User:Mandarbhanushe/Books/FYBSc/Maths 1/Sem1
Contents
Calculus and Analytical Geometry
Semester I
Unit I : Preliminaries
Objectives of the Unit
After going through this unit you shall be learn about:
- Meaning and pronunciation of frequently used symbols.
- Additive and multiplicative properties of real numbers
- Definition of absolute value and intervals
- Definition and graphs of function
INTRODUCTION
Calculus is the mathematics of motion and change. Where there is a motion or growth, where variable forces are at work producing acceleration, calculus is what is applied there and this was the beginning of the subject.
Before we start with the actual course, it is essential that we fulfil some prerequisites. In this chapter, we revise the properties of real numbers and absolute values. Further, we define the notion of intervals and neighbourhoods that are special subsets of real numbers. We shall also revise functions, their types and graphs.
SYMBOLS AND NOTATIONS
As Mathematics is a language of symbols, in the course of Calculus, we will come across many symbols and notations. So let us first tabulate those, which are frequently used.
Symbol | Meaning (+ pronunciation) |
---|---|
[math] \forall [/math] | for all, for every |
[math] \exists [/math] | there exists |
[math] \in [/math] | belongs to |
[math] \notin [/math] | does not belong to |
[math] \ni [/math] | such that |
iff or [math]\Leftrightarrow[/math] | if and only if |
[math] \Rightarrow [/math] | implies |
ε | epsilon |
δ | delta |
nbd | neighbourhod |
N(a,δ) | δ − nbd of a or (a − δ,a + δ) |
N^{ * }(a,δ) | deleted δ − nbd of a or (a − δ,a + δ) − {a} |
[math] \mathbb{R}^{+} [/math] | positive real numbers |