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Calculus I (Notes) / Review / Review : Exponential Functions   [Notes] [Practice Problems] [Assignment Problems]

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Paul
August 7, 2018


Calculus I - Notes

 Review : Exponential Functions

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 will 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.

 

x

f(x)

g(x)

-2

 

 

-1

 

 

0

 

 

1

 

 

2

 

 

 

Here’s the sketch of both of these functions.

ExpFunc_G1

 

This graph illustrates some very nice properties about exponential functions in general.

 

Properties of  

  1. .  The function will always take the value of 1 at .
  2. .  An exponential function will never be zero.
  3. .  An exponential function is always positive.
  4. The previous two properties can be summarized by saying that the range of an exponential function is .
  5. The domain of an exponential function is .  In other words, you can plug every x into an exponential function.
  6. If  then,
    1.  
    2.  
  7. If  then,
    1.  
    2.  

 

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.

 

Definition : The natural exponential function is  where, .

 

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.

ExpFunc_G2

 

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.


Calculus I (Notes) / Review / Review : Exponential Functions    [Notes] [Practice Problems] [Assignment Problems]

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