1. Derivatives of exponential functions
Exponential
functions are functions of the form 
. They play a
key role in the applications of mathematics. In a calculus course, we naturally wonder how to find 
. Experiments
with something as simple looking as 
quickly show
that first guesses are generally wrong.
a. Demonstrate the absurdity of thinking
that 
might be the
derivative of 
. For example,
have your computer plot the graphs of these two functions. Think about what the graph of 
would look like
if its derivative were negative.
Also look at x = 0 where 
changes sign.
b. We need to go back to the definition of
the derivative to derive a formula for 
where 
. Supply reasons
for each of the following three steps:
Use your computer to gather evidence that 
exists. You might evaluate the quotient for
small values of h
or zoom in on the graph of the quotient near the y-axis. Notice that the derivative of f is
simply this somewhat mysterious constant times the function itself.
c. Modify the process
in part b to find a formula for the derivative of 
. Notice again
that the derivative is a multiple of f, but with a different constant factor.
d. Show in general with any positive
number a as the base of an exponential function 
, that
e. We are interested
in locating a value of a for which the multiplier 
is equal to 1. Such a base will give an exponential function whose derivative
is exactly itself. Check back to
parts b and c to see that the base a = 2 gives a multiplier less than 1, while
a = 3 gives a multiplier greater than 1. Try to narrow in on a value of a
between 2 and 3 that gives an exponential function whose derivative is itself.
The number you have discovered in Problem 1e is commonly
designated by e. This symbol
was first introduced by the eighteenth century Swiss mathematician Leonhard
Euler. With this notation, your
discovery in Problem 1e is that if 
, then 
.
2. An exponential function in the world of banking
a. If you deposit money in a savings account paying interest at an
annual rate of r, your deposit will grow by a factor of 1 + r after
one year. (If the interest rate is
5%, then we use r = .05.
Thus, for example, a deposit of P dollars will grow to P + 0.05P = (1 +
0.05)P dollars after one year.)
If, however, the bank compounds the interest in the middle of the year,
your deposit will grow by a factor of 
after the first
six months, and another factor of 
after the second
six months. Thus the total growth
is 
. Verify that 
is slightly more
than 1 + r for all nonzero values of r.
b. By what factor will your deposit grow in one year if the bank
compounds quarterly? If it
compounds monthly? Daily? Hourly? The limiting value of the yearly growth factor as the number
of compounding periods increases to infinity is 
. This is the
growth factor used if the bank compounds continuously.
c. Substituting r = 1 into the limit in part b gives 
. Use your
computer to approximate the value of this limit. Where have you seen this number before?
d. In part b we introduced the limit 
, which is a function of r.
You evaluated this function for r = 1 in part c. If r = 2, we have 
. Use your
computer to approximate the value of this limit. Compare your answer with the number
. Repeat
for r = 5, comparing your answer with the number 
. In
fact, the function 
can be
defined by this limit for all values of x, namely, 
.
Created by:
Department of Mathematics and Computer
Science
The College of Wooster