Section 3.2 Exponential Functions

1. Use a calculator to evaluate, rounding to three decimal places.

1.a. e 2

b. e -2

c. e ½

1. a. ≈ 7.289

b. ≈ 0.135

c. ≈ 1.649

2. Express as a power of e

a. e 5 e -2

b.

c.

e5

e3

e5e 1

e 2e

a. = e 5+(-2) = e 3

b. e 3-2 = e 2

a. = = = e 4-(-1) = e 5

e5( 1)

e 21

e4

e 1

3. Graph y = 3 x on a graphing calculator.

- 5 < x < 5 and -1 < y < 100

4. Graph y = (1/3) x on a graphing calculator.

- 5 < x < 5 and -1 < y < 100

5. Evaluate e 1.74 using a calculator.

e 1.74 = 5.696

6. BUSINESS: Interest - Find the value of $1000 deposit in a bank at 10% interest for 8 years compounded

a. anuallyb. quarterlyc. continuously

a. For n = 1 m = 1 (annual compounding), P(1+r/n)nt simplifies to P(1+r)t

when P = 100, r = 0.1, and t = 8. The value is 1000(1 + 0.1)8 = 1000 (1.1) 8 = 2143.59The value is $2143.59

b. For quarterly compounding, n = 4, P = 1000, r = 0.1, and t = 8. Thus 1000(1+(0.1/4) 4x.8 = 1000(1 + 0.025) 4x.8

= 1000 (1.025) 32 = 2203.76The value is $2203.76

c. For continuous compounding P = 1000, r = 0.10, and t = 8. Thus 1000e 0.1x8 – 1000e 0.8 = 2225.54The value is $2225.54

7. Personal Finance: Interest - A loan shark lends you $100 at 2% compound interest per week (that is a weekly, not annual rate).

a.How much will you owe after 3 years?b.In “street” language, the profit on such a loan is known as the “vigorish” or the

“vig”. Fins the shark’s vig.

a. P = 100, r = 0.02 x 52 = 1.04 yearly, and n = 3.this gives a value of100(1 + (1.04/52)) 52x3

= 100(1.02) 52x3

= 100(1.02) 156

= $2196

b. The “vig” is equal to the amount owed after three years minus the amount loaned. This is $2196 - $100.00 = $2096

1n t

rA P

n

8. Personal Finance: Annual Percentage Rate (APR) - Find the error in the ad shown below, which appeared in a New York paper. [Hint: Check that the nominal rate is equivalent to the effective rate. For daily compounding, s some banks use 365 days and some use 360 days in the year. Try both ways.

The stated rate of 9.25% (compounded daily) is the normal rate of interest. To determine the effective rate of interest, use the compound interest formula, P (1 +r) n, with r = 9.25%/ number of days and n = number of days in a year. Since some banks use 365days and some use 360 in a year, we will try both ways. If n = 365 days then,

Then P(1+r) n = P(1.0002534) 365 ≈ 1.0969%.Subtracting 1 gives 0.0969, which expressed as a percent gives the effective rate of interest as 9.69%

If n = 360 days then Then P(1+r) n = P(1.0002569) 360 ≈ 1.0969% and the effective rate is also 9.69%

Thus, the error in advertisement is 9.825%. The annual yield should be 9.69% (based on the nominal rate of 9.25%)

r 9.25%

3650.0925

3650.0002543

r 0.0925

3600.0002569

At T&M Bank, flexibility is the key word. You can choose the length of time and the amount you deposit, which will earn an annual yield of 9.825% based on a rate of 9.25% compounded daily.

9. Personal Finance: Present Value - A rich uncle wants to make you a million. How much money must he deposit in a trust fund paying 8% compounded quarterly at the time of your birth to yield $1,000,000 when you retire at age 60?

If the amount of money P invested at 8% compounded quarterly yields $1,000,000in 60 years then and n = 604 = 240.

1,000,000 = P(1 +0.02) 240

r 0.08

40.02

P 1,000,000

(10.02)240$8629

10. Personal Finance: Zero-Coupon Bonds - FUJI Holding recently sold zero-coupon $1000 bonds maturing in 3 years with an annual yield of 10%. Find the price. [Hint: the price is the present value of $1000, 3 years from now at the stated interest rate]

For 10% compounded annually, r = 0.10 and n = 3.

Present value =

n 3

p 1000$751.31

(1 r) (1 0.10)

11. General: Compound Interest - Which is better 10% interest compounded quarterly or 9.8% compounded continuously?

To compare two interest rates that are compounded differently, convert them both to annual yields. 10% compounded quarterly: P(1+r) n = P(1.025) 4 ≈ P(1.1038)Subtracting 1, 1.1038 – 1 = 10.38%.

9.8% compounded continuously, Pe rn = Pe 0.098 ≈ P(1.1030)Subtracting 1: 1.130 – 1 = 0.1030The effective rate of interest is 10.30%.

Thus, 10% compounded quarterly is better than 9.8 compounded continuously.

12. Personal Finance: Depreciation - A Toyota Corolla automobile lists for $15,450, and depreciates by 35% per year. Find the values after:

a.4 years b. 6 months

Since the depreciation is 35% per year, r = 0.35.

a.P(1 +r) n = 15,450(1 – 0.35) 4 ≈ $2758

b.P(1 +r) n = 15,450(1 – 0.35) 0.5 ≈ $12,456

13. General Nuclear Meltdown: - The probability of a “severe core meltdown accident” at a nuclear reactor in the U.S. within the next n years is 1 – (0.9997) 100n.

Find the probability if a meltdown:

a.within 25 years. b. within 40 years

a. 1 = (0.9997) 100(25) ≈ 0.5277

b. 1 – (0.9997) 100(40) ≈ 0.6989

14. General: Population - As stated earlier, the most populous state is California, with Texas second but gaining. According to the Census Bureau, x years after 2005 the population of California will be 36e0.013x and the population of Texas will be 22e 0.019x (all in millions)

a. Graph these two functions on a calculator on the window [0,100] by [0,150].

b. In which year is Texas projected to overtake California as the most populous state?[hint: use INTERSECT]

a.

b. During the year 2087 (x ≈ 82.08)And 2005 + 82 = 2087.