作者sophiamag19 (秋人)
看板NTUBIME102HW
标题[转录][试题] 97暑 周青松 微积分甲上 期末考
时间Fri Jan 8 22:49:14 2010
※ [本文转录自 NTU-Exam 看板]
作者: ee0longsun (我是擂颗迷) 看板: NTU-Exam
标题: [试题] 97暑 周青松 微积分甲上 期末考
时间: Wed Jul 29 11:32:46 2009
课程名称︰微积分甲上
课程性质︰暑修
课程教师︰周青松
开课学院︰理学院
开课系所︰
考试日期（年月日）︰2009/7/29
考试时限（分钟）：110 min
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
1. Find f(x) from the information given.
(a) f'(x)＝ax^2 + bx + c , and f(0)＝0 ,where a,b,c are three constants.
(b) f''(x)＝cos x , f'(0)＝1 ,and f(0)＝2.
2. (a) Sketch the region Ω bounded by y＝x^2 and y＝2-x. Use the washer-method
to find the volume of the solid generated by revolving this region about
the x-axis.
(b) Sketch the region Ω bounded by x＝y^2 and x＝2-y. Use the shell-method
to find the volume of the solid generated by revolving this region about
the y-axis.
3. (a) Differentiate f(x)＝㏑(cos e^2x ) and calculate the integral
sin(e^-2x)
∫──────dx
e^2x
(b) Evaluate
____
√2㏑3
∫ xe^(-x^2 / 2) dx
0
4. (a) Show that for a > 0 ,we have
dx ｘ
∫────────── ＝ arcsin── + C
(a^2 － x^2)^(1/2) ａ
(b) Show that for a≠0 , we have
dx 1 ｘ
∫──────── ＝ ── arctan── ＋ C
a^2 ＋ x^2 a ａ
5. (a) Show that
d -1 1
──(sinh ｘ) ＝─────────
dx (1 ＋ x^2)^(1/2)
where x is a real number.
(b) Verifying the formula
1 -1
∫─────────dx ＝ sinh (x/a) ＋ C
(x^2 ＋ a^2)^(1/2)
where a is a positive constant.
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※ sophiamag19:转录至看板 NTUBIME102HW 01/08 22:44
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