Chapter 6.5, Problem 1PT

To determine

To choose: The appropriate option for the average value of $f\left(x\right)=3x^2+1$ on the interval $\left[2,4\right]$.

Answer to Problem 1PT

The appropriate option for the average value of $f\left(x\right)=3x^2+1$ on the interval $\left[2,4\right]$ is $\underset{\_}{\text{a})29}$.

Explanation of Solution

Formula used:

The formula to find the average value of the function $\left(f_{\text{avg}}\right)$ is, $f_{\text{avg}}=\frac{1}{b-a}{\displaystyle {\int }_a^{b}f\left(x\right)dx}$.

Calculation:

The function is $f\left(x\right)=3x^2+1$ on the interval $\left[2,4\right]$.

Compute the average value of the function by substituting $a=2$, $b=4$ and $f\left(x\right)=3x^2+1$ in the formula mentioned above.

$\begin{array}{c}{f}_{\text{avg}}=\frac{1}{4-2}{\displaystyle {\int }_2^4\left(3x^2+1\right)dx}\\ =\frac{1}{2}{\displaystyle {\int }_2^4\left(3x^2+1\right)dx}\end{array}$

Obtain the value of the above integral as follows.

$\begin{array}{c}{f}_{\text{avg}}=\frac{1}{2}{\left[\frac{3x^3}{3}+x\right]}_2^4\\ =\frac{1}{2}\left\{\left[\frac{3{\left(4\right)}^3}{3}+\left(4\right)\right]-\left[\frac{3{\left(2\right)}^3}{3}+2\right]\right\}\\ =\frac{1}{2}\left\{\left[64+4\right]-\left[8+2\right]\right\}\\ =\frac{1}{2}\left\{68-10\right\}\\ =\frac{58}{2}\\ =29\end{array}$

Thus, the average function value is 29.

Therefore, the appropriate option for the average value of $f\left(x\right)=3x^2+1$ on the interval $\left[2,4\right]$ is $\underset{\_}{\text{a})29}$.