A curve is defined parametrically by and

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A curve is defined parametrically by

x=t^{2}+1

and

y=t^{3}-3t

, where

t\in\mathbb R

. (i) Determine all real values of

t

at which the tangent to the curve is horizontal. (ii) Determine all real values of

t

at which the tangent to the curve is vertical. (iii) List the Cartesian points corresponding to all the

t

obtained in parts (i) and (ii). State whether the curve possesses any point where the tangent could be both horizontal and vertical, giving a brief reason for your answer.

Related theory

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

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