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Answer:
A. [1, 2} rate of change is 30; [3, 4] rate of change is 1080
B. the rate of change on [3, 4] is 36 times that on [1, 2]; an exponential function's rate of change is increasing with increasing x
Explanation:
A.
The average rate of change on the interval [a, b] is ...
m = (g(b) -g(a))/(b -a)
On the interval [1, 2], the average rate of change is ...
ma = (6^2 -6^1)/(2 -1) = (36 -6)/1 = 30 . . . . rate of change on [1, 2]
On the interval [3, 4], the average rate of change is ...
mb = (6^4 -6^3)/(4 -3) = (1296 -216)/1 = 1080 . . . . rate of change on [3, 4]
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B.
The rate of change in section B is 1080/30 = 36 times the rate of change in section A.
An exponential function has increasing slope everywhere. Consider the value of g(x+2):
g(x+2) = 6^(x+2) = (6^2)(6^x) = 36g(x)
Each interval will have 36 times the rate of change of the same width interval 2 units to its left because the value of the function at a point is 36 times the value 2 units to the left of that point.