MATH 2110 – EXAM 3 REVIEW PROBLEMS
This is not a comprehensive set of problems. Be sure to review your notes and
homework when preparing for the exam.
1. If z = f(x, y) =
2 2
2
x xy y
− +
, and x = r cos (t), y = r sin (t), find
z
r
∂
∂
and
z
t
∂
∂
via the
Chain Rule.
2. The radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and
the height is decreasing at a rate of 4 inches per minute. What is the rate of change of
the volume and surface area when the radius is 12 inches and the height is 36 inches?
3. a.) Find the directional derivative of f(x, y) =
2 3
4
x y y
−
at the point (2, -1) in the
direction of the vector v = 3i + 4j.
b.) What is the direction of maximum rate of change of f(x, y) at (2, -1)?
c.) What is the value of the maximum rate of change?
4. p.957 - #34
5. Let f(x, y) =
3 2
3 9 4
x y x y
+ − +
. Find the direction at the origin in which f(x, y) is
decreasing the fastest. At what rate is f(x, y) decreasing?
6. Locate and identify the local maximum, minimum, and saddle points of the function
f(x, y) =
2 2 2
2 2
x y x y
+ +
.
7. a.) Find the point on the plane x – y + z = 3 which is closest to the origin.
b.) What is the distance from that point to the origin?
8. Repeat #7 using Lagrange multipliers.
9. Use Lagrange multipliers to find the maximum and minimum values of
2 2
( , )
f x y x y
= +
subject to the constraint
4 4
1
x y
+ =
10. Use Lagrange multipliers to find the maximum and minimum values of
f(x, y) =
1 1
x y
+
subject to the constraint
2 2
1 1
1
x y
+ =
11. Estimate the volume of the solid that lies above the square R = [0, 8] x [0, 8] and
below the elliptic paraboloid f(x, y) =
2 2
120 3 4
x y
− −
. Divide R into four equal squares
and use the Midpoint Rule.
12. Evaluate
R
( sin ) dA
x y+
∫∫
, where R = {(x, y) | 0 ≤ x ≤ 2, 0 ≤ y ≤ π}
