[試題] 98上 王振男 微積分乙上 期中考

[liltwnboiz (TCL)]

課程名稱︰微積分乙上
課程性質︰必修
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）︰2009.11.XX
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:

1. (10%) Suppose that the derivative of the function y=f(x) is
   y'=[(x-1)^2](x-2)(x-4). At what points, if any, does the graph of f have
   a local minimum, local maximum, or point of inflection?


2. (20%) Find the centroid of the region bounded by the curve y=4-x^2 and
   the lines x=-1 and y=1.


3. (20%) Find the limit:

            ε/(1+ε)
      lim      ∫     [e^(-x^2)]/(x^2) dx
     ε→0-    ε

                       lim
      Hint: the limit ε→0  [e^(-x^2)-1]/(x^2) may be useful.

4. (20%) Let f:Ｒ→Ｒ be a differentiable and bounded function, i.e., there
   exists a positive constant M such that |f(x)|≦M for all x屬於Ｒ. Given
   a屬於Ｒ with a≠0. Assume that f'(x)≠a for all x屬於Ｒ. Show that there
   exists one and only one x_0屬於Ｒ such that f(x_0)=ax_0.

5. (10%) At points on the curve y=2[x^(1/2)], line segments of length h=y
   are drawn perpendicular to the xy-plane. (圖略) Find the area of the
   surface formed by these perpendiculars from (0,0) to (3,2[(3)^(1/2)]).

6. (20%) A rectangular sheet of 8.5-in.-by-11-in. paper is placed on a flat
   surface. One of the corners is placed on the opposite longer edge and held
   there as the paper is smoothed flat. The problem is to make the length of
   crease as small as possible. Call the length L.

   a. Show that L^2=(2x^3)/(2x-8.5)
   b. What value of x minimizes L^2?
   c. What is the minimum value of L?

             ＿＿＿＿＿＿＿＿
            |＼              |
            |\ ＼            |
            | \  ＼          |
            |  \ 　＼        |
            |   \  　＼      |
            |    \ 　　＼    | 11 in.
 √(L^2-x^2)|     \ L  　＼  |
            |　　　\     ／＼|
            |       \　  ＼／|
            |_       \　 ／  |
            |_|＿x＿＿\／＿＿|
　     　 　 　　 8.5 in.
　　　　　　　　　　　　

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