A curve is defined parametrically by and on the interval on and

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

Problem

A curve is defined parametrically by

x(t)

and

y(t)

on the interval

0<t<3

. Suppose

\dfrac{dy}{dx}>0

0<t<1

and

2<t<3

, but

\dfrac{dy}{dx}<0

1<t<2

. Further,

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

is negative on

0<t<2

and positive on

2<t<3

. Which statement below correctly describes the concavity of the curve in terms of

x

on the interval

0<t<3

? (a) Because

\dfrac{dy}{dx}

changes sign at

t=1

, the curve is concave down for

x

corresponding to

0<t<1

and concave up for

x

corresponding to

1<t<3

. (b) The curve is concave down for

x

corresponding to

0<t<2

and concave up for

x

corresponding to

2<t<3

. A point of inflection occurs where

t=2

. (c) The curve is concave up for

x

corresponding to

0<t<2

and concave down for

x

corresponding to

2<t<3

. A point of inflection occurs where

t=2

. (d) Concavity cannot be determined from the given information because

\dfrac{dy}{dx}

changes sign.