MAT272FinalS06.html

1. Calculate the given quantity if a = < 0, 2, 1 >, b = < -1, -2, 1 >, c = < 3, 4, 1 >. [8 points]

a) Dot product of a and b b)

c) A unit vector in the direction of a. d) b x c

2. For the line segment going from P₀(2, 1, -3) to the point P₁(5, -2, 0), find the vector equation, r(t), for the line segment. [5 points]

3. Find the length of the curve r(t) = < cos(3t), sin(3t), 5t > from t = 0 to t = 1. [8 points]

4. For the curve r(t) = < >, find: [8 points]

(a) The Unit Tangent vector, T(t)

(b) The Unit Normal vector, N(t) =

5. Find and classify all local extrema and/or saddle points for the given function. [10 points]

6. (Optional - Didn't teach in Spring 2007) Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. [10 points]

7. What are the spherical coordinates of the point whose rectangular coordinates are

(x = y = , z = )? [6 points]

8. For F = < [9 points]

a. Find curl F = ________________

b. Is F conservative? Explain why or why not.

c. Find div F = __________________

9. Given F = is conservative: [8 points]

(Hint: You can use the quicker method we have used in class)

b) Use the fundamental theorem of line integrals and your answer to part (a) to calculate the following integral where C is the curve going from t = 1 to t = 4.

10. Find the surface area, of the part of the plane

that lies above the cylinder [8 points]

11. Use Green's Theorem, = [10 points]

to evaluate the line integral along the given positively oriented curve, where C is the triangle with vertices (0, 0), (1, 2) and (0, 2):

12. [10 points] Use the Divergence theorem, to calculate the surface integral

where F and S is the surface of the box bounded by in the first octant.

div F = ___________________

__________________