A smooth curve in the plane is represented by the parametric equations and

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A smooth curve in the plane is represented by the parametric equations

x=f(t)

and

y=g(t)

, where both

f

and

g

possess continuous second derivatives on an interval

I

. Consider an arbitrary point on the curve corresponding to

t=t_0

. (i) State the condition, expressed in terms of derivatives with respect to

x

, that must be satisfied at

t=t_0

for the point to be considered a possible point of inflection. (ii) Explain why the additional requirement involving the sign of

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

on either side of

t_0

is necessary in deciding whether the point is actually an inflection point. (iii) Provide one illustrative example (without complete calculation) of how neglecting the requirement in part (ii) could lead to an incorrect conclusion about the presence of an inflection point.

Related theory

Curve Analysis

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