推 sophiachen26:原PO滿分 03/03 21:48
※ 發信站: 批踢踢實業坊(ptt.cc)
※ 轉錄者: fanif (118.167.36.15), 時間: 01/03/2012 23:24:08
→ fanif :如果真的原PO滿分就好了 01/03 23:33
※ [本文轉錄自 NTU-Exam 看板 #1DCVxtOU ]
作者: jesonk (東區) 看板: NTU-Exam
標題: [試題] 99上 周青松 微積分甲上 期末考
時間: Sun Jan 16 04:09:24 2011
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：管理學院、生農學院、理學院
開課系所︰工管系科管組、生工系、生機系、地質系
考試日期（年月日）︰2011/1/10
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
試題 :
Warning : Each part from I - V is of 20 points. Please write down your answer
in the answer sheets in details as well as possible.
I.
A. Let f be a continuous function on [a,b]. If G is any antiderivative for f
on [a,b], show that
b
∫ f(t)dt=G(b)-G(a).
a
B. Prove that for all positive numbers a and all rational numbers p/q,
p/q
a a
∫ dt/t = p/q∫ dt/t.
1 1
II.
A. Compare the following two statements,
d/dx[∫f(x)dx] and ∫d/dx[f(x)]dx.
B. Evaluate the integral
1/2
3 1/2
∫x^5(x^2+1) dx.
0
III.
-x^2/2
A. Let f(x)=e for all x.
(a)Determine the symmetry of the graph and find the asymptotes.
(b)On what intervals does f increases?decrases?
(c)Find the extreme values.
(d)Determine the concavity of the graph and find the points of inflection.
(e)Sketch the graph.
B. Find
3x
d/dx[(x^2+1) ].
IV.
A. Show that for a>0,
1/2
∫dx/(a^2-x^2) =arcsin(x/a)+c.
B.Show that
arctanhx=1/2ln(1+x/1-x),-1 < x < 1.
V.Prove the following equations.
n n-1 n-2
A. For n屬於N, ∫sin xdx = (-1/n)sin x cosx + (n-1/n)∫sin xdx.
n n-2 n-2
B. For n=>2, ∫sec xdx = (1/n-1)sec x tanx + (n-2/n-1)∫sec xdx.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.241.80