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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 1
Homework, Page 331Find the exact solution algebraically, and check by substituting into the original equation.
1.1 5
36 43
x
21 1 4 1 1 1 15 5 5 536 4
3 3 36 3 9 3 3
2 105
10 21 1 1536 36 36 4
3 3 9
x x x x
xx
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 2
Homework, Page 331Find the exact solution algebraically, and check by substituting into the original equation.
5.32 10 20
x
13 3 3
31 13
202 10 20 10 10 10
2
1 3 33
2 10 2 10 2 10 20
x x x
xx x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 3
Homework, Page 331Find the exact solution algebraically, and check by substituting into the original equation.
9. 4log 5 1x
14
4 4
1log 5 1 5 4 5
4
15
4
1 1log 5 5 log 1
4 4
x x x
x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 4
Homework, Page 331Solve each equation algebraically, and check by substituting into the original equation.
13. 0.03550 200xe
0.035 0.035 0.035
0.035
ln 40.035
ln 40.035
20050 200 4
50ln 4
ln ln 4 0.035 ln ln 4 39.6080.035
ln 450 50 50 4 200
0.035
x x x
x
e e e
e x e x
x e e
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 5
Homework, Page 331Solve each equation algebraically, and check by substituting into the original equation.
17. 3ln 3 4 5x
1 13 3
1 13 3
13ln 3 4 5 3ln 3 1 ln 3
3
3 3 4.396
13ln 3 3 4 3ln 4 3 4 53
x x x
x e x e x
e e
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 6
Homework, Page 331State the domain of each function. Match the function with its graph.
21. ln1
xf x
x
ln1
Domain : : , 1 0,
graph .
xf x
xx
d
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 7
Homework, Page 331Solve each equation, and support the solution by a second method.
25. 2log 6x 2 3log 6 2log 6 log 3 10 1000x x x x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 8
Homework, Page 331Solve each equation, and support the solution by a second method.
29. 2 24
3
x x
2 0
2
2 24 2 2 12 2 2 12 2
3
12 144 4 1 12 12 2 1 0 2
2 1
12 1482 6 37 ln 2 ln 6 37
2 1
ln 6 37ln 2 ln 6 37 3.595
ln 2
x xx x x x
x x x
x x
x x x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 9
Homework, Page 331Solve each equation, and support the solution by a second method.
29. 2 24
3
x x 3.595x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 10
Homework, Page 331Solve each equation, and support the solution by a second method.
33.0.3
500200
1 25 xe
0.3 0.3
0.3
0.3 0.3 0.3
500 500200 1 25 2.5 1 25
2001 251.5 1.5 1.5
25 1.5 ln ln 0.3 ln ln25 25 25
10 1.5ln 9.378
3 25
x xx
x x x
e ee
e e e x e
x x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 11
Homework, Page 331Solve each equation, and support the solution by a second method.
33.0.3
500200
1 25 xe
9.378x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 12
Homework, Page 331Solve each equation, and support the solution by a second method.
37. ln 3 ln 4 3ln 2x x
3
2 2
ln 3 ln 4 3ln 2 ln 3 4 ln 2
3 4 8 12 8 20 0
4 5 0 4
x x x x
x x x x x x
x x x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 13
Homework, Page 331Determine by how many orders of magnitude the quantities differ.
41. An earthquake rated 7 on the Richter scale and one rated 5.5
Order of magnitude 7 5.5 1.5
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 14
Homework, Page 33145. How many times more severe was the 1978 Mexico City earthquake (R = 7.9) than the 1994 Los Angeles earthquake (R = 6.6)?
1.3
log 7.9;log 6.6 log log 7.9 6.6
log 1.3 10 20 times greater
MC MCLA LA
MC MC
LA LA
a aa a
T T T Ta a
a a
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 15
Homework, Page 33149. A cup of coffee has cooled from 92ºC to 50ºC in 12 minutes in a room at 22ºC. How long will it take the coffee to cool to 30ºC?
12
12 12 12
0.0764 0.0764
0.0764 0.0764
50 22 92 22
28 428 70 ln ln
70 104 1 4
ln 12ln ln10 12 11
0.0764 30 22 70 8 70
8 4 4ln ln ln 0.0764 ln
70 35 351 4
ln 20.0764 35
kt km o m
k k k
t t
t t
T t T T T e e
e e e
k e k
k e e
e e t e
t t
8.407 minutes
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 16
Homework, Page 33153. The use of penicillin became so widespread in the 1980s in Hungary that it became practically useless against common sinus and ear infections. Now the use of more efficient antibiotics has caused a decline in penicillin resistance.
A. The data pairs are approximately (1, 11), (8, 6), (15, 4.8) (16, 4) and (17, 2.5) where t = 1 is 1976. construct a scatter plot of the data.
B. Discuss whether the bar graph in the text or the scatter plot you did best represents the data and why.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 17
Homework, Page 33153. A. The data pairs are approximately (1, 11), (8, 6), (15, 4.8) (16, 4) and (17, 2.5) where t = 1 is 1976. construct a scatter plot of the data.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 18
Homework, Page 33153. B. Discuss whether the bar graph in the text or the scatter plot you did best represents the data and why.
The scatter plot is a better representation because it more accurately shows the time interval between data points, giving a better sense of the plummeting use in the early 1990s.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 19
Homework, Page 331Data pairs are given. Determine whether a linear, logarithmic, exponential, or power regression equation is the best model for the data. Explain your choice, supporting it with tables and graphs.
57.
From the graphs, it is apparent that the graph of the exponential regression equation most closely fits the data. The table shows the graph passing through each data point.
1 2 3 4
3 6 12 24
x
y
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 20
Homework, Page 331Solve the problem without using a calculator.
61. Solve 3 12 32x
3 1 3 1 52 32 2 2 3 1 5 3 6 2x x x x x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 21
Homework, Page 331Solve the equation or inequality.
73. 5xe x
1.307x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 22
Homework, Page 331Solve the equation or inequality.
77. 2log 4log3 0x 2 4 2 4
2 4 2
2log 4log3 0 log log3 0 log log3
3 3 9
x x x
x x x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
3.6
Mathematics of Finance
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 24
What you’ll learn about
Interest Compounded Annually Interest Compounded n Times per Year Interest Compounded Continuously Annual Percentage Yield Annuities – Future Value Loans and Mortgages – Present Value
… and whyThe mathematics of finance is the science of letting your money work for you – valuable information indeed!
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Leading Questions
When financial institutions compound interest, they calculate the interest due and credit it to the account.An annuity is a sequence of equal periodic payments.Banks offer the highest interest they are able, while still making money.
Slide 3- 25
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 26
Interest Compounded Annually
If a principal is invested at a fixed annual interest
rate , calculated at the end of each year, then the value
of the investment after years is (1 ) , where
is expressed as a decimal.
t
P
r
t A P r r
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 27
Interest Compounded n Times per Year
Suppose a principal is invested at an annual rate
compounded times a year for years. Then / is
the interest rate per compounding period, and is
the number of compounding periods. The am
P r
n t r n
nt
ount
in the account after years is 1 .nt
A
rt A P
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 28
Example Compounding MonthlySuppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 29
Continuously Compounded Interest
Suppose a principal is invested at a fixed annual
interest rate , compounded continously. The value
of the investment after years is
in decimal formrt
P
r
t
A Pe r
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 30
Example Compounding Continuously
Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 31
Annual Percentage Yield
A common basis for comparing investments is the annual percentage yield (APY) – the percentage rate that, compounded annually, would yield the same return as the given interest rate with the given compounding period.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 32
Example Computing Annual Percentage Yield
Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 33
Future Value of an Annuity
The future value of an annuity, a sequence of equal,
periodic payments consisting of equal periodic payments
per year for years of dollars at an interest rate is
1
FV
n
t R r
iFV R
1,
ntr
ii n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 34
Example Computing Future Value of an Annuity
Andrew contributes $50 per month into the Hoffbrau Fund that earns 15.5% annual interest. What is the value of his investment after 20 years?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 35
Example Computing Future Value of an Annuity
Diego contributes to a Commercial National money market account that earns 4.5% annual interest. What should his monthly payment be if he wants to accumulate $120,000 in 30 years?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 36
Present Value of an Annuity
The present value of an annuity consisting of
equal payments per year for years of dollars at
an interest rate is
1 1
nt
PV n
t R
r
i rPV R i
i n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 37
Example Computing Present Value of an Annuity
An $86,000 mortgage for 30 years at 12% APR requires monthly payments of $884.61. Suppose you decide to make monthly payments of $1,050.00
a. When would the mortgage be completely paid?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 38
Example Computing Present Value of an Annuity
An $86,000 mortgage for 30 years at 12% APR requires monthly payments of $884.61. Suppose you decide to make monthly payments of $1,050.00
b. How much would you save compared to the original plan?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Following Questions
Exponential and logarithmic functions are inverse functions.
Logistic functions are only bounded above. Logarithmic functions grow very rapidly. Exponential functions grow very rapidly. Many exponential and logarithmic equations
may be solved by graphing. I need to understand the mathematics of
finance so I can better manage my money.
Slide 3- 39
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 40
Homework
Review Section 3.6 Page 341, Exercises: 1 – 69 (EOO), 19 Quiz next time