Chapter 2, Problem 11RCC

(a)

To determine

To express: The average rate of change of y with respect to x over the interval $\left[x_1,x_2\right]$

Answer to Problem 11RCC

The average rate of change of y with respect to x is $\frac{\Delta y}{\Delta x}=\frac{f\left(x_2\right)-f\left(x_1\right)}{x_2-x_1}$.

Explanation of Solution

Result used:

The average rate of change of y with respect to x is

$\frac{\Delta y}{\Delta x}=\frac{f\left(x_0+\Delta x\right)-f\left(x_0\right)}{\Delta x}$

Calculation:

The average rate of change of y with respect to x is

$\frac{\Delta y}{\Delta x}=\frac{f\left(x_0+\Delta x\right)-f\left(x_0\right)}{\Delta x}$

Where $\left[x_0,x\right]=\left[x_1,x_2\right]$ and $\Delta x=x_2-x_1$.

The average rate of change of y with respect to x over the interval $\left[x_1,x_2\right]$ is

$\begin{array}{c}\frac{\Delta y}{\Delta x}=\frac{f\left(x_1+\left(x_2-x_1\right)\right)-f\left(x_1\right)}{x_2-x_1}\\ =\frac{f\left(x_1+x_2-x_1\right)-f\left(x_1\right)}{x_2-x_1}\\ =\frac{f\left(x_2\right)-f\left(x_1\right)}{x_2-x_1}\end{array}$

Thus the average rate of change of y with respect to x is $\frac{\Delta y}{\Delta x}=\frac{f\left(x_2\right)-f\left(x_1\right)}{x_2-x_1}$.

(b)

To determine

To express: The instantaneous rate of change of y with respect to x at $x=x_1$

Answer to Problem 11RCC

The instantaneous rate of change of y with respect to x at $x=x_1$ is

$\underset{x\to x_1}{\lim }\frac{\Delta y}{\Delta x}=\underset{x\to x_1}{\lim }\frac{f\left(x\right)-f\left(x_1\right)}{x-x_1}$.

Explanation of Solution

Result used:

The instantaneous rate of change of the function $y=f\left(x\right)$ at the point $x_0$ in its domain is $\underset{x\to x_0}{\lim }\frac{\Delta y}{\Delta x}=\underset{x\to x_0}{\lim }\frac{f\left(x_0\right)-f\left(x\right)}{x_0-x}$

Now the instantaneous rate of change of the function $y=f\left(x\right)$ at $x=x_1$ is

$\begin{array}{c}\underset{x\to x_1}{\lim }\frac{\Delta y}{\Delta x}=\underset{x\to x_1}{\lim }\frac{f\left(x_1\right)-f\left(x\right)}{x_1-x}\\ =\underset{x\to x_1}{\lim }\frac{-\left(f\left(x\right)-f\left(x_1\right)\right)}{-\left(x-x_1\right)}\\ =\underset{x\to x_1}{\lim }\frac{f\left(x\right)-f\left(x_1\right)}{x-x_1}\end{array}$

Thus the instantaneous rate of change of y with respect to x at $x=x_1$ is

$\underset{x\to x_1}{\lim }\frac{\Delta y}{\Delta x}=\underset{x\to x_1}{\lim }\frac{f\left(x\right)-f\left(x_1\right)}{x-x_1}$.