For the curve defined implicitly by an expression for can be obtained by successive…

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

For the curve defined implicitly by

x^{2}+y^{2}-xy=7

an expression for

\dfrac{d^{2}y}{dx^{2}}

can be obtained by successive differentiation. (a) Describe the geometric information conveyed by the sign of

\dfrac{d^{2}y}{dx^{2}}

at a point on any differentiable curve. (b) Explain why an algebraic expression for

\dfrac{d^{2}y}{dx^{2}}

derived from an implicit relation often contains

\dfrac{dy}{dx}

and clarify what additional step is required before the sign of

\dfrac{d^{2}y}{dx^{2}}

can be analyzed. (c) At a point on the curve where

\dfrac{dy}{dx}=0

, discuss how one can still determine concavity and mention any special considerations that arise in such a case.