question a polar function is given by r = f(θ) = cos(7/4θ)+7. as θ increases on the interval 8π/7 < θ <…

question a polar function is given by r = f(θ) = cos(7/4θ)+7. as θ increases on the interval 8π/7 < θ < 10π/7, which of the following is true about the points of the graph of r = f(θ) on the xy - plane? answer attempt 2 out of 2 the points are negative because they lie below the x - axis and are decreasing because on the xy - plane from left to right, the graph is going down.

question a polar function is given by r = f(θ) = cos(7/4θ)+7. as θ increases on the interval 8π/7 < θ < 10π/7, which of the following is true about the points of the graph of r = f(θ) on the xy - plane? answer attempt 2 out of 2 the points are negative because they lie below the x - axis and are decreasing because on the xy - plane from left to right, the graph is going down.

Answer

Explanation:

Step1: Analyze the range of $\theta$

Let $t = \frac{7}{4}\theta$. When $\theta=\frac{8\pi}{7}$, $t=\frac{7}{4}\times\frac{8\pi}{7} = 2\pi$. When $\theta=\frac{10\pi}{7}$, $t=\frac{7}{4}\times\frac{10\pi}{7}=\frac{5\pi}{2}$.

Step2: Analyze the cosine - function

The function $r=\cos(\frac{7}{4}\theta)+7=\cos(t) + 7$. The range of $y = \cos(t)$ for $2\pi<t<\frac{5\pi}{2}$ is $[ - 1,1]$, and $r=\cos(t)+7$ has a range of $[6,8]$. In polar - coordinates, $x = r\cos\theta$ and $y = r\sin\theta$. We know that for $\frac{8\pi}{7}<\theta<\frac{10\pi}{7}$, $\sin\theta<0$. Since $r=\cos(\frac{7}{4}\theta)+7>0$, then $y = r\sin\theta<0$. So the points lie below the $x$ - axis. To check if it is increasing or decreasing, we find the derivative of $r$ with respect to $\theta$. Using the chain - rule, $r'=-\frac{7}{4}\sin(\frac{7}{4}\theta)$. When $\frac{8\pi}{7}<\theta<\frac{10\pi}{7}$, $\frac{7}{4}\theta\in(2\pi,\frac{5\pi}{2})$. $\sin(\frac{7}{4}\theta)>0$ in this sub - interval, so $r'=-\frac{7}{4}\sin(\frac{7}{4}\theta)<0$, which means $r$ is decreasing as $\theta$ increases in the given interval.

Answer:

The points are negative because they lie below the x - axis and are decreasing because on the xy - plane from left to right, the graph is going down.