Question

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.

Answer 1

A: 6 and 24

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

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.