Homework 1
Due Friday, January 6
Note : Only problems 1-4 involve something you actually need to submit, so please
dont freak out about having to do this by Friday.
Note : I highly recommend that you familiarize yourself with the notation. You
should be a

Homework 21
Due Friday, March 2
Reading exercie : Read page the last one-fourth of page 427 (starting
from The behavior of.) and the rst two-thirds of page 428 about the
remainder terms for log(1 + x) and arctan x. Especially note the point about
Taylor p

Homework 24
Never due, but highly recommended
1. Write a complete proof of Theorem 3 that I sketched in class. This means
you should prove that (i) Any Cauchy sequence is automatically bounded;
and (ii) If a subsequence of a Cauchy sequence cfw_an conver

Homework 23
Due Wednesday, March 7
Clarication regarding HW 22 : Perhaps I was sloppy in saying that #4
was to be proved using #3 - #3 can only be used as a hint for #4. And please
dont use Theorem 3 to do #4 : I want #4 to be something like a motivation

Homework 21
Due Wednesday, February 29
1. Reading : please read pages 424 and 425 on how to approximate values
of sin using Taylors theorem.
2. Page 436, #15.
Hint : This is easy. Just use Theorem 4, it is trivial to compute the
relevant derivatives - the

Homework 22
Due Wednesday, March 7
1. Prove that :
n517
lim
= 1.
n n517 + 1
Hint : Look at the computation on around the middle of page 454 of
the text book. Use the facts stated at the bottom of that page, which
I had stated in class.
2. Prove that :
lim

MA 162, Section 43
Final Examination
Winter 2011
Name:
Instructor: Vadakkumkoor Sandeep Varma
Instructions
1. There are 8 problems on 14 pages (including Rough/Extra sheets).
2. Anything you write needs justication.
3. No books or notes are allowed.
4. Un

Math 162, Section 21
Some problems to practise for the nal examination
These problems are intended to help you test and strengthen your conceptual grasp of
Lessons 13 - 22 (minus 16, 17 and 21), and there is no guarantee that the exam problems
will be sim

MA 162
Exam 1
Winter 2011
Name:
Student ID #:
Instructor: Vadakkumkoor Sandeep Varma
Instructions
1. Write your name on each page. Well, at least on the rst page.
2. There are 6 problems on 7 pages.
3. Anything you write needs justication, unless explicit

Math 162, Section 21
Some problems to practise for Hour Test 1
Note : These problems are intended to help you test and strengthen your
conceptual grasp of Lessons 13-14 and part of Lesson 15, and there is no
guarantee that the exam problems will be simila

MA 162
Hour Test 2
Winter 2011
Name:
Instructor: Vadakkumkoor Sandeep Varma
Instructions
1. Write your name on each page. Well, at least on the rst page.
2. There are 4 problems on 8 pages, including the rough sheet.
3. For problems from Chapters 15, 18 a

Math 162, Section 43
Some problems to practise for Hour Test 2
Note : These problems are intended to help you test and strengthen your
conceptual grasp of Lessons 15, 18 and 19 and part of Lesson 20, and there
is no guarantee that the exam problems will b

MATH 161, SHEET 5: CONTINUOUS FUNCTIONS
We recall from Script 1 that if f : A B is a function and Y B, then the preimage of
Y under f is the set
f 1 (Y ) = cfw_a A | f (a) Y .
We now give some basic properties of preimages.
Lemma 5.1. Let X C and f : X C.

SHEET 9: LIMITS and DERIVATIVES of FUNCTIONS
Throughout this sheet, we let f : A R be a real valued function with domain A R.
Definition 9.1. Let a R be such that there exists a region R containing a with R\cfw_a A.
A limit of f at a point a R is a number

MATH 162, SHEET 7: THE REAL NUMBERS
This sheet is concerned with proving that the continuum R is an ordered field. Addition
and multiplication on R are defined in terms of addition and multiplication on Q, so we will
use and for addition and multiplicatio

Homework 19
Due Wednesday, February 29
1. Let n N. Find P2n,(3/2),cos .
2. Let for all x, f (x) = x log x x + 1, and let n N. Find Pn,1,f .
Note : The pattern will start forming only from f (2) onwards.
3. Let f (x) = 1/(1 + x) for all x = 1. By now, you

Note : I typed this in a hurry, so there may be typos. Email me if
something is not clear. If I discover any problem by the end of problem
session tonight I will email you.
Homework 18
Due Wednesday, February 22
1. Suppose f is a function, a R, n N and Pn

Homework 15
Due Wednesday, February 15
Note : As I said, in thse kinds of questions you are allowed to be informal.
1. Page 381 #1 (x).
2. Page 381/382 #2 (viii), (ix).
3. Page 382 #3 (iii), (v), (vii), (viii).
4. Page 383 #4 (iv).
5. Page 388, #19.
2
Not

Homework 2
Due Wednesday, January 11
1. Prove the
pose f is
number I
such that
that :
following easy consequence of the proof of Theorem 2. Supa function bounded on [a, b]. Suppose there exists a real
such that : for all > 0, there exists a partition P of

Homework 3
Due Wednesday, January 11
1. Page 274, #14.
Hint : Of course, use Theorem 2; prove that U (f, P ) L(f, P ) is
equal to the corresponding expression for f (x c) and the partition P
given in the hint of the text book.
2. Prove Theorem 6 (of our c

Homework 4
Due on Friday, January 13
1. If a function f is bounded on a closed interval [a, b] and also continuous at all
but nitely many points in [a, b] show that f is integrable on [a, b].
Hint : It will help to use Theorem 4 to reduce to the case wher

Homework 6
Due Friday, January 20
Note :
Only four problems and none has parts, but learn these very
well. They are very important. Also, learn the exact statements of the two
fundamental theorems of calculus, and their informal statements - you should
kn

Homework 6
Due Wednesday, January 19
Note : This time more problems than usual, because of the long weekend.
Pre-Homework
Read the statement of Theorem 14.1 one hundred and twenty ve times,
with attention to detail. Remember, what the theorem First Fundam

Homework 7
Due Wednesday, January 25
Note : Hopefully this homework give you a partial review for some of
the topics for Hour Test 1. As I told you, we are discussing trigonometric
functions using area and not length, though the latter may be more natural

Homework 8
Due Wednesday, January 25
Note : We proved, or at least sketched a proof, that cos (x) = sin x and
sin (x) = cos x for all x. This is the basis for much in the chapter, you will
mostly not need how exactly we arrived at this. Give justications,

Homework 11
Due Friday, February 3
1. Do Page 319, #15 (a), (b), (c) - but no need to submit.
2. Page 319 #33 (a), (b), (c).
Note : I was just trying to give you an idea of these techniques in
class, and what I wrote down could contain mistakes.
3. Page 3

Homework 9
Due Wednesday, February 1
Note : The problems may be hard - not because of deeper mathematics,
but because these demand more intimate familiarity with the values of sin
and cos on various intervals. I am giving these since I think it is imperat

Homework 14
Due Wednesday, February 15
1. Dene the function f on R by dening, for x R,
2
f (x) =
e1/x , if x > 0, and
0,
if x 0.
Prove that for all k N, the function f is k-times dierentiable on
R (i.e., that f (k) has domain R) and f (k) (0) = 0. Do this

Homework 12
Due Wednesday, February 8
1. Page 351, #1 (i), (iv) (x).
Note : For (x), I havent yet dened xx for you - it is dened as xx =
exp(x log x) - this will become clear in Mondays class. For the time
being take this for granted. You cant use power r

MATH 162, SHEET 6: THE FIELD AXIOMS
We will formalize the notions of addition and multiplication in structures called fields. A
field with a compatible order is called an ordered field. We will see that Q and R are both
examples of ordered fields.
Definit