# The Exponential Function

**The Story**

Speed O’Fender, a TAMS student, just received his driver’s license. He talked
his father

into letting him drive a car back to TAMS after a closed weekend. He noticed
that just as

he passed the 60 mile marker, the speedometer read exactly 60 miles per hour. A
mile later

he passed the 61 mile marker and he happened to be traveling at 61 miles per
hour. Speed

decided that it would be fun to always travel at the speed corresponding to the
mile marker.

For example, when he was at the 61.5 mile point, his speed was 61.5 miles per
hour. When

he reached the 72 mile marker, he was traveling exactly 72 miles per hour. When
he was at

the 80 mile marker, he was introduced to a member of the Texas State Police who
did not

have enough mathematical or scientific curiosity to encourage Speed in this
experiment.

This experience suggested several interesting problems to Speed. First, is the
function

s(t) that describes Speed’s position at time t an exponential function ? Second,
could this

idea be used to define the exponential function instead of defining it as the
inverse of the

logarithm function ? Third, if we define the exponential function using this
idea, could we

derive the usual properties of the exponential function? Fourth, is a
minimization problem:

How should Speed tell his parents that he got a ticket for speeding in order to
minimize the

negative consequences ?

Speed went to you for help with these problems. Your job is to follow the
outline below

to help Speed understand how to define the exponential function using a simple
differential

equation.

**Background Information
**

It will be helpful to use a theorem from differential equations that we will not cover in class.

The theorem is a special case of an existence and uniqueness theorem that you will learn

when you take a differential equations class.

Theorem 1 For any real numbers a and b , there is exactly one solution to the initial value problem y' = y, y(a) = b. Furthermore, the domain of the solution is all real numbers. |

Example The function f(x) satisfies

f'(x) = f(x) for all x,

f(10) = 0.

Show that f(x) = 0 for every real number x.

Solution Since f'(x) = f(x), f(x) satisfies the
differential equation y' = y. So f(x) satisfies

the inital value problem:

y'= y

y(10) = 0.

Let g(x) = 0 for every x. Then g'(x) = 0 = g(x) for every x. We see that g(x)
satisfies

the same initial value problem:

y' = y

y(10) = 0.

The uniqueness part of Theorem 1 says that there is only one solution to this
inital value

problem. Therefore, f(x) = g(x) = 0 for every real number x.

Some initial value problems have more than one solution, while others have no
solution.

An example is the differential equation x^{3}y' = y. With the initial
condition y(0) = 1 there

is no solution, but with the initial condition y(0)=0 there are infinitely many
solutions.

**Directions**

Follow the outline below to give a reasonable definition of e^{x}. From
high school algebra, we

know how to define e^{x} for any rational number. The problem is that we
cannot define e^{x} for

irrational numbers using only precalculus mathematics . In the steps below , you
will define

the function E(x) and derive some of its basic properties. In the end you will
conclude that

it is reasonable to define e^{x} to be E(x) for all numbers, rational or
irrational. Be careful not

to assume anything about the function E(x) other than its definition and what
you have

already proved about it.

You do not have to follow the steps below in exactly the order given. The goal
is simply

to define E(x) as stated below and then derive the formula in step 10. If you
wish to do

things in a different order, that is fine. However, keep in mind that the
outline was designed

to allow you to break the bigger problem in smaller parts. For most of the
parts, you will

probably wish to use what you proved in a previous part.

1. Explain why Speed’s position is modeled by the differential equation y' = y.
We

normally use t as the variable representing time, but for the remainder of the
project

let’s use x. So y is a function of x.

2. Let’s define E(x) to be the unique solution to

y'= y,

y(0) = 1.

Explain why E' (x) = E(x). What is E(0)? Find a formula for E''(x).

3. Use the chain rule to determine the derivative
, where u is a function of x.

4. Prove that

E(a + x) = E(a)E(x),

where a and x are any numbers. (Hint: Think of a as a fixed number and x as a

variable. Use the uniqueness part of Theorem 1.)

5. Use induction to prove that E(nx) = (E(x))^{n} for any positive
integer n and any real

number x. (Hint: Fix x and then do an induction proof.)

6. Now, let’s define e = E(1). Prove that E(n) = e^{n} for all positive
integers n.

7. Prove that for any real number x.
Conclude that for

every positive integer n and every real number x. (Hint: This one is not hard!)

8. Prove that E(x) > 0 for all x. Where is E(x) increasing? Where is E(x)
concave up?

(Hint: Use the uniqueness part of Theorem 1 to show E(x) ≠ 0 for any value
of x. You

may wish to read the example after Theorem 1 again.)

9. Prove that for any positive integer n and any number x,
. Recall that

for r > 0 and n a positive integer, is
defined to be the unique positive number whose

n^{th} power is r.

10. Prove that for any integers p and q ≠
0.

11. Explain why it is reasonable to define e^{x} = E(x) for every real
number x.

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