Q. 24 - Precalculus Enhanced with Graphing Utilities

Question

Find the average rate of change of $f\left(x\right)=8x^2-x$.

(a) From $1$ to $2$ 

(b) From $0$ to $1$ 

(c) From $2$ to $4$ 

Step-by-Step Solution

Answer

Part a. The average rate of change is $23$.

Part b. The average rate of change is $7$.

Part c. The average rate of change is $47$.

Part (a) Step 1. Find f ( 1 ) and f ( 2 )

Consider the function $f\left(x\right)=8x^2-x$.

Now the function value for $x=1$ is given as

$$\begin{gathered} f\left(1\right)=8\times 1^2-1 \\ f\left(1\right)=8\times 1-1 \\ f\left(1\right)=8-1 \\ f\left(1\right)=7 \end{gathered}$$

Again, the function value for $x=2$ is given as

$$\begin{gathered} f\left(2\right)=8\times 2^2-2 \\ f\left(2\right)=8\times 4-2 \\ f\left(2\right)=32-2 \\ f\left(2\right)=30 \end{gathered}$$

Part (a) Step 2. Find the average rate of change

The average of change for the function $f\left(x\right)$ from $1$ to $2$ is given as

$\begin{array}{rcl}\frac{f\left(2\right)-f\left(1\right)}{2-1}& =& \frac{30-7}{1}\\ & =& 23\end{array}$

Part (b) Step 1. Find f ( 0 ) and f ( 1 )

Consider the function$f\left(x\right)=8x^2-x$.

Now the function value for $x=0$ is given as

$$\begin{gathered} f\left(0\right)=8\times 0^2-0 \\ f\left(0\right)=0 \end{gathered}$$

Again, the function value for $x=1$ is given as

$$\begin{gathered} f\left(1\right)=8\times 1^2-1 \\ f\left(1\right)=8\times 1-1 \\ f\left(1\right)=8-1 \\ f\left(1\right)=7 \end{gathered}$$

Part (b) Step 2. Find the average rate of change

The average of change for the function $f\left(x\right)$ from $0$ to $1$ is given as

$\begin{array}{rcl}\frac{f\left(1\right)-f\left(0\right)}{1-0}& =& \frac{7-0}{1}\\ & =& 7\end{array}$

Part (c) Step 1. Find f ( 2 ) and f ( 4 )

Consider the function $f\left(x\right)=8x^2-x$.

Now the function value for $x=2$ is given as

$$\begin{gathered} f\left(2\right)=8\times 2^2-2 \\ f\left(2\right)=8\times 4-2 \\ f\left(2\right)=32-2 \\ f\left(2\right)=30 \end{gathered}$$

Again, the function value for $x=4$ is given as

$$\begin{gathered} f\left(4\right)=8\times 4^2-4 \\ f\left(4\right)=8\times 16-4 \\ f\left(4\right)=128-4 \\ f\left(4\right)=124 \end{gathered}$$

Part (c) Step 2. Find the average rate of change

The average of change for the function $f\left(x\right)$ from $2$ to $4$ is given as

$\begin{array}{rcl}\frac{f\left(4\right)-f\left(2\right)}{4-2}& =& \frac{124-30}{2}\\ & =& \frac{94}{2}\\ & =& 47\end{array}$