課程名稱︰微積分乙下
課程性質︰共同必修
課程教師︰陳其誠
開課學院：管理學院
開課系所︰管院各系、地理系、經濟系
考試日期（年月日）︰2007.04.03
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
試題 :
Write down your answers on the answer sheet. You should include all the
necessary calculations and reasoning.
(1) Calculate the following (definite, indefinite, imporper) integrals
(6 points each):
2x √3 1 π/4 dx
(ａ) ∫────── dx (ｂ) ∫ ────── dx (ｃ) ∫ ───────
x^2 - 4x + 3 1 (1 + x^2)^2 0 (x^2 + 1)^3/2
π/4 1 2 √(x^2 - 1) π/2
(ｄ) ∫ ─── dx (ｅ) ∫ ────── dx (ｆ) ∫ cos^4 x dx
0 cos x 1 x^3 0
∞ 1 1 1
(ｇ) ∫ ── dx (ｈ) ∫ ── dx
3 x^2 -1 x^3
(2) Calculate the following limits (7 points each):
cos x - 1 4 ln (1 + x)
(ａ) lim ───── (ｂ) lim (1 + ──)^x (ｃ) lim ─────
x→0 x^2 x→∞ x x→0 x
(3) Let υ = (4, 0, 3).
(ａ) Find a vector μ = (a, b, c) such that
a + b + c = 1, ║μ║ = 5 and μ‧υ = 0 (5 points)
(ｂ) Find a vector ω =/= (0, 0, 0) such that ω‧μ = 0 (5 points)
(4) Calculate the area of the parallelogram having (2, 0, 7) and (4, 0, 3) as
adjacent sides (8 points).
(5) Suppose the volume of the parallepiped with (2, 0, 7) , (4, 0, 3) and
(8, x, 1) as adjacent edges equals 22. What could be the value of x
(7 points).
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