Question

What is the average rate of change of the function fx= 2x2+ 3 over the interval [1,4]?

Answer 1

Answer:

  10

Step-by-step explanation:

The average rate of change of f(x) on the interval [a, b] is found by ...

  m = (f(b) -f(a))/(b -a)

For your function, we have [a, b] = [1, 4}, so the rate of change is calculated as ...

  f(1) = 2·1² +3 = 5

  f(4) = 2·4² +3 = 35

  m = (35 -5)/(4 -1) = 30/3 = 10

The average rate of change on the interval is 10.

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Comment on quadratic slope

The slope of a quadratic curve changes continuously in linear fashion. The average slope on any interval is the slope at the midpoint of the interval. Using derivatives, we find the slope of the quadratic ax^2+bx+c to be given by 2ax+b. For the function given here, that is 4x. The midpoint of [1, 4] is x=2.5, so the slope at the midpoint is 4(2.5) = 10, as above.