[試題] 99上 康明昌 微積分甲上 期中考

[harveyhs (Hango)]

課程名稱：微積分甲上
課程性質：物理系必帶
課程教師：康明昌
開課學院：理學院
開課系所；物理系
考試日期：2010/11/16
考試時間：120分鐘
是否發放獎勵金：是

1.Define <an>=[((n^2)-1)^-0.5]+[((n^2)-4)^-0.5]+...+[((n^2)+(n-1)^2)^-0.5]
  Find lim<an>.                                                      (10%)
       n→∞

2.Find y=y(x) where y satisfies xy'=3y+x^3 and y(-exp(0.5))=exp(0.5)
                                                                     (10%)

3.Evaluate the integral ʃx*cosx*exp(x)dx                            (10%)
  (Hint: If you have a smart idea, go ahead.
  Otherwise, find ʃx*cosx*exp(x)dx+ʃx*sinx*exp(x)dx
  and ʃx*cosx*exp(x)dx-ʃx*sinx*exp(x)dx.)

4.Let y=y(x)
 (1)Find the general solution of y"+4y'+3y=0                         (5%)
 (2)Find y where satisfies y(0)=0,y'(0)=1/2, and
         y"+4y'+3y=x*cosx+exp(x)                                     (15%)
  (Hint: Even if you can not find ʃx*cosx*exp(x)dx,
  you may write something till the last step.)

5.Define a function f:R→R by f(x)=0 if x belongs to R\Q,
  and f(x)=1/p if x=q/p where p,q belong to Z, p≧1 and gcd{p,q}=1.
  Show that lim f(x)=0                                               (15%)
            x→1/2

6.Let f,g:[0,1]→R, be continuous functions. Define h(x)=max{f(x),g(x)}
  for all x belong to [0,1]. Show that h(x) is also a continuous function.
                                                                     (15%)
  (Hint: If you have no smart idea, how about considering
  the cases f(a)=g(a) and f(a)≠g(a) separately.)


7.Find all the values of c such that f(x)=(3/2)x^4-2x^3-6x^2+c=0
  has four distinct real roots.                                      (15%)
  (Hint: Sketch the graph y=(3/2)x^4-2x^3-6x^2+c)

8. Suppose x^(2/3)+y^(2/3)=1. Find y"(1/8) if y(1/8)=((27)^0.5)/8    (15%)


9.Discuss the convergence or divergence of the integrals
  ∞                 ∞
 ʃ(sinx/x)dx  and  ʃ(|sinx|/x)dx                                   (20%)
  0                  0


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