[試題] 98下 王振男 微積分乙下 期中考

[shokanshorin (上官薔凜)]

課程名稱︰微積分乙下
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）︰2010／4／27
考試時限（分鐘）：120min
是否需發放獎勵金：是
（如未明確表示，則不予發放）

1. For what values of x does the series below converge? What is its sum?

        1         1                  1
   1 - ─(x-1) + ─(x-1)^2 -... + (-─)^n(x-1)^n +...
        2         4                  2

   What series do you get if you differentiate the given series term by term?

   For what values of x does the new series converge? What is its sum? (10%)

2. Can you define f(0,0) such that function below is continuous at (0,0)? (10%)

                 3x^2 - (xy)^2 + 3y^2
   f(x,y) = ln (───────────)
                      x^2 + y^2

3. Find the shortest distance between the parabola y=x^2 and the line x-y=1.

   Also, specify the points where the shortest distance occurs. (20%)

   (Hint: method of Lagrange multiplier)

4. Find the critical point of  f(x,y)= xy + 2x - ln(yx^2)  in the open first

   quadrant (x＞0,y＞0) and show that f takes on a minimum there. (20%)

5. Show that the improper integral below converges and find its value, where

   0＜a＜b. (20%)

   ∞  e^(-ax) - e^(-bx)
   ∫ ───────── dx
   0          x
                                           e^(-ax) - e^(-bx)   b
   (Hint: to find its value, observe that ───────── = ∫e^(-xy)dy.)
                                                   x           a
                         2 (y+4)/2
6. Evaluate the integral ∫∫      y^3×(2x-y)× e^(2x-y)^2 dxdy (20%)
                         0 y/2

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