Let x(t)=e^ t

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

Problem

Let

x(t)=e^{t}, \qquad y(t)=e^{-t}+t, \qquad -2<t<3.

Assume the path is traced exactly once as

t

increases. (1) Show that

x

is a strictly increasing function of

t

on the open interval

(-2,3)

. What implication does this have for the relationship between extrema in

t

and extrema with respect to

x

? (2) Find all values of

t

(-2,3)

at which the particle’s

y

–coordinate attains a relative extremum with respect to

x

. Give exact values. (3) Classify each extremum obtained in part (b) as a relative minimum or a relative maximum of

y

with respect to

x

. Support your answer by examining the sign of the appropriate derivative on intervals of

t

. (4) State the coordinates

(x,y)

of each extremum, and interpret their locations in the context of the particle’s motion along the path.