School of Mathematics
ANALYSIS B
MATH10011
(Paper code MATH-10011)
加拿大MATH数学代写 On this examination, the marking scheme is indicative and is intended only as a guide to the relative weighting of the questions.
This paper contains two sections: Section A and Section B.
Each section should be answered in a separate booklet.
Section A contains FIVE questions and Section B contains TWO questions.
All SEVEN answers will be used for assessment.
Calculators are not permitted.
Candidates may bring four hand-written sheets of A4 notes written double-sided into the examination. Candidates must insert these four sheets into their answer booklet(s) for collection at the end of the examination.
On this examination, the marking scheme is indicative and is intended only as a guide to the relative weighting of the questions.
DO NOT REMOVE THIS PAPER FROM THE EXAMINATION ROOM.
Do not turn over until instructed.
Section A: Short Questions 加拿大MATH数学代写
A1. (a) (2+2 marks) Are the following statements true or false? You do not need to justify your answers.
(i) Any partition of [0, 1] contains at least two points.
(ii) Every step function is bounded.
(b) (4 marks) For each n ∈ N let ψn : [0, 1] → R be the step function given by
Show that
A2. (a) (4 marks) Let λ > 0. Prove that 
exp(λx) = ∞.
(b) (4 marks) Find the pointwise limit of the sequence (fn)n∈N where, for each n ∈ N,the function fn : (0, 1) → R is given by
A3. (a) (2+2 marks) Let f : [a, b] → R and c ∈ (a, b). Are the following statements true or false? You do not need to justify your answers.
(i) If f is regulated then 
f(x) exists.
(ii) If f(x) and 
f(x) both exist then f is continuous at c. 加拿大MATH数学代写
(b) (4 marks) Let f : [−1, 1] → R be given by
Show that 
f(x) does not exist.
A4. (a) (4 marks) State what it means to say that the power series
has radius of
convergence R where:
(i) R = ∞
(ii) 0 < R < ∞.
(b) (4 marks) Find the radius of convergence of the power series
A5. (a) (4 marks) State what it means for a series to be absolutely convergent.
(b) (4 marks) Give an example of a conditionally convergent series. You do not need to prove that your series is conditionally convergent.
Section B: Longer Questions 加拿大MATH数学代写
Please use a new booklet for this Part.
B1. (a) (5 marks) Let f : (0,∞) → R and suppose that f(x) is unbounded as x approaches 0.State what it means for the integral
to converge.
(b) (7 marks) Find t > 0 such that 
(c) (6 marks) State the integral test.
(d) (12 marks) Prove that
B2. (a) (4 marks) State the mean value theorem.
(b) (9 marks) Let f : R → R be differentiable on R and suppose there exists L ≥ 0
such that |f ′ (x)| ≤ L for all x ∈ R. Prove that f is uniformly continuous on R.
(c) (6 marks) Let f : R → R be twice differentiable on R and let a ∈ R. Write down the Taylor expansion of f around a of degree 1 with remainder R2written in Lagrange
form.
(d) (11 marks) Let f : R → R be given by
Prove that for any x > 1 there is ξ ∈ (1, x) such that f ′′(ξ) = xf(x).
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