Find the function with the given derivative whose graph passes through the point P.
-$\mathrm { r } ^ { \prime } ( \theta ) = 5 - \csc ^ { 2 } \theta , \mathrm { P } \left( \frac { \pi } { 4 } , 0 \right)$
A) $r ( \theta ) = 5 \theta + \cot \theta - \frac { 5 } { 4 } \pi - 1$
B) $r ( \theta ) = 5 \theta - \cot \theta - \frac { 5 } { 4 } \pi + 2$
C) $r ( \theta ) = 5 \theta - \cot \theta - \frac { 5 } { 4 } \pi - 2$
D) $r ( \theta ) = 5 \theta + \cot \theta - \frac { 5 } { 4 } \pi + 1$

Question

Quizplus · Accepted Answer

To find the function $r(\theta)$ given its derivative $r'(\theta) = 5 - \csc^2\theta$, we integrate the derivative. The integral of $5$ with respect to $\theta$ is $5\theta$, and the integral of $-\csc^2\theta$ is $+\cot\theta$, so the general form of $r(\theta)$ is $r(\theta) = 5\theta + \cot\theta + C$, where $C$ is the constant of integration. To find $C$, we use the point $P\left(\frac{\pi}{4}, 0\right)$, which gives us the equation $0 = 5\left(\frac{\pi}{4}\right) + \cot\left(\frac{\pi}{4}\right) + C$. Since $\cot\left(\frac{\pi}{4}\right) = 1$, we have $0 = \frac{5\pi}{4} + 1 + C$, solving for $C$ gives $C = -\frac{5\pi}{4} - 1$. Thus, the function is $r(\theta) = 5\theta + \cot\theta - \frac{5\pi}{4} - 1$.

Find the Function with the Given Derivative Whose Graph Passes r′(θ)=5−csc⁡2θ,P(π4,0)\mathrm { r } ^ { \prime } ( \theta ) = 5 - \csc ^ { 2 } \theta , \mathrm { P } \left( \frac { \pi } { 4 } , 0 \right)r′(θ)=5−csc2θ,P(4π​,0)

Find the Function with the Given Derivative Whose Graph Passes $\mathrm { r } ^ { \prime } ( \theta ) = 5 - \csc ^ { 2 } \theta , \mathrm { P } \left( \frac { \pi } { 4 } , 0 \right)$