Chapter 14.2, Problem 50AYU

In Problems 43-52, find the limit as $x$ approaches $c$ of the average rate of change of each function from $c$ to $x$.

$c=0$; $f\left(x\right)=4x^3-5x+8$

To determine

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

For, $c=0,ƒ\left(x\right)=4x^3-5x+8$.

Answer to Problem 50AYU

Solution:

$-5$

Explanation of Solution

Given:

$c=0,ƒ\left(x\right)=4x^3-5x+8$

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.

$\frac{\text{Δ}y}{\text{ Δ}x}=\frac{f\left(x\right)- f\left(c\right)}{x-c}=\frac{4x^3-5x+8-8}{x}=\frac{x\left(4x^2-5\right)}{x}=4x^2–5$

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

${\lim }_{x\to \text{ 0}}\frac{f\left(x\right)- f\left(0\right)}{x\text{ - 0}}={\lim }_{x\to \text{ 1}}4x^2-5=-5$