Consider the parametric curve defined for by x(t)= t

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

Problem

Consider the parametric curve defined for

0\le t\le \pi

x(t)=\sin t, \qquad y(t)=\ln(t+2).

A point

P

on the curve has coordinates

(0,\ln 2)

. (a) Find all

t

–values in the given domain that correspond to point

P

. Show your work. (b) Compute

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

at each

t

found in part (a). (c) State whether the curve is concave up or concave down at

P

. Justify your answer using the value(s) obtained in part (b).