# Differential Calculus

## Overview

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.

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.

## Syllabus

### COURSE LAYOUT

**Week 1**

1. Functions2. Inverse of Functions and Inverse Trigonometric Functions3. Limit of Functions – An Intuitive Approach

**Week 2**

4. Computing Limits - Limit laws5. The Precise Definition of Limit 6. Continuity

**Week 3**

7. Tangent Lines and Rate of Change8. The Derivative of a Function9. Techniques of differentiation and the Chain Rule

**Week 4**

10. Derivatives of trigonometric and inverse trigonometric functions11. Implicit Differentiation, Tangents and Normals and Logarithmic Differentiation12. Successive Differentiation, nth derivative of standard functions and Leibniz Theorem

**Week 5**

13.I – Local Linear Approximation of Functions of One VariableII - Differentials14. Increasing and Decreasing Functions and Concavity15. Extreme Values of Functions

**Week 6**

16. Extreme Values of Functions on Unbounded Intervals and Limit at Infinity17. The Rolles and mean value theorems18. Asymptotes

**Week 7**

19. Curve tracing in Cartesian Coordinates20. L’ Hospitals rule21. Sequences

**Week 8**

22. Techniques for Finding Limit of Sequences23. Infinite series24. Tests for Convergence of Series

**Week 9**

25. Alternating Series and Absolute Convergence of Series26. Power series and Radius of Converge27. Taylor and Maclaurin series of Functions

**Week 10**

28. Parametric Equations29. Differentiable Parametrized Curves, Second Derivative of Parametrized Curves30. Polar Coordinates

**Week 11**

31. Graphing Polar Curves32.I: Tracing of Cardiods and Families of RosesII: Tangent Lines to Polar Curves33.I: Curvature and Evolutes of Curves in Parametric EquationsII: Formula for Radius of Curvature of Cartesian Equations;III: Centre of Curvature and Circle of Curvature of Cartesian Curves

**Week 12**

34I: Curvature and Evolutes of Curves in Parametric EquationsII: Curvature in Polar coordinates35. Functions of Several Variables36. Limits and Continuity of Functions of Several Variables

**Week 13**

37. Partial Derivatives38. Partial Derivatives of Higher Order39. Differentiability

**Week 14**

40.I: The Chain Rule of functions of more than one variableII: Euler's theorem on homogeneous functions