Chapter 14.2, Problem 46AYU

To determine

To find: The limit as $x$ approaches $c$ of the average rate of change of the function from $c$ to $x$.

For, $c=3,ƒ\left(x\right)=x^3$.

Answer to Problem 46AYU

Solution:

27

Explanation of Solution

Given:

$3,ƒ\left(x\right)=x^3$

Formula used:

The average rate of change $\frac{\text{Δ}y}{\text{ Δ}x}=\frac{f\left(x\right)- f\left(c\right)}{x-c}$.

$a^3-b^3 =\left(a-b\right)\left(a^2+ab+b^2\right)$

Calculation:

The average rate of change from $c=3$ to $x$ is.

$\frac{\text{Δ}y}{\text{ Δ}x}=\frac{f\left(x\right)-f\left(3\right)}{x-3}=\frac{x^3-3^3}{x-3}=\frac{\left(x-3\right)\left(x+3x+9\right)}{x-3}=x^2+3x+9$

The limit as $x$ approaches 3 of the average rate of change is.

${\lim }_{x\to \text{ 3}}\frac{f\left(x\right)-f\left(3\right)}{x -\text{ 2}}={\lim }_{x\to 3}{x}^2+3x+9=9+9+9=27$