Differential Calculus

By Dr Bijumon Ramalayathil | Mahatma Gandhi College, Iritty, Keezhur PO, Kannur Dt – 670 703, Kerala

Learners enrolled: 2956

The course entitled “Differential Calculus” deals with the basic aspects differential calculus. The contents of this course are inevitable for many branches of sciences. The students of Mathematics, Physics, Chemistry, Computer Science, Statistics, etc., are equally benefited with this course as a stepping stone to the broad areas of Calculus.

The objective of this course is to familiarize students with important concepts coming under the branch “Differential Calculus” and to develop strong foundations on these concepts. Upon successful completion of this course, the students are expected to:

1. Familiarize with the concept of Limit and Continuity (Both informal and ε- δ definition)

2. Learn through examples Successive differentiation and Leibniz’s theorem.

3. Study Rolle’s theorem, Mean Value theorems through examples.

4. Study the concepts tangents and normals.

5. Familiarize with methods of finding Curvature, Asymptotes and Singular points.

6. Learn methods of Tracing of curves in Cartesian, Parametric and Polar forms

7. A study on sequences and series

8. A detailed study on Taylor’s theorem, Taylor’s series and Maclaurin’s series through examples.

9. Familiarise with functions of several variable.

10. Study Euler’s theorem on homogeneous functions and some of its mathematical applications.

Summary

Course Status :	Completed
Course Type :	Core
Duration :	15 weeks
Category :	Mathematics
Credit Points :	4
Level :	Undergraduate
Start Date :	15 Jun 2020
End Date :	28 Aug 2020
Enrollment Ends :	31 Aug 2020
Exam Date :	15 Nov 2020 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week 1

1. Functions

2. Inverse of Functions and Inverse Trigonometric Functions

3. Limit of Functions – An Intuitive Approach

Week 2

4. Computing Limits - Limit laws

5. The Precise Definition of Limit

6. Continuity

Week 3

7. Tangent Lines and Rate of Change

8. The Derivative of a Function

9. Techniques of differentiation and the Chain Rule

Week 4

10. Derivatives of trigonometric and inverse trigonometric functions

11. Implicit Differentiation, Tangents and Normals and Logarithmic Differentiation

12. Successive Differentiation, nth derivative of standard functions and Leibniz Theorem

Week 5

13.

I – Local Linear Approximation of Functions of One Variable

II - Differentials

14. Increasing and Decreasing Functions and Concavity

15. Extreme Values of Functions

Week 6

16. Extreme Values of Functions on Unbounded Intervals and Limit at Infinity

17. The Rolles and mean value theorems

18. Asymptotes

Week 7

19. Curve tracing in Cartesian Coordinates

20. L’ Hospitals rule

21. Sequences

Week 8

22. Techniques for Finding Limit of Sequences

23. Infinite series

24. Tests for Convergence of Series

Week 9

25. Alternating Series and Absolute Convergence of Series

26. Power series and Radius of Converge

27. Taylor and Maclaurin series of Functions

Week 10

28. Parametric Equations

29. Differentiable Parametrized Curves, Second Derivative of Parametrized Curves

30. Polar Coordinates

Week 11

31. Graphing Polar Curves

32.

I: Tracing of Cardiods and Families of Roses

II: Tangent Lines to Polar Curves

33.

I: Curvature and Evolutes of Curves in Parametric Equations

II: Formula for Radius of Curvature of Cartesian Equations;

III: Centre of Curvature and Circle of Curvature of Cartesian Curves

Week 12

I: Curvature and Evolutes of Curves in Parametric Equations

II: Curvature in Polar coordinates

35. Functions of Several Variables

36. Limits and Continuity of Functions of Several Variables

Week 13

37. Partial Derivatives

38. Partial Derivatives of Higher Order

39. Differentiability

Week 14

40.

I: The Chain Rule of functions of more than one variable

II: Euler's theorem on homogeneous functions

Books and references

Reference Books:

1. H. Anton, I. Birens and S. Davis, Calculus, John Wiley and Sons, Inc., 2002.

2. G.B. Thomas and R.L. Finney, Calculus, Pearson Education, 2007.

Web links will be provided along with modules.

Instructor bio

Dr Bijumon Ramalayathil has 19 years of teaching experience both in UG and PG level. He got First Rank in Kerala Public Service Examination for the recruitment of Lecturers in 2004. Presently he is the Associate Professor and Head of Post Graduate Department of Mathematics, Mahatma Gandhi College, Iritty, Kerala. He is the course coordinator of a couple of MOOC courses on SWAYAM platform.

He is the PI of the following Two MOOC (SWAYAM) Courses:

• “Algebra and Trigonometry” for CEC through EMMRC University of Calicut (July-September 2018)

• Differential Calculus for CEC through EMMRC University of Calicut (July-October 2019) . More than 4800 learners joined for t he course.

Also,

• Subject Expert of more than 80 e-content modules (2011-2012 for CEC) on the Topics Basic Algebra, Abstract Algebra and Vector Analysis and Geometry.

• Subject Expert of DTH programme (for CEC) on Complex Analysis

• Translation to Malayalam work of MOOC course on Algebra and Trigonometry (2020)

• Developed study materials (BSc and MSc level) for many Universities in Kerala for their School of Distance Education.

• Awarded Ph.D. from the University of Calicut in 2010 for thesis titled On the Study of Frame Multiresolution Anaysis in the Superspaces under the guidance of Prof Dr. M. S. Balasubramani - University of Calicut. His area of interest is Functional Analysis.