Consider the circle of radius represented by the parametric equations and for .

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

Problem

Consider the circle of radius

a>0

represented by the parametric equations

x(t)=a\cos t

and

y(t)=a\sin t

for

0\le t\le 2\pi

. (a) Show that the particle’s speed has constant magnitude and determine its value. (b) Compute the arc-length function

s(t)

measured from

t=0

to a general parameter value

t

. (c) Solve the result from part (b) for

t

as a function of

s

on the interval

0\le s\le 2\pi a

. (d) Express the coordinates of the particle as functions

x(s)

and

y(s)

of the arc-length parameter

s

. (e) Using your result in part (d), state the total range of the parameter

s

that traces the entire circle once and verify that the speed with respect to

s

is identically

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