Mathematics

Problem

A circle of radius

r

centered at the origin is described by

x^{2}+y^{2}=r^{2}

. Consider only the portion of the circle lying in the first quadrant (

x>0

y>0

). a. Find

\dfrac{dy}{dx}

in terms of

x

and

y

, and simplify your result. b. Describe qualitatively how the magnitude of

\dfrac{dy}{dx}

changes as a point moves along the arc from

(r,0)

toward

(0,r)

. Support your description with both algebraic reasoning and geometric insight. c. Explain why

\dfrac{dy}{dx}

approaches

-\infty

as the point approaches

(0,r)

. Relate this limiting behavior to the geometry of the circle.