Consider 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 parametric equations

x(\theta)=\sin\theta+\theta

and

y(\theta)=\cos\theta-\theta

for

0<\theta<2\pi

. (1) Determine all values of

\theta

in the interval

(0,2\pi)

where the tangent line to the curve is horizontal. Show all supporting work. (2) At

\theta=\frac{\pi}{3}

, compute the slope of the curve and decide whether the curve is increasing or decreasing with respect to

x

at that point. Clearly explain each step of your reasoning.