Question

find the average rate of change of f(x) = 15\sqrt{x}-18 over the interval 3,9. write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here $a = 3$, $b=9$, and $f(x)=15\sqrt{x}-18$.

Step2: Calculate $f(9)$

Substitute $x = 9$ into $f(x)$: $f(9)=15\sqrt{9}-18=15\times3 - 18=45 - 18=27$.

Step3: Calculate $f(3)$

Substitute $x = 3$ into $f(x)$: $f(3)=15\sqrt{3}-18\approx15\times1.732 - 18=25.98-18 = 7.98$.

Step4: Calculate average rate of change

$\frac{f(9)-f(3)}{9 - 3}=\frac{27-(15\sqrt{3}-18)}{6}=\frac{27 - 15\sqrt{3}+18}{6}=\frac{45 - 15\sqrt{3}}{6}=\frac{15(3-\sqrt{3})}{6}=\frac{5(3 - \sqrt{3})}{2}\approx\frac{5(3 - 1.732)}{2}=\frac{5\times1.268}{2}=3.17\approx3.2$.

Answer:

$3.2$