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MA131 ANALYSIS II: ASSIGNMENT 1

Solutions must be handed in by 14:00, Thursday, January 19th.

Each week homework will be handed out, due by noon on Thursday of the followingweek. Please submit your work to the supervisors pigeon loft in the Street before noon onThursdays. This work will be due in weeks 2 to 10.

Each assignment will consist of three sections. The problems in the first section arewarm-up problems which you should make sure you know how to do before attacking prob-lems in section B. You will notice that often there are hints in the questions from section Athat help in the later questions.

Your solutions should be easy to read and to the point. You should always write incomplete sentences. Your answers should include explanations and justifications, and 20%of your mark will be awarded for clarity. A clear, brief solution is best!

Section C problems consist of additional exercises which may be harder. You shouldalways spend some time thinking about these, as they may be referred to in the lectures oreven crop up in the exam!

A: Warm-up questions. (Not to hand in!)

Exercise 1. Sketch the graphs of the following functions, stating precisely where each graphcrosses the x-axis. In each case, calculate the image {f(x) : x R} of the function f.

(a) f : R R, f(x) = x2 3x + 2.(b) f : R R, f(x) = x2 4x + 4.(c) f : R R, f(x) = x2 + 1.(d) f : R R, f(x) = [x] = the largest integer n with n x (integer part of x)

Exercise 2. Find the value of the limits

(a) limm

cos()2m

(b) limm

cos(/3)2m

.

B: Problems to hand in.

Exercise 3. Sketch the graphs of the following functions, stating precisely where each graphcrosses the x-axis. In each case calculate the image {f(x) : x R} of the function f.

(a)

f(x) =x2(sin(1/x))2 if x = 0,0 if x = 0.

(b)

f(x) =

x

|x| if x = 0,0 if x = 0.

[ 6 marks]

Exercise 4. Show that for every x R, the limitlimm

cos(x)

2m

exists and find its value ( which depends on x). [ 3 marks]

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Exercise 5. (a) Define f : R R by

f(x) =

x2(sin(1/x))2 if x = 0,0 if x = 0.

Let = 1/100. Find a number > 0 such that |x| < implies |f(x) f(0)| < .Let = 1/1000. Find a number > 0 such that |x| < implies |f(x) f(0)| < .Determine for what values of > 0, it is possible to find a > 0 such that |x| < implies |f(x) f(0)| < .

(b) Define f : R R byf(x) =

x

|x| if x = 0,0 if x = 0.

Let = 1/100. Is it possible to find a > 0 such that|x| < implies |f(x) f(0)| < ?Justify your answer.

Let = 2. Is it possible to find a > 0 such that |x| < implies |f(x) f(0)| < ?Justify your answer.

Determine for what values of > 0, it is possible to find a > 0 such that |x| < implies |f(x) f(0)| < . [ 8 marks]

Exercise 6. Sometimes the words if...then are used in place of . For example,intead of saying For all > 0,there exists > 0 such that . . . , people sometimes writeIf > 0 then there exists > 0 such that . . . . Carefully write down the definitionof Cauchy sequence without using the word if. How many quantifiers are there? Nowcarefully write down the meaning of (xn) is not a Cauchy sequence without using theword not (see the discussion of negation on pages 17-19 of the online course lecture notes

at www.warwick.ac.uk/masbm/ana.html).[ 3 marks]

C: Extra Problems.

Exercise 7. Show that for every x R, the limit

limn

limm

cos(n!x)

2mexists and find its value ( which depends on x).

Exercise 8. Sketch the graphs of

x sin x and

x sin(1/x) paying special attention to thebehaviour of the graphs near the origin.

Exercise 9. Prove that the seriesn=1

sin(n2x)

n2

converges absolutely for all x R. If you have access to a computer or graphical calculator,plot the functions f1(x) = sin(x), then f2(x) =

2n=1

sin(n2x)n2 , and f3(x) =

3n=1

sin(n2x)n2 .

What about f(x) =n=1

sin(n2x)n2 ? (The series f(x) actually defines the famous Weier-

strass function which is everywhere continuous but nowhere differentiable.)