Mathematics A, Further H235: Surfaces and Partial Differentiation Topical Past Paper Questions

The surface S has equation $ z = f(x, y) $, where $ f(x, y) = 4x^2y - 6xy^2 - \frac{1}{12}x^4 $ for all real values of x and y. You are given that S has a stationary point at the origin, O, and a second stationary point at the point $ P(a, b, c) $, where $ c = f(a, b) $. (a) Dete…

A surface, C, is given by the equation $ z = f(x, y) $ for all real values of x and y. You are given that C has the following properties. - The surface is continuous for all x and y. • The contour $z = -1$ is a single point on the $z$-axis. • For -1 < a < 1, the contour $z = a$ i…

A trading company deals in two goods. The formula used to estimate z, the total weekly cost to the company of trading the two goods, in tens of thousands of pounds, is $$ z=0.9x+\frac{0.096y}{x}-x^{2}y^{2}, $$ where x and y are the masses, in thousands of tonnes, of the two goods…