Download Sample Lesson on Exponential Functions

Math 131 Lecture Notes - © Wiley Publishing, J. Whitfield
6/24/2017
Section 1-5
Page 1 of 3
Section 1.5 – Exponential Functions
●Introduction
The function f ( x)  2 x , where the exponent is a
, is called an
. The number 2 is called the
of the exponential function. Every
exponential function has
rate of change.
Exponential Growth Functions. - The table below contains approximate population data for the McAllen, Texas,
metropolitan area.
Year
2000
2001
2002
2003
Population (in thousands)
570
591
613
636
Is the population of McAllen growing linearly? Explain.
As a matter of fact, the population of McAllen is growing exponentially (increases slowly at first and eventually
climbs quickly). To show this models exponential growth we divide each year’s population by the previous year’s
population.
Population in 2001

Population in 2000
Population in 2002

Population in 2001
Population in 2003

Population in 2002
Since all calculations are near
, we can conclude the population grew by about
between
2000 and 2001, between 2001 and 2002, and between 2002 and 2003. Whenever we have
we have
. The population can be modeled by the function
where the base
, also called the
, represents the
factor by which the population grows each year.
Exponential Decay Functions - When a patient is given medication, the drug enters the bloodstream. The rate at
which the drug is metabolized and eliminated depends on the particular drug. For antibiotic ampicillin,
approximately 40% of the drug is eliminated every hour. A typical dose of ampicillin is 250 mg. The quantity, Q,
of ampicillin (in mg) in the bloodstream, is a function of the number of hours, t, since the drug was given. Find the
quantity remaining in the bloodstream at t = 0, t = 1, t = 2, and t =3.
Math 131 Lecture Notes - © Wiley Publishing, J. Whitfield
6/24/2017
Section 1-5
t
Page 2 of 3
Q
0
1
2
3
The amount of antibiotic Q can be modeled by the function
.
●The General Exponential Function
We say that P is an exponential function of t with base a if
where P0 is the
initial quantity (when t = 0) and a is the factor by which P changes when t increase by 1. If a > 1, we have
, if 0 < a < 1, we have
. The
factor a is given by
where r is the decimal representation of the percent rate of change, r
may be positive (for growth) or negative (for decay).
y
t
Properties of Exponential Growth
1)
2)
3)
4)
5)
6)
7)
P(t )  P0 a where
Domain:
Range:
Horizontal asymptote:
y – intercept:
Always
If P0  0 , the graph is
t
t
Properties of Exponential Decay
1)
2)
3)
4)
5)
6)
7)
P(t )  P0 at where
Domain:
Range:
Horizontal asymptote:
y – intercept:
Always
If P0  0 , the graph is
Example 1: There were approximately 5.4 million farms in the US in 1950 and approximately 2.8 million in 1970.
a) Assuming that the number of farms have been decreasing exponentially, find an equation of the form
P(t )  P0 at , where P is the number of farms (in millions) and t is the number of years since 1950.
b) Use the function found in part a) to estimate the number of farms in 2000.
Math 131 Lecture Notes - © Wiley Publishing, J. Whitfield
6/24/2017
Section 1-5
Page 3 of 3
●Alternate Formula for the Exponential Function
Using r (the percentage growth or decay rate where r = a – 1), is another way to write an exponential function.
P  P0 at  P0 (1  r )t
function. If r < 0, P is an
If r > 0, P is an e
function.
Example 2: The following functions give the population of four towns with time t in years.
i. P  600(1.12)t
ii. P  1000(1.03)t
iv. P  12(0.88)t
iii. P  200(1.08)t
a) Which town has the largest percentage growth rate? What is the percent growth rate?
b) Which town has the largest initial population? What is the initial population?
c) Are any of the towns decreasing in size? If so, which one?
Example 3: According the US Bureau of Statistics, the population of India will double in 35 years. What is the
percent growth rate?
●Comparison Between Linear and Exponential Functions
Every exponential function changes at a
.
. Every linear function changes at a
Example 4: Find a possible formula for each statement.
a) The city’s population increases 7% each year.
b) The number of cancer cells in a tumor decays 22% per day after radiation therapy.
c) As a drug is being injected into a patient’s bloodstream, the quantity in the blood increases by 0.2 mg per minute.
d) The price of houses in a town increases 15% per year since 1990.
e) The city’s population increases by 5000 people each year.
Example 5: Climbing health care costs are a continuing concern. The table below shows the average annual health
care expenditures per person for various years.
Year
1970
1975
1980
1985
1990
1995
Health expenditures
($ per person)
349
591
1055
1596
2714
3302
a) Does a linear or an exponential model fit the data best? Explain.
b) Find a formula for the regression function that you decided is the best.
c) Graph the function with the data on the same axes.