PRACTICE EXAM 1 - SOLUTIONS
General Exam Information
Note that the Practice Exam said here that you only get a calculator and something to write
with. This is incorrect - you also get one 8.5 x 11 sheet of paper for reference that you can
write whatever you want on.
Practice Problems
1. < 1, 10 >
2. 41.2 degrees (0.719 radians). (Graph at end)
3. Projection = 17
< 6, −1, 5 >. The vectors are not perpendicular (or orthogonal) since
62
the projection was not 0.
√
4. Area = 21 290
5. 21x + 13y + 3z = 12
6. Symmetric:
x
3
=
y−30
−10
=
z
1
Parametric: x = 3t, y = −10t + 30, z = t
Unit vector:
√1
110
< 3, −10, 1 >
7. x = 4 − 2t, y = 1 + 6t, −3 + t
√
8. Speed = 2 3, Acceleration = < 2, 1/2, 0 >, Curvature =
√
3
18
9. 0
10. y = 0 → z = −x2 + 9 (parabola)
z = 2 → 2 = y 2 /4 − x2 /9 (hyperbola)
The quadric surface is a hyperbolic parabola. (Graphs at end)
11. Note that the function is discontinuous at the line y = 0. Every level curve approaches
the points (−2, 0), (0, 0) but these points are not included in them. (Graphs at end)
√
√
12. Cylindrical: ( 17, 14o , 6) Spherical: ( 41, 14o , 20.4o )
13. Cylindrical: r2 = z, Spherical: ρ =
cos(φ)
sin2 (φ)
14. 6(x2 + y 2 + z 2 ) = xy
1
15. The limit does not exist since the expression is rational and evaluates as 8/0 when
plugging in the point. The limit cannot exist unless this evaluates as a real number or
sometimes 0/0.
16. Gradient (fastest increase direction) = < 0, 2 >
Direction of 2i + j: 2
The directional derivative in the direction of < 1, 0 > is 0.
17. T (x, y) = −4x + 8y − 4. T (2.1, 2.1) = 4.4 (Compare this to F(2.1, 2.1) = 4.41)
2