Calculus III

Review Test III

Summer 2006

1. a. Prove that ||u x v|| = ||u|| ||v||, where is the angle between the vectors u and v. 

     b. ||u x v|| = area of the parallelogram having the vectors u and v as adjacent sides.

c. u x v is orthogonal to both of the vectors u and v.

     d. If u and v are scalar multiples of each other then u x v = 0.

2. a. Find the area of the parallelogram with adjacent sides <1, 2,-3> and <-4, 1, 2>.

b. Find the volume of the parallelepiped having adjacent sides <3,2,1>, <1,1,2>, and <1,3,3>.

c. If you apply a downward force of 60 pounds on a pedal when the crank makes an angle of 40 degrees with the horizontal, find the torque where the crank attaches if it the crank is 8 inches in length.

3. a. Find the equation of the line in space the passes through the point (2, 3, 4) and is perpendicular to the plane given by 5x+4y-2z = 20.

      b. Find the equation of the plane passing through the points (3, 2, 1), (2, 1,-1), and (-1, 3, 2).

d. Find the distance between the point (1,-2, 3) and the plane 2x - 2y + z = 4.

4.

a. Sketch the graph of

b. Sketch the graph of

c. Sketch the graph of

d. Sketch the level curves of , for

5. Find

6. Find the slopes of the surface h(x, y) in both the x and y directions at the indicated point

7. Evaluate the expression and then use differentials to approximate it.

8. Find the directional derivative for the function f, at P, in the direction of Q: