A curve is originally described by the parameter with equations and

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

Problem

A curve is originally described by the parameter

u\ge 0

with equations

x(u)=u^{2}

and

y(u)=u^{3}

. A second description of the same curve is proposed using the parameter

s

defined implicitly by

u=\sqrt{s}

, giving

x(s)=s

and

y(s)=s^{3/2}

for

s\ge 0

. (a) Compute the speed of the particle when it is described by the parameter

u

. (b) Compute the speed of the particle when it is described by the parameter

s

. (c) Determine whether

s

measures arc length along the curve. Justify your conclusion using results from parts (a) and (b). (d) Find an explicit re-parameterization that does measure arc length starting from the point

(0,0)

, and state the resulting expressions for

x

and

y

in terms of your new parameter.

Related theory

Vector-Valued Functions
Parametric Functions

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