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Q. 3

Expert-verified
Found in: Page 136

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

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

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

See the step by step solution

## Step 1. Given Information

Given that $f\left(x\right)=3{x}^{2}-2,$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 $f\left(x\right)=3{x}^{2}-2$ from $x=2$ to $x=4$ is :-

$\frac{∆y}{∆x}=\frac{f\left(4\right)-f\left(2\right)}{4-2}\phantom{\rule{0ex}{0ex}}\frac{∆y}{∆x}=\frac{\left(3×{4}^{2}-2\right)-\left(3×{2}^{2}-2\right)}{2}\phantom{\rule{0ex}{0ex}}\frac{∆y}{∆x}=\frac{\left(48-2\right)-\left(12-2\right)}{2}\phantom{\rule{0ex}{0ex}}\frac{∆y}{∆x}=\frac{36}{2}\phantom{\rule{0ex}{0ex}}\frac{∆y}{∆x}=18.\phantom{\rule{0ex}{0ex}}$

Hence, $\frac{∆y}{∆x}=18.$