Chapter 14.2, Problem 49AYU

To determine

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

For, $c=0,ƒ\left(x\right)=\text{ 3}{x}^3-2x+4$.

Answer to Problem 49AYU

Solution:

0

Explanation of Solution

Given:

$c=0,ƒ\left(x\right)=\text{ 3}{x}^3-2x+4$

Formula used:

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

Calculation:

The average rate of change from $c=0$ to $x$ is.

$\frac{\text{Δ}y}{\text{ Δ}x}=\frac{f\left(x\right)- f \left(0\right)}{x-0}=\frac{3x^3-2x^2+4-4}{x}=\frac{x\left(3x^2-2x\right)}{x}=3x^2–2x$

The limit as $x$ approaches 0 of the average rate of change is.

${\lim }_{x\to 0}\frac{f\left(x\right)-f\left(0\right)}{x -\text{ 0}}={\lim }_{x\to 0}3x^2-2x=0$