Graphing Calculator Guide for Finite Mathematics

Graphing Calculator Guide for Finite Mathematics Second Edition

Berresford and Rockett Frank C. Wilson Green River Community College

Contents Chapter 1: Functions 1

Plot a function 1 Plot multiple functions simultaneously 2 Change the dimensions of the viewing rectangle 3 Determine points of intersection of two graphs 4 Determine intercepts graphically and numerically 5 Model data using linear regression 7 Model data using power regression 9

Chapter 2: Mathematics of Finance 11

Change table setup 12 Use STORE to assign a constant value to a variable 13 Use EQUATION SOLVER to solve an equation 14

Chapter 3: Matrices and Systems of Equations 17

Input a matrix 18 Find the inverse of a matrix 20 Store a matrix 21 Find the transpose of a matrix 22 Find the determinant of a matrix 23 Find the product of two matrices 24 Augment a matrix 26 Reduce a matrix to row echelon form 27 Reduce a matrix to reduced row echelon form 28

Chapter 4: Linear Programming 29

Graph an inequality 30 Clear a shaded region from the display 32 Graphically determine a linear programming problem solution 33

Chapter 5: Probability 35

Compute factorials 36 Compute permutations 37 Compute combinations 38 Determine a binomial distribution 39

Chapter 6: Statistics 41

Sort a list of numerical data 42 Calculate the mean, median, and standard deviation of a data list 43 Draw a box-and-whisker plot 44 Model data using a histogram 45 Calculate the cumulative values of the normal distribution 47 Shade the area under the normal distribution curve 49

Chapter 7: Markov Chains 51

Compute the product of a matrix and itself a finite number of times 52 Find the steady state distribution of a Markov chain transition matrix 54

Chapter L: Logic 55 THIS CHAPTER IS AVAILABLE ON THE CD/WEBSITES

Create truth tables to validate various laws of logic 56

Graphing Calculator Guide Finite Mathematics

Introduction

This guide is designed to help you learn how to use your TI-83 graphing calculator to solve the textbook problems in Finite Mathematics by Berresford and Rockett. In addition to covering essential keystrokes, this guide will show you how to use your calculator to solve problems using graphical, analytical, and numeric methods. Since each chapter builds upon previous material, you are encouraged to work through the guide rather than use it only as a keystroke reference guide. The chapters in this guide correspond with the chapters in Finite Mathematics. The examples in this guide are similar to the exercises in the text. You may find it helpful to review the calculator guide examples before attempting the text exercises.

Copyright © Houghton Mifflin Company. All rights reserved.

Chapter 1 Functions In this chapter you will learn how to do the following: 1. Plot a function 2. Plot multiple functions simultaneously 3. Change the dimensions of the viewing rectangle 4. Determine intercepts graphically and numerically 5. Determine points of intersection of two graphs 6. Model data using linear regression 7. Model data using power regression Plot a function Keystrokes:

Y = 1. Press . X,T,Ø,n 2. Type the function to be plotted using the key for the variable.

GRAPH 3. Press . Example: Plot y = sin(x2). Solution: )2 GRAPH ∧( X,T,Ø,nSIN Y =

Copyright © Houghton Mifflin Company. All rights reserved. 1

2 Chapter 1

Plot multiple functions simultaneously Keystrokes:

Y = 1. Press . X,T,Ø,n2. Type the function to be plotted using the key for the variable.

ENTER 3. Press .

4. Repeat steps (2) and (3) for subsequent functions. GRAPH 5. Press .

Example: Plot y = 2x + 3, y = 3x + 3, y = -2x + 3 and y = -x + 3 simultaneously. What is the effect of the coefficient of x on the graph?

Solution:

ENTER3+ X,T,Ø,n 2 Y =

ENTER3 + X,T,Ø,n 3

(—) ENTER3+ X,T,Ø,n 2

(—) GRAPH3 + X,T,Ø,n

TRACE The calculator will graph the lines in the order they were entered. Once the graphs are drawn you can use the key and horizontal arrow keys to move your cursor along a graph. To move between graphs, use the vertical arrow keys. The equation of the graph the cursor is on is shown in the upper left-hand corner of these calculator display. Lines with positive coefficients slope upward and lines with negative coefficients slope downward. The greater the absolute value of the coefficient of x the steeper the line.


Functions 3

Change the dimensions of the viewing rectangle

ZOOM

The default dimensions of the viewing rectangle are Xmin = -10, Xmax = 10, Ymin = -10, and Ymax =10. Frequently a graph may fall outside of the default viewing rectangle and, unless the viewing dimensions are changed, you will be unable to see it. The key brings up a menu that allows you to zoom in, zoom out, visually specify the viewing rectangle (zoom box), and perform many other zoom actions. Additionally, you can manually specify the viewing rectangle by following the steps below. Keystrokes:

WINDOW

GRAPH 3. Press to re-graph the function using the new viewing rectangle dimensions.

1. Press the key.

d dimensions, using the vertical arrow keys to move between the various

s between 1 (high detail) and 8 (low detail)

xample:

rom a waterfall x feet high will hit the ground with a speed of

2. Enter the desiredata inputs. The meaning of each of the data inputs is detailed below:

min: x-value of left side of viewing rectangle XXmax: x-value right side of viewing rectangle

sXscl: distance between tick marks on the x-axiYmin: y-value of bottom of viewing rectangle Ymax: y-value of top of viewing rectangle

axiYscl: distance between tick marks on the y-Xres: sets the pixel resolution of the graph and is

E

Water falling f 5.01160 x miles per

Waterfalls are no less than zero feet tall.) 281 feet tall.)

m.p.h.) occurs at the

hour (neglecting air resistance). The highest waterfall in the world (Angel Falls) is 3281 feet high. What dimensions of the viewing rectangle will allow the speeds of all the waterfalls in the world to be visually determined?

olution: S

Xmin = 0 (Xmax = 3281 (Waterfalls on earth are no more than 3Ymin = 0 (Falling water travels at a minimum of 0 miles per hour.) Ymax = 313 (The function is increasing so the largest y-value (312.4

largest x-value (3281 ft).)


4 Chapter 1


2nd

TRACE

5

TRACE

ENTER

TRACE

ENTER

ENTER

Keystrokes: 1. Plot the graphs.

2. Press the key.

3. Press the key. (Steps (2) and (3) activate the CALC menu.)

4. Press to activate the intersect function.

5. The calculator displays the graphs and asks, "First curve?". Use the and vertical arrow keys to select a curve. (The calculator defaults to the first curve defined.)

6. Press .

7. The calculator asks, "Second curve?". Use the and vertical arrow keys to select the second curve. (The calculator defaults to the second curve defined.)

8. Press .

9. The calculator asks, "Guess?". Type in an estimate of the point of intersection.

10. Press . The calculator returns the coordinates of the point of intersection.

Example: The median age at first marriage for men is y1 = 24.8 + 0.13x, and for women it is

is the number of years since 1980. According to this model, in t first marriage for men and women be the same?

e default curve values and picking 88 as our guess gives the after 1980), the median marriage age will be 36 years.

Determine points of intersection of two graphs

y2 = 22.2 + 0.16x, where xwhat year, will the median age a

Solution: Plotting the functions, using thgraph below. So in 2066 (86.67 years

Functions 5


TRACE 2. Press the key.

0 ENTER

function.

ress to set the x-value to zero. The y-value displayed is

2nd

TRACE

2

ENTER

ENTER

7. The calculator asks, "Right bound?". Use the horizontal arrow keys to move thethe right of the x-intercept.

8. Press .

ENTER

GRAPH 2nd

pts of a function may be determined graphically or numerically. Which matter of personal preference.

es

fun on u s you to see all intercepts.

3. Use the horizontal arrow keys to move the cursor to the x-intercept. Watch the y-value as r rc t. When the y-value becomes zero, the x-value is an x-intercept of the

4. P the t.

e Graphic 1. Plot the function using a viewing rectangle that allows you to see all intercepts.

2. Press the key.

3. Press the key. (Steps (2) and (3) activate the CALC menu.)

4. Press to activate the zero function.

5. The calculator displays the graphs and asks, "Left bound?". Use the horizontal arrow keys to move the cursor to the left of the x-intercept.

.

cursor to

Type in an estimate of the x-intercept.

and the calculator returns the value of the x-intercept closest to your

Numerical: 1. Plot the function using a viewing rectangle that allows you to see all intercepts.

2. Press to activate the table menu.

3. Use the vertical arrows to scroll through the table. X-intercepts occur where y1 = 0 and the y-intercept occurs where x = 0. (Changing the table setup will be covered in Chapter 2.)

Determine intercepts graphically and numerically X- and y-intercemethod you use is a Keystrok : Graphical:

1. Plot the sing a viewing rectangle that allowcti

you nea ep the inte

y-intercep

Alternativ al:

6. Press

9. The calculator asks, "Guess?".

10. Press guess. This process must be repeated for each x-intercept.

6 Chapter 1

Example: A company that installs car alarm systems finds that if it installs x systems per week, then its

. What are the fixed costs of the company? (Fixed costs typically include cost of building ent, utilities, etc.)

), is the difference between revenue and cost. Will the profit of the company nt.)

000, 75000] gives the graph (Figure A). Entering zero returns the y-intercept or fixed cost (Figure B).

r

seecompany i

costs will be C(x) = 210x + 72,000 and its revenue will be R(x) = -3x2 +1230x (both in dollars).

arental, equipm

b. Profit, P(xever be zero dollars? (The point where profit is zero is called the break-even poi

Solution: a. Fixed costs are costs incurred regardless of production level and are found by computing

the cost of installing 0 systems. Plotting C(x) = 210x + 72,000 with viewing rectangle [0,10] [70

b. The proectang

Using thex-intercep

that th

table valu

Figure A

TRACE

2

ey and

nstalls between 100 and 240 unitses 2 e see t

fit function P(x) = (-3x +1230x) -le [0, 250] [-1000, 15000] gives th

the horizontal

t x = 100, 24

hat the maxim

kts are between 98 and 101 and 239e x-intercepts occur a

) w (Table

C

Figure B

we

(210x + 72,000). Plotting P(x) with viewing e graph below (Fig

around keys we ca

0. (y1 = 0 at both places.

um profit occurs when

ure C).

n determine that the

) So as long as the ekly, it is profitable. Moving through the

170 units are installed.

and 242. Generating a table (Table 1) we can
F e igur Table 1
Copyright © Houghton Mifflin Compan

Table 2

y. All rights reserved.

Functions 7


STAT

4

3. Press the right horizontal arrow to move to the Calculate Menu.

Press to activate the LinReg (ax + b) function.

2nd 1

5. Press to inse

2nd 2

EN

ta using linear regression

. data plot appears to have a near constant change between consecutive values,

Keystrokes:

to activate the Statistics Menu.

4.

6. Press the comma key.

7. Press to insert the L2 list label. (To get L3 - L6, select 3 - 6 respectively.)

8. Press . The equation of the line that best fits the data entered is displayed.

Model daWhen analyzing real life data is it often useful to develop a mathematical model to represent the data. Regression is a method of finding the equation of a curve that best fits the dataWhen the linear regression is used to find the equation of the line that best fits the data.

1. Enter the data lists, L1 and L2, using the Stat List Editor. (See your manual for details.)

2. Press

rt the L1 list label.

TER

8 Chapter 1


Year (Years Since 1960) Expectancy Year (Years Since 1960) Male Life

Expectancy

Example: The tables below show human life expectancy by birth year. Use linear regression to develop a life expectancy model. Predict life expectancy for females and males born in 2000.

Birth Birth Year Female Life Birth Birth Year

1960 0 73.1 1960 0 66.61970 10 74.7 1970 10 67.11980 20 77.5 1980 20 70.01990 30 78.8 1990 30 71.8

Solution:

1 and {73.1, 74.7, 77.5, 78.8} in list L2 and following the ing:

e linear model that best fits the data. According to the r females born in 2000 is 81.0 years.

the nchange ente in L3 in cedures given earlier results in the following:

+ 66.1 is the linear model that best fits the data. According to the odel, life expectancy for males born in 2000 is 73.5 years.

Entering {0, 10, 20, 30} in list Lprocedures above results in the follow

So the line y = 0.199x + 73.04 is thmodel, life expectancy fo

Leaving ring {66.6, 67.1, 70.0, 71.8} in list L3 and substitutingfor L2 is the pro

L1 u d,

So the line y = 0.185xm

Functions 9


STAT

Model da

2nd 2

ENTER


7. Press respectively.)

8. Press

ENTER

2nd 1 5. Press to insert the L1 list label.

ta using power regression

sing verywhere, power regression is often used to find the equation of the function of the form

y hat be . Ot g es clude xponlogistic, s tic. It u heach of these curve types in order to determine which type of regression will give youm aningful m el. K okes: 1. Enter the data lists, L1 and L2, using the Stat List Editor. (See your manual for details.)

2. Press to activate the Statistics Menu.

to the Calculate Menu.

ll down to option A and press to activate the

to insert the L2 list label. (To get L3 - L6, select 3 - 6

. The equation of the power function that best fits the data is displayed.

When the data plot appears to be nonlinear and to be increasing everywhere or decreae = axn t st fits the data her forms of re r sion in

is usef logarithmic, e ential,

inusoidal, quadratic, cubic, and quar l to be familiar with t e shape of the

ost me od

eystr

3. Press the right horizontal arrow to move

4. Use the vertical arrows to scroPwrReg function.

10 Chapter 1


manufacturer of supercomputers finds that the profit on the sale of the first, the tenth, the

Example: Atwentieth, the thirtieth and the fortieth computer in a given month is as follows:

Supercomputer Number

Profit ($)

1 3500 10 6400 20 7400 30 8200 40 8500

Develop a power function model of profit per computer and use the model to predict the profit on the 50th supercomputer sold in a given month.

Entering {1, 10, 20, 30, 40} in list L1 and {3500, 6400, 7400, 8200, 8500} in list L2 and rlier results in the following:

.77x0.245 is the power function model that best fits the data. he sale of the fiftieth computer will create a $8714.10 profit.

fte aningfu see odel e data simultaneously, you can observe the accuracy of the model. To do this,

PLOT menu and specify the type of statistics plot you desire. (Procedures are covered in your .) In this example, our model very closely fit the data.

Solution:

following the procedures given ea

So the curve y = 3541According to the model, t

It is o how close your model fits the data. By plotting the mequation and th

n me l to

plot the power function using the procedures previously covered. Then activate the STAT

TI-83 Graphing Calculator Guidebook

Chapter 2

Mathematics of Finance

this chapter you will learn how to do the following:

. Use STORE to assign a constant value to a variable

ATION SOLVER to solve an equation

In

1. Change table setup

2

3. Use EQU


12 Chapter 2


WINDOW 2nd

GRAPH 2nd

Change table setup

is often meaningful to put data from a function into a table. This is the quickest way to look hange the table lues (∆TBL).

ystrokes: Press to activate the TABLE SETUP menu.

Enter the table starting point at TBLStart.

s at ∆TBL.

4. Press to draw the table with the new settings.

t x = 5 with ∆TBL = 10.

Solution:

It at a large number of domain and range values. The TI-83 allows you to cstarting point (TblStart) and the distance between consecutive domain va Ke1.

2.

3. Enter the distance between consecutive domain value

Example: Create a table for the function y = 23x + 10 starting a

Mathematics of Finance 13


Use STORE to assign a constant value to a variable

ple,

Most financial formulas contain more than one variable. Typically you choose which variables you want to set as constants and then solve for the remaining variable. For examin the amortization payment formula,

r

DP

+−

=

11mtr −

m

m

ENTER

,

st rate = number of times compounded per year

f years

Rather than entering this complex formula numerous times, you can enter it once and then change the value of each variable as needed.

Keystrokes: 1. Enter the value for the variable. 2. Press . 3. Enter the name of the variable. (Use the key to activate alphabetic characters.) 4. Press and the calculator confirms the value has been assigned. Example: Assign the value of 0.5 to the interest rate variable r. Solution:

P = payment D = debt r = interemt = number o

STO > ALPHA

0 . 5 STO > ALPHA ENTERX

14 Chapter 2


0

Use EQUATION SOLVER to solve an equation

ENTER 3. Press to display the d

ALPHA EN

useful

Keystrokes: 1. Press to activate the EQUATION SOLVER.

2. Enter the function. (The solver requires the function you enter to be equal to zero. You can make any equation equal zero by subtracting the left-hand side of the equation from both

is, if your equation is y = f(x) , 0 = f(x) - y. We'll use the amortization payment a in this example.)

ata entry screen.

a for all of the constants and move your cursor to the variable.

5. Press to solve. (In this exa e solved for the payment P given , debt, and frequency of payment.)

The SOLVER tool allows you to solve an equation for any variable. This is especially when using financial and other multivariable formulas.

MATH

sides. Thatformul

4. Input dat

mple, wTER

the interest rate, term

Mathematics of Finance 15


rank and Shelley have found the house they want to buy priced at $192,500. They can

ents per year.

rate must they obtain to have a $1400 monthly payment?

s a loan aym onth they will make n loan payments an w much would this

3.

life of their loan?

using the steps given earlier in this section. The amortization 1, we solved for r. In problem 2, we solved for p.

ive them a monthly payment of $1400.00.

ents to13 per year will reduce their payment to $1292.26.

rtened from 30 years (360 months) to 21.06 years onthly.

Example: Fafford $1400 per month for monthly principal and interest payments. They intend to take out a 30-year loan for the full purchase price. They will make twelve paym

1. What interest

2. If they ent every four weeks instead of once a mthirtee nually. If their interest rate is 7.907%, ho

ubmit p

reduce their payment?

The seller has offered to buy down their interest rate by paying points (1 point = 1% of the loan amount). The resultant rate is 6.5%. They intend to still pay $1400 monthly. How much will they shorten the

Solution:

These solutions were generated formula was only entered once. In problemIn problem 3, we solved for t.

1. An interest rate of 7.907% will g

2. Increasing the number of paym

3. The life of their loan will be sho(253 months) if they pay $1400 m

Chapter 3 Matrices and Systems of Equations In this chapter you will learn how to do the following:

1. Input a matrix

2. Find the inverse of a matrix

3. Store a matrix

4. Find the transpose of a matrix

5. Find the determinant of a matrix

6. Find the product of two matrices

7. Augment a matrix

8. Reduce a matrix to row echelon form

9. Reduce a matrix to reduced row echelon form

Matrices are commonly used to solve systems of linear equations. Many of the exercises your instructor will assign may be completed entirely on the TI-83 with limited understanding of the concepts. We recommend that the techniques covered in this chapter be used to check your work rather than do you work. A thorough understanding of the matrix concepts will allow you to further take advantage of versatility of the TI-83.


18 Chapter 4

All rights reserved.

MATRX

Use the horizontal arrows to

. ENTER 4. Press

ENTER

EN6. Type in the number of columns and press TER

MATRX

Input a matrix

Press to activate the Matrix Menu.

move to EDIT.

.

.

. Type in each of the matrix entries using the horizontal and vertical arrows to move

8. Press to return to the Matrix Menu.

eystrokes: K1.

2.

3. Select 1 to enter matrix A.

5. Type in the number of rows and press

7between columns and rows.

Copyright © Houghton Mifflin Company.

Matrices and Systems of Equations 19

Example:

=

1352321

= 35

B [ ]013 −=C Enter matrix A , , and

80 17Solution:

ns).

1 column).

Matrix C is a 1 x 3 (1 row and 3 columns).

Matrix A is a 3 x 3 (3 rows and 3 colum

Matrix B is a 3 x 1 (3 rows and


20 Chapter 4


Find the

MATRX

ENTER

x-1

ENTER

2. Use the vertical arrowpress .

3. Press .

4. Press to view the inverse.

inverse of a matrix

he inverse of a atrix is a l step in solvin a system of equations. Although computing the inverse of a matrix is challenging to do by hand, it is one of the

1. Press to display the saved matrices.

keys to scroll to the matrix whose inverse you want to find and

singular. If you try to compute the inverse of a e the error message below.

Determining t m critica g

easiest matrix operations on the TI-83.

Keystrokes:

If a matrix is not invertible, it is said to besingular matrix you will receiv


Store a matrix

X MATR

2nd (—)

ENTER

fter performing matrix arithmetic you will often want to store the resultant matrix for future

still on the calculator display, press

e you want to assign to the matrix.

4. Press twice to assign the name to the matrix.

x

Areference. Keystrokes: 1. With the matrix you want to store

2. Press .

3. Pres

STO >

s and scroll to the nam

We assigned the name D to the inverse of matrix A. We’ll use D when we introduce matrimultiplication.


22 Chapter 4


MATRX

Find the transp

ENTER

MATRX

ENTER

ENTER 5. Press

ose of a matrix

s: . Press to display the saved matrices.

roll to the matrix whose transpose you want to find and

nu, arrow down to “T”, and press .

to display the transpose of the matrix.

A matrix multiplied by its transpose always results in a symmetric matrix.

Keystroke1

2. Use the vertical arrow keys to scpress .

. Press .

to A H me

3

4. Scroll T the M


MATRX

ENTE R 2. Scroll to the MATH menu, arrow down to “det(”, and pres

ENTER

MATRX

press twice.

MATRX

x

ll invertible matrices have a non-zero determinant. Since the determinant is easier to find , it is often useful to compute the determinant first to ensure the matrix has an

Keystrokes: 1. Press to display the Matrix Menu.

s .

3. Press .

4. Use the vertical arrow keys to scroll to the matrix whose determinant you want to find and

Example:

Show that the matrix is not invertible.

Solution:

Find the determinant of a matri Athan the inverseinverse.

= 111424

E

313


24 Chapter 4

Find the product of two matrices

MATRX

ENTER

X 3. Press . MATRX

ENTER

to activate the Matrix Menu and display the dimensions of saved

to scroll to the matrix you want to be first in your matrix

4. Press and use the vertical arrow keys to scroll to the matrix you want to be second in your matrix product.

5. Press twice to display the product.

Keystrokes:

. Press 1matrices.

2. Use the vertical arrow keys product and press .




ind the solution to the system of equations using matrices.

A • X = B. The solution to the equation is X = A-1B.

and . We stored

D gives

, and x3 = 2 is the solution to the system of equations. This method works is invertib ue solution does not exist. For this reason,

anually using the techniques covered in the text. In practice, systems of equations rarely consist of just three equations and three unknowns. Oftentimes there are hundreds of equations and unknowns. When dealing with matrices of large dimensions, using the calculator or a computer to do matrix manipulations greatly reduces computation time and human error.

Example: F

++ 0 21

1

xx

++ 352

321

xxx===++

1783

532

3

2

x

xxx

3

Solution:

The equation may be represented as

We previously entered the matrix

=

801352321

A

puting D • B

=

1735

B

−25

3 as

−−

=− 51391640

1A . Com −1−

So x1 = 1, x2 = -1only if A le. If A is not invertible, a uniqit is important to know how to manipulate the matrices m

26 Chapter 4


Augment

MATRX 1. Press to display the Matrix Menu. ENTER

MATRX 3. Press you want to augment and press

ENTER

ENTER

MATRX


5. Press to display the Matrix Menu, use the arrow keys to scroll to the matrix you want to append to A, and press twice

a matrix

Instead of computing -1B to find the solution of a system of linear equations, AX = B, we can augment the matrix A with the matrix B and use Gaussian elimination to determine the solution. This section will demonstrate how to augment a matrix and the next two sections

ill illustrate how to put an augmented matrix in row echelon and reduced row echelon form.

es:

w down to “au .

to display the Matrix Menu, use the arrow keys to scroll to the matrix .

.

We’ll name the matrix E and store it for use in the next section using the procedures previously covered.

A

w

Keystrok

2. Scroll to the MATH menu, arro gment(”, and press



R

MATRX 1. Press to display the Matrix Menu. ENTER

ENTER

MATRX 3. Press to display th e

educe a matrix to row echelon form

chelon form, it is easy to deduce the solution to the system of equations.

k

w down to “ref(”, and press .

e M nu, use the arrow keys to scroll to the matrix you want to row reduce and press twice.

A matrix is said to be in row echelon if (1) the first nonzero entry in a row is a 1 (2) rows ofzeros (if any) are grouped at the bottom of the matrix, and (3) the leading one in a lower rowis further to the right of the leading one in a higher row. When an augmented matrix is in row e Keystro es:

2. Scroll to the MATH menu, arro

Matrix

The resultant matrix is

and is equivalent to the system of equations

From the third line, we see x3 = 2. Using back substitution, we find x2 = -1 and x1 = 1.

−− 2.66.2105.5.21

2100

15.1

=−=−

=++

212.66.21

5.15.15.2

3

32

321

xxxxxx

28 Chapter 4

A matrix in row echelon formcontaining a 1 has zeros everywheechelon form, the solution to the sys Keystrokes:

MATRX 1. Press to display the Matrix Menu.

2. Scroll to the MAT ENTER

ENTER

MATRX

=−=

=

21

1

3

2

1

xxx

Reduce a matrix to reduced row echelon form is said to be in reduced row echelon form if each column

re else. When an augmented matrix is in reduced row tem of equations is readily apparent.

H menu, arrow down to “rref(”, and press .

. Press to display the Matrix Menu, use the arrow keys to scroll to the matrix reduce and press twice.

The matrix above is equivalent to the system of equations . This solution is the same as the solution to the example in the Product of Two Matrices section. Although different, both me ield the same result. Which method is best is a

atter of personal preference.

3you want to row

thods ym



Chapter 4 Linear Programming In this chapter you will l rn how to do the following:

1. Graph an inequality

. Clear a shaded region from the display

. Graphically determine a linear programming problem solution

e covered in the graphical exploration portion of Finite Mathematics and will not be addressed in this manual. Techniques for solving linear programming problems using standard TI-83 functionality will be addressed herein.

ea

2

3

The optional programs ViewLP and PIVOT ar


30 Chapter 4

Graph an inequality The solution to an inequality may be drawn on the TI-83 using the Shade command; however, the process does require some analysis to determine what terms

PRGM 1. Press to activate the Draw2nd

ENTER 2. Scroll down to option 7 using the vertical arrow keys and press .

3. The syntax for the Shade function is Shade(lower function, upper function, mxel resolution). The lo

Ticance of the pattern and pixel reso

aphing Calculator Guidebook.

to input into the hade function.

menu.

inimum -value, maximum x-value, pattern, and pi wer function and upper

I-83 will use default values for e remaining parameters. The signif lution parameters are

ddressed in the TI-83 Gr

The TI-83 shaded the region bounded below by y = -x + 3 and above by y = 0. In other terms, -x + 3 ≤ y ≤ 0.

S

Keystrokes:

xfunction are the only parameters that must be specified: thetha

Linear Programming 31


Example: raph the solution to theG

ion are the domain values that solution on a num

ay be altered by changing the param

inequality –x2 + 4x ≥ 0.

atisfy the inequality ber line. In this case, we graphed

eters as shown below.

Solution:

The x-intercepts of the shaded reg

The appearance of the graph m

-x2 + 4x ≥ 0. Typically, we represent this sthe multivariable double inequality –x2 + 4x ≥ y ≥ 0.

32 Chapter 4

Clear a shaded regio

PRGM 2nd 1. Press

ENTER

n from the display

ecuted.

to activate the Draw menu.

cal arrow keys and press twice to clear

Example: Graph the inequality 0 < 2x < y . Clear the graphic display then graph 0 < –x + 5 < y. Then regraph the first inequality without clearing the graphic display. Solution:

he set of values that satisfy both inequalities.

A shaded region will remain on the graphic display until the ClrDraw command is ex Keystrokes:

2. Scroll down to option 1 using the vertie shaded region. th

The image below shows what happens when you don’t execute the ClrDraw command. The result is still useful: the intersection of the two regions is t

Linear Programming 33

Graphically determine a linear programming problem solution

The Fundamental Theorem of Linear Programming states that if a linear programming roblem has a solution, then it occurs at a vertex of the region determined by the constraints.

By graphing the constraints and using the Intersection or Trace features, we can determine e can also visually predict which vertex will yield the

w keystrokes are introduced in this section; however, culator skills to linear programming problems is worth a

Solution:

Replacing x1 with x and x2 with y and solving each inequality for y yields

p

the coordinates of the vertices. Wsolution before computing it. No neknowing how to apply existing calreview Example: Maximize P = 13x1 + 7x2

Subject to

≤+≤+126

2025

21

21

xxxx

≥

≥+

0

1234

2

21

x

xx

≥ 01x

≥≥

−≤

−≤

−≥

002104

61

25

34

yx

xyxy

xy

Graphing the three lines in the first quadrant yields

The bounded region satisfies all of the inequalities and has vertices (3, 0), (4, 0), (3.42, 1.42), and (1.71, 1.71). We are to maximize P = 13x1 + 7x2. The P values are, respectively: 39, 52, 54.4, 34.2. So P is maximized at vertex (3.42, 1.42) when subjected to the given constraints.


Chapter 5 Probability In this chapter you will learn how to do the following:

1. Compute factorials

2. Compute permutations

3. Compute combinations

4. Determine a binomial distribution


36 Chapter 5

Compute factorials Factorials are used to compute the number of ways a set of items may be ordered. n! read "n factorial" means n ⋅ (n-1) . . . 3 ⋅ 2 ⋅ 1 . The largest factorial the TI-83 can compute is 69!. Larger values result in an overflow error.

Keystrokes: 1. Enter the number of items you want to order on the home screen.

MATH 2. Press and scroll to PRB to activate the Probability Menu.

ENTER3. Scroll to item 4 and press .

Example: How many different four-digit PIN numbers can be made using 2, 4, 6, and 8? Solution:

4! = 4 ⋅ 3 ⋅ 2 ⋅ 1 = 24 so there are 24 four-digit PIN numbers that can be made.


Probability 37

Compute permutations The number of ways that n distinct objects taken r at a time may be ordered is called a permutation and is denoted by nPr. Keystrokes: 1. Enter the number of items you have to pick from on the home screen.



ENTER4. Enter the number of items you want to choose and press . Example: How many four-digit PIN numbers with distinct digits can be made from the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9? Solution: There are ten numbers to chose from and we want to pick four so we need to compute 10P4.

504078910!6!10

)!410(!10

410 =⋅⋅⋅==−

=P

So there are 5040 different four-digit PIN numbers with distinct digits.

In this example, the PIN numbers using a different ordering of the same digits are counted as distinct. That is, even though the following PIN numbers use the same four digits, they are each counted since their ordering is distinct.

1234, 1243, 1324, 1342, 1423, 1432 2134, 2143, 2314, 2341, 2413, 2431 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321

In the next section, we will discover how many four-digit PIN numbers can be made such that no two PIN numbers have the same digits.


38 Chapter 5

Computing combinations When order does not matter, combinations are used instead of permutations. The number of sets of n distinct objects taken r at a time is called a combination and is denoted by nCr. Keystrokes: 1. Enter the number of items you have to pick from on the home screen.



ENTER4. Enter the number of items you want to choose and press . Example: How many four-digit PIN numbers can be made without repetition and without regard to order from the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9? Solution: There are ten numbers to chose from and we want to pick four so we need to compute 10C4.

21024

5040123478910

!4!6!10

!4)!410(!10

410 ==⋅⋅⋅⋅⋅⋅

==−

=C

So there are 210 different four-digit PIN numbers with distinct digits. The number of combinations is significantly smaller than the number of permutations because PIN numbers containing the same digits in a different order are not counted. That is, all twenty-four PIN numbers shown in the previous section are considered to be the same PIN when using combinations.


Probability 39

Determine a binomial distribution Keystrokes:

2nd VARS 1. Press to activate the Distributions Menu.

ENTER 2. Scroll to 0 and press .

3. Enter the number of Bernoulli trials.

4. Press 'ENTER5. Enter the probability of success for each trial and press .

MATH

ENTER 6. To change the decimal approximations to fractions, press , scroll to 1, and press .


40 Chapter 5

Example: A family has six children. What the probability that they are all boys? What is the probability that there are three boys and three girls? What is the probability that there are less boys than girls? Solution: Let X be the number of boys and assume the probability a child is a boy is 0.50.

The probability distribution is

641,

323,

6415,

165,

6415,

323,

641 . The probability that they are

all boys is 641 or 0.015625. So there is roughly a 1.5% chance that they are all boys.

The same distribution may be used to answer the second and third question. The

probability that three of the children are boys is 165 or 0.3125. So there is roughly a 31%

chance that there are three boys and three girls in the family.

If there are 0, 1, or 2 boys there are less boys than girls. By adding their respective

probabilities (6415,

323,

641 ) we determine that the probability that there are less boys than

girls is 3211 or 0.34375. So there is roughly a 34% chance that the girls outnumber the boys.


Chapter 6 Statistics In this chapter you will learn how to do the following:

1. Sort a list of numerical data

2. Calculate the mean, median, and standard deviation of a data list

3. Draw a box-and-whisker plot

4. Model data using a histogram

5. Calculate the cumulative values of the normal distribution

6. Shade the area under the normal distribution curve


Statistics 42

Sort a list of numerical data Keystrokes: 1. Enter the data into list L1.

STAT 2. Press .

3. Pick item 2 to sort the data in ascending order or item 3 to sort the data in descending order.

2nd 1 4. Enter .

5. The list is sorted in ascending and descending order respectively.

This approach is useful when you want to determine the data value that occurs most frequently (mode). Unfortunately, the TI-83 does not have a mode command.

43 Chapter 6

Calculate the mean, median, and standard deviation of a data list Keystrokes:

1. Enter the data list into L1 2nd STAT 2. Press to activate the List menu.

3. Scroll to the MATH menu.

4. Select item 3 for mean, item 4 for median, or item 5 for standard deviation.

2nd 1 5. Enter .

6. Repeating steps 4 and 5 for each desired value results in the following display.

Example: A+ Airlines is hoping to attract customers to their Wednesday Seattle to Los Angeles route. To determine what ticket price to charge, their market researchers randomly collected the following ticket prices from 25 passengers traveling the route on Wednesday on competitor airlines: {215, 99, 135, 199, 215, 179, 199, 215, 173, 155, 107, 210, 159, 199, 199, 121, 299, 215, 189, 189, 209, 156, 209, 222, 215}. Conduct a statistical analysis to determine mean, median, and standard deviation. Solution:

Statistics 44

The average ticket price is $187.28 and the median ticket price is $199.00. Draw a box-and-whisker plot Keystrokes:

2nd Y= 1. Press to activate the Statistics Plot menu.

2. Select item 1 and press . ENTER

ENTER3. Move the cursor from Off to On and press .

ENTER

4. Scroll to the middle icon on the second row and press .

GRAPH 5. Press .

44(contineud) Ch 6

This is the box-and-whisker plot for the A+ Airlines data presented in the previous section. The first quartile is between 99 and 157.5, the second is between 157.5 and 199, the third is between 199 and 215, and the fourth is between 215 and 299. If A+ Airlines charges $157.50, their ticket will cost less than 75% of their competitors.

Statistics 45

Model data using a histogram

Keystrokes: 1. Press to activate the Statistics Plot menu.

ENTER2. Select item 1 and press .

ENTER

3. Move the cursor from Off to On and press . ENTER

4. Scroll to the middle icon on the second row and press .

GRAPH 5. Set the parameters for your graphing window as demonstrated in the following example and press .

46 Chapter 6

Example: To determine how severe the speeding problem was in a 15 mph zone near an elementary school, local law enforcement officials recorded the speed of the forty vehicles passing through the zone during the thirty minutes before the beginning of classes. The speeds were 20, 22, 10, 35, 23, 17, 15, 22, 30, 32, 29, 11, 13, 16, 14, 29, 23, 20, 31, 15, 5, 12, 13, 28, 34, 29, 19, 11, 28, 18, 12, 14, 18, 17, 19, 15, 16, 23, 22, and 19. Draw the histogram of the data with class width of 5. Solution: Entering the list of data into L1 and sorting the list in ascending order yields:

To guarantee that the entire histogram is visible it is important to set the window parameters to fit your data.

Since the speeds range from 5 mph to 35 mph, we set Xmin = 0 and Xmax = 40. The

Xscl parameter is the class width. Although the y-values will never be negative, it is wise to select Ymin = -5 to allow room on the display to show alphanumeric data. You may need to adjust the Ymax parameter a couple of times in order to display the entire histogram. Other parameters may remain unchanged.

Statistics 47

Twenty-five percent of the cars (10/40) of the cars were exceeding the speed limit by 10 mph or more. The information was presented to the City Council who approved a measure to double fines for drivers exceeding the speed limit by 10 mph or more in a school zone.

48 Chapter 6

Calculate the cumulative values of the normal distribution Keystrokes:

1. If you don’t know the mean and standard deviation of the normal distribution, you should compute them before using the normalcdf command.

2nd VARS 2. Press to activate the Distribution menu.

3. Select item 2.

ENTER 4. Enter the starting value, ending value, mean and standard deviation of the normal distribution and press .

The probability that the normal random variable is between 2.0 and 5.6 is 40.0%.

Statistics 49

Example: Members of the citizens’ group, Need for Speed, were outraged by the City Council’s approval of the fine doubling measure in the example in the previous section. They argued that the data sample was so small it did not accurately reflect the typical behavior of motorists in the area. The group hired an independent firm, We Fix Your Data, to measure motorist speeds in the area over a one-month period. The firm reported that the average speed was 17 mph with a 5 mph standard deviation. Determine the percentage of motorists traveling within one standard deviation of the mean. Solution: Twelve mph is one standard deviation below the 17 mph mean and 22 is one standard deviation above the mean.

Approximately 68% of drivers traveled at speed within one standard deviation of the mean. Most motorists travel through the area between 12 mph and 22 mph.

50 Chapter 6

Shade the area under the normal distribution curve Keystrokes:

1. If you don’t know the mean and standard deviation of the normal distribution, you should compute them before using the ShadeNorm command.

2nd VARS 2. Press to activate the Distribution menu.

3. Scroll to the Draw to activate the Draw menu.

4. Select item 1.

ENTER 5. Enter the starting value, ending value, mean and standard deviation of the normal distribution and press .

The area is the probability that the normal random variable is between 24.5 and 42. The data set used here is the same as that used in the example in the previous section. Twenty-four and one half mph is 1.5 standard deviations above the mean, 37 mph is four standard deviations above the mean, 17 mph is the mean, and 5 mph is the standard deviation. Approximately 6.7% of the motorists exceeded the speed limit by 10 mph or more. Since the We Fix Your Data group’s data was gathered over a period of a month, it more reliable than the police data that was gathered in a single day. The percentage of motorists daily exceeding the speed limit by 10 mph or more is closer to 6.7% than the 25% originally claimed.

Chapter 7 Markov Chains Markov chains rely heavily upon the matrix manipulation techniques developed in Chapter 3 of this guide. In this chapter we will demonstrate how to use those techniques and others to do the following:

1. Compute the product of a matrix and itself a finite number of times. (This technique is used to find the states of a Markov chain given the transition matrix.)

2. Find the steady state distribution of a Markov chain transition matrix.


52 Chapter 7

Compute the product of a matrix and itself a finite number of times Keystrokes: 1. Enter the matrix and display it on screen using techniques covered in Chapter 3.

∧ 2. Press the key and enter the number of times you want to multiply the matrix

by itself.

ENTER 3. Press .

The matrix displayed is the matrix that results from multiplying the matrix B by itself five times. If we want to multiply the resultant matrix by itself a designated number of times, we need only repeat steps 2 and 3. When calculating the states of a Markov chain, it is useful to use this technique as shown in the following example.


Markov Chains 53

Example: The property transfer records of a real estate agency in the Salt Lake City suburbs show that each year 96% of those who owned their house the entire previous year will remain for the next year while only 46% of those who moved into their house during the year will remain for the next year. Determine the Markov chain transition matrix and calculate the first four states. Explain the meaning of S4, the fourth state of the Markov chain. Solution: Letting the first column represent the probability of staying, the second column represent the probability of moving, the first row represent owners who owned their home for more than a year, and the second row represent owners who owned their home less than a year yields the following Markov chain transition matrix:

=

54.46.04.96.

B

Computing the first four states yields:

=

1375.8625.075.925.4B means that the probability that a home owner, after living in their home

for the past five years, will remain an additional year is 92.5%. The probability that a homeowner, who moved in between four and five years ago, will remain an additional year is 86.25%.


54 Chapter 7

Find the steady state distribution of a Markov chain Keystrokes: The keystrokes for this section are identical to the previous section. Example: Continuing with the example from the previous section, find the steady state distribution for

the Markov chain with transition matrix .

=

54.46.04.96.

B

Solution: We increase the number of transitions until a steady state is reached.

After 35 transitions, a steady state is reached indicating that 35 years after the initial observation both groups have a 92% chance of staying and an 8% chance of moving. The state distribution vector [ ]08.92.=D

54.46.04.96.

is a steady state distribution for the Markov

chain with transition matrix since D · B = D. =B

It follows that for n ≥ 35, .

=

08.92.08.92.nB


Chapter L (available on the CD/WEB) Logic Much of the symbolism used in logic is beyond the scope of most graphing calculators. However, truth tables may be represented on the TI-83. In this chapter, you will learn how to use the Boolean operators and, or, and not to create truth tables to validate various laws of logic.


56 Chapter L

Create truth tables to validate various laws of logic Keystrokes: 1. Enter the list L1 = (1 1 0 0) and L2 = (1 0 1 0). The “1” represents true and the “0” represents false.

This is symbolically equivalent to

p

q

Logic Statement

T T T F F T F F

The list L1 represents the possible choices for p and L2 represents the possible choices for q.

2. The Boolean operators and, or, and not are accessed by pressing MATH 2nd

and scrolling to the Logic menu.

ENTER

Enter the logic statement using L1 for p and L2 for q and press .


Logic 57

The result is symbolically equivalent to the truth table

p q ~ (p ∧ q) T T F T F T F T T F F T

Use of the calculator is especially meaningful when evaluating complex statements as demonstrated in the following example.

Example 1: Show that (p ∧ q) ∨ (~p ∧ q) is equivalent to q. Solution:

Using the L1 and L2 previously identified, the statement (p ∧ q) ∨ (~p ∧ q) may be represented as

The statement q may be represented as

Since both statements result in (1 0 1 0) they are equivalent. Use of the calculator in generating truth tables is especially useful as the complexity of the compound statement increases as shown in Example 2.


58 Chapter L

Example 2: Find the final column of the truth table for the compound statement: The stock market is down and the investors are confident or the investors are buying new stock offerings and the stock market is up. Solution: There are three basic statements:

p = The stock market is up. q = The investors are confident. r = The investors are buying new stock offerings.

The compound statement may be symbolically represented as (~p ∧ q ) ∨ (r ∧ p) .

We will use L1, L2, and L3 to represent p, q, and r.

The statement (~p ∧ q ) ∨ (r ∧ p) is equivalent to

The result may be displayed in the truth table as

p q r (~p ∧ q ) ∨ (r ∧ p)

T T T T T T F F T F T T T F F F F T T T F T F T F F T F F F F F


Date post:	12-Sep-2021
Category:	Documents
Upload:	others
View:	9 times
Download:	0 times

Graphing Calculator Guide for Finite Mathematics

Documents