1. (12%) Find an equation of the tangent plane to the surface y − x = 4 arctan (xz) at the point (1, 1, 0).
2. (12%) The temperature at a point (x, y) is given by T (x, y) = 100e−x2−3y2, where T is measured in ◦C and x, y in meters.
(a) In which direction does the temperature increase fastest at P (1, −1)?
(b) What is this fastest increasing rate?
(c) Evaluate the directional derivative ∂T
∂*u(1, −1) *u=(3
5,45)
3. (12%) Evaluate
1
0
1 x16
1 1 + y7dy
dx.
4. (12%) Find I =
Ω
(x − 2y)3/2(3x + y)1/2dA, where Ω is the region enclosed by 2x + 3y = 0, 3x + y = 0 and x − 2y = 1.
5. (15%) Find the critical points of z = ye−12(x2+y2), which give rise to local maxima? Local minima? Saddle points?
6. (10%) Let f (x, y) = x3y5− x2y − y3and x = x(u, v), y = y(u, v).
Suppose x(1, 3) = 2, y(1, 3) = 1 and
∂x
∂u(1, 3) = 1
5, ∂x
∂v(1, 3) =1 2
∂y
∂u(1, 3) = 1
11, ∂y
∂v(1, 3) =1 4 Find the value of ∂f
∂u and ∂f
∂v at (u, v) = (1, 3).
7. (12%) Find
Ω
px2+ y2dA, where Ω is the region inside the cardioid r = 1 − cos θ.
8. (15%) Find the extremal values of f (x, y) = x2+ xy + y2 subject to the constraint x2+ 2xy + 2y2= 1.
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