Download 3.5-Applications-of-Exponential-Functions

Lesson 3.5
3.5 Handout
#5-27 (ODD)
Exam (3.4-3.5): 12/4
Applications of Exponential Functions
 r 
At   P1 
 n 
nt
Objective
Students will know how to use exponential growth
models, exponential decay models, Gaussian models,
logistic growth models, and logarithmic models to solve
applications.

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.
x
5 
f (x)  4 
8 
Decay
f (x)  7  4
x
Growth


1 x
f (x)  2
3
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.
Compound Interest
nt
 r 
A  P1 
 n 
P = initial principal
r = annual rate (written as a decimal)
n = times compounded per year
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
Natural Base e
nt
 r 
A  P1 
 n 
n
 1 
1 
 n 

Continuously Compounded Interest
A  Pe
rt
P = principal

r = annual interest (as a decimal)
t = time (in years)
A function of the form f(x) = aerx
Natural Base Exponential Function
a > 0 and r > 0: exponential growth
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 tenthousandth.)
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?