Chapter 11.3, Problem 66E

To determine

To determine: The correct answer which represents the slope of the polar curve $r=f\left(\theta \right)$ graphed in the $xy$ -plane.

Answer to Problem 66E

The correct answer is (D).

Explanation of Solution

Given information:

The options:

(A) $\frac{dr}{d\theta }$

(B) $\frac{dy}{d\theta }$

(C) $\frac{dx}{d\theta }$

(D) $\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}$

(E) $\frac{dy}{dx}\frac{dr}{d\theta }$

Calculation:

It is known that in a polar coordinate system, if $r=f\left(\theta \right)$ is a polar curve, the coordinates of a point $\left(x,y\right)$ can be written in a rectangular system as:

  $x=r\cos \theta =f\left(\theta \right)\sin \theta$

  $y=r\sin \theta =f\left(\theta \right)\sin \theta$

Also, it is known that polar curves lie in the xy -plane, this means the slope of the tangent line is equal to the slope of the polar curve that is, $\frac{dy}{dx}$ .

So, $\frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}$

Hence, the correct answer is (D).