1. (15 pts) Let f(x, y, z) = sin(xy + z), and P be the point (0, −2,π3).
(a) (6 pts) Compute ∇f(x, y, z).
(b) (2 pts) At P , find the direction along which f obtains maximum directional derivative.
(c) (4 pts) Calculate the directional derivative ∂f∂u(P), where u is a unit vector making an angle
π
6 with the gradient ∇f(P).
(d) (3 pts) The level surface f(x, y, z) = √23 defines z implicitly as a function of x and y near P . Compute ∂x∂z at P .
2. (12 pts) Assume that f(x, y, z) and g(x, y, z) have continuous partial derivatives and (1, 2, −1) lies on the level surface f(x, y, z) = 3. Suppose the tangent plane of f(x, y, z) = 3 at (1, 2, −1) is 2x− y + 3z + 3 = 0 and fy(1, 2, −1) = 2.
(a) (4 pts) Find ∇f(1, 2, −1).
(b) (4 pts) Estimate f(1.1, 2.01, −0.98) by the linear approximation of f at (1, 2, −1).
(c) (4 pts) Suppose that when restricted on the surface f(x, y, z) = 3, g(x, y, z) obtains maxi- mum value at (1, 2, −1) and gx(1, 2, −1) = −2. Find ∇g(1, 2, −1) and the maximum directional derivative of g at the point (1, 2, −1).
3. (25 pts) f(x, y) = x2+ xy + y2+ 3x.
(a) (7 pts) Find critical point(s) of f(x, y) and determine whether it is a saddle point or f(x, y) obtains local maximum or local minimum at it.
(b) (15 pts) Find the maximum and minimum value of f(x, y) on the curve x2 + y2 = 9 by the method of Lagrange multiplies.
(c) (3 pts) Find the maximum value of f(x, y) on the region x2+ y2≤ 9.
4. (18 pts) (a) (8 pts) Reverse the order of integration and evaluate it. ∫04∫√1y
2
√x3+ 3 dx dy.
(b) (10 pts) Compute ∬
Ω
(ln y)−1dA, where Ω is bounded by y= ex and y= e√x.
1
5. (18 pts) (a) (8 pts) Evaluate∬
D
e−x2−y2dA, where D is the upper disc, x2+ y2 ≤ 25 and y ≥ 0.
(b) (10 pts) Calculate the area of the region inside the cardioid r= 1 − sin θ.
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-1.5 -1-1 -0.5-0.5 0.50.5 11 1.51.5
-2 -2 -1.5 -1.5 -1 -1 -0.5 -0.5 0.5 0.5
0 0
6. (12 pts) Evaluate ∬
D
exydxdy, where D is bounded by curves xy = 10, xy = 20, x2y = 20 and x2y= 40.
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