find the average rate of change for the function r = 13 on the interval π/6, π/5. then estimate the value of…

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, the function is $r(\theta)=13$ (a constant function), $a=\frac{\pi}{6}$, and $b = \frac{\pi}{5}$.

Step2: Calculate the average rate of change

Since $r(\theta)$ is a constant function, $r(\frac{\pi}{5})=13$ and $r(\frac{\pi}{6}) = 13$. Then the average rate of change is $\frac{r(\frac{\pi}{5})-r(\frac{\pi}{6})}{\frac{\pi}{5}-\frac{\pi}{6}}=\frac{13 - 13}{\frac{\pi}{5}-\frac{\pi}{6}}=0$.

Step3: Find the halfway - point of the interval

The halfway - point of the interval $[\frac{\pi}{6},\frac{\pi}{5}]$ is $\theta_0=\frac{\frac{\pi}{6}+\frac{\pi}{5}}{2}=\frac{\frac{5\pi + 6\pi}{30}}{2}=\frac{11\pi}{60}$.

Step4: Evaluate the function at the halfway - point

Since $r(\theta)=13$ for all $\theta$, $r(\frac{11\pi}{60}) = 13$.

Answer:

The average rate of change is $0$ and the value of $r(\theta)$ at the halfway point of the interval is $13$.