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



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
34
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.  


Course certificate

30% for in Course Assessment & 70% of End-term Proctored Exam.


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