※ [本文轉錄自 NTU-Exam 看板]
作者: jhs0909 (秉) 看板: NTU-Exam
標題: [試題] [試題] 94上 陳其誠 微積分乙 第三次考試
時間: Tue Jan 3 14:06:40 2006
課程名稱︰微積分乙
課程性質︰大一共同必修
課程教師︰陳其誠
開課系所︰醫學院、公衛學院等
考試時間︰2006.1.3
試題 :
Write down your answers on the answer sheet. You should include all the
necessary calculations and reasoning.
1. Calculate the following (definite, indenifite, improper) integrals (5 points
each):
∞ 1
(a) ∫e^x dx (b) ∫ xe^(-x) dx (c) ∫ (1+t)^100 dt
0 0
2 ∞
(d) ∫(x^7-9x) dx (e) ∫ 1/x dx (f) ∫ 1/x^2 dx
1 1
1 3 π
(g) ∫ 1/√x dx (h) ∫ √(9-x^2) dx (i) ∫ sinx dx
0 -3 0
1 1
(j) ∫ xf(x^2)dx if ∫ f(x)dx =2
0 0
2.
π
(1) Show that ∫ (x^2)(sinx^999) =0 (8 points)
-π
(2) Suppose f(x) is a continuous function defined on [0 1] and for every
b
a,bε[0 1] we have ∫ f(x)dx =0. Can we conclude that f(x)=0 for every x?
a
Why? (7 points)
3. Find the area bounded by the curves y = x and y = x^2 (10 points)
4. Suppose water is entering a tank at a rate of g(t) = 360t- 39t^2+ t^3
where g is measured in liters per hour and t is measured in hours.
(1) Find the total amount of water entering during the first 10 hours, from
t=0 to t=10, and find the average rate at which water entered during
this time (10 points)
(2) During the first hour, a chlorine leaking occurs and at time t,
1≧t≧0, there are totally h(t)=0.01+ 0.001t mini gram per liter of
chlorine leaking into the water entering the tank. In this situation,
during the first hour how much chlorine would be accumulated in the
tank? (5 points)
5. Explain why each of the following improper integrals converges or diverges
(5 points each):
∞ 1
(1)∫ e^(-x^2) dx (2) ∫ (sinx)^2 /x dx
0 -1
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