Find the function with the given derivative whose graph passes through the point P.
-$$\mathrm { r } ^ { \prime } ( \theta ) = 4 + \sec ^ { 2 } \theta , \mathrm { P } ( \pi , 0 )$$
A) $r ( \theta ) = 4 \theta + \frac { 1 } { 3 } \sec ^ { 3 } \theta$
B) $r ( \theta ) = 2 \theta ^ { 2 } + \tan \theta - 4 \pi$
C) $r ( \theta ) = 4 \theta + \tan \theta - 4 \pi$
D) $\mathrm { r } ^ { \prime } ( \theta ) = 2 \theta ^ { 2 } + \tan \theta + 4 \pi$

Question

Quizplus · Accepted Answer

We need to integrate the given derivative to find the function.

$$\int (4+\sec^2 \theta) d\theta = 4\theta + \tan \theta + C$$

Now we need to use the given point P to find the value of C.

At $\theta = \pi$, $r(\pi) = 4\pi + \tan \pi + C = 4\pi + 0 + C = 4\pi$

Therefore, $C=0$ and the function is

$$r(\theta) = 4\theta + \tan \theta - 4\pi$$

So the answer is choice C.

Find the Function with the Given Derivative Whose Graph Passes r′(θ)=4+sec⁡2θ,P(π,0)\mathrm { r } ^ { \prime } ( \theta ) = 4 + \sec ^ { 2 } \theta , \mathrm { P } ( \pi , 0 )r′(θ)=4+sec2θ,P(π,0)

Find the Function with the Given Derivative Whose Graph Passes $\mathrm { r } ^ { \prime } ( \theta ) = 4 + \sec ^ { 2 } \theta , \mathrm { P } ( \pi , 0 )$