MTH 209 Week 5 Third. Final Exam logistics Here is what I've found out about the final exam in MyMathLab (running from a week ago to 11:59pm five days

MTH 209 Week 5

Third

Final Exam logistics

Here is what I've found out about the final exam in MyMathLab (running from a week ago to 11:59pm five days after class tonight.

.

Slide 2Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Final Exam logistics There will be 50 questions. You have only one attempt to complete the exam. Once you start the exam, it must be completed in that sitting. (Don't start until you have

time to complete it that day or evening.) You may skip and get back to a question BUT return to it before you hit submit.

You must be in the same session to return to a question. There is no time limit to the exam (except for 11:59pm five nights after the last class). You will not have the following help that exists in homework:

Online sections of the textbook Animated help Step-by-step instructions Video explanations Links to similar exercises

You will be logged out of the exam automatically after 3 hours of inactivity. Your session will end.

IMPORTANT! You will also be logged out of the exam if you use your back button on your browser. You session will end.


Due for this week…

Homework 3 (on MyMathLab – via the Materials Link) The fifth night after class at 11:59pm.

Do the MyMathLab Self-Check for week 5. Learning team hardest problem assignment. Complete the Week 5 study plan after submitting

week 5 homework. Participate in the Chat Discussions in the OLS (yes,

one more time).


Section 12.1

Composite and Inverse

Functions

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Composition of Functions

• One-to-One Functions

• Inverse Functions

• Tables and Graphs of Inverse Functions

The compositions and represent evaluating functions f and g in two different ways. When evaluating function f is performed first followed by function g, whereas for functions g is performed first followed by function f.

g f f g

,g f

f g

Example

Evaluatea. b.Solutiona.

b.

(3).g f2( ) ; ( ) 2 3f x x g x x

2( ) 2 ; ( ) 2 1f x x g x x x

(3)g f )3( ( )g f

)9(g

15

2(3) 3 9f

2(6) 6 2(6) 1g

(3)g f )3( ( )g f)6(g

25

(3) 2(3) 6f

(9) 2(9) 3g

Try Q’s pg 806 13,15,19

Example

Use Table 12.1 and 12.2 to evaluate the expression. (3)f g

Table 9.1x 0 1 2 3

f(x) 1 2 5 7

x 0 1 2 3

g(x)

1 0 1 2

Table 9.2

(3)f g )3( ( )f g

)2(f

5

Try Q’s pg 806 23,25

Example

Use the graph below to evaluate ( )(2).g f

Try Q’s pg 806 29

Example

Determine whether each graph represents a one-to-one function.a. b.

The function is not one-to-one.

The function is one-to-one.

Try Q’s pg 806 37,39

Example

State the inverse operations for the statement. Then write a function f for the given statement and a function g for its inverse operations.

Multiply x by 4. SolutionThe inverse of multiplying by 4 is to divide by 4.

( ) 4f x x

( )4

xg x

Try Q’s pg 807 43,49

Example

Find the inverse of the one-to-one function.f(x) = 4x – 3 SolutionStep 1: Let y = 4x – 3 Step 2: Write the formula as x = 4y – 3Step 3: Solve for y.

4 3x y

3 4x y

3

4

xy

Try Q’s pg 807 63,71

Tables and Graphs of Inverse FunctionsInverse functions can be represented with tables and graphs. The table below shows a table of values for a function f.

x 1 2 3 4 5

f(x)

4 8 12 16 20

x 4 8 12 16 20

f -1(x) 1 2 3 4 5

The table below shows a table of values for the inverse of f.

Graphs of Inverse Functions

1( ) 2

( ) 2

f x x

f x x

Section 12.3

Logarithmic Functions


Objectives

• The Common Logarithmic Function

• The Inverse of the Common Logarithmic Function

• Logarithms with Other Bases

Common Logarithmic Functions

A common logarithm is an exponent having base 10.Denoted log or log10.

Example

Evaluate each expression, if possible.a. b.

Solutiona. Is x positive? No, x = –1000 is negative, log x is undefined.

b. Is x positive? Yes, x = 10,000 Write x as 10k for some real number k.10,000 = 104

If x = 10k, then log x = k; log x = log 10,000 = log 104 = 4

log( 1000) 2log100

Try Q’s pg 835 27,31

Example

Simplify each common logarithm. a. b. c. log 55

Solutiona.

log1000 1log

100

b.

31000 103log1000 10 3

2110

100

21log log10

002

1

log1000

1log

100

c. log 55

The power of 10 is not obvious, use a calculator.

Try Q’s pg 835 17,23, 29,43

Graphs• The graph of a common logarithm increases very slowly for

large values of x. For example, x must be 100 for log x to reach 2 and must be 1000 for log x to reach 3.

• The graph passes through the point (1, 0). Thus log 1 = 0.• The graph does not exist for negative values of x. The

domain of log x includes only positive numbers. The range of log x includes all real numbers.

• When 0 < x < 1, log x outputs negative values. The y-axis is a vertical asymptote, so as x approaches 0, log x approaches .

Graphs

The graph of y = log x shown is a one-to-one function because it passes the horizontal line test.

Example

Use inverse properties to simplify each expression. a. b.

Solution

a. b.

2 4log10x log810

log810 8

2 4log10x

2 4x

log810

Because 10logx = x for any positive real number x,

Try Q’s pg 835 25,35,37,41

Example

Graph each function f and compare its graph to y = log x.a. b.Solutiona. Use the knowledge of translations to sketch the graph. The graph of log(x – 3) is similar to the graph of log x, except it is translated 3 units to the right.

log( 3)x log( ) 2x

Example

Graph each function f and compare its graph to y = log x.a. b.Solutionb. Use the knowledge of translations to sketch the graph. The graph of log(x) + 2 is similar to the graph of log x, except it is translated 2 units upward.

log( 3)x log( ) 2x

Try Q’s pg 835 47,49

Example

Sound levels in decibels (dB) can be computed by f(x) = 160 + 10 log x, where x is the intensity of the sound in watts per square centimeter. Ordinary conversation has an intensity of 10-10 w/cm2. What decibel level is this?SolutionTo find the decibel level, evaluate f(10-10).

1 10 0log(1

( ) 160 10l

0

og

(10 ) 160 )10

f x x

f

160 10( 10) 60

Try Q’s pg 836 105

Logarithms with Other Bases

Common logarithms are base-10 logarithms, but we can define logarithms having other bases.

For example base-2 logarithms are frequently used in computer science.

A base-2 logarithm is an exponent having base 2.

Example

Simplify each logarithm.a. b.

Solution

2log 16 2

1log

32

4a. 16 251

b. 232

24

2log 16 log 2 4 2

52

1log log 2

325

Try Q’s pg 835 81,83

Natural Logarithms

The base-e logarithm is referred to as a natural logarithm and denoted either logex or ln x.

A natural logarithm is an exponent having base e.

To evaluate natural logarithms we usually use a calculator.

Example

Approximate to the nearest hundredth. a. b.

Solution

ln 201

ln4

a.ln 201

b. ln4

Try Q’s pg 836 61,63

Example

Simplify each logarithm.a. b.

Solution

6log 36 23log 9

6a. log 36 23b. log 9

6 62

236 6

log 36 log 26

2 2 2 4

43

9 (3 ) 3

log 3 4

Try Q’s pg 835-6 55,75,85,87

Example

Simplify each expression.a. b.

Solution

ln 6e log(4 16)10 x

6 4 16x ln6a. e

because elnk = k for all positive k.

log(4 16)b. 10 x

for x > 4 because 10logk = k for all positive k.

Try Q’s pg 835-6 39,57,59,89

Section 14.1

Sequences


Objectives

• Basic Concepts

• Representations of Sequences

• Models and Applications

, 0 and 1,xf x a a a

A finite sequence is a function whose domain is D = {1, 2, 3, …, n} for some fixed natural number n.

An infinite sequence is a function whose domain is the set of natural numbers.

SEQUENCES

The nth term, or general term, of a sequence is an = f(n).

Example

Write the first four terms of each sequence for n = 1, 2, 3, and 4.a. f(n) = 5n + 3 b. f(n) = (4)n-1 + 2Solutiona. f(n) = 5n + 3

a1 = f(1) = 5(1) + 3 = 8 a2 = f(2) = 5(2) + 3 = 13

a3 = f(3) = 5(3) + 3 = 18a4 = f(4) = 5(4) + 3 = 23

The first four terms are 8, 13, 18, and 23.

Example (cont)

Write the first four terms of each sequence for n = 1, 2, 3, and 4.a. f(n) = 5n + 3 b. f(n) = (4)n-1 + 2Solutionb. f(n) = (4)n-1 + 2

a1 = f(1) = (4)1-1 + 2 = 3 a2 = f(2) = (4)2-1 + 2 = 6 a3 = f(3) = (4)3-1 + 2 = 18 a4 = f(4) = (4)4-1 + 2 = 66

The first four terms are 3, 6, 18, and 66.

Try Q’s pg 915 9,11,13

Example

Use the graph to write the terms of the sequence.

SolutionThe points (1, 2), (2, 4),(3, −6), (4, 8), and (5, −10) are shown in the graph.

The terms of the sequence are 2, 4, −6, 8, and −10.

X

Y

1 2 3 4 5 6 7 8 9 10

-10

-8

-6

-4

-2

2

4

6

8

10

0

Example

An employee at a parcel delivery company has 20 hours of overtime each month. Give symbolic, numerical, and graphical representations for a sequence that models the total amount of overtime in a 6 month period. SolutionSymbolic RepresentationLet an = 20n for n = 1, 2, 3, …, 6Numerical Representation

n 1 2 3 4 5 6

an 20 40 60 80 100 120

Example (cont)

Graphical RepresentationPlot the points (1, 20), (2, 40), (3, 60), (4, 80), (5, 100), (6, 120).

X

Y

1 2 3 4 5 6 7 8 9 10

15

30

45

60

75

90

105

120

135

150

0

Months

Hours

Overtime

Example

Suppose that the initial population of adult female insects is 700 per acre and that r = 1.09. Then the average number of female insects per acre at the beginning of the year n is described by an = 700(1.09)n-1. (See Example 4.)SolutionNumerical Representation

n 1 2 3 4 5 6

an 700 763 831.67

906.52

988.11

1077.04

Example (cont)

Graphical RepresentationPlot the points (1, 700), (2, 763), (3, 831.67), (4, 906.52), (5, 988.11), and (6, 1077.04).

These results indicate that the insect population gradually increases. Because the growth factor is 1.09, the population is increasing by 9% each year.

X

Y

1 2 3 4 5 6 7 8 9 10

120

240

360

480

600

720

840

960

1080

1200

0

Year

Inse

ct P

opula

tion (

per

acr

e)

Try Q’s pg 916 29, 39,45

Section 14.2

Arithmetic and Geometric Sequences


Objectives

• Representations of Arithmetic Sequences

• Representations of Geometric Sequences

• Applications and Models


An arithmetic sequence is a linear function given by an = dn + c whose domain is the set of natural numbers. The value of d is called the common difference.

ARITHMETIC SEQUENCE

Example

Determine whether f is an arithmetic sequence. If it is, identify the common difference d. a. f(n) = 7n + 4

Solutiona. This sequence is arithmetic because f(x) = 7n + 4 defines a linear function. The common difference is d = 7.

Example (cont)

Determine whether f is an arithmetic sequence. If it is, identify the common difference d. b. c.n f(n)

1 −8

2 −5

3 −2

4 1

5 4X

Y

1 2 3 4 5 6 7 8

2

4

6

8

10

12

14

16

18

20

0

The table reveals that each term is found by adding +3 to the previous term. This represents an arithmetic sequence with the common difference of 3.

The sequence shown in the graph is not an arithmetic sequence because the points are not collinear. That is, there is no common difference.

Try Q’s pg 923-24 11,17,25

Example

Find the general term an for each arithmetic sequence.a. a1 = 4 and d = −3 b. a1 = 5 and a8 = 33 Solutiona. Let an = dn + c for d = −3, we write an = −3n + c, and to find c we use a1 = 4.

a1 = −3(1) + c = 4 or c = 7Thus, an = −3n + 7.

b. The common difference isTherefore, an = 4n + c. To find c we use a1 = 5.a1 = 4(1) + c = 5 or c = 1. Thus an = 4n + 1.

33 54.

8 1d

Try Q’s pg 924 29,31


The nth term an of an arithmetic sequence is given by

an = a1 + (n – 1)d,

where a1 is the first term and d is the common difference.

GENERAL TERM OF AN ARITHMETIC

SEQUENCE

Example

If a1 = 2 and d = 5, find a17

Solution

To find a17 apply the formula an = a1 + (n – 1)d. a17 = 2 + (17 – 1)5

= 82

Try Q’s pg 924 35


A geometric sequence is given by an = a1(r)n-1, where n is a natural number and r ≠ 0 or 1. The value of r is called the common ratio, and a1 is the first term of the sequence.

GEOMETRIC SEQUENCE

Example

Determine whether f is a geometric sequence. If it is, identify the common ratio. a. f(n) = 4(1.7)n-1

b. c. n f(n)

1 36

2 12

3 4

4 4/3

5 4/9 X

Y

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

9

10

0

Example

Determine whether f is a geometric sequence. If it is, identify the common ratio. a. f(n) = 4(1.7)n-1

b.n f(n)

1 36

2 12

3 4

4 4/3

5 4/9

This sequence is geometric because f(x) = 4(1.7)n -1 defines a exponential function. The common ratio is 1.7.

The table reveals that each successive term is one-third the previous. This sequence represents a geometric sequence with a common ration of r = 1/3.

Try Q’s pg 924 41,45,53

Example

Find a general term an for each geometric sequence.a. a1 = 4 and r = 5 b. a1 = 3, a3 = 12, and r < 0.Solutiona. Let an = a1(r)n-1.

Thus, an = 4(5)n-1

b. an = a1(r)n-1

a3 = a1(r)3-1

12 = 3r2

4 = r2

r = ±2

It is specified that r < 0, so r = −2

and an = 3(−2)n-1.

Example

If a1 = 2 and r = 4, find a9

SolutionTo find a9 apply the formula an = a1(r)n-1 with a1 = 2, r = 4, and n = 9.

a9 = 2(4)9-1

a9 = 2(4)8

a9 = 131,072

Try Q’s pg 925 61

Section 14.3

Series


Objectives

• Basic Concepts

• Arithmetic Series

• Geometric Series

• Summation Notation

Introduction

A series is the summation of the terms in a sequence.

Series are used to approximate functions that are too complicated to have a simple formula.

Series are instrumental in calculating accurate approximations of numbers like

and .e

Slide 65


A finite series is an expression of the form

a1 + a2 + a3+ ∙∙∙ + an.

FINITE SERIES

Example

The table represents the number of Lyme disease cases reported in Connecticut from 1999 – 2005, where n = 1 corresponds to 1999.

a. Write a series whose sum represents the total number

of Lyme Disease cases reported from 1999 to 2005. Find its sum.

b. Interpret the series a1 + a2 + a3+ ∙∙∙ + a9.

n 1 2 3 4 5 6 7

an 3215 3773 3597 4631 1403 1348 1810

Example (cont)

a. Write a series whose sum represents the total number

of Lyme Disease cases reported from 1999 to 2005. Find its sum.The required series and sum are given by: 3215 + 3773 + 3597 + 4631 + 1403 + 1348 + 1810 = 19,777.

b. Interpret the series a1 + a2 + a3+ ∙∙∙ + a9.This represents the total number of Lyme Disease

cases reported over 9 years from 1999 through 2007.

n 1 2 3 4 5 6 7

an 3215 3773 3597 4631 1403 1348 1810

Try Q’s pg 933 41


The sum of the first n terms of an arithmetic sequence denoted Sn, is found by averaging the first and nth terms and then multiplying by n. That is,

Sn = a1 + a2 + a3 + ∙∙∙ + an =

SUM OF THE FIRST n TERMS OF

AN ARITHMETIC SEQUENCE

1 .2

na an

Example

A worker has a starting annual salary of $45,000 and receives a $2500 raise each year. Calculate the total amount earned over 5 years.SolutionThe arithmetic sequence describing the salary during year n is computed by

an = 45,000 + 2500(n – 1).

The first and fifth years’ salaries area1 = 45,000 + 2500(1 – 1) = 45,000a5 = 45,000 + 2500(5 – 1) = 55,000

Example (cont)

Thus the total amount earned during this 5-year period is

The sum can also be found using

5

45,000 55,0005 $250,000.

2S

12 12n

nS a n d

5

52 45,000 5 1 2500 $250,000.

2S

Try Q’s pg 934 49

Example

Find the sum of the series

SolutionThe series has n = 19 terms with a1 = 4 and a19 = 58. We can then use the formula to find the sum.

4 7 10 58.

19

1

2

2

589

5419

8

nn

a aS n

S

Try Q’s pg 933 13


If its first term is a1 and its common ration is r, then the sum of the first n terms of a geometric sequence is given by

provided r ≠ 1.

SUM OF THE FIRST n TERMS OF A GEOMETRICSEQUENCE

1

1,

1

n

n

rS a

r

Example

Find the sum of the series 5 + 15 + 45 + 135 + 405.

Solution

The series is geometric with n = 5, a1 = 5, and r = 3,

so 5

5

1 35

1 3S

605.

Try Q’s pg 933 17,19

Example

A 30-year-old employee deposits $4000 into an account at the end of each year until age 65. If the interest rate is 8%, find the future value of the annuity.SolutionLet a1 = 4000, I = 0.08, and n = 35. The future value of the annuity is

1

1 1n

n

I

IS a

350.0

41 8

0.080 0

10

$689,267.

Try Q’s pg 934 23

SUMMATION NOTATION

1 2 31

n

k nk

a a a a a

Example

Find each sum.a. b. c.

Solution

a. b.

3

1

4k

k

3

1

4k

6

1

3 6k

k

3

1

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

k

k3

1

4 4 4 4 12k

= 4 8 12

24

Example (cont)

Find each sum.a. b. c.

Solution

c.

3

1

4k

k

3

1

4k

6

1

3 6k

k

9 12 15 18 21 24

99

6

1

3 6 3 1 6 3 2 6 3 3 6

3 4 6 3 5 6 3 6 6

k

k

Try Q’s pg 934 27,29,31

End of week 5

You again have the answers to those problems not assigned

Practice is SOOO important in this course. Work as much as you can with MyMathLab, the

materials in the text, and on my Webpage. Do everything you can scrape time up for, first the

hardest topics then the easiest. You are building a skill like typing, skiing, playing a

game, solving puzzles.

Documents

MTH 209 Week 5 Third. Final Exam logistics Here is what I've found out about the final exam in MyMathLab (running from a week ago to 11:59pm five days