Question

find the average rate of change of ( f(x) = x^3 ) over the interval (-4, 2). write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). Here, \( a = -4 \), \( b = 2 \), and \( f(x)=x^3 \).

Step2: Calculate \( f(-4) \) and \( f(2) \)

First, find \( f(-4) \): \( f(-4)=(-4)^3=-64 \).
Then, find \( f(2) \): \( f(2)=(2)^3 = 8 \).

Step3: Substitute into the formula

Substitute \( f(-4)=-64 \), \( f(2)=8 \), \( a = -4 \), and \( b = 2 \) into the average rate of change formula:
\(\frac{f(2)-f(-4)}{2 - (-4)}=\frac{8 - (-64)}{2 + 4}=\frac{8 + 64}{6}=\frac{72}{6}\).

Step4: Simplify the fraction

Simplify \(\frac{72}{6}\) to get \( 12 \).

Answer:

\( 12 \)