Let and define a smooth planar curve

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Let

x=f(t)

and

y=g(t)

define a smooth planar curve. Assume

f

and

g

are differentiable on an open interval containing

t=t_{0}

, with

g'(t_{0})\ne 0

and

f'(t_{0})=0

. (a) Explain why the curve possesses a vertical tangent line at the point corresponding to

t=t_{0}

. (b) Derive an expression for the equation of this vertical tangent line in terms of

f

g

, and

t_{0}

, and justify each step of your derivation. (c) Describe how your argument would change if instead

g'(t_{0})=0

and

f'(t_{0})\ne 0

, and state the geometric consequence for the curve at

t=t_{0}

Related theory

Advanced Differentiation
Implicit and Parametric Differentiation
Derivatives of parametric functions
Continuity and Differentiability
Higher-Order Derivatives

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