Consider the curve defined parametrically by and for .

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

Problem

Consider the curve defined parametrically by

x=e^{3t}

and

y=\ln(4t)

for

t>0

. (i) Show that

\dfrac{dx}{dt}\neq 0

for all

t>0

. (ii) Explain why the result in part (i) guarantees the existence of

\dfrac{dy}{dx}

for the curve for all

t>0

. (iii) Suppose that

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

is evaluated at

t=2

. State whether knowing only the sign of

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

at that point is sufficient to determine the concavity of the curve there. Give a brief reason.