→ YSClaire:抱歉第37行有誤@@ 已經更正了~ 02/21 01:30
→ YSClaire:版主收精華區收太快了XD 02/21 01:32
課程名稱︰微積分乙
課程性質︰共同必修
課程教師︰陳其誠
開課系所︰管院、地理、經濟各系
考試日期（年月日）︰2006.12.05
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是 謝謝^^
試題 :
Write down your answers on the answer sheet. You should include all the
necessary calculaitons and reasoning. The exam contains two sections written
in two pages.
The total number of points is 150. However the maximum score is 100.
◎First Part--This part contains totally 100 points.
(1) Calculate the following definite, indefinite intergrals (5 points each):
1
(a) ∫(x^12 + 5x) dx (b) ∫ √(1 - x^2) dx
-1
e π
(c) ∫ (1 / x) dx (d) ∫ sinχ dx
1 0
1 ln 2
(e) ∫ x(x^2 + 1)^99 dx (f) ∫ e^x dx
-1 0
(g) ∫ tanχ dx (h) ∫[(x^2 + 2x +3) / (x^3 + 3x^2 + 9x)] dx
(i) ∫[1 / √(x + 1)] dx (j) ∫[(ln x)^2 / x] dx
(2) Calculate the follwoing limits:
1+h n
(a) lim (1/h) ∫ (e^x)^2 dx (b) lim Σ (π/n) sin(kπ/n)
h→0 1 n→∞ k=1
(8 points) (7 points)
(3) Calculate the derivatives of the following functions (5 points each):
(a) f(x) = 2^x (b) f(x) = log x
10
x
(c) f(x) = ln(sinχ) (d) f(x) = ∫ √(1 - t^4) dt
1
x^2
(e) f(x) = ∫ √(1 - t^4) dt
1
(4) Suppose that y = f(x) and x = g(y) are inverse functions of each other.
(a) Suppose that f'(x) > 0 for all x. Show that g'(y) > 0 for all y.
(b) Suppose that f"(x) > 0 for all x. Then what can we say about the sign
of g"(y) ? Prove your assertion. (5 points each)
◎Second Part--This part contains 50 points.
(1) Solve the follwoing differential equations (8 points each) :
(a) y' = -3y +4
(b) y' +y = x
(c) xdy - (x + y) dx = 0
(2) Calculate the following intergrals (5 points each) :
(a) ∫[1 / (1 + x^2) dx (b) ∫[1 / √(4 - x^2)] dx
(3) A group of 1200 Formosan Landlocked Salmon are released into a mountain
lake. It is estimated that the capacity of the lake for the species is
20400. After the first year, there are 2000 Formosan Landlocked Salmon
in the lake.
(a) Write a logistic equation that models the number of Formosan
Landlocked Salmon in the lake (8 points).
(b) Find the number of Formosan Landlocked Salmon in the lake after
8 years (8 points).
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 61.228.212.206
※ 編輯: YSClaire 來自: 61.228.210.173 (02/21 01:30)