0
answers
0
watching
145
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 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.)
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 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.)0
answers
0
watching
145
views
For unlimited access to Homework Help, a Homework+ subscription is required.