※ [本文轉錄自 NTU-Exam 看板]
作者: Nzing (黑色神話) 看板: NTU-Exam
標題: [試題] 周青松 微積分甲上（2005秋 期末考）
時間: Mon Jan 9 16:05:06 2006
課程名稱︰微積分甲
課程性質︰共同必修
課程教師︰周青松
開課系所︰數學系
考試時間︰2005.01.09 13:20~15:00
試題 :
Ⅰ.A) Find f from th information given:
f''(x) = sin x, f'(0) = -2, f(0) = 1
B) Calculate the derivative:
d 2x
─ (∫ t√(1+t^2) dt)
dx tanx
Ⅱ.A) The base of a solid is the region between the parabolas
x = y^2 and x = 3 - 2 y^2
Find the volume of the solid given that the caves section perpendicular to
the x-axis are squares.
B) Find the volume of the solid generated by revoluting the region between
y = x^2 and y = 2x about the y-axis.
Ⅲ.A) a. Find the derivative:
d cos x
─ [(sin x) ]
dx
b. Evaluate the integral:
1 1+x^2
∫ x 10 dx
0
kx
B) Let f'(x) = kf(x) for all x in some interval. Prove that f(x) = Ce ,
where C is an arbitrary constant.
Ⅳ.A) Show that for a＞0
dx -1 x+b
∫────────── = sin (──) + C
√[a^2 - (x+b)^2] a
B) Determine A, B and c so that y = A cosh cx + B sinh cx satisfies the
conditions y''- 9y = 0, y(0) = 2, y'(0) = 1. Take c＞0
Ⅴ.A) Show that
d -1 1
─ (cosh x) = ────── , x＞1
dx √(x^2 - 1)
B) Prove that 1 -1 x
∫ ─────── dx = cosh (──) + C , a＞0
√(x^2 - a^2) a
（每大題均20分）
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