Exam 2 Review
1. Consider the function f (x, y) = x cos(xy)
(a) What is ∇f (x, y)?
(b) What is Dv f (2, π) if v = ⟨1, 1⟩?
(c) In what direction should is the function increasing most rapidly at the point (2, π)?
Decreasing most rapidly?
(d) Suppose you’re traveling along the curve c(t) = ⟨t, 1 − t2 ⟩. What is the rate in change
in height when t = 1?
2. Classify all critical points of the function
2
f (x, y) = xye−x −y
2
3. Find the absolute maximum and minimum of f (x, y) = 14 − 2x + 4y on the closed triangular
region with vertices (0, 0), (4, 0), and (4.6).
List the maximum/minimum values as well as the points at which they occur.
4. Find the global maximum and minimum values of f (x, y) = x2 + 2y 2 − 4y subject to the
constraint x2 + y 2 = 9.
5. Set up, but do not evaluate, the integral that computes the arc length of the curve parameterized by:
x(t) = 4t2 − t
y(t) = et
6. Set up the double integral
ZZ
f (x, y)dA
D
twice, once for each order of integration, where D is the region bounded by the curves y = x2
and y = x3 .
7. Set up the triple integral
ZZZ
f (x, y, z)dV
E
where E is the solid bounded by y 2 + z 2 = 49 and y = 2x, where x ≥ 0, y ≥ 0, and z ≥ 0.
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