MATH 208 Section 003
Practice Test 1
Name:
You must show me enough work that I can reproduce your results.
1. (10 points) Let f (x, y) = y 2 − 2x. Draw a contour diagram of f using at least 4 labeled contour lines.
[Do not use a calculator for this problem.]
y
x
Page 1 of 7
2. (10 points) Show that
lim
x3 y
does not exist.
+ y2
(x,y)→(0,0) x6
Page 2 of 7
3. Let ~v =
√
11~i − ~j − 2~k and let w
~ = 4~j + 3~k.
(a) (4 points) Compute ~v · w.
~
(b) (6 points) Find the angle θ between ~v and w.
~ (Remember that, by definition, 0 ≤ θ ≤ π.)
4. Let f (x, y) = (3x + x2 y 2 + 6y)4 .
(a) (6 points) Compute fx ( 13 , 0).
(b) (6 points) Compute fy (0, − 61 ).
Page 3 of 7
5. (12 points) A 100-meter race is run on a track in the direction of the vector ~v = 2~i + 6~j. The wind
velocity w
~ is 5~i + ~j. The rules say that the race is only considered legal if the wind speed in the direction
of the dash is at most 5 km/h. Is the race legal? Justify your answer.
Page 4 of 7
6. (12 points) Find an equation of the plane containing the points P = (3, 5, 6), Q = (−2, 6, 0), and
R = (1, 1, 1). [Hint: start by finding two vectors among the three points.]
Page 5 of 7
7. (10 points) Find the equation of the plane tangent to the surface ln (xy) + 2eyz = 2 at the point (1, 1, 0).
8. (12 points) Two straight roads meet at a right angle. If car A is approaching the intersection from the
west at 25 miles per hour and car B is approaching the intersection from the south at 30 miles per hour,
how fast is the distance between the two cars decreasing when car A is 0.3 miles from the intersection
and car B is 0.4 miles from the intersection?
Page 6 of 7
9. Let g(x, y, z) = ln(xz) + 2xy 3 + ex−z .
(a) (6 points) Find the direction of the maximum rate of increase of g at the point (1, 1, 1).
(b) (2 points) Find the magnitude of the maximum rate of increase of g at the point (1, 1, 1).
(c) (2 points) Find the direction of the minimum rate of increase of g at the point (1, 1, 1).
(d) (2 points) Find a vector that points in a direction where the rate of change of g is 0 at the
point (1, 1, 1). [Many answers are possible.]
Question:
1
2
3
4
5
6
7
8
9
Total
Points:
10
10
10
12
12
12
10
12
12
100
Score:
Page 7 of 7