Consider the curve in the plane defined by the equation x^ 2 +xy+y^ 2 =7
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Consider the curve in the plane defined by the equation
x2+xy+y2=7.
Answer the following questions about points where the curve changes direction most noticeably.
(a) Explain, in words, what conditions on dxdy characterize a critical point for a curve given by an equation involving both x and y. Include both situations that may arise.
(b) Verify whether the point (2,1) on the curve is a critical point. Show all necessary work based on the equation of the curve.
(c) Describe geometrically what it means for dxdy to be undefined at a point on the curve, and contrast this with the case dxdy=0. Relate your discussion to the curve above.