作者  ajenny13 (~~~)                                          看板  NTU-Exam 
 標題  [試題] 103-1 周青松 微積分甲上 第三次小考                              
 時間  Sun Mar 22 11:51:36 2015                                               
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課程名稱︰微積分甲上
課程性質︰必修
課程教師︰周青松
開課學院：理學院
開課系所︰地質科學系 生物環境系統工程系 生物產業機電工程系 工管系科管組
考試日期（年月日）︰103/12/1
考試時限（分鐘）：50分鐘

試題 :

(1) (20 pts)
    Find the integral of f(x) = 2sin^3x + 3sinx from [-π, π]
    and sketch the graph on [-π, π]

(2) (20 pts)
    Let f be continuous on [a,b] ,
    define F(x) = ∫x f(t) dt and G(x) = ∫x f(t) dt, for c,d in [a,b]
                    c                      d
Show that F and G differ by a constant.
Show that F(x) – G(x) = ∫d f(t) dt
                           c

(3) (20 pts)
    (a) Find g(x) = ∫x f(t) dt, (b) Sketch the graphs of f, g
                      -2
    (c) Where is f continuous, differentiable? Where is g differentiable?

(4) (20 pts)
    Find f from the information given: f''(x) = cosx, f'(0) =1, f(0)=2

(5) (20 pts)
    Calculation d/dx ∫(x^2+2)  1/(2+√t ) dt and d/dx∫2x  t√(1+t^2) dt
                         √x                           tanx

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