Answer Key 1
          MATH225: Calculus III                     Name:
          Practice Exam 1.2 February 15, 2007   Instructor:
          Record your answers to the multiple choice problems by placing an × through one letter
          for each problem on this page. There are 9 multiple choice questions worth 6 points each
          and 3 partial credits problems worth 10 points each. You start with 16 points.
                 1. •   b  c   d   e            6.  a  b   c  •   e
                 2. •   b  c   d   e            7.  a  b   c  •   e
                 3. a   b  c   d   •            8.  a  •   c  d   e
                 4. a   b  c   d   •            9.  a  •   c  d   e
                 5. a   b  c   d   •
                                                1
          MATH225: Calculus III                    Name:
          Practice Exam 1.2 February 15,2007    Instructor:
          Record your answers to the multiple choice problems by placing an × through one letter
          for each problem on this page. There are 9 multiple choice questions worth 6 points each
          and 3 partial credits problems worth 10 points each. You start with 16 points.
                 1. a   b  c   d  e             6. a   b  c   d  e
                 2. a   b  c   d  e             7. a   b  c   d  e
                 3. a   b  c   d  e             8. a   b  c   d  e
                 4. a   b  c   d  e             9. a   b  c   d  e
                 5. a   b  c   d  e
                                                1
            1. Find the distance from the origin to the plane 2x + 3y − 6z = 14.
               (a) 2                   (b) 14                   (c) 1                   (d) 7                   (e) 7
                                                                                             3
                                                                                     q       q
                                      2                                                             
                                   −x          2            2
            2. Let f(x;y) = e          (sin(x ) +cos(y )). Compute f                   π=2;     π=2 .
                                                                                 xy
                         −π=2                                   q         −π=2                              −π=2
               (a) 2πe                                 (b) −4 π=2 e                             (c) −4πe
                    q         −π=2
               (d)     π=2 e                           (e) 0
                                                                                                      q 2          2
            3. Determine which of the following is the contour map of f(x;y) =                           x +3y .
                     6                                       6                                       6
                     4                                       4                                       4
                     2                                       2                                       2
                     0                                       0                                       0
                    -2                                       -2                                      -2
                    -4                                       -4                                      -4
                    -6                                       -6                                      -6
               (a)    -6 -4  -2 0  2   4  6            (b)    -6  -4 -2  0  2  4   6            (c)   -6  -4 -2 0   2  4  6
                     6                                       6
                     4                                       4
                     2                                       2
                     0                                       0
                    -2                                      -2
                    -4                                      -4
                    -6                                      -6
               (d)    -6 -4  -2 0   2  4  6            (e)    -6 -4  -2 0   2  4  6
            4. Find a unit vector that has the same direction as h−4;−7;4i.
                                                                  4         7       4                   2     7 2
               (a) h0;−1;0i;                           (b) h−√15;−√15;√15i                      (c) h−3;−6; 3i
               (d) h−2;− 7 ; 2i                        (e) h−4;−7; 4i;
                        9     18 9                              9     9 9
                                                                           2
                                                                                                   3       2  3     2
           5. Find the equation of the line tangent to the curve defined by r(t) = ht − t;t ;t + t i at
              the point (0;1;2).
              (a) x = 2, y = 1 + 2t, z = 5 + 2t
                          3                   2              3      2
              (b) x = 3t −t, y = 1+2t , z = 2+3t +2t
              (c) x = 3t, y = 1 + 2t, z = 2 + 3t
                          2                        2
              (d) x = 3t −1, y = 2t, z = 3t +2t
              (e) x = 2t, y = 1 + 2t, z = 2 + 5t
           6. Determine which of the following curves is defined by the vector function
              r(t) = ht;cos(t);sin(t)i.
              (a)                                  (b)                                  (c)
              (d)                                  (e)
           7. Find the area of the triangle with vertices (1;1;−1), (2;1;1), and (0;2;−1).
              (a) 3                 (b) 2                 (c) 1                  (d) 3                 (e) 1
                                                               2                     2
           8. Find the cosine of the angle between the vectors h2;2;−1i and h1;2;3i.
                     1                     1                                           3                    1
              (a) −2                (b) √                 (c) 0                  (d) √                 (e) 42
                                           14                                           14
                                                                     3