Find the average rate of change of the function from $$x _ { 1 } = 1 \text { to } x _ { 2 } = 3$$ $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } - x$$
A) The average rate of change from $x _ { 1 } = 1$ to $x _ { 2 } = 3$ is 19 .
B) The average rate of change from $x _ { 1 } = 1$ to $x _ { 2 } = 3$ is $- 12$.
C) The average rate of change from $x _ { 1 } = 1$ to $x _ { 2 } = 3$ is 6 .
D) The average rate of change from $x _ { 1 } = 1$ to $x _ { 2 } = 3$ is $- 15$.
E) The average rate of change from $x _ { 1 } = 1$ to $x _ { 2 } = 3$ is 15 .

Question

Quizplus · Accepted Answer

The formula for the average rate of change between two points $(x_1, f(x_1))$ and $(x_2, f(x_2))$ is:

$$\frac{f(x_2)-f(x_1)}{x_2-x_1}$$

For $x _ { 1 } = 1$ and $x _ { 2 } = 3$, we have:

$$\frac{f(3)-f(1)}{3-1} = \frac{(3^3-6(3^2)-3)-(1^3-6(1^2)-1)}{2} = \frac{19}{2} = 9.5$$

Therefore, the average rate of change from $x _ { 1 } = 1$ to $x _ { 2 } = 3$ is $- 9.5$, which corresponds to answer choice B.

Find the Average Rate of Change of the Function From x