Consider the parametric equations and where is a constant.

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

Problem

Consider the parametric equations

x=a t^{2}

and

y=2a t^{3}

where

a>0

is a constant. (i) Obtain

\dfrac{dy}{dx}

in terms of

t

. (ii) Without eliminating the parameter, find

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

in terms of

t

and simplify your answer as far as possible. (iii) Now eliminate

t

to express the curve explicitly as

y

in terms of

x

, and then compute

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

by ordinary differentiation. (iv) Show that the results of parts (ii) and (iii) are identical, thereby confirming the correctness of the second–derivative formula for this curve.

Related theory

Advanced Differentiation
Parametric Functions
Implicit and Parametric Differentiation
Derivatives of parametric functions
Continuity and Differentiability

Browse the problem library