Mathematics

Problem

A curve is defined by the parametric equations

x=te^{t}

and

y=\ln t+t^{2}

, where

t>0

. a. (i) Find

\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

. b. (ii) Hence obtain a simplified expression for

\dfrac{dy}{dx}

in terms of

t

. c. (iii) Evaluate

\dfrac{dy}{dx}

t=1

. d. (iv) Using the value from part (iii), state whether the curve is rising or falling with respect to

x

at the point corresponding to

t=1

. Justify your answer briefly.