## Details of MA1101 (Autumn 2019)

 Level: 1 Type: Theory Credits: 3.0

Course CodeCourse NameInstructor(s)
MA1101 Mathematics I Saugata Bandyopadhyay

Syllabus

LaTeX4Web 1.4 OUTPUT

\textcolordtePart I: Set Theory I

Standard Operations: unions \& intersections, various laws, complementation, cartesian product, symmetric difference.

Relations: various relations, equivalence classes, partition.

Mappings: injective, surjective, bijective, inverse of a function, examples, characteristic functions, step functions.

\textcolordtePart II: Number Systems

Construction of natural numbers: may mention Peanos Axioms.

Construction of integers: via equivalence relation on \mathbbN×\mathbbN.

Construction of rational numbers: via equivalence relation on \mathbbZ× \mathbbZ.

\textcolordtePart III: Propositional Calculus

Logical quantifiers: examples from set theory.

Negation: contrapositive statements involving various quantifiers.

\textcolordtePart IV: Methods of Mathematical Proof

Mathematical induction: examples including AM GM, partial sum of a geometric or arithmetic progression, tower of Hanoi, derangement.

Pigeonhole principle: examples including Dirichlets Approximation Theorem, arrangement of points on a square or sphere.

Proof by contradiction: examples including infinitude of primes, 2 is irrational, if 0 a <e for every e>0 then a=0.

\textcolordtePart V: Theory of Equations

Polynomials: degree of a polynomial, examples.

Properties of roots: definition, repeated roots, Fundamental Theorem of Algebra (statement only), relation between roots and coefficients, number and location of real roots (Descartess rule of sign, Sturms Theorem (statement only)), conjugate root Theorems, roots of cyclotomic polynomials and geometric description.

Methods of solving equations: Cardanos method, Ferraris method.

\textcolordtePart VI: Calculus

Basic notions: limit, continuity, differentiability, chain rule, Leibniz rule.

Mean Value Theorems: Rolles Theorem (statement only), Mean Value Theorem, Taylors Theorem of order 2, LHospitals rule.

Applications of derivatives: monotone function, maxima and minima, convex function.

\textcolordtePart VII: Geometry of Curves

Graphing curves: curve tracing, asymptotes.

Tangent \& normal: derivative and its geometric/physical meaning.

Geometric notions: radius of curvature, points of inflexion.

\textcolordtePart VIII: Basic Inequalities

Examples including, triangle inequality, p-power inequality, Youngs inequality, Cauchy-Schwarz inequality, H\"olders, Jensens, Minkowskis inequalities, AM-GM-HM.

References

Suggested Texts:

1. Apostol, T.M., Calculus I, Wiley India Pvt Ltd.

2. Apostol, T.M., Calculus II, Wiley India Pvt Ltd.

3. Artin, M., Algebra, Prentice-Hall of India, 2007.

4. Barnard, S. and Child, J.M., Higher Algebra, Macmillan, 1936.

5. Bartle, R.G., Sherbert, D.R., Introduction to Real Analysis, John Wiley \& Sons, 1992.

6. Denlinger, C.G., Elements of Real Analysis, Jones \& Bartlett Learning, 2010.

7. Halmos, P.R., Na\"ive Set Theory, Springer

8. Kreyszig, E., Advanced Engineering Mathematics (8th Edition), Wiley India Pvt Ltd, 2010.

9. Piskunov, N., Differential and Integral Calculus: Volume 1, CBS, 1996.

10. Polya, G., How to Solve It, Princeton University Press, 2004.

#### Course Credit Options

Sl. No.ProgrammeSemester NoCourse Choice
1 IP 1 Not Allowed
2 IP 3 Not Allowed
3 IP 5 Not Allowed
4 MR 1 Not Allowed
5 MR 3 Not Allowed
6 MS 1 Core
7 RS 1 Not Allowed
8 RS 2 Not Allowed