Anyone can solve these questions?
Anyone can solve these questions? (4) Let if (z, y)メ(0,0) if (x, y) (0,0) f(z, y)...
*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
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...
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).
Question 2 (20 points): Consider the functions f(x, y)-xe y sin y and g(x, y)-ys 1. Show f is differentiable in its domain 2. Compute the partial derivatives of g at (0,0) 3. Show that g is not differentiable at (0,0) 4. You are told that there is a function F : R2 → R with partial derivatives F(x,y) = x2 +4y and Fy(x, y 3x - y. Should you believe it? Explain why. (Hint: use Clairaut's theorem) Question 2...
= 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.
b) i. Using e-8 definition show that f is continuous at (0,0), where f(x,y) = {aš sin () + yś sin () if xy + 0 242ADES if xy = 0 ii. Prove that every linear transformation T:R" - R" is continuous on R". iii. Let f:R" → R and a ER" Define Dis (a), the i-th partial derivative of f at a, 1 sisn. Determine whether the partial derivatives of f exist at (0,0) for the following function. In...
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...
Problem 3. Define the function: 2+_ 0 if (z,y)#10.0) if (a,y)-(0,0) f(x, v)= (a) Graph the top portion of the function using Geogebra. Does the function appear to be continuus at 0? (b) Find fz(z, y) and fy(z, y) when (z, y) #10.0) (c) Find f(0,0) and s,(0,0) using the limit definitions of partial derivatives and f,(0,0)-lim rah) - f(O,0) d) Use these limit definitions to show that fay(0,0)--1, while x(0,0)-1 (e) Can we conclude from Clairaut's theorem that()-yr(x,y) for...
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).
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