MTH 101 Single-variable Calculus
(August-November 2018)
Instructor: Dr Anupam Singh
Audience: 1st year BS-MS students at IISER Pune
Tutors: Dilpreet Kaur, S. Taruni, Debaprasanna Kar, Kartik Roy and Rahul Deshmukh
Schedule: Monday and Tuesday 10:15 - 11:15
Tutorial: Thursday (B3, B4) 10:15 - 11:15, (B1, B2) 3:15 - 4:15
Quizzes will be held at 11 AM on Thursdays. Please see the "weekly topics covered" section below.
Evaluation: Continuous evaluation! Mid sem 30%, end-sem 30%, Quiz (total 8) 5% each.
Prerequisite: None from mathematics but a lot from individuals such as willingness to work hard, pay attention to what is going on in the class, do home-works etc.
Goal of the course: To learn functions of one-variable and its application. We also plan to learn how to use SAGEmath to learn calculus and other structures in mathematics.
Proposed content:
Properties of real numbers, least upper bound axiom, convergent sequences, limits of functions.
Continuity, intermediate value theorem, differentiability, product and chain rules, mean value theorem, Taylor's Theorem and Taylor's Expansion, maxima and minima.
Riemann integration, fundamental theorem of calculus, integration by parts and change of variables, applications to area and volume.
Textbooks:
Calculus Vol. 1 and 2: T.M. Apostol
Calculus: M. Spivak
Calculus: J. Stewart
Calculus and Analytic Geometry: G.B. Thomas, R. Finney
References: Those students who plan to take Mathematics as their major, it is strongly recommended that they follow one or more of the following books.
Principles of mathematical analysis : Rudin
Problems and Solutions for undergraduate analysis : Shakarchi
Mathematical analysis : Apostol
The elements of real analysis : Bartle
Computational Softwares : We plan to learn how to use SAGEmath. http://www.sagemath.org/
Weekly topics covered
06 August 2018 - Introduction, Natural numbers, Integers and Arithmetic
07 August 2018 - Rational numbers, concept of "small", How close are rationals?
09 August 2018 - SAGEmath
13 August 2018 - Field axioms, Order axiom on reals
14 August 2018 - Completness axiom and real numbers
16 August 2018 - Tutorials
20 August 2018 - Functions, Neighbourhood of a point, Limit of a function at a point.
21 August 2108 - Examples.
21 August 6 PM: extra class for the students who have joined late.
23 August 2018 - Tutorial and QUIZ at 11 AM
27 August 2018 - Continuous function at a point, examples
28 August 2018 - sum and product of limits, continuous functions, examples
30 August 2018 - Tutorial and QUIZ at 11 AM
3 September 2018 - Composition of functions, examples
4 September 2018 - Sign preserving property of continuous functions, Balzano's theorem, Intermediate value theorem
6 September 2018 - Tutorial and QUIZ at 11 AM
10 September 2018 - Derivative of a function at a point, examples
11 September 2018 - Interpretation: geometric, tangent etc
13 September 2018 - Holiday
17 September 2018 - No lecture due to Dean's office declaring taime-table change
18 September 2018 - Algebra (arithmetic operations) of functions
20 September 2018 - Tutorial and Quiz at 11 AM
Mid Sem exam 25 September 10AM
02 October 2018 - Holiday
04 October 2018 - Tutorial
08 October 2018 - Chain rule
09 October 2018 - Trailer to multi-variable differentiation, maxima, minima
11 October 2018 - Tutorial and Quiz at 11 AM
Interesting documentary "The birth of Calculus"
Clarification: These theorems and it's applications are part of the course. Although you won't be required to write proofs in the exams.
15 October 208 - Continuous functions on closed interval, boundedness theorem, extreme-value theorem (with proofs)
Rolle's theorem
16 October 2018 - Mean-value theorem, derivative-test for maxima and minima (with proofs)
18 October 2018 - Holiday
17-21 mid-sem break
22 October 2018 - Axiomatic definition of area, partitions of [a,b], step functions
23 October 2018 - Integral of a step function, refinement of a partition, properties
25 October 2018 - Tutorial and Quiz at 11 AM
29 October 2018 - Integral of a bounded function on closed interval, lower integral and upper integral, Riemann sum
30 October 2018 - Examples, integrability of monotone functions
01 November 2018 - Tutorial and Quiz at 11 AM
05 November 2018 - small-span theorem/uniform continuity, integrability of continuous functions on [a,b]
06 November 2018 - first and second fundamental theorem
08 November 2018 - Tutorial and Quiz at 11 AM
12 November 2018 - Taylor's polynomial, approximation of functions by polynomial
13 November 2018 - Power series, Taylor's series.
15 November 2018 - Tutorial and Quiz at 11 AM
End Sem Exam - 30 November 2018 at 3PM