Section 4.1Exponential Functions

Example

The exponential equation 13.49 .967 1 predicts the number of O-rings

that are expected to fail at the temperature x F on the space shuttles. The

O-rings were used to seal the connections between d

x

o

f x

ifferent sections of the shuttle

engines. Use a calculator to find the number expected to fail at the temperature of

40 degrees.

Graphing Exponential Functions

Example 1Graph the following two equations: f(x)= , f(x)= 4

4

Draw the asymptotes.

xx

Transformations of Exponential Functions

Example Use the graph of f(x)=4 to obtain the graph of g(x)=4 3.

What is the domain and range of each function?

x x

Example 2Use the graph of f(x)=4 to obtain the graph of g(x)=4

Find the domain and range for the g(x) function.

x x

Example Use the graph of f(x)=4 to obtain the graph of g(x)=2 4

Find the domain and range for the g(x) function.

x x

The Natural Base e

1The values of 1+ for increasingly

large values of n. As n the

approximate value of e to nine decimal

places is e 2.718281827.

The irrational number e, approximately

2.72, is called the natura

n

n

l base. The

function f(x)=e is called the natural

exponential function.

x

Example

.8

The population of a small polynesian island can be modeled by

the equation f(x)=8e . Round off your answers to the

nearest integer.

a. How many people originally went to this small island? (x=0)

b. H

x

ow many people will be living on this island 10 years later

due to general population growth with no new immigration.

Compound Interest

Compound interest is interest computed on your original investment

as well as on any accumulated interest. The sum of money is called

the principal, P. It is invested at an annual percentage rate r, and

compounded once a year.

Sometimes interest is compounded semiannually, monthly, quarterly, etc. This introduces

the variable for the number of compounding periods per year. The formula is then adjusted

to take this into acc

n

ount the number of compunding periods in a year. If there are compuonding

periods per year, in each time period the interest rate is = and there are time periods in years.

The results in the f

n

ri nt tn

ollowing formula for the balance, A, after t years is

A=P 1nt

r

n

Continuous Compounding of Interest

Example

Find the accumulated value of an investment of $15,000

for 2 years at an interest rate of 3.2% if the money is

a. compounded semiannually

b. compounded quarterly

c. compounded monthly

d. compounded continuously

(a)

(b)

(c)

(d)

3For the graph of f(x)=4 1, What is the domain and range?x

: ( 1, ), : ( 1, )

: ( , ), : ( 1, )

: ( , ), : ( , )

:[ 1, ), :[ 1, )

D R

D R

D R

D R

(a)

(b)

(c)

(d)

Find the accumulated value of a CD of $20,000 for 3 years

at an interest rate of 3.1% if the money is compounded continuously?

50,690

218,760,384

21,949

21,860