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4.8.10. Recall the function f : R² → R whose graph is the crimped sheet, if (x, y) # (0,0), if (x, y) = (0,0). f (x, y) (a) Show that f is continuous at (0,0). (b) Find the partial derivatives Dif(0,0) and D2f(0,0). (c) Let d be any unit vector in R? (thus d takes the form d for some 0 € R). Show that Daf(0,0) exists by finding it. (d) Show that in spite of (c), ƒ is not differentiable at (0, 0). (Use your re- sults from parts (b) and (c) to contradict Theorem 4.8.2.) Thus, the existence of every directional derivative at a point is not sufficient for differentiability at the point. (cos 0, sin 0) %3D

4.8.10. Recall the function f : R² → R whose graph is the crimped sheet, if (x, y) # (0,0), if (x, y) = (0,0). f (x, y) (a) Show that f is continuous at (0,0). (b) Find the partial derivatives Dif(0,0) and D2f(0,0). (c) Let d be any unit vector in R? (thus d takes the form d for some 0 € R). Show that Daf(0,0) exists by finding it. (d) Show that in spite of (c), ƒ is not differentiable at (0, 0). (Use your re- sults from parts (b) and (c) to contradict Theorem 4.8.2.) Thus, the existence of every directional derivative at a point is not sufficient for differentiability at the point. (cos 0, sin 0) %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve problem 4.8.10

4.8.10. Recall the function f : R² → R whose graph is the crimped sheet, if (x, y) # (0,0), if (x, y) = (0,0). f (x, y) (a) Show that f is continuous at (0,0). (b) Find the partial derivatives Dif(0,0) and D2f(0,0). (c) Let d be any unit vector in R? (thus d takes the form d for some 0 € R). Show that Daf(0,0) exists by finding it. (d) Show that in spite of (c), ƒ is not differentiable at (0, 0). (Use your re- sults from parts (b) and (c) to contradict Theorem 4.8.2.) Thus, the existence of every directional derivative at a point is not sufficient for differentiability at the point. (cos 0, sin 0) %3D
Transcribed Image Text:4.8.10. Recall the function f : R² → R whose graph is the crimped sheet, if (x, y) # (0,0), if (x, y) = (0,0). f (x, y) (a) Show that f is continuous at (0,0). (b) Find the partial derivatives Dif(0,0) and D2f(0,0). (c) Let d be any unit vector in R? (thus d takes the form d for some 0 € R). Show that Daf(0,0) exists by finding it. (d) Show that in spite of (c), ƒ is not differentiable at (0, 0). (Use your re- sults from parts (b) and (c) to contradict Theorem 4.8.2.) Thus, the existence of every directional derivative at a point is not sufficient for differentiability at the point. (cos 0, sin 0) %3D
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