af af. and 1. Fоr f(x, y) = ?у? + 5xy?, find Әх ду 1 Әf af 2. Fоr f (x, y) = ln(x? + 3у?), find and Әr ду N 1
,y)-3x2-5xy + y2 find F 3. or the function (x a) f (x, y) b) fy,(xr, y) c) f(x, y) ,y)-3x2-5xy + y2 find F 3. or the function (x a) f (x, y) b) fy,(xr, y) c) f(x, y)
find fxx(x,y), fxy(x,y), fyx(x,y) and fyy(x,y) for the function f. f(x,y)=8xe^5xy 19. Find fxx (x,y), fxy(x,y), fyx(x,y), and fyy(x,y) for the function f. f(x,y) = 8x e 5xy fx(x,y)= fxy(x,y)= fyx (x,y) = fyy(x,y) =
2. The force F(x, y) = (y + 2x) sin(xy + x)i + x sin(xy + x2) is conservative. (a) Find a potential V such that F = -VV. [2 marks] (b) Is F central? Provide a reason for your answer. [2 marks]
sin x aln3' nena) xln tan1 (x - Vx2 +1) Q1/Find the Derivatives: 1) y 2) = y ln3 c logs x2 b lnx 3) y ae* + sin x aln3' nena) xln tan1 (x - Vx2 +1) Q1/Find the Derivatives: 1) y 2) = y ln3 c logs x2 b lnx 3) y ae* +
QUESTION 5 a) Find and sketch the domain of the function f(x,y) = \n(x2 - y +1) + VÝ +1. (5 marks) b) Evaluate eży sin(3x +2y) lim (x,y) (-2,3) 3x +2y (6 marks)
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.
3)If w = x2 + y2 + z2 ; x = cos st, y = sin st , z = sat find 4)Find the minimum of the function f(x,y) = x2 + y2 subject to the constraint g(x, y) = xy - 3 = 0 5)Find the first and second order Taylor polynomials to the function f(x,y) = ex+y at (0,0). 6) Let f(x, y, z) = x2 – 3xy + 2z, find Vf and Curl(f)
Find the derivative of each one. a. y = (tan(x2 + 1))4 + 5 In Vx b. с. У-(sin x)cos x a. y = (tan(x2 + 1))4 + 5 In Vx b. с. У-(sin x)cos x
Problem 1.11 Find the gradients of the following functions: (a) f(x, y, z) x2 + y3 + z4. (b) f(x, y,-x23-4 (c) f(x, y, z) e* sin(y) In(z).