A plane curve is defined by the differentiable parametric equations and

MATHEMATICS · Subjective · A Level · Mathematics · Differentiation · Differentiation

Problem

A plane curve is defined by the differentiable parametric equations

x=f(t)

and

y=g(t)

. (i) Explain the conceptual difference between

\dfrac{dy}{dt}

and

\dfrac{dy}{dx}

. (ii) State the precise conditions, in terms of

\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

, under which the tangent to the curve at a point is (a) horizontal, and (b) vertical. (iii) Give a brief justification to show why the simultaneous vanishing of

\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

at the same value of

t

implies that the direction of the tangent at the corresponding point in the

xy

–plane is not uniquely defined.

Related theory

Advanced Differentiation
Implicit and parametric differentiation
Derivatives of parametric functions
Continuity and Differentiability

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