For and , , define…

MATHEMATICS · Multi-part · ISC · Mathematics · Calculus · Calculus

Problem

For

x\neq0

and

x\neq\dfrac{\pi}{2}+n\pi

n\in\mathbb{Z}

, define

p(x)=\dfrac{\tan x}{x^{2}}

. (i) Obtain the derivative

p'(x)

and write it in its simplest form. (ii) Calculate

p'\!\left(\dfrac{\pi}{6}\right)

and

p'\!\left(\dfrac{\pi}{4}\right)

, leaving each answer in exact form. (iii) Between

0

and

\dfrac{\pi}{2}

, at which of the two points considered in part (ii) is

p(x)

changing more rapidly? Give a brief justification based on the numerical values you have obtained.