Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
Elementary Mathematical Modeling
Chapter 4. Exponential Functions
4.1 Exponential Growth and Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Da Zheng
University of Houston
March 24, 2014
Introduction-A Glimpse at Exponential
Functions
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
An exponential function is a function of the following
form:
f (x) = P × ax
where P and a are constants. For example, the following
functions are all exponential functions:
f (x) = 5 × 2x
f (x) = ex
f (x) = 3 × ( 21 )x
Introduction-A Glimpse at Exponential
Functions
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Let f (x) = 5 × 2x so that f is an exponential function.
Suppose we are at x0 and then move ahead to x0 + ∆x.
Then the function value changes from f (x0 ) to
f (x0 + ∆x). Also, we observe that
5 × 2x0 +∆x
5 × 2x0 × 2∆x
f (x0 + ∆x)
=
=
= 2∆x .
f (x0 )
5 × 2x0
5 × 2x0
We can see that this ratio has nothing to do with x0 .
That is, it does not care about where we start. It just
cares about how much the increment is, i.e. ∆x.
Introduction-A Glimpse at Exponential
Functions
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
This means, for an exponential function, regardless of the
original input, as long as we increase the input by ∆x,
then the new function value will be a∆x multiplying the
original function value.
This is also a characterization of exponential functions.
One can compare this with that of linear functions.
Exponential Growth
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Now let’s see how exponential functions arise.
Exponential functions occur quite often in population
growth. The following example is a typical one.
Suppose there are initially 3000 bacteria in a culture and
the number of bacteria doubles every hour. Then, we
have the following table
Hour
0
1
2
3
Number of Bacteria
3000
2 × 3000 = 6000
2 × 6000 = 12000
2 × 12000 = 24000
Exponential Growth
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
So far, maybe you cannot see that these data can be
modeled by an exponential function. However, let N (t)
be the number of bacteria after t hours, and we observe
the following,
N (0) = 3000 = 3000 × 20
N (1) = 3000 × 2 = 3000 × 21
N (2) = 3000 × 2 × 2 = 3000 × 22
N (3) = 3000 × 2 × 2 × 2 = 3000 × 23
Exponential Growth
Elementary
Mathematical
Modeling
Hence, from this pattern, we know we can write H(t) as
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
N (t) = 3000 × 2t ,
which is clearly of the form N (t) = P at . So N (t) is an
exponential function.
The advantage to know the formula for N (t) is that if we
want to know the number of bacteria for any t, we don’t
have to multiple 3000 by 2 t times. For instance, we can
calculate N (6) = 3000 × 26 = 192, 000. Moreover, this
formula allows us to calculate the value for non-integer
inputs, such as N (1.5) = 3000 × 21.5 = 8485.
Base, Growth Factor, Initial value
Elementary
Mathematical
Modeling
Let’s take a look at the formula for an exponential
function: f (x) = P ax . Here, a is called the base.
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Sometimes, if a > 1, we call it the growth factor, since
you can see from the example above,the function has
base 2 and it gets larger and larger as we increase the
inputs. If a < 1, we call it the decay factor, since you
can check that the function value gets smaller and
smaller as we increase the inputs. The following is one
example for an exponential function with decay factor.
t
1
H(t) = 3000 ×
,
2
and it is easy to see that H(t) is decreasing.
Base, Growth Factor, Initial value
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
What if we view the variable of an exponential function
as time and input x = 0? We have f (0) = P a0 = P . So
P is called the initial value.
We can also see their graphs. It will convince you why
we the name the base as growth factor or decay factor.
Base, Growth Factor, Initial value
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
We can also conclude that for growth factors, the graph
is concave up and increasing rapidly; for decay factors,
the graph is also concave up but decreasing rapidly.
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Problem
Suppose an amount has initial value 10. For the next
period, we multiply the amount by 3. Find the
exponential function for this amount.
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Problem
Suppose that f is an exponential function with growth
factor 2.4 and that f (0) = 3. Find f (2) and a formula
for f (x).
Unit Conversion
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Sometime, it is better if can we represent the same
exponential function with a different base. For example,
suppose from 1790 to 1860, the U.S. population grew
exponentially with yearly growth factor 1.03. So we can
model this as f (x) = P × 1.03x , where x is the years
since 1790 and P is the initial value equal to the
population in 1790.
Now what if we just want the variable to represent
decades since 1790? To achieve this, we must calculate
the decade growth factor, which is 1.0310 = 1.344.
Then, our exponential function is g(x) = P × 1.344x ,
where x is the decades since 1790.
Unit Conversion
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
In fact, we can check that f and g are the same model,
just the variables represents different periods. In more
details, we have
x
f (x) = g( ).
10
So if we want to know the population in 1800, we plug
40 in f (x) = P × 1.03x or 4 in g(x) = P × 1.344x . Both
will give us the same answer.
Unit Conversion
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Now, we reverse the above problem. Suppose we first
model the data as h(x) = P × 1.344x , where x is the
decades since 1790. Then, if we want the variable to be
the years since 1790, we can calculate the yearly growth
factor as 1.3441/10 = 1.03. Then the new exponential
function is k(x) = P × 1.03x .
In deed, we have the following formula to change
between the growth factors.
a =growth factor for one period −→
ak = growth factor for k periods
b =growth factor for k periods −→
b1/k = growth factor for one period
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Problem
A certain quantity has a yearly growth factor of 1.17.
What is its monthly growth factor?
Percentage Growth Rate and Growth
Factor
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Exponential functions also arise from constant percentage
change, which occurs quite common in our daily life.
Let’s consider the following case:
Suppose Mike gets a job in some company. Initially his
salary is P , and his salary increases by 5% each year. So
what will be his salary after t years? Let S(t) denotes his
salary after t years. Then we have:
S(0) = P = P × (1 + 5%)0
S(1) = P × (1 + 5%) = P × (1 + 5%)1
S(2) = P × (1 + 5%) × (1 + 5%) = P × (1 + 5%)2
... ...
Percentage Growth Rate and Growth
Factor
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Hence, following this pattern, we can model the data as
an exponential function,
S(t) = P (1 + 5%)t .
Here, we see that the base (or the growth factor) is
1 + 5%. We say that our salary has a yearly percentage
growth rate of 5%.
Maybe you have already known how to write an
exponential function given the initial value P and yearly
percentage growth rate r, that is,
f (x) = P (1 + r)x .
Percentage Decay Rate and Decay Factor
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Similarly, if we know some quantity decreases by r each
year, we know that the exponential function should be
f (x) = P (1 − r)x .
Here, 1 − r is the decay factor and we call r the yearly
percentage decay rate.
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Problem
An exponential function has a growth factor of 1.06.
What is the percentage growth rate?
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Problem
Suppose f is an exponential function with percentage
growth rate of 5% and f (0) = 8. Find a formula for
f (x).
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Problem
A certain population has a yearly growth rate of 2.3%,
and the initial value is 3 million.
a. Use a formula to express the population as an
exponential function.
b. Express using functional notation the population
after 4 years, and then calculate that value.
Percentage Change and Unit Conversion
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Suppose a quantity has yearly growth rate of 5%, then
what is its decade growth rate? Is it 5% × 10? NO!
This problem is not that simple. Let’s take a look at the
correct way to solve it. First, the yearly growth factor is
1 + 5% = 1.05. So use the unit conversion technique we
learned in last section, we can calculate that the decade
growth factor is 1.0510 = 1.63. This mean the decade
growth rate is 1.63 − 1 = 0.63.
Hence, you can see that if we calculate wrongly and
obtain 5% × 10 = 0.5, then it is much smaller than the
actual decade growth rate.
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Problem
A quantity increases by 4% for each of 8 years. What is
its percentage increase over the 8-year period?
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Problem
A quantity decreases by 3% for each of 5 years. What is
its percentage decrease over the 5-year period?
Recognizing Exponential Data
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Remember that in the last chapter, we learned how to
model the given data by a linear function. Now, we have
a new tool, i.e. exponential functions, so let’s see how to
model given data by exponential functions. To explain
the method, consider the following example:
The table below shows how the balance in a saving
account grows over time since the initial investment:
Time(month)
Balance($)
0
3500
1
3542
2
3584.5
3
3627.52
4
3671.05
Recognizing Exponential Data
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Can we model the data by some exponential function?
That is, we are going to find a function of the following
form:
B(t) = P × at ,
where t denotes the months, and B(t) denotes the
balance, such that
B(0) = 3500
B(1) = 3542
B(2) = 3584.5
B(3) = 3627.52
B(4) = 3671.05
Recognizing Exponential Data
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Like the case for linear modeling, we need to test
whether there does exist such an exponential function
before we try to construct it.
Recall the characterization of an exponential function,
after each period, we multiply the old value by the same
number to obtain the new value. That is, if the table of
data can be modeled by an exponential function, then we
must show common quotients if we divide each data
entry by the one preceding it. So we create the following
table of quotients.
Recognizing Exponential Data
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
∆t
0 to 1
3542
Ratios of B 3500 = 1.012
∆t
2 to 3
3627.52
Ratios of B 3584.5 = 1.012
1 to 2
3584.5
=1.012
3542
3 to 4
= 1.012
3671.05
3627.52
Construct an Exponential Model
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
From this table of quotients, we can see that we have a
common quotient, so we can try to seek for a
exponential model.
Notice that our function should be of the form:
B(t) = P × at , hence we need to obtain the base
(growth factor) a and initial value P first.
P is easy to obtain, since it is the value when t = 0 and
we can read the table and have P = 3500.
Construct an Exponential Model
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Next, to obtain a, let’s use the table of quotients we
have just created. The quotients are all the same, so
let’s just look at the first entry. This quotient is
P × a1
B(1)
=
= a.
B(0)
P × a0
So the base is exactly the common quotient. So
a = 1.012.
Thus, for the above example, we have
B(t) = 3500 × 1.012t .
When data are measured in increment not
equal to one
Elementary
Mathematical
Modeling
Da Zheng
Consider the following table of data:
x
y
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
0
64
3
22.91
6
8.2
9
2.93
To see whether they can be modeled by an exponential
function, we create the table of quotients.
∆x
Ratio of y
0 to 3
= 0.36
22.91
64
3 to 6
= 0.36
8.2
22.91
6 to 9
= 0.36
2.93
8.2
When data are measured in increment not
equal to one
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
So these data can be modeled by an exponential,
function, even though the increment is not 1, it does not
matter, since for equal difference in period, we have
equal quotients.
The problem is how to obtain the base (decay factor in
this case). Notice that the common quotient 0.36 seems
to be the base. But it is indeed the quotient
corresponding to a period of 3 units. Hence, recall the
unit conversion technique, we perform the following to
obtain the correct quotient.
1
a = (0.36) 3 = 0.71.
So the exponential model for these data is
f (x) = 64 × 0.71x .
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Problem
The value of a coin collection increases as new coins are
added and the value of some rare coins in the collection
increases. The value V , in dollars, of the collection t
years after the collection was started is given by the
following table.
t=time in years
0
1
2
3
4
V=value in dollars
130
156
187.2
224.64
269.57
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Problem
a. Show that these data are exponential.
b. Find an exponential model for the data.
c. According to the model, when will the collection
have a value of $500?
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
4.1.
Exponential
Growth and
Decay
4.2. Constant
Percentage
Change
4.3. Modeling
Exponential
Data
Problem
Test the given data to see whether it is exponential. If it
is, find an exponential model for it.
x
y
0
6
2
18
4
54
6
162