6 multiple choice 1 point what is the slope of the line tangent to the polar curve r = 2θ at the point θ =…

Answer:

B. $-\frac{2}{\pi}$

Explanation:

Step1: Recall slope formula for polar curves

The slope of the tangent line to a polar curve $r = f(\theta)$ is given by $\frac{dy}{dx}=\frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}$.

Step2: Differentiate $r$ with respect to $\theta$

Given $r = 2\theta$, then $\frac{dr}{d\theta}=2$.

Step3: Substitute $\theta=\frac{\pi}{2}$, $r = 2\theta$ and $\frac{dr}{d\theta}=2$ into the slope - formula

When $\theta=\frac{\pi}{2}$, $r = 2\times\frac{\pi}{2}=\pi$. Substitute into $\frac{dy}{dx}=\frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}$: $\frac{dy}{dx}=\frac{2\times\sin\frac{\pi}{2}+\pi\times\cos\frac{\pi}{2}}{2\times\cos\frac{\pi}{2}-\pi\times\sin\frac{\pi}{2}}$.

Step4: Evaluate the trigonometric values

We know that $\sin\frac{\pi}{2} = 1$ and $\cos\frac{\pi}{2}=0$. So $\frac{dy}{dx}=\frac{2\times1+\pi\times0}{2\times0 - \pi\times1}=\frac{2}{-\pi}=-\frac{2}{\pi}$.