MA 105 Theorems / Proofs
Manish Goregaokar
August 15, 2012
1
Sequences
Definition: A sequence in a set X is a function f : N X, that is, a function from the
natural numbers to X
Definition: A sequence is said to be a monotonically increasing sequence if a
Chapter 2
Tutorial sheets
2.1. Tutorial sheet No. 0:
Revision material on Real numbers
Mark the following statements as True/False:
(1) + and are both real numbers.
(2) The set of all even natural numbers is bounded.
(3) The set cfw_x is an open interval
MA 105 Proofs
Manish Goregaokar
August 15, 2012
1
Sequences
Lemma: Every convergent sequence is bounded
Proof: Suppose an converges to l. Choose = 1. N N such that |an l| < 1 n > N .
In other words, l 1 < an < l + 1 n > N , which gives |an | < l + 1 n > N
MA 105 Theorems
Manish Goregaokar
August 15, 2012
1
Sequences
Definition: A sequence in a set X is a function f : N X, that is, a function from the
natural numbers to X
Definition: A sequence is said to be a monotonically increasing sequence if an an+1 n
Problems
September 17, 2013
1. If an an+1
P 0 is a sequences of real numbers, prove that
verges iff n 2n a2n coverges.
P
n
an con-
2. If an+1 = sin(an ), prove that limn an = 0
Rp
3. Find limn 02 sinn (x) + cosn (x)dx.
4. Let (an ) and (bn ) be two sequen
STUDENTS OF MA 105:
Here is some information and a list of do's and dont's regarding the final
exam in addition to the ones we have regularly been following (no calculators,
no cell phones).
1. The exam starts at 1400 and will terminate at 1630 hrs. (2-4:
MA 105 D4 Lecture 10
Sandip Singh
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
Recap: Sequential criterion for continuity
Recap: The ratio test for the convergence of a series
Towards Taylors Theorem - higher derivatives
Darboux integration
S
MA 105 D4 Lecture 1
Sandip Singh
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
The course objectives
Sequences
Limits of sequences
Aims of the course
First, welcome to IIT Bombay.
I
To briefly review the calculus of functions of one variable a
MA 105 D4 Lecture 6
Sandip Singh
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
Continuity
More about continuous functions
Continuity - the definition
Definition: If f : [a, b] R is a function and c [a, b], then f is
said to be continuous at th
MA 105 D4 Lecture 8
Sandip Singh
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
Maxima and minima
Properties of differentiable functions
Beyond the first derivative
The definition
Recall that f : (a, b) R is said to be differentiable at a point
MA 105 D4 Lecture 2
Sandip Singh
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
Recap
More examples and formul for limits
Sequences and limits
Definition: A sequence in a set X is a function f : N X , that is,
a function from the natural number
MA 105 D4 Lecture 4
Sandip Singh
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
Cauchy sequences: the definition
Limits of functions
Cauchy sequences
As we saw last time, it is not easy to tell whether a sequence
converges or not because we hav
MA 105 D4 Lecture 9
Sandip Singh
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
The Second Derivative Test
Concavity and convexity
Towards Taylors Theorem - higher derivatives
Back to maxima and minima
We will assume that f : [a, b] R is a cont
MA 105 D4 Lecture 11
Sandip Singh
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
Darboux integration
Riemann integration
Partitions
Definition: Given a closed interval [a, b], a partition P of [a, b] is
simply a collections of points
P = cfw_a
MA 105 D4 Lecture 7
Sandip Singh
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
Sequential continuity
Functions of several variables
Differentiability
The function sin x1
Let us look at Exercise 1.13 part (i).
Consider the function defined as f
MA 105 D4 Lecture 3
Sandip Singh
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
Recap
Convergence without explicit knowledge of the limit
Cauchy sequences: the definition
Formul for limits
If an and bn are two convergent sequences then
1. limn
MA 105 D4 Lecture 5
Sandip Singh
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
Recap
Odds and ends about limits
The rigourous definition of the limit of a function
Since we have already defined the limit of a sequence rigourously,
it will not
MA 105 D1 Lecture 5
Ravi Raghunathan
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
The derivative
Arnolds problem
Maxima and minima
Properties of differentiable functions
Beyond the first derivative
The defintion
For now, if you did not unders
MA 105 D1 Lecture 3
Ravi Raghunathan
Department of Mathematics
July 31, 2017
Odds and ends
Limits of functions
Odds and ends about limits of functions of a real variable
Continuity
Zenos paradox animated
Achilles and the Tortoise
Formul for limits
If an a
MA 105 D1 Lecture 2
Ravi Raghunathan
Department of Mathematics
July 27, 2017
Tutorial problems for August 2
The numbers refer to the tutorial sheet. E.g. 1.1 (iii) means
Problem no. 1 part (iii) of the first tutorial sheet.
1.1(iii), 1.2(iv), 1.3(i), 1.5(
MA 105 D1 Lecture 4
Ravi Raghunathan
Department of Mathematics
August 3, 2017
Recap
More about continuous functions
Arnolds problem
Functions of several variables
Set Cardinality
Continuity - the definition
Definition: If f : [a, b] R is a function and c
MA 105 D1 Lecture 1
Ravi Raghunathan
Department of Mathematics
July 24, 2017
About the course
Sequences
Limits of sequences
Course objectives
Welcome to IIT Bombay.
I
To help the students achieve a better and more rigourous
understanding of the calculus o
MA 105 D1 Lecture 6
Ravi Raghunathan
Department of Mathematics
Autumn 2017, IIT Bombay, Mumbai
Maxima and Minima
1. The Extreme Value Theorem
2. Fermats theorem
3. The second derivative test
Local and Global extrema.
The continuity of the first derivative
0
1a
1b
1
2a
2b
1
3a
3b
Absolute value function
4a
D: (-,+)
R: [0,+)
4b
Adding or subtracting
antiderivatives
5a
5b
Alternative Definition of a
Derivative
6a
f '(x) is the limit of the following difference quotient
as x approaches c
6b
Antiderivative of f
0
1a
1b
1
2a
2b
1
3a
3b
Absolute value function
4a
D: (-,+)
R: [0,+)
4b
Adding or subtracting
antiderivatives
5a
5b
Alternative Definition of a
Derivative
6a
f '(x) is the limit of the following difference quotient
as x approaches c
6b
Antiderivative of f
0
1a
1b
1
2a
2b
1
3a
3b
Absolute value function
4a
D: (-,+)
R: [0,+)
4b
Adding or subtracting
antiderivatives
5a
5b
Alternative Definition of a
Derivative
6a
f '(x) is the limit of the following difference quotient
as x approaches c
6b
Antiderivative of f
0
1a
1b
1
2a
2b
1
3a
3b
Absolute value function
4a
D: (-,+)
R: [0,+)
4b
Alternative Definition of a
Derivative
5a
f '(x) is the limit of the following difference quotient
as x approaches c
5b
cf'(x)
6a
6b
Cosine function
7a
D: (-,+)
R: [-1,1]
7b
cos(x)
8a