Section 2.2
Lecture 2
MTH 124
Exponential Functions
“The greatest shortcoming of the human race is our inability to understand the exponential
function.” - Albert A. Bartlett
The function f (x) = 2x , where the power is a variable x, is an example of an exponential
function. Exponential functions take the general form of
A(t) = P bt
where b is positive and P is some constant. We call b the base of this exponential function. You
may remember the exponential function as P ert , we’ll discuss this ambiguity later.
Exponent Review
1. (a) Create an exponential function with coefficients P = 0.7, and b = 2.5.
(b) Evaluate 3 · 23 by hand, how about 3 · 2−2 ?
(c) Identify the base of the following exponential functions:
1
(5.5)x , 0.34(0.2)x .
2
(d) Suppose f (x) = −0.75ex , find (f (2).
Review the rules of exponents on page 138 of the text. Please make sure you’re completely
comfortable with algebraically manipulating exponents as we will use exponential functions
regularly.
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Section 2.2
Lecture 2
MTH 124
Given a general exponential function of the form P bt
• the constant P represents
• the constant b represents
In particular, for:
• b > 1 we have
,
• and for b < 1 we have
.
Plot some examples of functions to understand how values b effect exponential growth
and decay.
2. (a) The figure below gives the plot of 5x and ex . Label the two functions. Note e ≈ 2.72.
(b) The figure below gives the plot of 0.7x and 0.9x . Label the two functions.
Exponential functions change by a constant percent unlike linear functions which change
by a constant amount.
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Section 2.2
Lecture 2
MTH 124
3. Consider the following table. Is this growth linear? If not, what type of function fits this
data? How can you tell?
Year
Population(millions)
2006
2007
2008
2009
14.225 14.721 15.324 15.757
Fitting an Exponential Function
Example 1 The U.S. population was 180 million in 1960 and 309 million in 2010.
Use the data to give an exponential growth model showing the U.S. population A as
a function of time t in years since 1960. Use your model to predict the population in
2020.
Solution We are asked to fit this data to an exponential function so we’re looking
for a function of the form
A(t) = P bt ,
and thus we need to determine P and b. Note that t is years since 1960 so t = 0 in
1960 and t = 50 in 2010. This gives us the two data points (0, 180) and (50, 309).
Our model should contain these points so we have that
A(0) = 180,
and
A(50) = 309.
If we use our general equation for A(t) we have
A(0) = P b0 = P = 180, and
So we immediately find that P = 180.
now solve for b.
180 · b50
b50
(b50 )1/50
b
A(50) = P b50 = 309
By plugging P into our last equation we can
= 309
= 309/180
= (309/180)1/50
≈ 1.01
So our population model is given by A(t) = 180(1.01)t and the population in 2020 is
A(60) ≈ 327 million people. Yikes!
4. The amount of a hormone in the body can change rapidly. Suppose the initial amount is
20 mg. Find a formula for H, the amount in mg, at a time t minutes later if H is
(a) Increasing by 0.4 mg per minute.
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Section 2.2
Lecture 2
MTH 124
(b) Increasing by 3% per minute.
(c) Decreasing by 0.4 mg per minute.
(d) Decreasing by 3% per minute.
As we’ve seen, an exponential function of the form Abt is completely determined by the two
parameters, A and b.
Compound and Continuous Exponential Growth
Exponential functions also characterize many financial applications. For example we may have
an account which earns interest at a certain percentage at the end of some period of time(e.g.
once every month). For these situations it helps to generalize our formula Abt to so we can
modify the terms for interest rate and number of times we are compounding. The standard
equation that describes this type of exponential growth is given by
mt
r
A(t) = P 1 +
m
where A(t) represents the future amount given
• a principal amount P ,
• compounded m times per year
• at a rate r given as decimal.
Note: If we choose m = 1 we get our original form P bt . Thus we can think of our original
formula as compounding yearly, or annually.
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Section 2.2
Lecture 2
MTH 124
5. Shari invests $5000 into an account that pays an interest rate of 2.15% compounded
quarterly. What is the value of the account after 5 years?
Continuously Compounded Growth
In many cases, such as in biological organisms, it is unatural to describe growth( or “interest”)
that compounds at fixed times. Certain types of growth occur not at fixed times but at
every moment in time. This type of growth is known as continuously compounding growth.
Continuous compounding can be described by the formula
A(t) = P ert
where A(t) is the future amount given
• a principal amount P ,
• compounded continuously
• at a rate r given as decimal.
Notice that this form looks different than our original general form P bt however using exponent
rules we have that
A(t) = P ert = P (er )t
so we see er is equivalent to b in our original formula P bt . We chose to use either form depending
on the problem we want to solve.
Some problems may ask for you to convert an annual percent growth to a continuous percent
growth rate. In that case simply set the two formulas equal and solve for the unknown rate.
Example 2 What continuous percent decay rate is equivalent to an annual percent
decay rate of 11%?
Solution An annual decay of 11% can be described by A(t) = P (1−0.11)t = P (0.89)t .
Continuous compounding decay has the form P ert so we want to rewrite our original
equation in this new form. Thus we have
P (0.89)t = P ert
0.89t = ert
(0.89)t = (er )t
So we see 0.89 = er , which can be solved using logarithm properties. We’ll cover
these in the next section!
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Section 2.2
Lecture 2
MTH 124
How can I keep it all straight!?
At this point we’ve discussed several different formulas for exponential functions so it can be
confusing to know what to use in any particular situation. Below you will find some general
tips for determining which form is most appropriate.
General Exponential(Annual Growth)
A(t) = P bt
• Use this form for problems specify exponential growth but give no further details.
Discrete Compounding Growth
mt
r
A(t) = P 1 +
m
• Use this form for problems specify a discrete time that an amount is compounded.
e.g. an account compounds monthly.
Continuously Compounding Growth
A(t) = P ert
• Use this form for problems specify the exponential growth is continuous or provide a
context where growth should be happening all the time.
e.g. the exponential growth of a disease.
6. Suppose that the population of starfish in a pond decreased exponentially from 573 in
1980 to 300 in 1995 and continued to decrease at the same percentage rate between 1995
and 2000, calculate what the starfish population would be in 2000.
7. The population of a city is decaying according to the formula P (t) = 43, 021e−0.2t where t
is years after 2010. Determine the annual decay rate.
Ideas From Today... exponent rules, exponential function, base, initial value, exponential
decay, exponential growth, types of compounding
For Next Time... review the logarithm and the natural logarithm
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