(14,-9) (7,-10) (14,-9) Show that f(fdx+ gdy = f(10) 12x²y dx + 16y3x³dy is independent of path: (7,-10) af = (14,-9) J (7,-10) 12x2y+dx+16y3x³ dy =

Evaluate S(z² + 3z)dz along the straight line from (2,0) to (2, 2) and then from (2, 2) to (0, 2).

Find Duf at P. = f(x, y, z) = 6x³y³z³; P(2, -2, 2); u Duf = i i + 2 - 2 k

Determine Duf (4, – 1,6) for f(x,y,z) = e*³y + 4zx³ in the direction of (-3,2,6).

Find the orthogonal trajectories of the family of 2 curves x² + y²=cy A B C none of the choices x² + y² = cx x² - y² = cy (D) x² + y²=c

Which of the following is true? * f (x, y) = x³ - 2xy + 3y² fz = 3x² -2y+y³; fy = 6x Option 1 f = -2x+6y + 3xy²; fy = 3x² - 2y + y² Option 3 2 points faz = 6x; fzy = −2+3y² Option 2 fax=6+6xy; fyy = 6x Option 4

r(2,4) Sa (2y+x) dx +(3x– y)dy} along the straight line x-2t and Evaluate J(0,3) y=P+3.

7 FInd the linear approximatron near (2,0) of

Solve for the orthogonal trasctories x* ( 3k-x) - ky2 = xy2

Find dx (9x² + 16 1 1 - In |3x + V16 + 9x²| + c -In |3x – V16 + 9x²| + c Option 2 Option 3 1 In |3x + V16 + 9x²| + c 1 In |3x – V16 + 9x²|+c - ie Option 1 Option 4