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10 October, 08:14

Given the function h (x) = 3 (2) x, Section A is from x = 1 to x = 2

and Section B is from x = 3 to x = 4.

Part A: Find the average rate of change of each section.

Part B: How many times greater is the average rate of change of Section B than Section A? Explain why one rate of change is greater than the other.

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  1. 10 October, 10:17
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    A: 6 and 24

    B: 4 times as great; the rate of change increases exponentially

    Step-by-step explanation:

    Part A: The average rate of change on the interval [a, b] is given by ...

    average rate of change = (h (b) - h (a)) / (b - a)

    On the interval [1, 2], the rate of change is ...

    (h (2) - h (1)) / (2 - 1) = (12 - 6) / 1 = 6

    On the interval [3, 4], the rate of change is ...

    (h (4) - h (3)) / (4 - 3) = (48 - 24) / 1 = 24

    For Section A, the average rate of change is 6; for Section B, the average rate of change is 24.

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    Part B: The ratio of the rates of change on the two intervals is ...

    (RoC on [3,4]) / (RoC on [1,2]) = 24/6 = 4

    The average rate of change of Section B is 4 times that of Section A.

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    The rate of change is exponentially increasing, so an interval of the same width that starts at "d" units more than the previous one will have a rate of change that is 2^d times as much.
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