Let be a positive constant

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Let

k

be a positive constant. The variables

x

and

y

are related by

\tan y = kx,

where

0<y<\dfrac{\pi}{2}

. (a) Without substituting a specific value for

k

, explain why

\dfrac{dy}{dx}>0

for every point of the curve in the given interval for

y

. Your explanation should reference the behavior of both sides of the defining equation. (b) Derive an inequality that describes how the magnitude of

\dfrac{dy}{dx}

changes as

y

increases from

0

\dfrac{\pi}{2}

. Provide a clear verbal interpretation of this result.