MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Consider the curve defined implicitly by x2+xy+y2=3.
(i) Obtain dxdy in terms of x and y.
(ii) State clearly the algebraic condition under which the tangent to any curve becomes (a) horizontal, (b) vertical.
(iii) Using your answer to part (i), explain why the given curve cannot possess a vertical tangent at any point located in the first quadrant (x>0,y>0).
(iv) For points lying in the first quadrant, determine the condition on x and y for which the tangent is horizontal, and interpret your result geometrically in not more than three sentences.