新加坡国立大学06-07数学分析考试.pdf

NATIONAL UNIVERSITY OF SINGAPORE FACULTY OF SCIENCE SEMESTER 2 EXAMINATION 2006-2007 MA3110SMathematical Analysis II (Version S) April/May 2007 — Time allowed : 2 hours INSTRUCTIONS TO CANDIDATES 1. This is a closed book examination. 2. This examination paper contains FIVE (5) questions and comprises FOUR (4) printed pages. 3. Answer ALL questions in this paper. Marks for each question are indicated at the beginning of the question. PAGE 2MA3110S Answer all the questions. Marks for each question are indicated at the beginning of the question. Question 1 [15 marks] (a) Let f be continuous on [a,b] and diff erentiable on (a,b). Suppose that f(a) = f(b) = 0. Prove that there exists α ∈ (a,b) such that f(α) + f0(α) = 0. (b) Let f and g be bounded real-valued functions on [a,b]. If f is integrable with respect to g on [a,b], prove that f is integrable with respect to g on [a,c] for all c ∈ (a,b). Question 2 [20 marks] (a) Let f be diff erentiable on [0,1] such that f(0) = 0. Suppose that there exists M 0 such that |f0(x)| ≤ M|f(x)| for all x ∈ [0,1]. Find f(x) for all x ∈ [0,1]. (b) Let (an) be a decreasing sequence of positive numbers such that limn→∞nan= 0. Prove that the series ∞ X n=1 ansin(nx) converges uniformly on [0, π 2 ]. (You may assume that | k X n=1 sin(nx)| ≤ π x for k = 1,2,··· and all x ∈ (0, π 2 ].) . . . – 3 – PAGE 3MA3110S Question 3 [20 marks] (a) Prove the following Abel’s theorem. Let (an) be a sequence of real numbers such that ∞ X n=0 anis convergent. Let f be the function defi ned on (−1,1) by f(x) = ∞ X n=0 anxn. Then lim x→1− f(x) = ∞ X n=0 an. (b) Let p and q be positive integers. Prove that Z 1 0 xp−1 1 + xq dx = 1 p − 1 p + q + 1 p + 2q − 1 p + 3q + ··· Question 4 [20 marks] (a) Let f,g,p be Riemann integrable on [a,b]. Suppose that f and g are both strictly increasing on [a,b] and p(x) 0 for all x ∈ [a,b]. Prove that ¡ Z b a p(x)f(x)dx¢¡ Z b a p(x)g(x)dx¢≤ ¡ Z b a p(x)dx¢¡ Z b a p(x)f(x)g(x)dx¢. (b) Let f be a real-valued function defi ned on a neighbourhood U of a point (a,b) ∈ R2. Suppose that the partial derivatives fx,fyand fxyexist in U and that fxyis continuous at (a,b). Prove that the partial derivative fyxexists at (a,b) and fyx(a,b) = fxy(a,b). . . . – 4 – PAGE 4MA3110S Question 5 [25 marks] (a) Let T : Rn→ Rnbe a linear operator with norm kTk 0. Let E = {x ∈ [0,1] : f(x) ≥ α 3 }. Prove that E contains fi nitely many mutually disjoint intervals I1,I2,···,Insuch that n X k=1 ‘(Ik) ≥ α 3M , where M = sup{|f(x)| : x ∈ [0,1]} and ‘(I) denotes the length of the interval I. (c) Evaluate lim t→1− (1 − t)1/2(1 + t + t4+ ··· + tn 2 + ···). (You may assume that Z ∞ 0 e−x 2 dx = √π 2 .) END OF PAPER 。