For a parametric curve given by and

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

For a parametric curve given by

x=\phi(t)

and

y=\psi(t)

, which of the following conditions must hold at

t=t_{0}

for

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

to be defined at that point? (a)

\dfrac{d^{2}\psi}{dt^{2}}\bigl|_{t_{0}}\neq 0

(b)

\dfrac{d\psi}{dt}\bigl|_{t_{0}}\neq 0

(c)

\dfrac{d\phi}{dt}\bigl|_{t_{0}}\neq 0

(d)

\dfrac{d\phi}{dt}\bigl|_{t_{0}}\neq 0 \text{ and } \dfrac{d\psi}{dt}\bigl|_{t_{0}}\neq 0