question 5(multiple choice worth 1 points) (r...
question 5(multiple choice worth 1 points) (rates of change in polar functions mc) the table gives selected arocs of a polar function r = f(θ). interval π/4, π/3 π/3, π/2 2π/3, 5π/6 5π/6, π aroc -3.164 -3.820 -2.796 -1.023 given that f(2π/3)=2 and f(3π/4)=1.172, if f(3π/4) is estimated using the values from the table, which of the following is the correct estimated value for f(3π/4)? 1.268 0.096 -0.096 -1.268
Answer
# Explanation:
## Step1: Recall average - rate - of - change formula
The average rate of change (AROC) formula is $\text{AROC}=\frac{f(b)-f(a)}{b - a}$. We want to estimate $f(\frac{3\pi}{4})$ given $f(\frac{2\pi}{3}) = 2$ and we need to use the appropriate AROC from the table. The interval that contains $\frac{2\pi}{3}$ and $\frac{3\pi}{4}$ is $[\frac{2\pi}{3},\frac{5\pi}{6}]$ with an AROC of $- 2.796$. Here $a=\frac{2\pi}{3}$, $b = \frac{3\pi}{4}$, and we know $f(a)=2$.
## Step2: Rearrange the AROC formula
From $\text{AROC}=\frac{f(b)-f(a)}{b - a}$, we can solve for $f(b)$: $f(b)=f(a)+\text{AROC}\times(b - a)$. First, calculate $b - a=\frac{3\pi}{4}-\frac{2\pi}{3}=\frac{9\pi - 8\pi}{12}=\frac{\pi}{12}$.
## Step3: Substitute values
Substitute $f(a) = 2$, $\text{AROC}=-2.796$, and $b - a=\frac{\pi}{12}$ into the formula $f(b)=f(a)+\text{AROC}\times(b - a)$. $f(\frac{3\pi}{4})=2+( - 2.796)\times\frac{\pi}{12}$.
\[
\begin{align*}
f(\frac{3\pi}{4})&=2-2.796\times\frac{\pi}{12}\\
&=2 - 2.796\times\frac{3.14159}{12}\\
&=2-2.796\times0.2618\\
&=2 - 0.732\\
&=1.268
\end{align*}
\]
# Answer:
A. 1.268