Rates of Change in Polar Functions Practice Test

A marine radar system tracks a vessel whose path is modeled by the polar function $r(\theta)=15-3\cos(\theta)$, where $r$ (nautical miles) is the vessel’s distance from the radar and $\theta$ (radians) is the bearing angle. The derivative $\dfrac{dr}{d\theta}$ gives the instantaneous change in distance per radian as the bearing increases. Using $\dfrac{d}{d\theta}\cos\theta=-\sin\theta$, differentiate to find how quickly the vessel’s distance changes at a specific bearing.

Given $r(\theta)=15-3\cos(\theta)$, what is the rate of change of $r$ with respect to $\theta$ at $\theta=\dfrac{\pi}{4}$?​

A robotics lab uses a rotating sensor to track a small rover whose planned path is modeled by $r(\theta)=5-2\sin(\theta)$, where $r$ (in meters) is the rover’s distance from the sensor and $\theta$ (in radians) is the sensor’s angle. The derivative $\dfrac{dr}{d\theta}$ represents the rate of change of the rover’s distance per radian as the angle changes, found by differentiating the polar function with respect to $\theta$. Using $\dfrac{d}{d\theta}(\sin\theta)=\cos\theta$, the lab can determine whether the rover is moving inward or outward relative to the sensor as the scan angle increases. This helps tune the tracking algorithm so it can respond appropriately to rapid radial changes.

Given $r(\theta)=5-2\sin(\theta)$, what is $\dfrac{dr}{d\theta}$ at $\theta=\pi/4$?