課程名稱︰高等微積分二
課程性質︰數學系必修
課程教師︰陳俊全
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2011/4/26
考試時限（分鐘）：170分
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Choose 5 from the following 6 problems
1. Discuss the pointwise convergence and uniform convergence of
x
(a) fn(x) = ─────── , 0 < x < ∞
1 + n^2 x^2
n x
(b) gn(x) = Σ ─────── , 0 < x < ∞
k=1 1 + k^2 x^2
2. Suppose {fn} is an equicontinuous sequence of functions from a compact
set Ａ to Ｒ(實數), and {fn} converges pointwise on Ａ. Prove that {fn}
converges uniformly on Ａ.
3. Let f: [0,1] → Ｒ(實數) be continuous.
1
(a) Assume ∫ f(x) x^k dx = 0 for k = 0,1,2,...
0
Prove that f(x) = 0 on [0,1]
(b) Assume f(0) = f(1) = 0. Prove that there are polynomials Ｐn such that
sup |f(x) - x(1-x)Ｐn(x)| → 0 as n → ∞
x屬於[0,1]
∞ ∞
4. Prove that Σ a_k = Ａ(Ｃ,1) implies Σ a_k = Ａ(Abel).
k=0 k=0
5.
(a) What is the contraction mapping principle?
(b) Let Ａ = {(x,y)| 0≦x≦1, 0≦y≦1}, k(x,y) : Ａ→Ｒ(實數) be continuous,
and a(x) : [0,1]→Ｒ(實數) be continuous. Assume |k(x,y)| < 1 for each
(x,y) 屬於 Ａ. Prove that there is a unique continuous real-valued
function f(x) on [0,1] such that
1
f(x) = a(x) + ∫ k(x,y) f(y) dy.
0
6. Let f : Ｒ^2(實數^2)→Ｒ^3(實數^3) be defined by
f(x,y) = (x^3 y, g(x,y), x^4 y^2). Assume f is differentiable at (1,2).
(a) Prove that g(x,y) is continuous at (1,2).
dg dg
(b) Assume further that ─(1,2) = 5 and ─(1,2) = 1. (partial不好打 用d代替)
dx dy
Compute Ｄf(1,2).
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