Chapter 3.1, Problem 3AYU

To determine

The average rate of change of $f\left(x\right)=3x^2-2,\text{From}\text{2}\text{to}\text{4}$ .

Answer to Problem 3AYU

The average rate of change of $f\left(x\right)=3x^2-2,\text{From}\text{2}\text{to}\text{4}$ is $18$ .

Explanation of Solution

Given:

  $f\left(x\right)=3x^2-2,\text{From}\text{2}\text{to}\text{4}$ .

Concept Used:

Average rate of change:

  $=\frac{\text{change}\text{in}\text{output}}{\text{change}\text{in}\text{input}}$ .

Calculation:

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

So, plug the terms into the equation,

  $3{\left(2\right)}^2-2$ and $3{\left(4\right)}^2-2$

Now subtract the two new terms to get an average of the $2$ ,

  $\begin{array}{l}12-2=10\\ 48-2=46\\ 46-10=36\end{array}$

Subtract the $2$ terms because it is the distance traveled in terms of $x$

  $4-2=2$

Now divide the average by the amount traveled to reach an average per movement,

  $36/2=18$

To check this let’s see if $y$ values look close to it.

  $\begin{array}{l}f\left(2\right)=10\\ f\left(3\right)=25\\ f\left(4\right)=46\\ 10+18=28\\ 28+18=46\end{array}$

Conclusion:

Hence,the average rate of change of

  $f\left(x\right)=3x^2-2,\text{From}\text{2}\text{to}\text{4}$ is $18$ .