Express g′(0) and g′′(0) in terms of ∂f ∂x(1, 3), ∂f ∂y(1, 3), ∂2f ∂x2(1, 3), ∂2f ∂x∂y(1, 3), ∂2f ∂y2(1, 3)

(1)

1. (13%) Evaluate ∬

Ω

(x − y)²⁰dA, where Ω is the parallelogram enclosed by 2x + y = 0, 2x + y = 1, x + 2y = 0 and x + 2y = 1.

2. (12%) Evaluate

∬Ω

e^−(x²^+y²⁾dA, Ω = {(x, y) ∶ 1 ≤ x²+y²≤2, y ≥ 0, y ≥ x ≥ −y}.

3. (13%) Find the points on the curve 10x²+12xy + 5y²=14 that are closest to and farthest from (0, 0).

4. (12%) Given a function f (x, y) in two variables x, y, consider the function g(t) = f (1 + 2t, 3 + 4t). Express g^′(0) and g^′′(0) in terms of ∂f

∂x(1, 3), ∂f

∂y(1, 3), ∂²f

∂x²(1, 3), ∂²f

∂x∂y(1, 3), ∂²f

∂y²(1, 3).

5. (12%) Find an equation for the tangent plan to the surface xe^yz+ln(y³z²) =tan⁻¹( x

z)at the point (0, 1, 1).

6. (13%) Let f (x, y) = (x²

4 +y²−1)

2

+x²y². (a) Find ∇f (2, 1).

(b) Find ∂f

∂ ⃗u(2, 1), where ⃗u is the unit vector in the direction of the vector (3, 1).

(c) Find the unit vector ⃗u such that ∂f

∂ ⃗u(2, 1) attains its maximum. Also find this maximum.

(d) Find the equation of the tangent line to the level curve of f (x, y) through the point (2, 1) at the point (2, 1).

7. (13%) Find the critical points of f (x, y) = e^y²^−x²(y²+x²)and determine it is local maximum, local minimum or saddle points.

8. (12%) Find ∬

Ω

sin x

x dA where Ω is the triangle in the xy-plane bounded by the x-axis, the line y = x, and the line x = 1.

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