[試題] 99上 王振男 微積分乙 期中考

[liltwnboiz (TCL)]

課程名稱︰微積分乙
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）︰2010/11/09
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. Find the following limits.

 (a)        (x^2)cos(x)
      lim  ───────
      x→0    1-sec(x)

 (b)        (x^2)
             ∫  te^(-t^2)dt
             (0)
      lim  ─────────
      x→∞     e^(-x^4)

2. For these two functions y=e^(x^x) and y=x^(e^x), the value of y approaches
infinity as x approaches infinity. Determine which function grows faster and
justify your answer.

3. Graph y=e^x-2e^(-x)-3x. Find out all the extremal points, where the function
is increasing or decreasing, the concavity of the function, and the point of
inflection, etc.

4. The function f(x) is continuous on [a,b]. 1, 2, ..., n are postive integers
and x1, x2, ...., xn are points on [a,b]. Show that there exists a point c on
[a,b] such that
           f(x1)+f(x2)+...+f(xn)
   f(c)=  ────────────
                     n

5. The function f(x) is a strictly increasing function on [a,b]. Suppose its
inverse function f^(-1)(x) is continuous on [f(a),f(b)]. Now define
xj=a+(j/n)(b-a), j=1, 2, ..., n. Using the result from the last problem, we
know that there exists a number cn in [a,b] such that
            f(x1)+f(x2)+...+f(xn)
   f(cn)=  ────────────
                      n
Does the limit of cn as n approaches infinity exist? If yes, find the limit.

6. 2y=x+4, y=x, and y-axis together form a triangular area. Find the volume of
rotation of the are under the circumstances below.
(a) Use the disk(washer) method, rotaion about x-axis.
(b) Use the shell method, rotation about y-axis.
(c) Use the shell method, rotation about the line x=4.
(d) Use the disk(washer) method, rotation about y=8.



(因為是憑記憶，所以有些用詞不精確...歡迎補充)
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◆ From: 140.112.7.59
※ 編輯: liltwnboiz      來自: 140.112.7.59         (11/09 19:34)
※ liltwnboiz:轉錄至某隱形看板                                     11/09 19:51
→ shokanshorin:1.(b) x approaches +(infinity)
※ 編輯: liltwnboiz      來自: 140.112.7.59         (11/09 20:46)
推 raymond123  :好像比去年難耶XD
※ 編輯: liltwnboiz      來自: 140.112.7.59         (11/09 20:55)
推 woieyufan   :助教表示：＂老師今年也太狠＂
→ shokanshorin:弟二題一定是(e^x)^x!!!一定是!!!(洗腦中)
推 raymond123  :XDDDDD那期末應該會稍微好過一點(應該吧)