A local movie theater with a 30-foot-high screen that is 10 feet above a person’s eye level when seated has a viewing angle $\theta$ $\theta=\cot ^{-1} \frac{x}{40}-\cot ^{-1} \frac{x}{10},$ where $x$ is the distance in feet away from the movie screen that the person is sitting, as shown in the following figure.
a. Find $\frac{d \theta}{d x}$
b. Evaluate $\frac{d \theta}{d x}$ for $x=5,10,15,$ and $20.$
c. Interpret the results in $\mathrm{b}$.
d. Evaluate $\frac{d \theta}{d x}$ for $x=25,30,35,$ and 40
e. Interpret the results in d. At what distance $x$ should the person stand to maximize his or her viewing angle?
Derivatives
Derivatives of Inverse Functions