1. (14%) Find the maximum and the minimum of f (x, y) = xy subject to the constraint x2+xy+y2=1.
2. (12%) Let f (x, y) = x3−3λxy + y3, where λ ≠ 0 is a real number. Find all the critical points of f . Determine which give rise to local maxima, local minima, saddle points. (Note: It depends on the value of λ.)
3. (12%) Evaluate∬
Ω
x2
x2+y2dA, where Ω is the region 1 ≤ x2+y2≤2.
4. (12%) Evaluate∬
Ω
(y −x)(2x+y)dA, where Ω is the region enclosed by y −x = 1, y −x = 2, 2x+y = 0, and 2x + y = 2.
5. (12%) Let f (x, y) be differentiable. Suppose that ∂f
∂x(2, −2) =
√
2, and ∂f
∂y(2, −2) =
√
5. Let x = u − v and y = v − u. Find the value of ∂f
∂u +∂f
∂v at (u, v) = (1, −1).
6. (14%) Let f (x, y) = excos y + a sin y, where a is a constant.
(a) (7%) Find an equation of the tangent plane to the level curve f (x, y) = −1 at the point (0, π).
(The equation may contain a.)
(b) (7%) Suppose the maximum of the directional derivative ∂f
∂⇀u(0, 0) occur at ⇀u = (3 5,4
5), find the value of a.
7. (12%) Evaluate∬
Ω
3x2 (x3+y2)2
dA, where Ω = [0, 1] × [1, 3].
8. (12%) Evaluate∬
R
xey
y dA, where R ∶ 0 ≤ x ≤ 1 and x2 ≤y ≤ x.
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