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）︰2007.1.9
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 calculations and resoning.
(1) Calculate the following (definite, indefinite) intergrals (6 points each):
1/√2
(a) ∫[1 / (1 + x^2)] dx (b) ∫[ 1 / √1 - x^2)] dx
0
(c) ∫[(1 + x) / (1 + x^2)] dx (d) ∫xe^x dx
(e) ∫arc sinχ dx (f) ∫[1 / √x(1 - 2√x)] dx
(2) Solve the following differential equations (8 points each):
(a) dy / dx = x / y^2
(b) dy / dx = (x + y) / 2x
(c) y' - (y / 2) = x^2 - 1, y(1) = 0
(3) Let R be the region bounded by the curves y = x and y = x^2.
(a) Find the area of R (10 points).
(b) Find the volume of the solid obtained by rotating the region R
about the x-axis (10 points).
(4) Find the center of mass of the thin flat plate (lamina) of uniform
density bounded by the graph of f(x) = 1 - x^2 and the x-axis (6 points).
(5) At time t = 0, a bacterial culture weighs 1 gram. Two hours later,
the cuweighs 2 grams. The maximum weight of the culture is 10 grams.
(a) Write a logistic equation that models the weight of the bacterial
culture (4 points).
(b) Find the culture's weight after 4 hours (5 points).
(6) Let y = f(x) = tanχ^2, x in (0, π/2) and let x = g(y) be its inverse
function. Find the derivative g'(y) = dx / dy of g(y) (5 points).
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 61.228.212.206