solutions for question (i)
b) i. Using e-8 definition show that f is continuous at (0,0), where f(x,y) = {aš...
Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f continuous at (0,0). 2. Compute the partial derivatives of f at any (x, y) E R2. Are the partial derivatives continuous (0,0). at (0,0) (0,0) and 3. Compute the second derivatives 4. Compute the linear approzimant of f at (0,0). Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f...
Consider the function f: RR given by if (x, y)メ(0,0) íf (z, y) = (0,0). f(z, y)=(0+r R2 does the directional derivative Duf(0,0) exist? Evaluate the For which vectors 0メu directional derivative wherever it exists. 4 marks
if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R defined by f(r,y)-+ (a) Show by explicit computation that the directional derivative exists at (x, y)- (0,0) for all oi rections u є R2 with 1 11-1, but that its value %(0.0) (Vf(0,0).u), fr at least one sucli u. (b) Show that the partial derivatives of f are not continuous at (0,0) if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R...
Anyone can solve these questions? (4) Let if (z, y)メ(0,0) if (x, y) (0,0) f(z, y) / 0 a) Show that f is a continuous function b) Show that f has partial derivatives at (0,0) and find (0,0) as well as c) Is f differentiable at (0, 0)? d) Are the partial derivatives r tinuous at (0,0)? (0,0) (5) Let A E M(mx n,R) and f : RnRm be the linear map f(x)-Ax. Show that f is a differentiable function...
= f ху Consider the function f(x,y) if x² + y² (x, y) + (0,0) = 0 if (x, y) = (0,0). Which one of the statement is incorrect. Select one: a. f(x,y) is differentiable everywhere. b. f(x,y) is differentiable everywhere except at the origin. c. f(x,y) is not continuous d. First partial derivatives f(x,y) exist. ху e. lim(x,y)—(0,0) x2 + y2 does not exist.
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y) = if (x, y) (0, 0) (a) Prove that f is continuous at (0,0) (b) Calculate the partial derivatives (0,0) and (0,0) directly from the definition of partial derivatives. (c) Prove that f is not differentiable at (0,0).
3. Find lim f(,y) if it exists, and determine if f is continuous at (0,0. (x,y)--(0,0) (a) f(1,y) = (b) f(x,y) = { 0 1-y if(x, y) + (0,0) if(x,y) = (0,0) 4. Find y (a) 3.c- 5xy + tan xy = 0. (b) In y + sin(x - y) = 1.
*Let f : R2 -R be given by z, y)(0,0 r, y)- 2y and f(0,0) = 0. (a) Decide if both partial derivatives of f exist at (0, 0) (b) Decide if f has directional derivatives along all v R2 and if so compute these. (c) Decide if f is Fréchet differentiable at (0, 0)? (d) What can you infer about the continuity of the partial derivatives at (0, 0)? て
Please answer fully and clearly. Thank you. )if (e-) (0.0)) if (z, y)#10, 0), 2. Consider f R2R defined by f(,) plHP, 0, if (x,y) (0,0)J (a) Show by explicit computation that the directional derivative exists at (x, y) - (0,0) for all di rections u R2 with lull-1, but that its value (0,0)メ(Vf(0,0), u), for at least one such u. b) Show that the partial derivatives of f are not continuous at (0,0) )if (e-) (0.0)) if (z, y)#10,...
Let f(x,y) = (x" + 2?y?)!. compute all second-order partial derivatives of fat (0,0), if they exist, and determien wheter dæðyəyər at (0,0).