Consider the function f : R2 → R, ={* (r, u) + (0,0) (x, y) = (0,0) f(x, y) 0, (a) Argue that f is differentiable on R²\{(0,0)} and compute f, and fy away from the origin. (b) Check that f is continuous at (0, 0) (and hence everywhere). (c) Check whether all directional derivatives Dy f(0,0), v = (v1, v2), exist at (0,0). In particular, check if either of the partials f-(0,0) and fu(0, 0) exist. If your answer is "yes", find the respective derivative.
Consider the function f : R2 → R, ={* (r, u) + (0,0) (x, y) = (0,0) f(x, y) 0, (a) Argue that f is differentiable on R²\{(0,0)} and compute f, and fy away from the origin. (b) Check that f is continuous at (0, 0) (and hence everywhere). (c) Check whether all directional derivatives Dy f(0,0), v = (v1, v2), exist at (0,0). In particular, check if either of the partials f-(0,0) and fu(0, 0) exist. If your answer is "yes", find the respective derivative.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Consider the function f : R2 → R,
={ (2, u) # (0,0)
(x, y) = (0,0)
f(r, y)
0,
(a) Argue that f is differentiable on R²\{(0,0)} and compute f, and fy
away from the origin.
(b) Check that f is continuous at (0,0) (and hence everywhere).
(c) Check whether all directional derivatives Dy f(0,0), v = (v1, v2), exist
at (0,0). In particular, check if either of the partials fa(0,0) and
fy (0, 0) exist. If your answer is "yes", find the respective derivative.
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