question 7(multiple choice worth 1 points) (r...
question 7(multiple choice worth 1 points) (rates of change in polar functions mc) determine the average rate of change for the polar function r = f(θ) = 5 sin θ on the interval 7π/6, 5π/3. -5√3 + 5/π 5√3 - 5/π 10√3 - 10/π -10√3 + 10/π
Answer
# 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 $\frac{f(b)-f(a)}{b - a}$. For a polar function $r = f(\theta)$ over the interval $[\alpha,\beta]$, the average rate of change is $\frac{f(\beta)-f(\alpha)}{\beta-\alpha}$. Here, $\alpha=\frac{7\pi}{6}$, $\beta = \frac{5\pi}{3}$, and $f(\theta)=5\sin\theta$.
## Step2: Calculate $f(\beta)$ and $f(\alpha)$
First, find $f(\frac{5\pi}{3})$:
\[
\begin{align*}
f\left(\frac{5\pi}{3}\right)&=5\sin\left(\frac{5\pi}{3}\right)\\
&=5\times\left(-\frac{\sqrt{3}}{2}\right)\\
&=-\frac{5\sqrt{3}}{2}
\end{align*}
\]
Next, find $f(\frac{7\pi}{6})$:
\[
\begin{align*}
f\left(\frac{7\pi}{6}\right)&=5\sin\left(\frac{7\pi}{6}\right)\\
&=5\times\left(-\frac{1}{2}\right)\\
&=-\frac{5}{2}
\end{align*}
\]
## Step3: Calculate the average rate of change
\[
\begin{align*}
\frac{f\left(\frac{5\pi}{3}\right)-f\left(\frac{7\pi}{6}\right)}{\frac{5\pi}{3}-\frac{7\pi}{6}}&=\frac{-\frac{5\sqrt{3}}{2}-\left(-\frac{5}{2}\right)}{\frac{10\pi - 7\pi}{6}}\\
&=\frac{-\frac{5\sqrt{3}}{2}+\frac{5}{2}}{\frac{3\pi}{6}}\\
&=\frac{\frac{- 5\sqrt{3}+5}{2}}{\frac{\pi}{2}}\\
&=\frac{-5\sqrt{3}+5}{\pi}
\end{align*}
\]
# Answer:
A. $\frac{-5\sqrt{3}+5}{\pi}$