[試題] 98上 王振男 微積分乙上 期末考

[shokanshorin (上官薔凜)]

課程名稱︰微積分乙上
課程性質︰必帶
課程教師︰王振男
開課學院：醫學院
開課系所︰醫學系
考試日期（年月日）︰2010／1／12
考試時限（分鐘）：140min(?)
是否需發放獎勵金：Yes!
（如未明確表示，則不予發放）

1. Find ∫sin(ln x)dx (10%)


                 x^2 dx
2. Evaluate∫───────, x＞1 (10%)
              (x^2-1)^5/2

        2π/3
         ∫      cosθdθ
3. Find  ∫ ───────── (10%)
         ∫  sinθcosθ+sinθ
        π/2

4. Test for conditional and absolute convergence of

   ∞     (-1)^n
   Σ ───────, p＞0 (20%)
   n=2  n^p+(-1)^n

5. Test for the convergence fo the following series. (20%)

      ∞     1×3×...×(2n-1)
  (a) Σ ─────────────
      n=1 [2×4×...×(2n)](3^n+1)

      ∞  n(ln n)
  (b) Σ ─────
      n=1 1+n-n^2

6. A 200-gal tank is half full of distilled water. At time t=0, a solution

   containing 0.5 lb/gal of concentrate enters the tank at the rate of

   5 gal/min, and the well-stirred mixture is withdrawn at the rate of

   3 gal/min. (10%)

  (a) At what time will the tank be full?

  (b) At the time the tank is full, how many pounds of concentrate will

      it contain?

7. Test the convergence of the improper integral (20%)

    ∞  cos x
    ∫ ──── dx
   π/2   x
                                       (3/2+n)π
                   ∞  cos x        ∞   ∫  cos x
   Hint: Consider  ∫ ──── dx = Σ   ∫ ──── dx
                  π/2   x          n=0  ∫    x
                                       (1/2+n)π

         and identify the series you get.

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