College

Find the average rate of change for [tex]f(x) = 7x^6[/tex] over the interval [2, 6]. Be sure to simplify your answer.

Answer :

Answer:

Absolutely, the average rate of change of a function represents the slope of the secant line that intersects the graph of the function at the interval's endpoints. Let's find the average rate of change for f(x) = 7x - 6 over the interval [2, 6].

Formula for Average Rate of Change:

The average rate of change of a function f(x) over the interval [a, b] is calculated as:

Average Rate of Change = (f(b) - f(a)) / (b - a)

Applying the formula:

Identify the function: f(x) = 7x - 6

Identify the interval: [a, b] = [2, 6]

Find f(b) and f(a):

f(b) = f(6) = (7 * 6) - 6 = 42 - 6 = 36

f(a) = f(2) = (7 * 2) - 6 = 14 - 6 = 8

Calculate the average rate of change:

Average Rate of Change = (f(b) - f(a)) / (b - a)

= (36 - 8) / (6 - 2)

= 28 / 4

= 7

Answer:

The average rate of change of f(x) = 7x - 6 over the interval [2, 6] is 7. This means the secant line intersecting the graph of the function at x = 2 and x = 6 has a slope of 7.