consider the graph of the polar function r = f(θ), where f(θ)=θ(θ - 2)(θ - 4), in the polar coordinate…

consider the graph of the polar function r = f(θ), where f(θ)=θ(θ - 2)(θ - 4), in the polar coordinate system for 0 ≤ θ ≤ 4. which of the following statements is true? a on the interval 2 < θ < 2.1, the distance between (f(θ),θ) and the origin is increasing because the values of f(θ) are negative and decreasing. b on the interval 2 < θ < 2.1, the distance between (f(θ),θ) and the origin is decreasing because the values of f(θ) are negative and decreasing. c on the interval 2 < θ < 2.1, the distance between (f(θ),θ) and the origin is increasing because the values of f(θ) are negative and increasing. d on the interval 2 < θ < 2.1, the distance between (f(θ),θ) and the origin is decreasing because the values of f(θ) are negative and increasing.

consider the graph of the polar function r = f(θ), where f(θ)=θ(θ - 2)(θ - 4), in the polar coordinate system for 0 ≤ θ ≤ 4. which of the following statements is true? a on the interval 2 < θ < 2.1, the distance between (f(θ),θ) and the origin is increasing because the values of f(θ) are negative and decreasing. b on the interval 2 < θ < 2.1, the distance between (f(θ),θ) and the origin is decreasing because the values of f(θ) are negative and decreasing. c on the interval 2 < θ < 2.1, the distance between (f(θ),θ) and the origin is increasing because the values of f(θ) are negative and increasing. d on the interval 2 < θ < 2.1, the distance between (f(θ),θ) and the origin is decreasing because the values of f(θ) are negative and increasing.

Answer

Explanation:

Step1: Recall polar - coordinate concept

In polar coordinates, the distance between a point $(r,\theta)$ and the origin is given by $|r|$. Here $r = f(\theta)=\theta(\theta - 2)(\theta - 4)=\theta^3-6\theta^2 + 8\theta$.

Step2: Analyze the sign and behavior of $f(\theta)$ on the interval $(2,2.1)$

First, find the derivative of $f(\theta)$ using the power - rule. $f'(\theta)=3\theta^2-12\theta + 8$. Evaluate $f(\theta)$ at $\theta = 2$ and $\theta=2.1$. $f(2)=2\times(2 - 2)\times(2 - 4)=0$. $f(2.1)=2.1\times(2.1 - 2)\times(2.1 - 4)=2.1\times0.1\times(-1.9)=- 0.399$. To check if $f(\theta)$ is increasing or decreasing on $(2,2.1)$, evaluate $f'(\theta)$ at a value in the interval, say $\theta = 2.05$. $f'(2.05)=3\times(2.05)^2-12\times2.05 + 8=3\times4.2025-24.6 + 8=12.6075-24.6 + 8=-4.9925<0$. So $f(\theta)$ is decreasing on $(2,2.1)$. The distance between the point $(f(\theta),\theta)$ and the origin is $|f(\theta)|$. Since $f(\theta)<0$ on $(2,2.1)$ and $f(\theta)$ is decreasing (becoming more negative), $|f(\theta)|$ is increasing.

Answer:

A. On the interval $2<\theta<2.1$, the distance between $(f(\theta),\theta)$ and the origin is increasing because the values of $f(\theta)$ are negative and decreasing.