Q 96. - Precalculus Enhanced with Graphing Utilities

Question

Show that a constant function $f\left(x\right)=b$ has an average rate of change of $0$. Evaluate the average rate of change of $y=\sqrt{4-x^2}$ on the interval $[-2,2]$. Explain how this can happen.

Step-by-Step Solution

Answer

It is proved that a constant function $f\left(x\right)=b$ has an average rate of change of $0$ and the average rate of change of the function $y=\sqrt{4-x^2}$ is $0$ on $[-2,2]$. Zero average rate of change shows that the ending point and starting point of the $y$-coordiante are the same.

Step 1. Given information

The functions are $f\left(x\right)=b$ and $f\left(x\right)=\sqrt{4-x^2}$.

Step 2. Prove that a f ( x )   =   b has an average rate of change of 0 , where f ( x ) = b is constant function.

The average rate of change from $a$ to $b$ is defined as folllows:

$\frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}\dots \left(1\right)$, where $a\ne b$.

Substitute $x_1$ for $a$ and $x_2$ for $b$ into $\left(1\right)$.

$\frac{∆y}{∆x}=\frac{f\left(x_2\right)-f\left(x_1\right)}{x_2-x_1}\dots \left(2\right)$

Since $f\left(x\right)=b$, the values are as follows:

$\begin{array}{rcl}f\left(x_2\right)& =& b\\ f\left(x_1\right)& =& b\end{array}$

From $\left(2\right)$, the average rate of change is as follows:

$\begin{array}{rcl}\frac{∆y}{∆x}& =& \frac{b-b}{x_2-x_1}\\ & =& \frac{0}{x_2-x_1}\\ & =& 0\end{array}$

Thus, it is proved that a constant function $f\left(x\right)=b$ has an average rate of change of $0$.

Step 3. Find the value of y = 4 - x 2 on the interval [ - 2 , 2 ] .

Assume $y=f\left(x\right)$.

So, the function is $f\left(x\right)=\sqrt{4-x^2}\dots \left(3\right)$.

To find the value of $f\left(x\right)$on the given interval, substitute $x=-2$ and $x=2$ into the function $f\left(x\right)$.

So, substitute $x=-2$ into the function $\left(3\right)$.

The value of $f(-2)$ is as follows:

$\begin{array}{rcl}f(-2)& =& \sqrt{4-{(-2)}^2}\\ & =& \sqrt{4-4}\\ & =& 0\end{array}$

Now, substitute $x=2$ into the function $\left(3\right)$.

The value of $f\left(2\right)$ is as follows:

$\begin{array}{rcl}f\left(2\right)& =& \sqrt{4-{\left(2\right)}^2}\\ & =& \sqrt{4-4}\\ & =& 0\end{array}$

Step 4. Find the average rate of change of the function y = 4 - x 2 on [ - 2 , 2 ] .

Substitute $-2$ for $a$ and $2$ for $b$ into $\left(1\right)$.

$\frac{∆y}{∆x}=\frac{f\left(2\right)-f(-2)}{-2-2}\dots \left(4\right)$

Substitute the value of $f\left(2\right)$ and $f(-2)$ into $\left(4\right)$.

$\begin{array}{rcl}\frac{∆y}{∆x}& =& \frac{0-0}{-2-2}\\ & =& \frac{0}{-4}\\ & =& 0\end{array}$

Thus, the average rate of change of the function $y=\sqrt{4-x^2}$ is $0$.

Hence, from the average rate of change of $f\left(x\right)$ conclude that ending point and starting point of the $y$-coordiante are the same.