[轉錄][轉錄]周輕鬆考古題

[st3001 (連死狀態)]

※ [本文轉錄自 NTUcourse 看板]

作者: chukenren (龍潭內灣尖石竹東N￾Nꌩ 看板: NTUcourse
標題: [轉錄]周輕鬆考古題
時間: Mon Sep  8 21:40:17 2003

※ [本文轉錄自 NTUBA95study 看板]

作者: shoestring (洗衣精≠沐浴乳) 看板: NTUBA95study
標題: 周輕鬆考古題
時間: Mon Sep  8 17:17:22 2003

Ⅰ.

A. Determine the discontinuities,if any of the following function:

           ┌ 2x+1    , x≦0
     f(x)= │    1    , 0<x≦1
           └ x平方+1 , x>1

B.Show that
          sinx                       1-cosx
  (a)lim ─── =1           (b)lim ──── = 0
     x→0   x                   x→0    x

Ⅱ.

A.Find f'(1) give that

         ┌ x平方 , x≦1
   f(x)= │
         └ 2x-1  , x>1

B.Let

      ┌ x平方-x  ,x≦2
f(x)= │
      └ 2x-1     ,x>2

(a)Show that f is continuous at 2

(b)Find f'_(2) and f'+(2)

Ⅲ.

A.Two functions f and g are said to be inverse functions if
   (f。g)(x)=x   and (g。f)(x)=x

  Let f and g be differentiable. Show that if f and g are inverse functions,
                1
  then f'(y)= ─── , where y=g(x) ,provided g'(x)≠0
              g'(x)

             d    r      r-1
B.Show that ── X =  rX      ,  r any rational number.
             dx

Ⅳ. Let f be differentiable on (a,b) and continuous on 〔a,b〕.

A.Prove that if there is a constant M such that f'(x)≦M for all xε(a,b),
  then f(b)≦ f(a)+M(b-a).

B.Prove that if there is a constant m such that f'(x)≧m for all xε(a,b),
  then f(b)≧ f(a)+m(b-a).

Ⅴ.Sketch the grapf of the function
   f(x)=2sin三次方x+3sinx, xε〔0,π〕.

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