A curve is given by and where .

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

Problem

A curve is given by

x=e^{t}+t

and

y=\ln(5t^{2}+1)

where

t>0

. (i) Find

\dfrac{dy}{dx}

in terms of

t

. (ii) Obtain

\dfrac{d^{2}y}{dx^{2}}

in its simplest form. (iii) Evaluate

\dfrac{d^{2}y}{dx^{2}}

t=1

and hence comment on the concavity of the curve at the corresponding point.