[轉錄][試題] 94上 微積分乙 張鎮華 第二次期中考
※ [本文轉錄自 NTU-Exam 看板]
作者: Roychess (I Love Avril) 看板: NTU-Exam
標題: [試題] 94上 微積分乙 張鎮華 第二次期中考
時間: Tue Dec 6 13:21:39 2005
課程名稱︰微積分乙
課程性質︰大一共同必修
課程教師︰張鎮華
開課系所︰醫學院，生命科學院，公衛學院等
考試時間︰2005/12/06 10:20~12:10
試題 :
Each problem weights 10 points, except problem 9 weights 20 points.
2 3 2
1.Find the normal line of the curve x y +4y =xy+x passing through the point
(2，-1).
2 2 1/2
2.Find the derivative of the function f(x)=sin(exp(3x -1))*ln(cos(1+x ) )
2x 4/3 2 1/2
e x (5+x )
3.Differentiate y=----------------------
2 4 5
(x +3) (tanx+secx)
4.Denote by y=arcsinx,-1≦x≦1, the inverse of y=sinx,-π/2≦x≦π/2.
d 1
Show that ---arcsinx=-------------- -1<x<1.
2 1/2
dx (1-x )
x
X x x
5.Differentiate y=X and y=(X )
6.Use statement(A) to prove statement(B),and use statment(B)
to prove statement(C).
(A:Rolle's theorem) If f is continuous on the closed interval [a,b],
differentiable on the open interval (a,b),
and f(a)=f(b), then there exists a number c屬於(a,b)
such that f'(c)= 0.
(B:Mean Value Theorem) If f is continuous on the closed interval [a,b]
and differentiable on the open interval (a,b),
then there exists a number c屬於(a,b) such that
f(b)-f(a)
------------=f'(c).
b-a
(C) If f is continuous on the closed interval [a,b] and differentiable on
the open interval (a,b), with f'(x)=0 for all x屬於(a,b), then f is a
constant on [a,b].
2 1/2
7. A trapezoid is bounded by the x-axis and the semicircle y=(36-x )
using the vertices (-6,0),(6,0),(x,y),(-x,y), which are points on the
semicircle. What is the maximum area of this trapezoid?
2x
e sin3x
8.Evaluate lim ---- and lim (sin2x)
x→∞ 3 x→0+
x
2
3(x +4x+5)
9.Analyze and sketch the graph of f(x) =------------ by indicating the
2
X +4x
intervals of increasing, intervals of decreasing, inrervals of concavity,
local maximum, local extrema, global extrema, points of inflection,
asymptotes (vertical, horizontal, or oblique one if there is any);
and any others if you think they are helpful to understand the graph.
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※ 編輯: Roychess 來自: 140.112.240.96 (12/06 21:17)
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