1. (12%) Let g(t) = f(x(t), y(t)), where {x(t) = a + th
y(t) = b + tk a, b, h, k∈ R. Find g′′(0).
2. (14%) Let f(x, y, z) = z tan−1(xy).
(a) (6%) Find∇f(2,1 2, 4).
(b) (4%) Suppose that at point(2,1
2, 4) the directional derivative of f in the direction of vector (a, 1, 0) is 0. Find the value of a.
(c) (4%) Find the tangent plane of the surface f(x, y, z) = π at point (2,1 2, 4).
3. (18%) Suppose that(−2, 1) is a critical point of f(x, y) = x2+ 4x + y3− 3ay.
(a) (2%) Find the value of a.
(b) (6%) Find and classify all critical points of f(x, y).
(c) (10%) Find the maximum and minimum value of f(x, y) on the rectangle R = {(x, y)∣ − 3 ≤ x ≤ 0, −2 ≤ y ≤ 2}.
4. (16%) On the ellipse x2+ 4y2= 1, find the maximum and minimum value of −x2+ 4xy + 2y2. 5. (14%) Compute the integrals.
(a) (6%) ∫
1 0 ∫
1
√3
y
1
x(1 + x3)dxdy.
(b) (8%) ∬Ωy2exy2+xdA, where Ω is the region bounded by x= 0, y = 0, y =√
3 and x= 1 1+ y2.
6. (14%) Evaluate the double integral ∬ΩxdA, where Ω is given in terms of polar coordinates by 0 ≤ θ ≤ π 2 and 0≤ r ≤ sin 2θ.
7. (12%) Evaluate the double integral∬ΩexydA, where Ω is the region enclosed by y= 1, y = 3, xy = 1 and xy = 4.
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