[試題] 98下 曾琇瑱 微積分甲下 第一次期中考
課程名稱︰微積分甲
課程性質︰大一共同必修
課程教師︰曾琇瑱
開課學院：
開課系所︰地質、生機、工管等
考試日期（年月日）︰99.04.12
考試時限（分鐘）：110mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.
y^2 ln y
Find the arc length of x= ——— - ——— (1 < y < e)
4 2
2. Solve the differential equations
dy
(a)—— + (3x^2)y = 6x^2 y(0)=3
dx
(b)y'=x(e^-sinx)-ycosx
3.
Suppose the popularity of a kind of animal P can be fitted
by the follow model:
dP 2
—— = 2cos (3t-1) , P(0)=100
dt
(a) Find the solution to the above equation.
(b) What can we say about the popularity as the years go by?
4.
Explain why the functions with the given graphs can't be solution
of the differential equation
dy t 2
—— = e (y-1) (fig in 統一教學課本p571 ex11)
dt
5.
What is wrong with the following equation?
0 = 0 + 0 + 0 + 0 +....
= (1-1) + (1-1) + (1-1) + (1-1)+....
= 1 + (-1+1) + (-1+1) + (-1+1) +.....
= 1 + 0 + 0 + 0 + 0
= 1
6.
Evaluate the series
∞ n
(a) Σ ———
n=1 (n+1)!
∞ 1
(b) Σ ln(1 + —)
n=1 n
7.
r = 1+2sinθ
(a) Sketch the curve
(b) Find the area of the region enclosed by the curve
8.
x=t^2
y=t^3-3t
(a) Sketch the curve
(b) Find the tangent at point C (3.0)
9.
Populations of birds and insects are modeled by the equations
dx
—— = 0.4x - 0.002xy
dt
dy
—— = -0.2y + 0.000008xy
dt
(a) Which of the varible,x or y, represents the bird population
and which represent the insect population ? Explain.
(b) Find the equilibrium solutions and explain their significance
(c) Sketch the phase trajectory corresponding to initial populations
of 100 birds and 40000 insects
(d) Sketch the two populations as function of time
10.
Show the equation defined by
1
a1=2 a(n+1) = ——
3-an
(a) Show the sequence is convergent
(b) Find it limit
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.239.243
推
04/13 19:34, , 1F
04/13 19:34, 1F
中間打錯 謝謝指正
※ 編輯: a3225737 來自: 140.112.239.243 (04/13 23:55)