MATB44H3 Lecture 10: Oneclass.

Find fx(1, 0) and fy( 1, 0) and interpret these numbers as slopes for the following equation. f(x, y) = - x2 - 9 y2 fx(1, 0) = fy(1, 0) =

Let g : [0, 1] R rightarrow R2 be given by g(t) = (x(t), y(t)), and let f : R2 rightarrow R be a C1 function (all partial derivatives exist and are continuous). Assume that (dx/dt)fx + (dy/dt)fy 0. Show that f(x(1), y(1)) f((x(0), y(0)).

Explain why the function is differentiable at the given point. f(x, y) = 2 + x ln(xy - 5), (3, 2) The partial derivatives are fx(X, Y) = and fy(X, Y) = , so fx(3, 2) = and fy(3, 2) = . Both fx and fy are continuous functions for xy > [ ] and f is differentiable at (3, 2). Find the linearization L(x, y) of f(x, y) at (3, 2).