推 t0444564 :模擬考 01/12 22:56
→ rod24574575 :模擬考怎麼了嗎@@? 01/12 22:59
推 liltwnboiz :酷噎 01/12 23:42
課程名稱︰微積分甲上
課程性質︰必修
課程教師︰薛克民
開課學院：電資學院、工學院、管理學院
開課系所︰電機系、資工系、材料系、資管系
考試日期（年月日）︰99/12/31
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. y^3 = x^2 defines a curve on xy—plane.
(a) Find the arc length of this curve from (-1,1) to (8,4).
(b) Find the area of the surface generated by revolving about x-axis this
curve from (0,0) to (1,1).
2. (a) Solve the differential equation
x
y' - [tan(x)]*y = ──────
[sin(x)]^2
(b) Solve the differential equation
x
y' - [tan(x)]*y = ────── * (y^3)
[sin(x)]^2
3. Calculate the following integral
(a)
1
∫ ──────dx
√(1+e^x)
(b)
x^3 + x^2
∫ ───────── dx
(x^2 + 2x + 2)^2
4. (a) Find all intersection points of the curve y^2 = x^3 - 2xy and y = mx.
(b) Find a parametrization of the curve y^2 = x^3 - 2xy, and find all lines
tangent to the curve at (0,0).
5. The curve defined by x^2 + 2xy - y^2 = 1 is a hyperbola(雙曲線)
(a) Write this curve in polar coordinates, and use it to determine the
asymptotes of this curve.
(b) Using polar coordinate to compute the area of the region bounded by
this hyperbola and a straight line passing through origin θ = c lying
between x-axis and the asymptotes in first quadrant(第一象限).
(c) What is the area of the region bounded by the hyperbola and its
asymptotes in first quadrant?
6. (a) Find the area of the surface generated by revolving about x-axis the
curve y = 1/x from x = 1 to x = c for some c > 1. Use it to show that
the surface area generated by rotating {(x,y)│y = 1/x, x≧1} is
infinite.
(b) Is the surface area generated by rotating {(x,y)│y = (1/x)^2, x≧1}
finite?
Is the surface area generated by rotating {(x,y)│y = (1/x)^3, x≧1}
finite?
In general, for what n is the surface area generated by rotating
{(x,y)│y = (1/x)^n, x≧1} finite?
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 218.167.191.32
※ 編輯: rod24574575 來自: 218.167.191.32 (01/12 22:35)