0
answers
0
watching
186
views
6 Nov 2019
Parts b and d
Let f (0 ,0 ) = 0, f(x, y) = 2xy/ x2 + y2 for (x, y) (0,0). Sketch the graph of z = f(x, y) near the origin. Use a polar coordinate overlay to clarify the picture. Show that the partial derivatives .f x(0.0) and f y (0 ,0 ) exist, and determine their values. Show that the directional derivative D u f (0,0) does not exist if u is not an axis direction. Explain this result in terms of the graph of f . Conclude that f , like the "manta ray" counterexample, fails to be differentiable at the origin. (In fact, for both this function and the "manta ray," f(x, y) = O (1) is true but f(x, y) = o (l) is false.) Show transcribed image text
Parts b and d
Let f (0 ,0 ) = 0, f(x, y) = 2xy/ x2 + y2 for (x, y) (0,0). Sketch the graph of z = f(x, y) near the origin. Use a polar coordinate overlay to clarify the picture. Show that the partial derivatives .f x(0.0) and f y (0 ,0 ) exist, and determine their values. Show that the directional derivative D u f (0,0) does not exist if u is not an axis direction. Explain this result in terms of the graph of f . Conclude that f , like the "manta ray" counterexample, fails to be differentiable at the origin. (In fact, for both this function and the "manta ray," f(x, y) = O (1) is true but f(x, y) = o (l) is false.)
Show transcribed image text