question 6(multiple choice worth 1 points) (r...
question 6(multiple choice worth 1 points) (rates of change in polar functions mc) consider the graph of the polar function r = f(θ) = 4 cos θ + 4, in the polar coordinate system. which of the following descriptions is true? as θ increases from 7π/6 to 3π/2, the polar function r = f(θ) increases and the distance between the point with polar coordinates (f(θ), θ) and the origin increases. as θ increases from 7π/6 to 3π/2, the polar function r = f(θ) increases and the distance between the point with polar coordinates (f(θ), θ) and the origin decreases. as θ increases from 7π/6 to 3π/2, the polar function r = f(θ) decreases and the distance between the point with polar coordinates (f(θ), θ) and the origin increases. as θ increases from 7π/6 to 3π/2, the polar function r = f(θ) decreases and the distance between the point with polar coordinates (f(θ), θ) and the origin decreases.
Answer
# Explanation:
## Step1: Analyze the polar - function
The polar function is \(r = f(\theta)=4\cos\theta + 4\).
## Step2: Find the derivative of \(r\) with respect to \(\theta\)
We know that \(\frac{dr}{d\theta}=-4\sin\theta\).
## Step3: Evaluate the derivative in the interval \(\left[\frac{7\pi}{6},\frac{3\pi}{2}\right]\)
When \(\theta\in\left[\frac{7\pi}{6},\frac{3\pi}{2}\right]\), \(\sin\theta\lt0\), so \(\frac{dr}{d\theta}=- 4\sin\theta>0\). This means the function \(r = f(\theta)\) is increasing in the interval \(\left[\frac{7\pi}{6},\frac{3\pi}{2}\right]\).
## Step4: Recall the meaning of \(r\) in polar - coordinates
In polar coordinates, the distance between the point \((r,\theta)\) and the origin is given by \(r\). Since \(r = f(\theta)\) is increasing in the interval \(\left[\frac{7\pi}{6},\frac{3\pi}{2}\right]\), the distance between the point \((f(\theta),\theta)\) and the origin is increasing.
# Answer:
As \(\theta\) increases from \(\frac{7\pi}{6}\) to \(\frac{3\pi}{2}\), the polar function \(r = f(\theta)\) increases and the distance between the point with polar coordinates \((f(\theta),\theta)\) and the origin increases.