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In this section we’re going to review one of the more common
functions in both calculus and the sciences.
However, before getting to this function let’s take a much more general
approach to things.
Let’s start with 
,
. An exponential function is then a function in
the form,
Note that we avoid because that would give the constant function,
. We avoid since this would also give a constant function
and we avoid negative values of b for
the following reason. Let’s, for a
second, suppose that we did allow b
to be negative and look at the following function.
Let’s do some evaluation.
So, for some values of x
we will get real numbers and for other values of x we well get complex numbers.
We want to avoid this and so if we require this will not be a problem.
Let’s take a look at a couple of exponential functions.
|
Example 1 Sketch
the graph of and
Solution
Let’s first get a table of values for these two functions.
Here’s the sketch of both of these functions.
|
This graph illustrates some very nice properties about
exponential functions in general.
Properties of
- . The function will always take the
value of 1 at .
- . An exponential function will never be
zero.
- .
An exponential function is always positive.
- The
previous two properties can be summarized by saying that the range of an
exponential function is .
- The
domain of an exponential function is . In other words, you can plug every x into an exponential function.
- If then,
-
-
- If then,
-
-
|
These will all be very useful properties to recall at times
as we move throughout this course (and later Calculus courses for that
matter…).
There is a very important exponential function that arises
naturally in many places. This function
is called the natural exponential
function. However, for most people
this is simply the exponential function.
So, since we also know that and .
Let’s take a quick look at an example.
|
Example 2 Sketch the graph of
Solution
Let’s first get a table of values for this function.
|
t
|
-2
|
-1
|
0
|
1
|
2
|
3
|
|
|
-35.9453
|
-21.4084
|
-12.5914
|
-7.2436
|
-4
|
-2.0327
|
Here is the sketch.
|
The main point behind this problem is to make sure you can
do this type of evaluation so make sure that you can get the values that we
graphed in this example. You will be
asked to do this kind of evaluation on occasion in this class.
You will be seeing exponential functions in pretty much
every chapter in this class so make sure that you are comfortable with them.