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8.3 The number e

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Objectives:

1. Manipulate e with exponents

2.Use e in calculations with a calculator

3.Graph exponential functions involving e

4.Use e in the real world

Vocabulary:

Euler’s number, natural base

We have learned the compounded interest rate problem in 8.1/8.2. Interest of a bank AccountYou deposit P dollars in the bank and receive an interest rate of r compounded annually for t year.Now if the interest rate r is given by compounded monthly or daily, what is the asset after t year?

Example 1 You deposit $1500 in an account that pays 7.5% interest compounded annually. What is the balance after 3 year? How does this compare to the same investment compounded monthly and daily?

If the interest rate is compounded annually, at

the end of 3rd year, the asset value is:

A3= P(1 + r )3 = 1500 (1 + 0.075)3 = $1863.45

Example 1 You deposit $1500 in an account that pays 7.5% interest compounded annually. What is the balance after 3 year? How does this compare to the same investment compounded monthly and daily?

If the interest rate is compounded monthly, at

the end of 3rd year, the asset value is:

A3= P (1 + r / 12)12·3 = 1500 (1 + 0.075 / 12)36

= $1877.17

Example 1 You deposit $1500 in an account that pays 7.5% interest compounded annually. What is the balance after 3 year? How does this compare to the same investment compounded monthly and daily?

If the interest rate is compounded daily, at

the end of 3rd year, the asset value is:

A3= P (1 + r / 365)365·3

= 1500 (1 + 0.075 / 365)1095

= $1878.44

From the above example, we see that the total

asset is affected by the factor

(1 + r / 12)12·3 = [(1 + r / 12)12]3

(1 + r / 365)365·3 = [(1 + r / 365)365]3

Or in general, the factor of

(1 + r / n)n

or simply,

(1 + 1 / n)n

Mathematician Euler found that and used the

(1 + 1 / n)n is approaching a fixed decimal

number, 2.7182818284590…

The number

was named after Euler as Euler number or

natural base.

This number possesses some properties:

1) The number increases when the n is increases.

2) The number is approaching to 2.7182818284590… or approximate to 2.718281828459.

3) The number is an irrational number (non-terminating and non-repeating), like .

4)

Example 2 Simply the expression.

- e4· e–2
- (2e– 5x) –2
Practice P. 483 Q 17 ~ 20

Example 3 Graph the function.

- y = 3e0.5x
- y =e0.4(x-2) – 2

Example 3 Graph the function.

c) y = -2e0.75x

- y = e–x– 2 – 1
= e–(x+ 2) – 1

Practice Graph the function.

P484 Q74 ~ 75

Example 4 (P484 Q78)You deposit P dollars in an bank account that pays interest rate of r per year compounded n times for t years. Then the balance at the end of t years is

If the interest rate is compounded continuously,

Then at the end of t-th years, the asset value is:

Example 5You deposit $1500 dollars in an bank account that pays 7.5% per year interest compounded

a) 2 times

b) 12 times

c) continously

for 3 years. Find the balance.

a)

b)

c)

PracticeThe atmospheric pressure P (in psi – pounds per square inch) of an object d miles above seal level can modeled by P = 14.7e -0.21d. How much pressure in psi would you experience at the summit of Mount Washington, 6,288 feet above sea level? Graph the model and find how high above sea level you must be to experience 13.25 psi.

Practice The radioactive decay of radon-222 can be modeled as the function A = Ce -0.1813t where A is the amount remaining, C is the original amount, and t is the time in days. If there are 15 mg of radon-222 sealed in a glass tube, how much will remain in the tube after 8 days? If 10 mg of radon-222 remain after 5 days, how much was originally there?

Assignment:

P 483

#21-32

33 – 48 (multiples of 3)

49 – 60 (even)

61 – 66

67 -75 (even)

79 - 80