A smooth curve is defined parametrically by differentiable functions and on an open…

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

Problem

A smooth curve

C

is defined parametrically by differentiable functions

x=f(t)

and

y=g(t)

on an open interval containing

t=a

. Assume that

\dfrac{dx}{dt}\big|_{t=a}=0

while

\dfrac{dy}{dt}\big|_{t=a}\neq 0

, and that

\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

are not simultaneously zero at any parameter value. Answer each part, providing full justification in complete sentences. (a) Describe the orientation of the tangent line to

C

at the point

(f(a),g(a))

. Explain your reasoning using the information given about the derivatives. (b) State and justify the condition, expressed in terms of

\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

, that guarantees the tangent line to

C

is horizontal at a parameter value

t=b

. (c) Both derivatives occasionally vanish at the same parameter value. Explain why additional analysis is required to classify the tangent line when

\dfrac{dx}{dt}=\dfrac{dy}{dt}=0

. Give one example of a possible geometric outcome in this situation.

Related theory

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