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# PART III Average rate of change of exponential functions near zero. (5) 1. Given f(x) = 2*, find the average rate of change of

## PART III Average rate of change of exponential functions near zero. (5) 1. Given f(x) = 2*, find the average rate of change of f(x) from x = 0 to x = 0.1, so that Ax = 0.1. Recall that the average rate of change is A. Show your steps and write your answer on the first line of the table below. Ax (5) 2. Continue finding the average rate of change using 0 as your initial x-value and letting Ax get smaller. Write the results in the table below and copy to your paper. Interval [0, 0.1] [0, 0.01] 0.1 0.01 [0, 0.001] 0.001 [[0, 0.0001] 0.0001 Average rate of change (2) 3. Using your observations from the table, complete this sentence on your paper: As Ax gets smaller and smaller, gets Ay Ax (5) 4. Make a similar table for g(x) = 3*, using the same intervals, and copy to your paper. (2) 5. Using your observations from this new table, complete this sentence on your paper: As Ay Ax Ax gets smaller and smaller, gets (5) 6. For both functions it seems that as Ax gets smaller, the average rate of change approaches a particular decimal. One of the decimals is greater than 1 and the other is less than 1. Using trial and error, try to find an exponential function whose average rate of change, using the same intervals as above, approaches 1. You should present at least 3 other exponential functions and their tables and each table should be accompanied by an observation similar to those in parts 3 and 5. (5) 7. Write your conjecture about the exponential function whose average rate of change on the interval [0, Ax], as Ax gets smaller and smaller, comes the closest to 1.

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