Given:
[tex]f(x)=x^3+3x[/tex]Let's determine the average rate of change with respect to x over the interval:
[tex]2\leq x\leq4[/tex]To find the average rate of change, apply the formula:
[tex]avg=\frac{f(b)-f(a)}{b-a}[/tex]Where the closed interval is [a, b].
Thus, we have:
(a, b) ==> (2, 4)
Let's solve for f(2) and f(4).
We have:
[tex]\begin{gathered} f(2)=2^3+3(2) \\ f(2)=8+6 \\ f(2)=14 \\ \\ \\ f(4)=4^3+3(4) \\ f(4)=64+12 \\ f(4)=76 \end{gathered}[/tex]To find the average rate of change where f(a) = 14 and f(b) = 76, we have:
[tex]\begin{gathered} \text{avg}=\frac{f(b)-f(a)}{b-a} \\ \\ \text{avg}=\frac{76-14}{4-2} \\ \\ \text{avg}=\frac{62}{2} \\ \\ \text{avg}=31 \end{gathered}[/tex]Therefore, the average rate of change of f(x) over the given interval is 31
ANSWER:
31
