課程名稱︰微積分甲上
課程性質︰暑修
課程教師︰周青松
開課學院︰理學院
開課系所︰
考試日期（年月日）︰2009/7/29
考試時限（分鐘）：110 min
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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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