A circle of radius centered at the origin is described by
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A circle of radius r centered at the origin is described by x2+y2=r2. Consider only the portion of the circle lying in the first quadrant (x>0, y>0).
(1) Find dxdy in terms of x and y, and simplify your result.
(2) Describe qualitatively how the magnitude of dxdy changes as a point moves along the arc from (r,0) toward (0,r). Support your description with both algebraic reasoning and geometric insight.
(3) Explain why dxdy approaches −∞ as the point approaches (0,r). Relate this limiting behavior to the geometry of the circle.