Find the limit as x approaches c of the average rate of change of the function from c to x.
-$$c = - 4 ; \quad f ( x ) = x ^ { 3 }$$
A) 0
B) 16
C) 48
D) 32

Question

Quizplus · Accepted Answer

The average rate of change of a function $f(x)$ from $c$ to $x$ is given by the formula $\frac{f(x) - f(c)}{x - c}$. For the given function $f(x) = x^3$ and $c = -4$, we substitute these values into the formula to get $\frac{x^3 - (-4)^3}{x - (-4)} = \frac{x^3 - 64}{x + 4}$. Simplifying this expression as $x$ approaches $-4$, we factor the numerator as a difference of cubes: $\frac{(x - 4)(x^2 + 4x + 16)}{x + 4}$. Canceling $x + 4$ in the numerator and denominator, we are left with the limit of $x^2 + 4x + 16$ as $x$ approaches $-4$. Substituting $-4$ into this expression gives $(-4)^2 + 4(-4) + 16 = 16 - 16 + 16 = 16$. However, the correct calculation should actually result in evaluating the derivative of $f(x) = x^3$ at $x = c$, which is $f'(x) = 3x^2$, and then substituting $c = -4$ to get $3(-4)^2 = 3 \cdot 16 = 48$, which correctly reflects the instantaneous rate of change of the function at $x = c$.

Find the Limit as X Approaches C of the Average c=−4;f(x)=x3c = - 4 ; \quad f ( x ) = x ^ { 3 }c=−4;f(x)=x3

Find the Limit as X Approaches C of the Average $c = - 4 ; \quad f ( x ) = x ^ { 3 }$