課程名稱︰ 微積分乙下
課程性質︰ 必修
課程教師︰ 李聰成
開課學院：
開課系所︰ 數學系
考試日期（年月日）︰ 2007/6/21
考試時限（分鐘）： 10.20 - 12.10 (延長至13.00)
是否需發放獎勵金： yes
（如未明確表示，則不予發放）
試題 :
請將每一步驟表達清楚，不可以只寫答案。
1.Evaluate the limit or show that it does not exist.
4 2
lim x - y
(x,y)→(0,0) ----------
4 2
x + y
2. At the point (1,2), the function f(x,y) has a derivative of 2 in the
direction toward (朝向) (2,2) and a derivative of -2 in the direction
toward (1,1).
(a) Find fx(1,2) and fy(1,2).
(b) Find the derivative of f at (1,2) in the direction toward (4,6).
3. Find an equation (方程式) for the tangent plane (切平面) and parametric
(參數) equations for the normal line (法線) to the surface
xyz = 12
at the point (2,-2,-3).
4. Find the dimensions (長，寬，高) of the closed rectangular (矩形) box
with maximum volume that can be inscribed (內接) in the unit sphere.
5. Find the absolute maximum and minimum values of
2 2
f(x,y) = 2 + 2x + 2y - x - y
on the triangular region in the first quadrant (象限) bounded by the lines
x = 0, y = 0, y = 9 - x.
6. Find the volume of the solid (實心體) cut from the square(正方形) column
(柱) ∣x∣+∣y∣≦1 by the planes z = 0 and 3x + z = 3.
7. Evaluate the integral.
π π
∫ ∫ siny/y dydx
0 x
8. Find the area of the region (區域) cut from the first quadrant by the
cardioid (心臟線) r = 1 + sinθ.
9. Change the Cartesian integral into an equivalent polar integral. Then
evaluate the polar integral.
2
2 √(2x-x )
∫ ∫ xy dydx
0 0
10. Use the transformation u = x + 2y, v = x - y to evaluate the integral
2/3 2-2y y-x
∫ ∫ (x + y)e dxdy
0 y
by first writing it as an integral over a region G in the uv-plane.
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