課程名稱︰微積分甲上
課程性質︰必修
課程教師︰康明昌
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2011/01/14
考試時限（分鐘）：150(15:34~18:04)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
滿分135
∞
1.(15%)Let a_1≧a_2≧．．．≧a_n≧．．．＞0 Show that Σa_n converges if
∞ n=1
and only if Σ(2^k．a_2^k) converges
k=0
2.(15%)For -1<x<1, consider the Taylor's expansion
∞
arc sinx/(1-x)^0.5 = Σ(a_n．x^n). Find a_1 a_2 a_3 a_4
n=0
3.(15%)Evaluate lim x^2[(1+1/x)^x-e㏑(1+1/x)^x]
x→∞
4.(15%)Find the area of the loop bounded by the folium of Descartes
x^3+y^3 = 3axy where a>0 (請GOOGLE圖片 folium of descartes)
5.(15%)Let γ=f(θ) be the equation of a curve in polar coordinates
Show that the curvature κ is given by the formula
κ = (2γ'^2-γγ"+γ^2)/(γ'^2+γ^2)^1.5
where γ'=df/dθ γ"=d^2f/dθ^2
6.(15%)Let 0<ε<N Show that
∞
Σ[(1/2n+1)．(x-1/x+1)^(2n+1)] converges uniformly for ε≦x≦N
n=0
7.Let α be any real number. Define f_n(x) by
╓n^α．x if 0≦x≦1/n
║
f_n(x)=╟n^α(2/n - x) if 1/n≦x≦2/n (※n≧2)
║
╚0 if 2/n≦x≦1
(i)(15%)For 0≦x≦1. Show that lim f_n(x) = 0
n→∞
(ii)(15%)Determine all the values α such that f_n(x)→0 as n→∞
uniformly for 0≦x≦1
∞
8.(15%)Let x_0≠0. Assume that Σ(a_k．X_0^k) converges.
∞ k=0
Show that Σa_k．x^k converges for all x satisfying 0≦|x|≦|x_0|
k=0
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※ 編輯: tonyh24613 來自: 59.115.21.199 (01/15 16:22)