enter coefficient matrix in spreadsheet.

443107121825126125

zyxzyxzyx

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

Questions:

1) Find the inverse, transpose, determinant, eigenvalues,eigenvectors, and LU decomposition of the matrix

18833114518126

m

2) Find the dot and cross products of the vectors

jiujiu 432 21

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

Questions: (cont)

4) The method of least squares (for 4 data points). Given a set of data points (xi,yi), 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 foundby solving the equation

4

3

2

1

4

3

2

1

yyyy

Ybm

X

1x1x1x1x

A

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.

Hours Conc. (ppm)

2 2.14 1.66 1.48 1.0

ATAX = ATY for

56

Answers:1) Use the method in the example to enter the matrix m.

From the math menu (matrix submenu), or catalog, m^-1, mT, det(m), eigVl(m), eigVc(m), andLU m, m1, m2, m3 give

24)det(18311881412356

12/112/144/98/1

536/1T1

mmm

100010001

3100

164018126

212/12/1016/5001

1

.112.420-.068-.556-.725-.227-

.824.547-.971)eigVc(

.348426}35.7234{1.92818)eigVl(

mmm

m

m

57

Answers:2) Use the method in the example to enter u1, u2 (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, ABCWe 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)

1000)u,crossP(u5)u,dotP(u

21

21

58

Answers:3) (cont) The transition matrix P is given below.

28.28.22.022.000042.0006.42.008.0055.011.028.06.00033.033.17.17.00001000000001000000001000000001

ABCBCACABCBA

ABCBCACABCBA

P

Note: P can be blocked off into four 4X4 pieces as

QSI

P04

59

Answers:3) (cont) From the theory of absorbing Markov chains, we definethe matrix

14 )( QIT

The information we need is included in the matrices T and T*S.

4.166.69.007.1000025.200005.1

ABCBCACAB

ABCBCACAB

T

44.27.19.09.14.71.014.25.063.13.05.25.25.

ABCBCACAB

CBA

ST

60

Answers:3) (cont) The sum of the entries in the last row of T givesthe expected number of shots fired

75.24.166.69.0

The last row of T*S gives the probabilities of the battle endingin the state associated with that column.

%44survives)(%27survives)(%19survives)(

%9survive) None(

CPBPAP

P

61

Answers:4) Enter the matrices A, X, and Y as before. The solution to ATAX = ATY is X = (ATA)-1ATYComputations:

4.2175.

YAA)A(Xbm

1.01.41.62.1

Y3/21/4-1/4-1/20

A)A(,42020120

AA

T1-T

1-TT

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

Homework Assignment #41) Solve the system of equations

5321043

1245232

wyxzyxwzy

wzyx

2) Find the inverse, transpose, determinant, eigenvalues, andeigenvectors for

14371218126125

mUse 4 point decimalapproximation

63


4) Find the (Householder) QR factorization of the matrix

18833114518126

m

(use 4 point decimal approximations)

3) Find a unit vector that points in the same direction as u1 andthe dot and cross products of the vectors

jiukjiu 332 21

64


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)

12.9 62.5

7.2 28.7

11.3 52.2

18.6 80.6

8.8 41.6

10.3 44.5

15.9 71.3

13.1 54.4

65

HW Assignment #4 Solutions1) Enter the matrix using the matrix editor, then use therref 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:

.9434.0906-.0166

.2674.3143-.6786-.1958-.9449-.7342

)eigVc(

}488942.,1376.44,6486.31{)(eigVl,683)det(

17124126131825

,0469.21215.1581.

5725.0161.0571.4143.019.0234.

)1(^ T

m

mm

mm

66


3) Input the vectors as a single row matrix and compute.

066)u,crossP(u6)u,dotP(u

36

66

66)unitV(

21

21

1

u

2,1, mmmQR

4) Input the matrix into the editorgives

892.001395.132160.308887.378408.193667.8

2,892.2754.3586.446.6663.5976.0743.693.7171.

1 mm

67


5) Input

4.543.715.446.416.802.527.285.62

Y

11.1319.1513.1018.816.1813.1112.719.12

A

Solution:

22919.42372.4

YAA)A(Xbm T1-T

The line is y = 4.42372 x + .22919. If x = 22(inches), y = 97.55(bushels/acre) (approx.)

68

The TI89 can solve first and second order differentialequations using the deSolve() function.

It can plot the solutions of higher order differential equations by transforming an nth order differential equation into a system of n 1st order equations.

Initial conditions can be entered as well to give a specific solution (instead of a general solution).

Differential Equations

69

Solve: 2' xyy

Solution: deSolve(y’ + y = x^2, x, y) :

Example:

2221 xxecy x

Note: deSolve() is under the Calc menu, prime is 2nd

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

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

Example: Solve: 1)0('',1)0(',0)0(

),sin('2''2'''

yyy

xyyyy

Since this is not 1st or 2nd order, the best we can do isgraph the solution. This is always useful to get an ideaof the behavior of the solution.

First, we must transform the equation into a system of four 1st orderequations by introducing the intermediate variables y1, y2, y3 as

''''',''

,

123

12

1

yyyyyyy

yy

transforming the equation to the system 1233

32

21

22)sin(',','

yyytyyyyy

Now we can enter the equation in the calculator.

72

Example:(cont)

In the y= screen, enter the system of equations withinitial conditions yi1=0, yi2=1, yi3=1.

A few adjustments have to be made before graphing the solution.

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=TIME2nd F2

In the window screen enter t0=0, tmax=10, tstep=.1, tplot=0xmin=-1, xmax=10, xscl=1, ymin=-3, ymax=3, yscl=1ncurves=0, diftol=.001

74

Example:(cont)

A solution (using MA311 tools) can be found to be

)sin()12.1sin(367.)12.1cos(554.554. 28.228.254.1 xxexeey xxx

Now graph. F3

75


1) Solve: )cos()sin('2'' xxyyy

2) Solve: 2/1)32(')32( xyyx

3) In the solution to number 2), find the value of the constantwhen y=0 and x= –1. (Hint: See pg 184)

76

4) Follow our example for third order differential equationsto plot the solution to

3)0('',2)0(',1)0(,013'9''3'''

yyy

yyyy


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 is9

)3sin(49

)3cos(49

4)(22 xexeexy

xxx

77

HW Assignment #5 Solutions1) ),),cos()sin('2''deSolve( yxxxyyy

xecxcxx )21(

50)2sin(3

25)2cos(2

gives

2) ),,)32(')32(deSolve( 2/1 yxxyyx

gives

3212

)32ln(32

xcxx

78


3) 3212

)32ln(32y Define

xcxx

then -1x|1)0,solve(y c gives 01c

4) Follow the example to get the graph:

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.

62nd 5

(MATH menu) (Statistics submenu)

Note: from the home screen,

Statistics

80

Example:Input is similar to method for Matrices.

APPS 36

Keep as “data” and input Variable name: Example: set1 ( or - not case sensitive)alpha

- now on data entry screen (“New”)

81

In column c1, type 1, 4, 7, 7, 10, 39

From screenF5

Calculation Type : OneVarX : c1

ENTER

gives mean, x, x2, Sx, # entries, minX, quartiles, maxX

82

Example: APPS 36

Input data name: set2 C1:1,2,3,4,5,6,10C2:1,8,27,64,125,216,1000

Press F5

Change Calculation Type: LinReg X: C1 Y:C2Store ReGEQ to y1(x)

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

F1 - brings up the “y=” screen

F3 - graphs it all

F2 F1 X: C1 Y: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

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

86

Homework Assignment #61) 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


62 37 49 56 89 52 41 70 80 28 54 45 95 52 6643 59 56 70 64 55 62 79 48 26 61 56 62 49 7158 77 74 63 37 68 41 52 60 69 58 73 14 60 8455 44 63 47 28 83 46 55 53 72 54 83 70 61 3646 50 35 56 43 61 76 63 66 42 50 65 41 62 7445 60 47 72 87 54 67 45 76 52 57 32 55 70 4481 72 54 57 92 61 42 30 57 58 62 86 45 63 2857 40 44 55 36 55 44 40 57 28 63 45 86 61 5168 56 47 86 52 70 59 40 71 56 34 62 81 58 4346 60 45 69 74 42 55 46 50 53 77 70 49 58 63

2) The following 150 data points are scores from a recentgovernment 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


3) The following table holds the scores obtained by 44 cadets firing at a targetfrom a kneeling position, X and from a standing position, Y.

X Y X Y X Y X Y81 83 81 76 94 86 77 8393 88 96 81 86 76 97 8676 78 86 91 91 90 83 7886 83 91 76 85 87 86 8999 94 90 81 93 84 98 9198 87 87 85 83 87 93 8282 77 90 89 83 81 88 7892 94 98 91 99 97 90 9395 94 94 94 90 96 97 9298 84 75 76 96 86 89 8791 83 88 88 85 84 88 92

Create a scatter plot and describe the relationship between the scoresin the two positions.

89


1) APPS 36

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 + .2292F2 F1 X : c1, Y : c2.

F2 X : 0 to 23, Y : 0 to 100.

F3 Graphs it all.

90


1) (cont)

F5 Select value and input x=22. Result y=97.55

91


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 = 15.02Min = 14, Q1 = 46, Median = 56.5, Q3 = 67, Max = 95

92


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.

F2 F1(Highlight Plot 2)

93


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.

F2 F1(Highlight Plot 2)

(Shown with y=x to indicate standing scoresaren’t as good as kneeling)

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

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

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

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 overthe time horizon. • No inflation is assumed.

• Amount invested per interest period (equal per period). Variable name: P.

98

Formulas:

The basic formula for the future value of a one timeinvestment P, at rate r, for t years, with n compoundingperiods is:

nt

nrPF

1

To get the formula we are interested in, we usea finite sum over the entire time horizon. (next page)

99

Formulas: (cont)

nt

nrP

nrP

nrPP

111

2

Using a simple formula on partial sums of geometricseries, we have

11Value1nt

nr

rnP

We now use this in our function, program, and the numeric solver.

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 usethe store command here.

Type:

p*n/r*((1+r/n)^(t*n+1)-1) value1(p,n,r,t)STO

ENTERpress

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 dataabove and press enter.

102

Program: (ch. 17)

We use the program editor to enter our program which prompts the user for values of the variablesand returns the value of the investment.

APPS 7 3 - opens the program editor

Type: Program, Folder: main, Variable: value2

103

Program: (cont)

Using the key,enter the commands on theline under “Prgm”.

CATALOG

value2()PrgmClrIODisp "Enter P":Prompt pDisp "Enter n":Prompt nDisp "Enter r":Prompt rDisp "Enter t":Prompt tp*n/r*((1+r/n)^(t*n+1)-1)->valDisp "Value is"Disp valEndPrgm

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 theparenthesis and press enter.

105

Numeric Solver: (ch. 19)

At this point you may be wondering about the wholemillionaire part. The trouble is that there are four variables that you can adjust to meet your goal. That’swhere the numeric solver can help since you get tochoose which variable to solve for.

APPS 9

opens the numeric solver

106

Numeric Solver: (cont.)

Enter the equation:

1000000=p*n/r*((1+r/n)^(t*n+1)-1) ENTERpress

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 top = and press F2

p = 15971 (note: this is dollars per quarter)

107

1) Calculate and .

What does this say about ? (Note: The exact value is unknown)

14

1n n

16

1n n

15

1n n

Final Problem SetInstructions: The solutions to these problems MUST includedetails about how the calculator was used in addition toyour final answers. Graph link software may be used for printouts. I can supply the software for installation on yourcomputer.

108

Final Problem Set (cont)

2a) Compute dyy

y

0

)sin(2

2b) Plot

13)13sin(

5)5sin(

3)3sin()sin(4 xxxxy

(Note: This problem is related to something calledGibb’s phenomenon in signal processing)

109


3a) Find the determinant of the following matrices. These special matrices are called Vandermond matrices.

2221

333322321

4444433343224321

3b) Find the formula for the determinant of

nnnnnnnnn

333322321

110


4) The Hessian of a function is defined to

be the determinant yyyx

xyxx

ffff

(Note: means )

xf

yxyf2

Find the Hessian of the function)ln(2),( 2 yxxyxyxf

),( yxf

xyf

111


5) The given chart represents mile run times (in seconds) by world class runners in the given year (after 1900).

Year Time Year Time Year Time Year Time54 239.4 58 236.2 64 234.1 68 231.854 238.0 60 235.3 64 234.9 70 232.056 238.1 60 234.8 66 231.3 70 231.956 238.5 62 235.1 66 232.7 72 231.458 234.5 62 234.4 68 231.4 72 231.5

a) Find the linear, cubic, and logarithmic regression curvesfor 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


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

22 11

1'x

x

xxy

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.

)(xfy

21 1)/1(cosh xxy

113


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

0,)!1(

)( 1

xexr

xf xrr

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

7) (cont) The mean μ and variance σ2 formulas for any distribution are

0

222

0

)(,)( dxxfxdxxfx

Compute the mean and standard deviation for theErlang distribution in the case that λ=3 and r=2.


115


8) The center of mass of an object is

MM

MM

MM zyx ,,

where b

a

xf

xf

yxg

yxg

dzdydxzyxM)(

)(

),(

),(

2

1

2

1

),,(

b

a

xf

xf

yxg

yxgz dzdydxzyxzM

)(

)(

),(

),(

2

1

2

1

),,(

b

a

xf

xf

yxg

yxgx dzdydxzyxxM

)(

)(

),(

),(

2

1

2

1

),,( b

a

xf

xf

yxg

yxgy dzdydxzyxyM

)(

)(

),(

),(

2

1

2

1

),,(

and δ(x,y,z) is the density of the object.

116


8)(cont) Find the centroid of the object defined by

)2),(,0),((20

24)(,

24)(

24

24

)2,2(22

1),,(

21

2

2

2

1

22

xyxgyxgxz

xxfxxfxyx

bax

zyx

117


9) For what values of λ does the following system ofequations have a solution? For those λ, give the solution.

44310718251261

zyxzyxzyx

118


1 r2 r3 r4 r5

1r2

r3

r4

r5

10) One of the roots of x5 – 1 = 0 is 1. Find the other (complex) roots of the equation x5 – 1 = 0. Lable the roots 1, r2, r3, r4, r5 and complete the chart below (where the entry in the ith row and jth column is ri * rj).

119

Final Problem Set Solutions1)

90),1,,4^/1(1 4

14

nnnn

9451

90

6

15

4

n n

945),1,,6^/1(1 6

16

nnnn

So,

120

2) 17898.1),0,,/)(sin(*)/2( yyy

13)13sin(

5)5sin(

3)3sin()sin(4)(1 xxxxxy

Final Problem Set Solutions (cont)

Store

Plot with x between –2π and 2π, y between –2 and 2

121


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 ofrows in the matrix.

122


4) Define

)ln(2),( 2 yxxyxyxf

To get the Hessian, use the command

)]])),),,(((),),),,((([)]),),,(((),),),,(((det([[

yyyxfddxyyxfddyxyxfddxxyxfdd

(the three dots indicate that this is entered on one line)

Answer: 1222

yxffff

yyyx

xyxx

123


5a) Store the information in the data editor as “mile”.Using the calc menu, compute the linreg, cubicregand lnreg with x:c1, y:c2.

)ln(*4370.267471.343:lnreg9988.201*4711.3*0791.*)40742.5(:cubicreg

7218.260*4190.:linreg23

yTyyyET

yT

5b) Using the value command on the graph (or direct calc)with y = 102 (for year 2002).

579.221)102(:lnreg024.272)102(:cubicreg

981.217)102(:linreg

TT

T

Best approximation

124


6)

yx

x

x

xxy ,

11

1'deSolve22

gives 122 111ln)ln( cxxxy

This cannot be true since the domain of this function is empty. Look at the second term’s domain.Plot

50:10:

1)/1(cosh)( 212

yx

xxxy

125


7)

3/2

DeviationStandard

9/2,0,,!1

3

:Variance

3/2,0,,!1

3

:Mean

2312

22

312

xexx

xexx

x

x

126

Final Problem Set Solutions (cont)8)

5664.122,2,,2

4,2

4,,2,0,,122

xxxyxzM

708.152,2,,2

4,2

4,,2,0,,22

xxxyxzzM

28319.62,2,,2

4,2

4,,2,0,,22

xxxyxzxM

02,2,,2

4,2

4,,2,0,,22

xxxyxzyM

So, center of mass is (-1/2,0,5/4)

127


9) ]])4,1,4,3][10,7,,18][25,12,61,([[rref

gives

C100B010A001

with

151564)(

)(802354

)()7336(3

)(227873

2

2

p

pC

pB

pA

So the solution is x = A, y = B, z = C ifi49132 gives ) 1515, 64–^2czeros(

i49132

128


10) In approximate mode, ),1(czeros 5 xx

gives 1 and i9511.30917.i,9511.30917.i5868.8090.i,5868.8090.

54

32

rrrrrrrr

(Note, you can’t use the system variables r2 to r5)

1 rr2 rr3 rr4 rr5

1 1 rr2 rr3 rr4 rr5

rr2 rr2 rr5 1 rr3 rr4

rr3 rr3 1 rr4 rr5 rr2

rr4 rr4 rr3 rr5 rr2 1

rr5 rr5 rr4 rr2 1 rr3

TI-89 Manual With Solutions

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