Transcript 3.2 Related Rates

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.
e5
e3
c.


e 5e 1
e 2e
a. = e 5+(-2) = e 3
b. e 3-2 = e 2
a. =

e 5(1)
e 21
=

e4
e 1
= e 4-(-1) = e 5
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. anually
b. quarterly
c. 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 = 2143.59
The 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 = 2203.76
The value is $2203.76
c. For continuous compounding P = 1000, r = 0.10, and t = 8. Thus
1000e 0.1x8 – 1000e 0.8 = 2225.54
The 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.
n t
r

A  P 1  
n

a. P = 100, r = 0.02 x 52 = 1.04 yearly, and n = 3.
this gives a value of
100(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
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.
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.
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 365
days and some use 360 in a year, we will try both ways. If n = 365 days then,
r
9.25% 0.0925

 0.0002543
365
365
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%
r
0.0925
 0.0002569
360
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%)
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 $1,000,000
in 60 years then r  0.08  0.02 and n = 604 = 240.
4
1,000,000 = P(1 +0.02) 240


P
1,000,000
240  $8629
(1 0.02)
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 =
p
1000

 $751.31
(1  r)n (1  0.10)3
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 = 0.1030
The 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.