The function is defined on each open interval that avoids its vertical asymptotes

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

The function

p(x)=\cot x

is defined on each open interval that avoids its vertical asymptotes. Focus on the interval

\left(0,\pi\right)

. (1) Without explicit calculation of

p'(x)

, explain why the derivative of

p

must be negative throughout

(0,\pi)

. Base your reasoning on the geometric behavior of

p

. (2) Describe how the slope of the graph of

p

changes as

x

approaches each endpoint of the interval

(0,\pi)

. Connect your description to the limiting behavior of

p'(x)

near the vertical asymptotes.