3 mark for review consider the graph of the polar function r = f(θ), where f(θ) = 2 sin θ - 1, in the polar…

3 mark for review consider the graph of the polar function r = f(θ), where f(θ) = 2 sin θ - 1, in the polar coordinate system. which of the following descriptions is true? a as θ increases from 0 to π/6, the polar function r = f(θ) is increasing, and the distance between the point (f(θ),θ) on the curve and the origin is increasing. b as θ increases from 0 to π/6, the polar function r = f(θ) is increasing, and the distance between the point (f(θ),θ) on the curve and the origin is decreasing. c as θ increases from 0 to π/6, the polar function r = f(θ) is decreasing, and the distance between the point (f(θ),θ) on the curve and the origin is increasing. d as θ increases from 0 to π/6, the polar function r = f(θ) is decreasing, and the distance between the point (f(θ),θ) on the curve and the origin is decreasing.

3 mark for review consider the graph of the polar function r = f(θ), where f(θ) = 2 sin θ - 1, in the polar coordinate system. which of the following descriptions is true? a as θ increases from 0 to π/6, the polar function r = f(θ) is increasing, and the distance between the point (f(θ),θ) on the curve and the origin is increasing. b as θ increases from 0 to π/6, the polar function r = f(θ) is increasing, and the distance between the point (f(θ),θ) on the curve and the origin is decreasing. c as θ increases from 0 to π/6, the polar function r = f(θ) is decreasing, and the distance between the point (f(θ),θ) on the curve and the origin is increasing. d as θ increases from 0 to π/6, the polar function r = f(θ) is decreasing, and the distance between the point (f(θ),θ) on the curve and the origin is decreasing.

Answer

Explanation:

Step1: Find the derivative of $r = f(\theta)$

We have $r=f(\theta)=2\sin\theta - 1$. Using the derivative formula for $\sin\theta$ ($\frac{d}{d\theta}\sin\theta=\cos\theta$), the derivative $r'=f'(\theta)=2\cos\theta$.

Step2: Analyze the sign of the derivative on the interval $\theta\in[0,\frac{\pi}{6}]$

When $\theta\in[0,\frac{\pi}{6}]$, $\cos\theta> 0$. So $r' = 2\cos\theta>0$. This means the function $r = f(\theta)$ is increasing on the interval $[0,\frac{\pi}{6}]$.

Step3: Understand the meaning of $r$ in polar - coordinates

In polar coordinates, the distance between the point $(r,\theta)=(f(\theta),\theta)$ on the curve and the origin is given by $|r|$. Since $r = 2\sin\theta - 1$ and it is increasing on $[0,\frac{\pi}{6}]$, the distance between the point $(f(\theta),\theta)$ on the curve and the origin is increasing.

Answer:

A. As $\theta$ increases from $0$ to $\frac{\pi}{6}$, the polar function $r = f(\theta)$ is increasing, and the distance between the point $(f(\theta),\theta)$ on the curve and the origin is increasing.