Question

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

Short answer

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

Step-by-step solution

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,$\frac{∆y}{∆x}=18.$