2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y)...
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...
= 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.
*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
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...
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...
(1 point) Consider the function defined by F(x, y) = x2 + y2 except at (r, y) - (0, 0) where F(0,0)0 Then we have (0,0) = (0,0) = ax dy Note that the answers are different. The existence and continuity of all second partials in a region around a point guarantees the equality of the two mixed second derivatives at the point. In the above case, continuity fails at (0,0) Note: You can earn partial credit on this problem...
the function of two real variables defined below: 1 –9x + 2y“ (x, y) + (0,0), f(x, y) = { 6x + 3y 10 (x, y) = (0,0). Use the limit definition of partial derivatives to compute the following partial derivatives. Enter "DNE" if the derivative does not exist. fx(0,0) = DNE fy(0,0) = 0
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,...
2. [1 mark] Calculate the limit of the vector valued function f: ACRY-R lim G logy) 3. Consider the function :R? - R. given by Flv = 0 if if (,y) (0,0): (x,y) -(0,0) (a) (1 mark] State the definition of continuity of a function at the point. (1 mark] Then calculating the limit (by any technique of your choice) show that f is continuous at (0,0). (b) [2 marks] Find the partial derivatives and at (x,y) + (0,0). and...
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...