Calculus III Exam 1 Review Questions:

1. Find the projection of a onto b, and find the angle between a and b: a = , b =

2. Find the volume of the parallelepiped having adjacent edges a, b, and c: a = , b = , c =

3. Find the equation of the plane that passes through the points (3, 2, 1) and (3, 1, -5) and is perpendicular to the plane 6x + 7y + 2z = 10.

4. Find the distance between the planes 4x 4y + 9z = 7 and 4x 4y + 9z = 18.

5. Identify and sketch the quadric surface: 16x2 + 9y2 + 16z2 32x 36y + 36 = 0

6. Sketch the curve represented by the vector-valued function and give the orientation of the curve: r(t) =

7. Evaluate the limit:

8. Find r(t) for the given conditions: r`(t) = , r(1) =

9. If r(t) = and u(t) = , find Dt[r(t)u(t)] and Dt[r(t) u(t)]

10. Sketch the space curve and find its length over the given interval: r(t) = in the interval [0, /2].

11. Find the curvature of the curve r(t) =