Question 4: Is the fuction f(x,y) Continuous? Explain why. if (x,y) + (0,0), if (x,y) =...
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...
x?sin’y If the function f(x y)= { x2 +2V2' (x,y) (0,0) is continuous at (0,0), then (x y) = (0,0) AK) Ox= 1/2 0-1 .k=0 ok=2 ok=1
3. (1,7) (0,0) { 0 f (x, y) = 23r²ty (z,y) = (0,0) (a) (10 pts) Evaluate the limit, or use a two-path test to show that it does not exist: lim f(x,y). (sy)(0,0) Math 250 Final Exam, CSU Northridge (b) (4 pts) Is f(x,y) continuous? Explain why or why not. 4. (17 pts) Use Lagrange multipliers to find the absolute maximum and absolute min- imum of f(x,y) = ry", subject to the constraint 3x² + y = 12. 5/12/20
Define f: R2R by 224V2 y) (0,0) 0 if (x, y)-(0,0) if (z, f(z, y) (a) Prove that Dif(z, y) and D2f (x, y) exist for each (x, y) E R2. (b) Prove that f is not continuous at (0,0).
Given the function f(r.y) lim f(x, y) (ry)-+(0,0) a. Evaluate iii. Along the line y= r: i. Along the r-axis: iv. Along y12 ii. Along the gy-axis: ,f(x, y) exist? If yes, find the limit. If no, explain why not. lim (a.)-(0,0) b. Does (0,0)? Why or why not? c. Is f continuous at d. The graphs below show the surface and contour plots of f (graphed using WolframAlpha). Explain how the graphs explain your answers to parts (a)-(c) above....
' + 5y6 if (x, y) = (0,0) Which of the following functions is continuous at (0,0)? (i) f(x,y) = if (x, y) = (0,0) Xy5 0 (ii) g(x, y) = xo + 2y if (x, y) + (0,0) if (x, y) = (0,0) 0 V2 + y2 +9 - 3 *++y? (iii) h(x, y) = if (x, y) + (0,0) 3 if (x, y) = (0,0) (A) (i) only (B) (ii) only (C) all of them (D) (i) and...
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...
y? - 2xy x + y2 if (x, y) + (0,0) 7. Given the piecewise function: f(x,y) 0 if (x, y) = (0,0) a) Show that: limf(x,y) does not exist. *(x,y) (0,0) b) Find: fy(0,0). c) Where is f continuous? Where is f differentiable? Explain.
21. Is the following function continuous at (0,0)? Hint:lim 1-cos T (1-cos(x2 +y2) f(x, y)=11-cosztym if(x,y) (0,0) if (x,y) = (0,0)