The Exponential Function
Figure 6.1.1 Graph of y = ex
Example The integral may also be handled by integration by parts, although
with a little more work than in the previous example. Here we will let
We now perform another integration by parts by choosing
At first glance it may seem that we are back to where we
started; however, all we need to
do now is solve for . That is, we have
Note that we have added an arbitrary constant c since we are seeking the general antiderivative.
Figure 6.1.2 Graph of y = e-x
We now have sufficient information about the exponential function to understand the
geometry of its graph. Since e > 0 , we know that ex > 0 for all rational values of x, and
hence, by continuity, for all values of x. Since e > 1, it follows that
for all x, the graph of y = ex is always
increasing and always concave up. Moreover,
(6.1.30) and (6.1.31) indicate that as x increases, the graph is not only increasing, but its
slope is increasing at the same rate that y is increasing. Thus we should expect y to grow
at a very rapid rate, as we see in Figure 6.1.1. This rate of growth is characterized as
exponential growth. Figure 6.1.2 shows the graph of y = e-x , which is the graph of y = ex
reflected about the y-axis. In this case y decreases asymptotically toward 0 as x increases;
this is known as exponential decay
We will close this section with an application to the problem of uninhibited population
growth, a problem we first considered in Section 1.4.
Uninhibited population growth
Recall from Section 1.4 that if represents the size of a population after n units of time
and the population grows at a constant rate of α100% per unit of time, then the sequence
must satisfy the linear difference equation
for n = 0, 1, 2, . . .. At that time we saw that the solution of this equation is given by
The crucial aspect of (6.1.32) is the statement that
amount of change in the size of the
population over any unit of time is proportional to the current size of the population.
Hence if x(t) represents the size of a population at time t, where the population can
change continuously over time, then the continuous time analog of (6.1.32) is the differential
for all time t. If is
the size of the population at time t = 0, then we know from our
work in this section that the only solution to this equation is the function
Hence if the size of a population is growing at a rate
which is proportional to itself, an
assumption which, as we noted in Section 1.4, is often reasonable over short periods of
time, then the population will grow exponentially. As in Section 1.4, we refer to such
growth as uninhibited population growth.
Example In 1970 the population of the United States was 203.3 million and in 1980
the population was 226.5 million. Assuming an uninhibited growth model and letting x(t)
represent the population t years after 1970, by (6.1.35) we should have
for some constant α. Since x(10) = 226.5, we can find α by solving
That is, we need to find a value for α such that
Unfortunately, solving this equation exactly requires
being able to reverse the process of
applying the exponential function. In other words, we need an inverse for the exponential
function. We shall take up that problem in the next section; for now we may use a
numerical approximation. You should verify that α = 0.0108 satisfies the equation. Thus
this model would predict the population of the United States t years after 1970 to be
For example, this model would predict a 1990 population of
Figure 6.1.3 Uninhibited growth model for the United
and a population in the year 2000 of
While the prediction for 1990 is fairly accurate (the
actual population was approximately
249.6 million), the second prediction differs significantly from the Census Bureau’s own
prediction of a population of 268.3 million for the year 2000. As we discussed in Sections
1.4 and 1.5, an uninhibited growth model is a simple model which cannot be expected to
be accurate for predictions too far into the future.
We shall have more to say about population models in Section 6.3, where we will also
see another example of a differential equation. We will have a much fuller discussion of
differential equations in Chapter 8.
1. Find the derivative of each of the following functions.
2. Evaluate each of the following integrals.
3. Find the maximum value of
on the interval (0,∞).
4. (a) Use the Taylor series for e-x to show that . Hence conclude that e < 3.
(b) Show that if is the nth order Taylor polynomial for ex at 0, then
for any value of x.
(c) Use (b) to find an approximation for e with an error of less than 0.000005.
5. Find the following limits.
6. (a) Show that for
any positive integer n.
(b) Use (a) to show that if p is any polynomial, then . This shows
that ex grows faster as x → ∞ than any polynomial function.
7. Graph the following functions on the specified intervals.
8. Evaluate the following improper integrals
9. Use the integral test to show that the infinite series converges.
10. (a) Find the Taylor series for
(b) Use (a) to find the the Taylor series for
known as the error function.
(c) Use your result in (b) to approximate erf(1) with an error less than 0.0001.
11. Suppose x(t) is the population of a certain country t years after 1985, x(0) = 23.4
(a) What will the population of the country be in the year 2000?
(b) In what year will the population be twice what it was in 1985?
(a) Graph f on the interval [−5, 5].
(b) Show that f'(0) = 0.
(c) Show that for n = 0, 1, 2, . . ..
(d) Show that f is on (−∞,∞).
(e) Note that the Taylor series for f about 0 converges for all x in (−∞,∞), but does
not converge to f(x) except at 0. Thus f is on (−∞,∞), but not analytic at