Name:

Directions: Clearly mark your answer and show your work for full credit.

1. If u = < 1,2,-1 > and v = < 0,3,1 > , then determine

a.)

u ·v

b.)

u ×v

c.)

projection of u onto the direction of v

d.)

2u -3v

2. Compute the equation of the plane that passes through the points (1,0,-1), (2,3,1), and (3,2,0).
3. Consider the function f(x,y) = xy²-6x²-3y²

a.)

Determine all critical points for f.

b.)

Apply the second derivative test to classify all critical points as local maxima, minima, or saddle points.

4. Let f(x,y) = [(2x-3y)/(x+1)].

a.)

Plot the level curve of f for c = 1.

b.)

Compute the gradient of f at the point (4,1) and sketch it on the graph in part (a).

5. Show that the sphere x²+y²+z²-4y-2z+2 = 0 is perpendicular to the paraboloid 3x²+2y²-2z = 1 at the point (1,1,2). (i.e. their tangent planes are perpendicular.)
6. Let S be the tetrahedron determined by the coordinate planes and the plane x+y+2z = 4.

a.)

Sketch and parameterize the region S.

b.)

Compute the integral    _S  (y-x)  dx  dy  dz

7. Use cylindrical coordinates to compute the volume of the solid above the xy-plane, below the surface z = xy and within the cylinder x²+y² = 2x.
8. Use the chain rule to compute [du/dt] where u = 2x²-xy+y³, x = sin(t), and y = cos(t). (Do not simplify.)
9. Show that curl ( f) º 0.
10. Compute the line integral of f(x,y,z) = 2x+9z along the curve C given by x = t, y = t², and z = t³, 0 £ t £ 1.
11. Let C be the semi-circle formed by the upper half of a circle of radius 2 and center the origin and diameter along the x-axis.

a.)

Compute   _C (y+1) dx+2x dy   directly, using the definition of the line integral, where C is oriented in the counter-clockwise direction.

b.)

Use Green's theorem to evaluate the integral in part (a).