Q. 3 - Precalculus Enhanced with Graphing Utilities

Question

Find the average rate of change of $f\left(x\right)=3x^2-2$ from 2 to 4.

Step-by-Step Solution

Answer

The average rate of change is $\frac{∆y}{∆x}=18.$

Step 1. Given Information

Given that $f\left(x\right)=3x^2-2,$and $x=2$ to $x=4.$

Step 2. Solution

The average rate of change of function $f\left(x\right)$ from $x=x_1$ to $x=x_2$is: -

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

Now, the average rate of function $f\left(x\right)=3x^2-2$ from $x=2$ to $x=4$ is :-

$$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(4\right)-f\left(2\right)}{4-2} \\ \frac{∆y}{∆x}=\frac{(3\times 4^2-2)-(3\times 2^2-2)}{2} \\ \frac{∆y}{∆x}=\frac{(48-2)-(12-2)}{2} \\ \frac{∆y}{∆x}=\frac{36}{2} \\ \frac{∆y}{∆x}=18. \end{gathered}$$

Step 3. Final Answer

Hence, $\frac{∆y}{∆x}=18.$