For , a particle follows the path described by the polar equation while its angular…

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

Problem

For

0\le t\le 2

, a particle follows the path described by the polar equation

r=\theta^{2}+1

while its angular position is

\theta(t)=t^{2}

. (a) Find the Cartesian coordinates of the particle at

t=1

. (b) Derive the particle’s velocity vector in Cartesian coordinates as a function of

t

. (c) Obtain an explicit expression for the speed

v(t)

, and verify that it is always positive on the given interval. (d) Determine the value of

t

0\le t\le2

at which the particle attains its maximum speed. Give a complete justification, including any necessary derivative tests.