M102 - Mathematics II

Concept of ordered field, Bounds of a set, ordered completeness axiom and characterization of R as a complete ordered field. Archimedean property of real numbers. Modulus of real numbers, intervals, neighbourhood of a point.  Sequences of Real Numbers: Definition and examples, Bounded sequences, Convergence of sequences, Uniqueness of limit, Algebra of limits, Monotone sequences and their convergence, Sandwich rule.  Series: Definition and convergence, Telescopic series, Series with non-negative terms. Tests for convergence [without proof]: Cauchy condensation test, Comparison test, Ratio test, Root test, Absolute and conditional convergence, Alternating series and Leibnitz test.  Limit of a function at a point, Sequential criterion for the limit of a function at a point. Algebra of limits, Sandwich theorem, Continuity at a point and on intervals, Algebra of continuous functions. Discontinuous functions, Types of discontinuity.  Differentiability: Definition and examples, Geometric and physical interpretations, Algebra of differentiation, Chain rule, Darboux Theorem, Rolle’s Theorem, Mean Value Theorems of Lagrange and Cauchy. Application of derivatives: Increasing and decreasing functions, Maxima and minima of functions. Higher order derivatives, Leibnitz rule, L’Hopital rule.