' + 5y6 if (x, y) = (0,0) Which of the following functions is continuous at...
Problem #9: Which of the following functions is continuous at (0,0)? () S(x, y) = x + 3yt if (x,y) # (0,0) if (x, y) = (0,0) x2 394 (ii) g(x, y) to + if (x, y) + (0,0) if (x, y) = (0,0) (iii) h(x, y) V22 + y2 + 1 - 1 x² + y² if (x, y) + (0,0) if (x,y) = (0,0) (A) none of them (B) (iii) only (C) (ii) only (D) all of them...
Q1. Let z = f(x,y) -√4x² – 2y² Find (i). domain of f(x,y) (ii). range of f(x,y) (iii). f(1,1) (iv). The level curves of f(x,y) for k = 0,1,2 4x2y Q3. Let f(x,y) = x2+y2 if (x,y) = (0,0) 1 if (x,y) = (0,0) Find (i) lim limf(x,y) (x,y)-(0,0) (ii). Is f(x, y) continuous at (0,0)? (iii). Find the largest set S on which f(x,y) is continuous.
Problem #2: Which of the following sets of functions are linearly independent on the interval (-0, c.)? [2 marks] (i) f1(x) = x, f2(x) = 4x, 13(x) = = x2 +6 (ii) f1(x) = 2e2x, 12(x) = 4e4x, f3(x) = 8e8x (iii) f1(x) = 8sinx, 12(x) = 4cos 2x, f3(x) 9 (A) (i) and (iii) only (B) (iii) only (C) none of them (D) (ii) only (E) all of them (F) (i) only (G) (i) and (ii) only (H) (ii)...
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...
Q2. x+y (a). Let f(x,y) = x²+y²+1 Find (i). lim (x,y)-(1,1) f(x,y) (ii). lim f(x,y) (x,y)-(-1,1) (iii). lim f(x,y) (x,y)-(1,-1) (iv). lim f(x,y) (x,y)-(0,0) ( 4x²y if (x, y) = (0,0) Q3. Let f(x,y) = x2 + y2 1 if (x,y) = (0,0) Find (i). lim f(x,y) (x,y)--(0,0) (ii). Is f(x,y) continuous at (0,0)? (iii). Find the largest set S on which f(x,y) is continuous.
Please answer part a and b :)
Which of the following vector fields are conservative? (i) F(x, y) = (9y8 +3) i + (8x8y' +7) j (ii) F(x,y) = (8ye8x + cos 3ji + (e8x + 3x sin 3jj (iii) F(x,y)-7y2e7xyİ + (7 +xy) e7xyj (A) all of them (B) (iii) only (C) (i) and (ii) only (D) (i) and (iii) only (E) none of them (F) (ii) and (iii) only (G) (ii) only (H) (i) only st Save Submit...
2. Define f : RR by - y 1(x) = { "2+2 (ay) (0,0); (z,y) = (0,0). (i) Isf continuous at (0,0)? Justify your answer. (ii) Show that Daf(3,0) = x for all x and D.f(0,y) = -y for all y (iii) D2f(0,0) + D2,1f(0,0). (iv) Is f differentiable at (0,0)? Justify your answer.
I need to answer a, c, and d
1. Find the limit at xo if it exists. (a) f(x, y) = xy/(x2 + y2), Xo = el + e2 (b) f(x, y) = xy/(x2 + y2), Xo = (0,0). (c) f(x) = (1-cos x)/x2, X,-0. [Hint: limx-o (sin x)/x = 1 .] (d) f(x)-|x-2le, + lx + 2le_, xo = 3. (e) f(x, y) = ye, + (xy)리[(xy)2 + (x-y)2je2, Xo = (0,0). At which points is each of these...
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.
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...