find the average rate of change for the function r = 5 sin 3θ on the interval π/3, π/2. then estimate the…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=\frac{\pi}{3}$, $b = \frac{\pi}{2}$, and $r=f(\theta)=5\sin(3\theta)$.

Step2: Calculate $f(\frac{\pi}{3})$

Substitute $\theta=\frac{\pi}{3}$ into $r = 5\sin(3\theta)$: $f(\frac{\pi}{3})=5\sin(3\times\frac{\pi}{3})=5\sin(\pi)=0$.

Step3: Calculate $f(\frac{\pi}{2})$

Substitute $\theta=\frac{\pi}{2}$ into $r = 5\sin(3\theta)$: $f(\frac{\pi}{2})=5\sin(3\times\frac{\pi}{2})=5\times(- 1)=-5$.

Step4: Calculate the average rate of change

$\frac{f(\frac{\pi}{2})-f(\frac{\pi}{3})}{\frac{\pi}{2}-\frac{\pi}{3}}=\frac{-5 - 0}{\frac{\pi}{2}-\frac{\pi}{3}}=\frac{-5}{\frac{3\pi - 2\pi}{6}}=\frac{-5}{\frac{\pi}{6}}=-\frac{30}{\pi}$.

Step5: Find the halfway - point of the interval

The halfway - point of the interval $[\frac{\pi}{3},\frac{\pi}{2}]$ is $\theta_0=\frac{\frac{\pi}{3}+\frac{\pi}{2}}{2}=\frac{\frac{2\pi + 3\pi}{6}}{2}=\frac{5\pi}{12}$.

Step6: Estimate $f(\frac{5\pi}{12})$

$f(\frac{5\pi}{12})=5\sin(3\times\frac{5\pi}{12})=5\sin(\frac{5\pi}{4})=5\times(-\frac{\sqrt{2}}{2})=-\frac{5\sqrt{2}}{2}$.

Answer:

Average rate of change: $-\frac{30}{\pi}$; Value at halfway - point: $-\frac{5\sqrt{2}}{2}$