question 2(multiple choice worth 1 points) (r...
question 2(multiple choice worth 1 points) (rates of change in polar functions mc) the table gives selected values from a polar function r = f(θ) on the polar coordinate system. θ 5π/6 π 5π/4 r = f(θ) 8 0 -8√2 if f(π) is estimated using the average rate of change of the function on 5π/6, 5π/4, which of the following is true? the estimated value overestimated the actual value by approximately 0.172. the estimated value overestimated the actual value by approximately 0.275. the estimated value underestimated the actual value by approximately 0.172. the estimated value underestimated the actual value by approximately 0.275.
Answer
# 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 $\frac{f(b)-f(a)}{b - a}$. For the polar function $r = f(\theta)$ on the interval $\left[\frac{5\pi}{6},\frac{5\pi}{4}\right]$, the average rate of change is $\frac{f\left(\frac{5\pi}{4}\right)-f\left(\frac{5\pi}{6}\right)}{\frac{5\pi}{4}-\frac{5\pi}{6}}$.
We know that $f\left(\frac{5\pi}{6}\right)=8$, $f\left(\frac{5\pi}{4}\right)=- 8\sqrt{2}$.
First, calculate the denominator: $\frac{5\pi}{4}-\frac{5\pi}{6}=\frac{15\pi - 10\pi}{12}=\frac{5\pi}{12}$.
The numerator is $-8\sqrt{2}-8=-8(\sqrt{2} + 1)$.
So the average rate of change $m=\frac{-8(\sqrt{2}+1)}{\frac{5\pi}{12}}=\frac{-96(\sqrt{2}+1)}{5\pi}$.
## Step2: Use linear - approximation formula
The linear - approximation formula is $L(\theta)=f(a)+m(\theta - a)$. Let $a=\frac{5\pi}{6}$ and $\theta=\pi$. Then $\theta - a=\pi-\frac{5\pi}{6}=\frac{\pi}{6}$.
$L(\pi)=f\left(\frac{5\pi}{6}\right)+m\left(\pi-\frac{5\pi}{6}\right)$.
Substitute $f\left(\frac{5\pi}{6}\right)=8$ and $m = \frac{-96(\sqrt{2}+1)}{5\pi}$ into the formula:
$L(\pi)=8+\frac{-96(\sqrt{2}+1)}{5\pi}\times\frac{\pi}{6}=8-\frac{16(\sqrt{2}+1)}{5}$.
Calculate $8-\frac{16(\sqrt{2}+1)}{5}=8-\frac{16\sqrt{2}+16}{5}=\frac{40-(16\sqrt{2}+16)}{5}=\frac{24 - 16\sqrt{2}}{5}\approx\frac{24-16\times1.414}{5}=\frac{24 - 22.624}{5}=\frac{1.376}{5}=0.2752$.
The actual value of $f(\pi) = 0$. Since the estimated value $L(\pi)\approx0.275$ and the actual value is $0$, the estimated value overestimated the actual value by approximately $0.275$.
# Answer:
The estimated value overestimated the actual value by approximately 0.275.