Calculus I | Department of Mathematics

Calculus I

Core course for B.Sc. (Research) programs in Mathematics, Physics and Economics. Optional course for B.Sc. (Research) Chemistry.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Class XII mathematics or MAT 020 (Elementary Calculus)

Overview:  This course covers one variable calculus and applications. It provides a base for subsequent courses in advanced vector calculus and real analysis as well as for applications in probability, differential equations, optimization, etc. One of the themes of the course is to bring more rigour to the formulas and techniques students may have learned in school.

Detailed Syllabus:

  1. Real Number System: The axioms for N and R, mathematical induction.
  2. Integration: Area as a set function, integration of step functions, upper and lower integrals, integrability of bounded monotone functions, basic properties of integration, polynomials, trigonometric functions.
  3. Continuous Functions: Functions, limits, continuity, Intermediate Value Theorem, Extreme Value Theorem, integrability of continuous functions, Mean Value Theorem for integrals.
  4. Differentiation: Tangent line, rates of change, derivative as function, algebra of derivatives, implicit differentiation, related rates, linear approximation, differentiation of inverse functions, derivatives of standard functions (polynomials, rational functions, trigonometric and inverse trigonometric functions), absolute and local extrema, First Derivative Test, Rolle's Theorem, Mean Value Theorem, concavity, Second Derivative Test, curve sketching.
  5. Fundamental Theorem of Calculus: Antiderivatives, Indefinite Integrals, Fundamental Theorem of Calculus, Logarithm and Exponential functions, techniques of integration.
  6. Polynomial Approximations: Taylor polynomials, remainder formula, indeterminate forms and L'Hopital's rule, limits involving infinity, improper integrals.
  7. Ordinary Differential Equations: 1st order and separable, logistic growth, 1st order and linear.

References:

  1. Calculus, Volume I, by Tom M Apostol, Wiley.
  2. Introduction to Calculus and Analysis I by Richard Courant and Fritz John, Springer
  3. Essential Calculus – Early Transcendentals, by James Stewart. Cengage, India Edition.
  4. Calculus with Analytic Geometry by G F Simmons, McGraw-Hill

Past Instructors: Amber Habib, Debashish Bose

Course Code: 
MAT101
Course Credits: 
4.00
Department: 
Course Level: