Reference no: EM13867041

Problem 1. In each of the following, determine whether F is the gradient of a function f . If it is, find f .

(a)   F(x, y) = .3x2y2  + 3y + x. i + .2x3y + 3xy - √y. j.

(b) F(x, y) = .2xey + e2x - 3. i + .x2 ey + cos 2y + 1. j.

Problem 2.

(a)     Determine the minimum value of  f (x, y) = 2x2 + xy - y2 + 1  subject to the constraint  2x + 3y = 16.

(b)     Determine the maximum value of  f (x, y, z) = 3x - 2y + z  subject to the constraint  x2 + y2 + z2 = 14.

Problem 3.    Estimate (√119)(√5  34) using differentials.

Problem 4. Use a double integral to find the volume of the solid S in the first octant that is bounded above  by  the surface   z = 4 - x2 - y2, below by  the x, y-plane, and on the sides by  the planes   y = 0   and   y =   x.

Problem 5.

(a)   Evaluate 

2yz dx dy dz   where T  is the solid in the first octant bounded above by the  cylinder z = 4 - x2   below by the x, y-plane, and on the sides by the planes  z = 0,  x = 0,  y = 2x,  and  y =  4.

(b)  Set up a triple integral in cylindrical coordinates that gives the volume of the solid in the first  octant

that is bounded above by the hemisphere z = ,2 - x2 - y2,  below by the paraboloid  z = x2 + y2  and on the sides by the x, z- and y, z-planes.

(c)   Set up a triple integral in spherical coordinates that gives the volume of the solid that lies outside the cone  z = ,x2 + y2   and inside the hemisphere  z = ,1 - x2 - y2.

Problem 6.   Let  h(x, y, z) = xy i + y j - yx k, and let C be the curve given by

r(u) = u i + u2 j + 2u k,  0 ≤ u ≤ 1.

Calculate

h(r) · dr.C

Problem 7.

(a)  Find the Jacobian of the transformation:  x = u ln v, y = uv 

(b)  Take ? as the parallelogram bounded by   x + y = 0,  x + y = 1,  x - y = 0,  x - y = 2

and evaluate 2xy dx dy?

From pages 940-941 in text (Skills Mastery Review at the end of chapter 15): problems 53, 55, 57, 63, 68a, 69,  71

From pages 1094-1095 in text (Skills Mastery Review at the end of chapter 17): problems 1-11odd, 19, 24, 28 (just be able  to set up problems 24 and 28)