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You may notice something about values of b between 2 and 3, that for a certain value of b the graph of F and its derivative seem to coincide. For what value of b does this occur?
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Answer: 2.7
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Fix this value of b. Do the graphs of F and its derivative still seem to be the same when you change the value of C? Try to explain your observation: if they stay the same, why? If they start to look different, why?
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Answer: When the value of C changes, the graphs stay the same. They stay the same because when the coefficient is multiplied by the same base, the product would also be the same in the derivative.
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The next function, R(x), is the ratio of a power function x^p and the exponential function F(x), defined for non-negative values of x.
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Set b=2 in the definition of F(x). What do you notice about the behavior of R(x) as x goes to infinity (i.e., becomes very large)?
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Answer: It stays the same at 0.
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negative 10
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Now change the value of p in the definition of R(x). What happens to the behavior of R(x) as x goes to infinity? Does your answer depend on the value of p? (Remember that you may have to use the zoom and scroll features.)
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Answer: Yes, the answer depends on the value of p. As p increases, the x and y values also increase.
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When you have finished answering the above questions, save this graph and share it with me.
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