# A Little Bit of Calculus

#### Author:**Stephen Vadakkan**

This is not a Text Book, Primer or Guide Book on Calculus. It is just an introduction to the fundamental concepts. Particular words and symbols are used to express these concepts. These concepts and words take time to sink in. It is hoped that in studying this booklet the student will become familiar with the concepts, terminology, notation and the kind of calculations that can be done: what to look for and what to expect.

The fundamental concepts in Calculus are INFINITESIMAL and LIMIT which are used to develop the concepts of INSTANT, INSTANTANEOUS, CONTINUITY and DIFFERENTIABLITY. Once these concepts are in place we can talk of functions that are WELL BEHAVED: SINGLE VALUED, CONTINUOUS and DIFFERENTIABLE. We can then do two very beautiful calculations or operations:

- Given a SINGLE VALUED, CONTINUOUS and DIFFERENTIABLE function that expresses CHANGE we can DIFFERENTIATE the function and get the new function that expresses the INSTANTANEOUS RATE OF CHANGE.
- Given the function that expresses the INSTANTANEOUS RATE OF CHANGE, we can INTEGRATE the function and get the new function that expresses CHANGE.

Working with polynomials is easy since they are CONTINUOUS everywhere. The fundamental concepts and calculations (Differentiation and Integration) can be taught with ease and clarity. Getting more information about the function (increasing, decreasing, maximum, minimum, inflexion) from its Higher Order Derivatives can also be shown.

Simple examples have been chosen to illustrate the concepts. All rigorous details under which calculations are done have been avoided. The theory of Real Analysis and Calculus deals with this. Rigour does not necessarily mean clarity and ease of understanding.

## Contents

### Part 1 : CONTINUITY

- NUMBERS AND THE NUMBER LINE
- TENDS TO and LIMIT
- REALS, COMPLETE, CONTINUOUS
- INFINITESIMALS
- δx and INSTANT
- SINGLE VALUED FUNCTIONS
- LIMIT of a Function
- VALUE versus LIMIT
- Analysis of Limits
- CONTINUITY of a Function

### Part 2 : DIFFERENTIATION

- INSTANTANEOUS RATE OF CHANGE of y(t)
- INSTANTANEOUS RATE OF CHANGE of f(x)
- INSTANTANEOUS RATE OF CHANGE of x^n
- Differentiation from FIRST PRINCIPLES
- Tables and Rules
- Units of Measure
- DIFFERENTIABILITY

### Part 3 : ANALYTICAL GEOMETRY

- FIRST DERIVATIVE = Slope of Tangent
- Angle of Intersection
- Decreasing, MINIMUM , Increasing
- Increasing, MA XIM UM , Decreasing
- MAXIMA and MINIMA
- Points of INFLEXION

### Part 4 : INTEGRATION

- F(x) = ANTIDERIVATIVE {f(x)}
- F(x) = ANTIDERIVATIVE {f(x)} = ∫f(x)dx
- F(x) = area under f(x) = ∫f(x)dx
- Direction of Integrationī and CHANGE in F(x)
- Constant of Integration
- Area under f(x) and Plotting FC (x)
- INDEFINITE and DEFINITE INTEGRAL
- INTEGRABLE
- Applications of Integration