Basic Exponential Functions
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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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3. List as some basic properties for .
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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.
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.