dlscrib.com_review.pdf - The Jones family drove to...

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The Jones family drove to Disneyland at 50 miles per hour and returned on the same route at 30 miles per hour. Find the distance to Disneyland if the total driving time was 7.2 hours. (distance)=(rate)(time) or d=rt rate to Disnseyland:50 rate from Disneyland:30 time to Disneyland:t time to Disneyland+time frome Disneyland=7.2 therefore time from Disneyland:7.2-t Distance to Disneyland=50t Distance from Disneyland=30(7.2-t) These two distances are equal, therefore: 50t=30(7.2-t) 50t=216-30t 50t+30t=216-30t+30t 80t=216 80t/80=216/80 t=2.7 hrs Plug this t into the distance equation going to Disneyland. Distance to Disneyland=50(2.7) miles Distance to Disneyland=135 miles
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Exponential Functions Let’s start off this section with the definition of an exponential function. If b is any number such that and then an exponential function is a function in the form, where b is called the base and x can be any real number. Notice that the x is now in the exponent and the base is a fixed number. This is exactly the opposite from what we’ve seen to this point. To this point the base has been the variable, x in most cases, and the exponent was a fixed number. However, despite these differences these functions evaluate in exactly the same way as those that we are used to. We will see some examples of exponential functions shortly. Before we get too far into this section we should address the restrictions on b . We avoid one and zero because in this case the function would be,
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and these are constant functions and won’t have many of the same properties that general exponential functions have. Next, we avoid negative numbers so that we don’t get any complex values out of the function evaluation. For instance if we allowed the function would be, and as you can see there are some function evaluations that will give complex numbers. We only want real numbers to arise from function evaluation and so to make sure of this we require that b not be a negative number. Now, let’s take a look at a couple of graphs. We will be able to get most of the properties of exponential functions from these graphs. Example 1 Sketch the graph of and on the same axis system. Solution Okay, since we don’t have any knowledge on what these graphs look like we’re going to have to pick some values of x and do some function evaluations. Function evaluation with
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exponential functions works in exactly the same manner that all function evaluation has worked to this point. Whatever is in the parenthesis on the left we substitute into all the x ’s on the right side. Here are some evaluations for these two functions, x -2 -1 0 1
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2 Here is the sketch of the two graphs. Note as well that we could have written in the following way, Sometimes we’ll see this kind of exponential function and so it’s important to be able to go between these two forms.
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