- Use the functions f(x) = 4x + 3 and g(x) = 2x2 + 2 to answer parts (a)-(c). a. Which function has a greater rate of change from x = 0 to x = 1?
b. Which function has a greater rate of change from x = 2 and x = 3?
C. Does g(x) ever have a greater rate of change than f(x)? Explain.

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  a) f(x)

  b) g(x)

  c) for all values of x > 1, g(x) has a greater rate of change

Step-by-step explanation:

a) The rate of change of f(x) is the x-coefficient: 4. The average rate of change of g(x) on an interval can be found by dividing the change by the interval width. Here, the width of the interval of interest is 1-0 = 1, so the average rate of change is g(1) -g(0) = 4-2 = 2.

On the interval [0, 1], f(x) has the greater rate of change.

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b) The interval [2, 3] also has a width of 1, so the rate of change of g(x) on that interval is g(3) -g(2) = 20 -10 = 10. This value is greater than 4, so ...

on the interval [2, 3], g(x) has the greater rate of change.

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c) Part (b) demonstrates that g(x) will have a greater rate of change on some intervals. In fact, for any interval whose center is greater than x=1, g(x) will have a greater rate of change than f(x).

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In calculus terms, the rate of change of g(x) is g'(x) = 4x. This will be greater than 4 for any x > 1. The graph shows the rate of change is 4 at x=1, and is higher for x > 1. The average rate of change will be greater than 4 on any interval whose center is greater than x=1.

We can figure the average rate of change on the interval [a, b] as ...

  m = (g(b) -g(a))/(b -a)

  m = ((2b² +2) -(2a² +2))/(b -a) = 2(b² -a²)/(b-a) = 2(b+a)

For the average rate of change to exceed 4, the sum of the ends of the interval must exceed 2, which is to say the midpoint must exceed 1.

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