[試題] 96上 陳金次 高等微積分 期末考

[zephurl (星雲沉淪)]

課程名稱︰ 高等微積分一
課程性質︰ 系定必修
課程教師︰ 陳金次
開課學院： 理學院
開課系所︰ 數學系
考試日期（年月日）︰  2008/1/19(六)
考試時限（分鐘）： 180分鐘
是否需發放獎勵金： 是
（如未明確表示，則不予發放）

試題 :

WARNING! You have to choose 8 problems to answer. Please draw a score table
on the first page of your examination paper, and write down explicitly what
you choose to answer.

1. Let f and g be Riemannian integrable on [a,b], show that fg is also

   Riemannian integrable on [a,b].

2. Let f be continuous on [0,π], try to find the value of

               π
        lim   ∫ f(x)|sin(nx)|dx.
        n→∞  0

3. Given 0 < x < 1, try to order tan(x^2), (tan(x))^2, xtan(x).

4. Let f be convex on R and assume that

         f(x) ≦ K  ( x屬於R )

   for some constant K. Prove that f ≡ constant.

                                            n
5. Let a_i,x_i > 0 for i = 1,2,3,...,n. If Σ (x_i)^2 = 1
                                           i=1

                               n    a_i
   try to find the minimum of Σ  ------- .
                              i=1   x_i

6. Prove that there is no function f defined on R such that f is continuous

   at all rational points but discontinuous at all irrational points.

7. Let               xy(x^2-y^2)
                    -------------    (x,y)≠(0,0)
               ╭      x^2+y^2
        f(x) = │
               ╰         0          (x,y)＝(0,0)

   (a) Prove that   f  (0,0)≠f  (0,0)
                     xy        yx

   (b) Is f continuous at (0,0) ?

8. Let f be continuous on [0,1].

   (a) Prove that
                           1
                    lim   ∫ (x^n)f(x)dx = 0
                    n→∞  0

   (b) Try to find the value of

                            1
                    lim  n*∫ (x^n)f(x)dx
                    n→∞   0

                                              ?
9. (a) f^2 is Riemannian integrable on [a,b] ---> f is Riemannian integrable

       on [a,b].?
                                              ?
   (b) f^3 is Riemannian integrable on [a,b] ---> f is Riemannian integrable

       on [a,b].?

10.Let               ╭ (x^α)*sin(1/x)   x屬於(0,1]
              f(x) = │
                     ╰       0           x = 0

   try to determine the value of α such that f has finite arc length

   over [0,1].

11.(a) Let           ╭ 1     x ≧ 1
              f(x) = │
                     ╰ 0     x ≦ 0    ,

       try to define the value of f on (0,1) such that f屬於C^∞(R).

   (b) Let
                      ╭ 1      x^2+y^2 ≦ (0.5)^2
              g(x) =  │
                      ╰ 0      (x,y) 不屬於 [-1,1] X [-1,1]  ,
                                                          _____
       try to define the value of g on [-1,1] X [-1,1] － N  (0)
                                                          1/2
       such that g 屬於 C^∞(R^2).

12. 0 < α < 2. Let          ╭  (x^α)*sin(1/x)   x屬於(0,1]
                      f(x) = │
                             ╰        0           x = 0      ,

    prove that
                   |f(x) - f(y)| ＜ K|x-y|^(α/2)  所有的 x ≠ y 屬於 [0,1] ,

                    where K is constant.

--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 220.135.131.55
推 rabbitmix:原來他也會出英文的唷...普通的微積分他都出中文XD
→ zephurl:不是~好像是助教打的 期中考兩次都是中文
※ 編輯: zephurl         來自: 220.135.131.55       (01/31 20:41)
推 dogiko:第二題怪怪...是n到無限大吧?
※ 編輯: zephurl         來自: 220.135.131.55       (02/01 06:35)
→ zephurl:thx