Chapter 14.2, Problem 51AYU

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, $c=1,ƒ\left(x\right)=\frac{1}{x}$.

Answer to Problem 51AYU

Solution:

$-1$

Explanation of Solution

Given:

$c=1,ƒ\left(x\right)=\frac{1}{x}$

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 $c=1$ to $x$ is.

$\frac{\text{Δ}y}{\text{Δ}x}\text{ = }\frac{f\left(x\right)-f\left(1\right)}{x-1}=\frac{\frac{1}{x}-1}{x-1}=\frac{-\left(x-1\right)}{x\left(x-1\right)}=-\frac{1}{x}$

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

${\lim }_{x\to \text{ 1}}\frac{f\left(x\right)- f\left(1\right)}{x\text{ - 1}}={\lim }_{x\to \text{ 1}}-\frac{1}{x}=-1$