For Exam 1, I recommend that you do as many problems from the book (including review problems) for the sections we have covered in class. Below are a few problems you can practice with (though they do not cover everything).

1. We use the points P = (1,2,3);     Q = (3,2,2) and the vectors v = i+ 3 j+ k;    w = 2i- 3j+ k. Calculate the following:

   1. [PQ] ·v
   2. v×(w+ [PQ])
   3. proj_wv
   4. |v-[PQ]|
   5. a unit vector in the direction of w.

2. Show, using vector operations, that the diagonals of a parallelogram bisect each other.
3. find the area of the triangle \triangle PQR where P = (1,2,3);     Q = (3,2,2) and R = (1,0,1).

4. Find the symmetric equations of the line given by the vector presentation r(t) = a+ tv, where a = i+ 2j-k and v = 2i+ 4j-2k. Does this line pass through the origin?

5. Find the distance between the points P = (1,2,3) and Q = (3,2,2).

6. Find the distance between the line r(t) = a+ tv, where a = i+ 2j-k and v = 2i+ 4j- 2k, and the point P = (1,2,3).

7. Find the distance between the line in the previous problem and the x-axis.

8. Find the distance between the point P = (1,2,3) and the plane given by the equation 2x + 3y + 4 z = 12.

9. Find the equation of the plane containing the three points P = (1,2,3);    Q = (3,2,2) and R = (1,0,1).
10. Consider the functions:

    f(t) = sint i+ t2 j+ k g(t) = (1+t) i+ cost k h(t) = et.

    Calculate the derivatives of

    1. (f·g)(t)
    2. (f×g)(t)
    3. g(h(t))

11. Find a tangent vector to the vector valued function r(t) = (e^t+ t)i+ lnt j when t = 1.

12. Find a unit tangent vector to the vector valued function r(t) = sin (e^t) i+ cos(e^t)j at an arbitrary time t.

13. Describe the quadric surfaces:

    1. 9y² - 6x² = 54
    2. x² - 4y² + 4 z² = 0
    3. x² - 4y² + 4 z² = 4
    4. 6x² +9 y² + 36 y - 54 z + 36 = 0

14. Write an equation for the surface x² + y² = 9z in cylindrical coordinates.

15. Write an equation for the surface x² + y² = 9z² in spherecal coordinates.

16. Find the arc length of the curve traced by r(t) = cos t i+ sin t j+ t k over the interval 0 £ t £ 2p.

17. Write an integral describing the arc length of the curve traced by

    r(t) = t i+ t² j+ t³ k over the interval 0 £ t £ 2.

18. Find the limit, if exists:

    lim (x,y) -> (0,0)  xy2 / (x2 + y2)

19. Find the limit, if exists:

    lim (x,y) -> (0,0)  xy2 / (x2 + y4)

20. Let f(x,y) = [(x² - 2)/( y+2)] and P = (2,0,1).

    1. find an equation for the line normal to the graph of f(x,y) at P.
    2. find an equation for the plane tangent to the graph of f(x,y) at P.

21. Find the linear approximation formula for the function f(x,y) = x² y +y² x around the point (x₀,y₀) = (2,1) and use it to give the approximation for f(1.9, 1.2).