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INTRODUCTION

For the study of Calculus, some of the important topics from Algebra, Geometry, Trigonometry and Analytical Geometry are essential. They are briefly summarized below.

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Inequalities

It has been aptly remarked that the fundamental results of mathematics are often inequalities rather than equalities. Inequalities are basically related to positivity.

1. If a, b are positive, so are the product of ab and the sum of a + b.

2. If a is a number, then either a is positive or a is negative or a = 0, and these possibilities are mutually exclusive.

If a number is not positive and not 0, then it is negative. So if a is negative then - a is positive.

If a is positive and not zero, we write a > 0

If a is positive or zero, we write a ³ 0

If a - b > 0 it implies that a > b

We write a < 0 if - a > 0 and a < b if b > a

We write a ³ b if we want to say that a is greater than or equal to b

Thus 6 ³ 4 and 6 ³ 6 both are true inequalities.

We give below, in brief, some simple rules concerning inequalities.

1. If a > b and b > c then a > c

2. If a > 0 and b > 0 then ab > 0

3. If a > b and c > 0 then ac > bc

4. If a > b and c < 0 then ab < bc

5. If a < b and a, b, c Î R then a + c < b + c

6. If a > b and c > d then a + c > b + d but a - c b - d

7. If a > b then - a < - b

8. If a > b > 0 and c > d > 0 then ac > bd

9. If c > b > a > 0 then b - a > 0 and c - a > c - b

10. If b > a > 0 then 11. If x , y Î R then xy £ Absolute Value Or Modulus Of A Real Number

Notation:

The absolute value of any number a is denoted by | a|.

Definition 1

The absolute value |a| of a ÎR is defined to be a if a is positive or zero, and to be -a if a is negative.

i.e. | a | = Then | 2 | = 2. Here a = 2 which is greater than 0

| 0 | = 0. Here a = 0

and |- 2 | = - (- 2) ; Here a = - 2 which is less than 0

Definition 2

The absolute value |a | of a Î R is 0 if a Î R = 0. Otherwise |a | is the positive number of the set {a, -a}. Thus, if a = 4 then |a | is the positive of {4, - 4} i.e. 4 ; if a = -3 then |a | is the positive of the set {- 3, - (-3)} i.e. 3

For algebraic manipulations we give still a suitable definition.

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