-s) Define the function FiR³>R by34+if (x,y) + (0,0)if (x, y) = (0,0).O(a) Determine with proof of Fis differentiable at (0,0) or not.(b) Determinu withproof if F is differentiable at (1,0) or not.1F(x,y) = { x = 1/²2x² +

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Answered: ( +5) Define the function F: R ²³>R by…

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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Consider i = (1,0,0), j= (0, 1, 0), k = (0,0, 1) and, f (x, y) = (x, ²y°, sin x, cos y, xyean +3 f(x, y, z) = x² + y°e2z+3y=5z f (x) = (x, sin x, cos x, xe) f (x, y) = 3x – 8y + sin æ + 5xye f (x, y) = sin x ·i – 3xy · k • 6. • 1. • 2. • 3. • 4. • 5. f(x, y, z) = (x + y+ z + sin æ, cos y, xye") %3D Make the appropriate selection. 1. f:R^2
In the complex plane C (a) Find singular points of the functions • f1(²) = z²+i) • f₂(2) = √√√₂² + 1, • f3(~)= tanh(z). (b) Representing z = x+iy, where x, y ≤ R, find the real u(x, y) and the imaginary v(x, y) parts of the function fi(x + iy) = u(x,y) + iv(x, y). Verify the Cauchy- Riemann equations.
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Exercice 1: Sit = Devir Libre f(x) = x-3+x+3 3) Dobrowing fat thanking que I out police. a) Determing Df, et b) Ecrire & Sans valeur absolue sur [0;3] et [3,+x[. 2) Tray & An R. On consistère la fonction of diffinie sun. g(x) = 2/21. IR par : sur le même repère pricedent. Tracez G Exercice 2 Seit = f(x) 2-2 - 1-pl 1- 2- a) Determine D. b) Etudiez la parité She la fonction of. Sritent a et b soux éléments destincts De R$1} Montrez que Tf(a;b) - (a-1) (-b) - 1 ↳ Déduire les variations de f sur [0,1 [et] 1; +h[ SJ Déduire le Tableau the variation de fx/Dg. Exercia 3: Seit & la cambre présentative d'une function f représenté ci-dessous = 1 2 1- Par lecture graphique, Déterming: 2- a) Df. b) bs images de -2, -3 et 0. c) les antécédents des ©, et 1. le Tableau de signe de four dj soit m D8. un réel. Determines, selon As valeurs m le nombre she solutions de l'équation. f(x) xm. 3- és graphiquement f(x) > + ; f(x) > •
Use the chain rule to find дz Əs дz at || z = tan 3v - 5u n(=), 5v + 3u 12 ² sec² əz Əs and əz at u = 3s + 5t, v= 5s 3t 2(=) Your answer cannot be understood X 2 sec²() * Your answer cannot be understood

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( +5) Define the function F: R ²³>R by F(x,y) = { *+/+3+² of (x,y)+ (0,0) x²y²³²-34² 4 2 y O if (x, y) = (0₂0)- (a) Determine with proof of Fis differentiable at (0,0) or not. (b) Determion with proof of F is differentiable at (1,0) or not.