Find the average rate of change of f(x)=x^3−9x+6 over the following intervals:

Question

Find the average rate of change of f(x)=x^3−9x+6 over the following intervals:

(a) = From -6 to -4

(b) = From -2 to 3

(c) = From 3 to 9

Answer 1

The formula for the average rate of change is [f(b) - f(a)] / b - a where the interval is [a, b].

f(x) = x³ − 9x + 6

a) a = -6 and b = -4

f(a) = (-6)³ - 9(-6) + 6 = -216 + 54 + 6 = -156

f(b) = (-4)³ - 9(-4) + 6 = -64 + 36 + 6 = -22

Avg. ROC = [-22 - (-156)] / [-4 - (-6)] = 134/2 = 67

b) a = -2 and b = 3

f(a) = (-2)³ - 9(-2) + 6 = -8 + 18 + 6 = 16

f(b) = (3)³ - 9(3) + 6 = 27 - 27 + 6 = 6

Avg. ROC = (6 - 16) / [3 - (-2)] = -10/5 = -2

c) a = 3 and b = 9

f(a) = (3)³ - 9(3) + 6 = 27 - 27 + 6 = 6

f(b) = (9)³ - 9(9) + 6 = 729 - 81 + 6 = 654

Avg. ROC = (654 - 6) / (9 - 3) = 648/6 = 108