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Basic Exponential Functions    [Show All Solutions]

First, let’s recall that for  and  an exponential function is any function that is in the form

 

We require  to avoid the following situation,

 

 

So, if we allowed  we would just get the constant function, 1.

 

We require  to avoid the following situation,

 

 

By requiring  we don’t have to worry about the possibility of square roots of negative numbers.

 

1. Evaluate ,  and  at .

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The point here is mostly to make sure you can evaluate these kinds of functions.  So, here’s a quick table with the answers.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Notice that the last two rows give exactly the same answer.  If you think about it that should make sense because,

 

 

2. Sketch the graph of ,  and  on the same axis system.

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Note that we only really need to graph  and  since we showed in the previous Problem that .  Note as well that there really isn’t too much to do here.  We found a set of values in Problem 1 so all we need to do is plot the points and then sketch the graph.  Here is the sketch,

 

3. List as some basic properties for .

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Most of these properties can be seen in the sketch in the previous Problem.

(a)  for every x.  This is a direct consequence of the requirement that

 

(b) For any b we have .

 

(c) If  (  above, for example) we see that  is an increasing function and that,

 

 

 

(d) If  (  above, for example) we see that  is an decreasing function and that,

 

 

 

Note that the last two properties are very important properties in many Calculus topics and so you should always remember them!

 

4. Evaluate ,  and  at .

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Again, the point of this problem is to make sure you can evaluate these kinds of functions.  Recall that in these problems e is not a variable it is a number!  In fact,  

 

When computing  make sure that you do the exponentiation BEFORE multiplying by 5.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13.59141

 

 

 

5. Sketch the graph of  and .

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As with the other “sketching” problem there isn’t much to do here other than use the numbers we found in the previous example to make the sketch.  Here it is,

 Note that from these graphs we can see the following important properties about  and .

 

 

 

 

These properties show up with some regularity in a Calculus course and so should be remembered.

 

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