ABOUT THE COURSE:Calculus is known to enhance and develop calculations skills using abstract concepts and logics. This course is intendent to develop logic reasoning and enhance student skills. In this course I will focus on functions and its limiting values, slopes and their various forms used in daily life. This course will consist of basic facts about mathematics and their notations and terminology, basic geometrical figures and graphical displays and important facts. It also covers Taylors theorem which is viatl for function approximations and its uses.INTENDED AUDIENCE: BSc. (Mathematics Major/Minor) Sem-I and Sem-II. Engineering B.S./ (B.Tech. ) Sem-I and IIPREREQUISITES: Upto 12 Std Basic Maths

Syllabus

Week 1: Introduction to functions, Domain and Range, Definition of real sequence, Bounded sequence, Limit of a sequence, convergent sequence. Every convergent sequence is bounded. A sequence can have at most one limit. Algebra of limits of sequences.
Week 2:Monotone sequence: Least upper bound and greatest lower bound(definition and example only). Monotone sequence. Every monotonic increasing and bounded above sequence is convergent. Every monotonic decreasing and bounded below sequence is convergent. Cauchy sequence (definition and example only).
Week 3:Subsequence: Definition and examples. If a sequence converges to a limit, then every subsequence of that sequence also converges to the same limit. Every subsequence of a monotone increasing (decreasing) sequence of real numbers is monotone increasing (decreasing). A monotone sequence of real numbers having convergent subsequence with a limit, is convergent with the same limit. Every sequence of real numbers has a monotone subsequence. Bolzano-Weierstrass theorem.
Week 4:Infinite Series, Comparison Test (First and Second Type), D’Alembert’s ratio test, Cauchy’s root test, Raabe’s test.
Week 5:Limit and limit points. ε- δ definition of Limit and Continuity (ε- δ definition), types of Discontinuity, Examples and Problems on Continuity
Week 6:Differentiability of functions, Darboux Theorem, Rolle’s theorem
Week 7:Lagrange’s and Cauchy’s mean value theorems and their geometrical interpretations.
Week 8:Successive differentiation, Leibnitz’s theorem
Week 9:Taylor’s theorem with Lagrange’s and Cauchy’s forms ofremainder, Taylor’s series, Maclaurin’s series of sin x, cos x, exp(x) , log(l+x), (1+x)m
Week 10:Maxima and minima: global minima and maxima, local minima and maxima. First derivative test for extrema. Higher order derivative test for extrema.
Week 11:Integral Calculus: Definite integral as a limit of sum.
Week 12:Asymptotes, Curve tracing: Definition of asymptote, asymptotes of the general algebraic curve, two parallel asymptotes, asymptotes parallel to x-axis and y-axis. Curve tracing: Curve tracing of Cartesian and polar curves using symmetry, concavity and convexity, point inflection, asymptotes, singular points, tangent at origin, multiple points, position and nature of double points.