[試題] 96下 王昭男 工程數學二 1st期中考

[pattrick (派特瑞克)]

課程名稱︰工程數學二
課程性質︰必修
課程教師︰王昭男/謝傳璋
開課學院：工學院
開課系所︰工程科學與海洋工程學系
考試日期（年月日）︰2008/4/7
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :

工程數學(二) 期中考一
                                                        ┌-2  2 -3┐
1.Find the eigen values and eigen vectors of matrix A = │ 2  1 -6│
                                                        └-1 -2  0┘

  and then diagonalize it. (20%)

2.Find out what type of conic section the following quadratic form represents

  and transform it to principal axes: (15%)

        2             2
  Q = 4x1 +12x1x2 +13x2 = 16

             3    2                ┌2  2┐
3.If p(x) = x - 2x +3x +1  and A = │    │ , find p(A) (15%)
                                   └2 -1┘

                                       4
4.Find the fourier series of f(x) = cos x (15%)


                                        2  if |x|<1
5.Find the fourier integral of f(x) =                and then prove that
                                        0  if |x|>1

   ∞  cosωxsinω         π/2  if 0≦x＜1
  ∫ ───────dω =   π/4  if x = 1          (15%)
   0        ω               0   if x ＞ 1


6.Find the fourier series of f(x) = x^2 , -1<x<1, f(x+2) = f(x)

                    2
                  π       1    1    1    1
  and prove that  ─ = 1 - ─ + ─ - ─ + ─ - .....   (20%)
                  12       4    9    16   25


   hint :∫p(x)f(x)dx = p(x)F1(x) - p'(x)F2(x) + p''(x)F3(x) -

                  m  (m)
          +...(-1)  p  (x) Fm+1(x)



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