Lesson 3.5

3.5 Handout#5-27 (ODD)

Exam (3.4-3.5): 12/4

Applications of Exponential Functions

Objective

Students will know how to use exponential growth models, exponential decay models, Gaussian models, logistic growth models, and logarithmic models to solve applications.

A t P 1r

n

nt

Exponential Growth: f(x) = abx, where b > 1

Exponential Decay: f(x) = abx, where 0 < b < 1

Tell whether the function represents exponential growth or exponential decay.

f (x) 458

x

Decay

f (x) 74 x

Growth

f (x) 13

2 x

Decay

Compound Interest - is interest paid on the initial investment, called the principal, and on previously earned interest.

Interest paid on only the principal is simple interest.

Interest is usually compounded more frequently than once per year.

P = initial principal

r = annual rate (written as a decimal)

n = times compounded per year

AP 1 rn

nt

Compound Interest

A = amount

t = time (in years)

If $18,000 is invested at an interest rate of 4% per year, find the value of the investment after 15 years

if the interest is compounded:

a) annually

b) quarterly

c) semiannually

How can we use a natural base exponential function in real-life?

Compound Interest

AP 1 rn

nt

Natural Base e

1 1n

n

Continuously Compounded Interest

APert

P = principal

r = annual interest (as a decimal)

t = time (in years)

A function of the form f(x) = aerx

a > 0 and r > 0: exponential growth

Natural Base Exponential Function

a > 0 and r < 0: exponential decay

If $18,000 is invested at an interest rate of 4% per year, find the value of the investment after 15 years

if the interest is compounded:

d) continuously

A certain culture of bacteria has a relative growth rate of 135% per hour. Two hours after the culture is formed,

the count shows approximately 10,000 bacteria.

a) Find the initial number of bacteria in the culture.

b) Estimate the number of bacteria in the culture 9 hours after the culture was started. Round your answer to the nearest million.

c) How long after the culture is formed will it take for the bacteria population to grow to 1,000,000?

If $7,000 is invested at an interest rate of 6.5% per year, compounded monthly,

a) Find the value of the investment after 4 years.

b) How long will it take for the investment to triple in value.

A certain radioactive substance has a half-life of 120 years. If 850 milligrams are present initially,

a) Write a function that models the mass of the substance remaining after t years. (Within the formula, round the rate to the nearest ten-thousandth.)

b) Use the function found in part (a) to estimate how much of the mass remains after 500 years.

c) How many years will it take for the substance to decay to a mass of 200 milligrams?