Transcript Power Point Version

MTH 209 Week 5
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.
.
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2
Final Exam logistics
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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.
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3
Due for this week…
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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).
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4
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 compositionsg f andf g
represent
evaluating functions f and g in two different
g f,
ways. When evaluating
function f is
f g g,
performed first followed by function
whereas for
functions g is performed first
followed by function f.
Example
Evaluate  g f  (3).
a. f ( x)  x 2 ; g ( x)  2 x  3
Solution
a.  g f  (3)  g ( f (3))
b.
g
Try Q’s pg 806
2
f
(
x
)

2
x
;
g
(
x
)

x
 2x 1
b.
 g (9 )
f (3)  32  9
 15
g (9)  2(9)  3
f  (3)  g ( f (3))
 g ( 6)
 25
13,15,19
f (3)  2(3)  6
g (6)  62  2(6)  1
Example
Try Q’s pg 806
23,25
Use Table 12.1 and 12.2 to evaluate the expression.
f
g  (3)
Table 9.2
Table 9.1
x
0
1
2
3
x
0
1
2
3
f(x)
1
2
5
7
g(x
)
1
0
1
2
f
g  (3)  f ( g (3))
 f ( 2)
5
Example
Try Q’s pg 806
29
Use the graph below to evaluate ( g f )(2).
Example
Try Q’s pg 806
37,39
Determine whether each graph represents a one-toone function.
a.
b.
The function is
not one-to-one.
The function is
one-to-one.
Example
Try Q’s pg 807
43,49
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.
Solution
The inverse of multiplying by 4 is to divide by 4.
f ( x)  4 x
x
g ( x) 
4
Example
Try Q’s pg 807 63,71
Find the inverse of the one-to-one function.
f(x) = 4x – 3
Solution
Step 1: Let y = 4x – 3
Step 2: Write the formula as x = 4y – 3
Step 3: Solve for y. x  4 y  3
x  3  4y
x3
y
4
Tables and Graphs of Inverse
Functions
Inverse 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
The table below shows a table of values for the
inverse of f.
x
4
8
12
16
20
f1(x)
1
2
3
4
5
Graphs of Inverse Functions
f ( x)  x  2
f 1 ( x)  x  2
Section 12.3
Logarithmic
Functions
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
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
Try Q’s pg 835 27,31
Evaluate each expression, if possible.
2
log(

1000)
log100
a.
b.
Solution
a. 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
Example
Try Q’s pg 835 17,23, 29,43
Simplify each common logarithm.
1
log1000
log
a.
b.
c. log 55
100
Solution
a. log1000
c. log 55
1000  103
log1000  103  3
b. log 1
100
1
 10 2
100
1
log
 log102  2
100
The power of 10 is
not obvious, use a
calculator.
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
Try Q’s pg 835 25,35,37,41
Use inverse properties to simplify each expression.
x 4
log10
a.
b.10log8
2
Solution
a. log10x 4
2
 x2  4
b.10log8
Because 10logx = x
for any positive real
number x,
10log8  8
Example
Graph each function f and compare its graph to
y = log x.
a. log( x  3)
b. log( x)  2
Solution
a. 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.
Example
Try Q’s pg 835 47,49
Graph each function f and compare its graph to
y = log x.
a. log( x  3)
b. log( x)  2
Solution
b. 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.
Example
Try Q’s pg 836 105
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?
Solution
To find the decibel level, evaluate f(10-10).
f ( x)  160  10log x
f (1010 )  160  10 log(1010 )
 160  10(10)
 60
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
Try Q’s pg 835 81,83
Simplify each logarithm.
1
a. log 2 16
b. log 2 32
Solution
a.
16  2
4
log2 16  log2 24  4
b.
1
 2 5
32
1
log2
 log2 25  5
32
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
Try Q’s pg 836 61,63
Approximate to the nearest hundredth.
1
ln
a. ln 20
b.
4
Solution
a.ln 20
b.
ln
1
4
Example
Try Q’s pg 835-6 55,75,85,87
Simplify each logarithm.
a. log6 36
b. log 3 9 2
Solution
a.
log 6 36
36  62
log 6 36  log 6 62  2
b.
log 3 92
92  (32 )2  34
log3 34  4
Example
Try Q’s pg 835-6 39,57,59,89
Simplify each expression.
a. eln 6
b. 10log(4 x16)
Solution
a. e
ln 6
6
because elnk = k
for all positive k.
b. 10log( 4 x 16)  4x  16
for x > 4
because 10logk =
k for all positive
k.
Section 14.1
Sequences
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
Objectives
•
Basic Concepts
•
Representations of Sequences
•
Models and Applications
f  x   a , a  0 and a  1,
SEQUENCES
x
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.
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 + 2
Solution
a. f(n) = 5n + 3
a1 = f(1) = 5(1) + 3 = 8
a2 = f(2) = 5(2) + 3 = 13
a3 = f(3) = 5(3) + 3 = 18
a4 = f(4) = 5(4) + 3 = 23
The first four terms are 8, 13, 18, and 23.
Example (cont)
Try Q’s pg 915 9,11,13
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 + 2
Solution
b. 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.
Example
Use the graph to write the terms of the sequence.
Solution
The points (1, 2), (2, 4),
(3, −6), (4, 8), and (5, −10) are
shown in the graph.
10
8
6
4
2
X
0
The terms of the sequence
are 2, 4, −6, 8, and −10.
Y
-2
-4
-6
-8
-10
1
2
3
4
5
6
7
8
9
10
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.
Solution
Symbolic Representation
Let an = 20n for n = 1, 2, 3, …, 6
Numerical Representation
n
an
1
20
2
40
3
60
4
80
5
6
100 120
Example (cont)
Graphical Representation
Plot the points (1, 20), (2, 40), (3, 60), (4, 80), (5,
100),
(6, 120).
150
Y
135
120
Hours
105
90
75
60
45
30
15
0
X
1
2
3
4
5
6
7
Months
Overtime
8
9 10
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.)
Solution
Numerical Representation
n
an
1
2
700 763
3
831.6
7
4
906.5
2
5
6
988.1 1077.04
1
Example (cont)
Try Q’s pg 916 29, 39,45
1200
Insect Population (per acre)
Graphical Representation
Plot the points (1, 700), (2, 763),
(3, 831.67), (4, 906.52),
(5, 988.11), and (6, 1077.04).
Y
1080
960
840
720
600
480
360
240
120
0
X
1
2
3
4
5
6
7
8
9 10
Year
These results indicate that the insect population
gradually increases. Because the growth factor is
1.09, the population is increasing by 9% each year.
Section 14.2
Arithmetic and
Geometric
Sequences
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
Objectives
•
Representations of Arithmetic Sequences
•
Representations of Geometric Sequences
•
Applications and Models
f  x   aSEQUENCE
, a  0 and a  1,
ARITHMETIC
x
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.
Example
Determine whether f is an arithmetic sequence. If it
is, identify the common difference d.
a. f(n) = 7n + 4
Solution
a. This sequence is arithmetic because f(x) = 7n + 4
defines a linear function. The common difference
is
d = 7.
Example (cont)
Try Q’s pg 923-24 11,17,25
Determine whether f is an arithmetic sequence. If it
is, identify the common difference d.
b. n f(n)
c.
20
Y
18
1
2
3
4
5
−8
−5
−2
1
4
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.
16
14
12
10
8
6
4
2
0
X
1
2
3
4
5
6
7
8
The sequence shown in the
graph is not an arithmetic
sequence because the points
are not collinear. That is, there
is no common difference.
Example
Try Q’s pg 924 29,31
Find the general term an for each arithmetic
sequence.
a. a1 = 4 and d = −3
b. a1 = 5 and a8 = 33
Solution
a. 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 = 7
33  5
Thus, an = −3n + 7.
d
 4.
8 1
b. The common difference is
Therefore, an = 4n + c. To find c we use a1 = 5.
a1 = 4(1) + c = 5 or c = 1. Thus an = 4n + 1.
GENERAL TERM OF AN ARITHMETIC
f
x

a
,
a

0
and
a

1,


SEQUENCE
x
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.
Example
Try Q’s pg 924 35
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
f  x   SEQUENCE
a , a  0 and a  1,
GEOMETRIC
x
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.
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
3
4
5
12
4
4/3
4/9
10
Y
9
8
7
6
5
4
3
2
1
0
X
1
2
3
4
5
6
7
8
Example
Try Q’s pg 924 41,45,53
Determine whether f is a geometric sequence. If it is,
identify the common ratio.
a. f(n) = 4(1.7)n-1 This sequence is geometric because
f(x) = 4(1.7)n -1 defines a exponential
function. The common ratio is 1.7.
b.
n
f(n)
1
2
3
36
12
4
4
5
4/3
4/9
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.
Example
Find a general term an for each geometric sequence.
a. a1 = 4 and r = 5 b. a1 = 3, a3 = 12, and r < 0.
Solution
a. 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
Try Q’s pg 925 61
If a1 = 2 and r = 4, find a9
Solution
To 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
Section 14.3
Series
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
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.
 and e.accurate
Series are instrumental in calculating
approximations of numbers like
Slide 65
 x   a , a  0 and a  1,
FINITEf SERIES
x
A finite series is an expression of the
form
a1 + a2 + a3+ ∙∙∙ + an.
Example
The table represents the number of Lyme disease
cases reported in Connecticut from 1999 – 2005, where
n = 1 corresponds to 1999.
n
an
1
2
3
4
5
6
7
3215 3773 3597 4631 1403 1348 1810
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.
Example (cont)
n
an
Try Q’s pg 933 41
1
2
3
4
5
6
7
3215 3773 3597 4631 1403 1348 1810
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.
SUM OF THE FIRST n TERMS OF
f
x

a
,
a

0
and
a

1,


AN ARITHMETIC SEQUENCE
x
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,
 a1  an
Sn = a1 + a2 + a3 + ∙∙∙ + an n= 2

.

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.
Solution
The arithmetic sequence describing the salary during
year n is computed by
an = 45,000 + 2500(n – 1).
The first and fifth years’ salaries are
a1 = 45,000 + 2500(1 – 1) = 45,000
a5 = 45,000 + 2500(5 – 1) = 55,000
Example (cont)
Try Q’s pg 934 49
Thus the total amount earned during this 5-year
period is
 45, 000  55, 000 
S5  5 

2
  $250, 000.

The sum can also be found using
n
S n   2a1   n  1 d 
2
5
S5   2 45, 000   5  1 2500   $250, 000.
2
Example
Try Q’s pg 933 13
Find the sum of the series 4  7  10 
 58.
Solution
The series has n = 19 terms with a1 = 4 and a19 = 58.
We can then use the formula to find the sum.
 a1  an 
Sn  n 

 2 
 4  58 
S19  19 

 2 
 589
 
SUM OF THE FIRST
x n TERMS OF A GEOMETRIC
f x  a , a  0 and a  1,
SEQUENCE
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
 1 rn 
Sn  a1 
,
 1 r 
provided r ≠ 1.
Example
Try Q’s pg 933 17,19
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
 1  35 
S5  5 
  605.
 1 3 
Example
Try Q’s pg 934 23
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.
Solution
 1  I n  1 
Let a1 = 4000, I = 0.08, and n = 35. Sn  a1 



I
The future value of the annuity is


 1  0.08 35  1 
 4000 



0.08


 $689, 267.
SUMMATION NOTATION
n
a
k 1
k
 a1  a2  a3    an
Example
Find3 each sum.
a.  4k
k 1
6
3
c.   3k  6 
b.  4
k 1
k 1
Solution
3
a.  4k  4(1)  4(2)  4(3)
k 1
= 4  8  12
 24
3
b.
 4  4  4  4  12
k 1
Example (cont)
Try Q’s pg 934 27,29,31
Find3 each sum.
a.  4k
k 1
3
b.  4
k 1
6
c.   3k  6 
k 1
Solution
6
c.
  3k  6  3 1  6  3  2  6   3 3  6  
 3  4   6    3  5  6    3  6   6 
k 1
 9  12  15  18  21  24
 99
End of week 5
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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.