1. (15%) Find an equation for the plane tangent to the surface z = ex2y−1 at the point (1, 1, 1).
2. (10%) Compute f (x, y) = y4+ 2xy3+ x2y2 (a) (5%) the gradient of f at (0, 1);
(b) (5%) the directional derivatives at (0, 1) in the direction (1, 2).
3. (12%) Let the function f (x, y) = xy − x2y − xy2. Find (a) critical point(s), and (b) discuss the property of extreme value(including saddle points).
4. (12%) Find the distance from (0, 0) to the curve y = x2 − 5
4 by using the method of Lagrange multiplier.
5. (15%) Find Z Z
Ω
1
(1 + x + y)2 dA, where Ω = [0, 2] × [0, 3].
6. (12%) Compute Z 1
0
Z 1 x14
1
1 + y5 dydx.
7. (12%) Let the figure r = sin 2θ be as below. Find Z Z
Ω
xy dA, where Ω is a leaf in the first quadrant.
(Hint: cos θ sin θ = sin 2θ 2 ) 8. (12%) Find
Z Z
Ω
(3x+y)6dA, where Ω is the parallelogram enclosed by x+y = ±1 and 3x+y = ±1.
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