Homework, Page 331

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



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



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


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


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


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


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



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



29. 2 24

3

x x 3.595x



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



33.0.3

500200

1 25 xe

9.378x



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


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


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


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


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.


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.


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.


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


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


Homework, Page 331Solve the equation or inequality.

73. 5xe x

1.307x


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


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!


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


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


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


Example Compounding MonthlySuppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.


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


Example Compounding Continuously

Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.


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.


Example Computing Annual Percentage Yield

Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?


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


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?


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?


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


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?


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?


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


Homework

Review Section 3.6 Page 341, Exercises: 1 – 69 (EOO), 19 Quiz next time

Documents

Homework, Page 331