Calculus I Essentials

CHAPTER 1

FUNDAMENTALS

1.1 NUMBER SYSTEMS

The real number system can be broken down into several parts and each of these parts have certain operations which can be performed on them. First, let us define the components of the real number system.

The natural numbers, denoted N, are 1, 2, 3, 4, .... The integers, denoted Z, are ...-3,-2,-1,0,1,2,3,.... The rational numbers, denoted Q, are all numbers of the form p/q where p and q are integers and q#0. A real number x is a non-terminating decimal (with a sign + or -).

Six basic algebraic properties of rational, numbers:

The closure property: If x and y are rational numbers, then x+y and x·y are also rational numbers.

Additive and multiplicative identity elements: If x is a rational number, then x+0=x and x · 1=x .

Associative property: If x, y and z are rational numbers, then x+(y+z)=(x+y)+z, x(y·z)=(x·y)z.

Additive and multiplicative inverses: For each rational number, x, such that x+(-x)=0; if x#0, there exists a rational number x-1 such that x·x-1=1.

Commutative property: If x and y are rational numbers, then x+y=y+x , x·y=y·x.

Distributive property: If x, y, and, z are rational numbers, then

x · (y+z) = (x · y) + (x · z)

If q and p are rational numbers and p-q is negative, then q is greater than p, (q>p) or p is less than q, (p

1.1.1 PROPERTIES OF RATIONAL NUMBERS

Trichotomy property: If p and q are rational numbers, then one and only one of the relations q=p, q>p or q

Transitive property: If p, q, and r are rational numbers, and if p

If p, q, and r are rational numbers and p

If p, q, and r are rational numbers and if p

1.2 INEQUALITIES

To solve a linear inequality

ax+b>0 or x>–b/a, where a>0

draw a number line, dashed for x<-b/a and solid for x>-b/a.

To solve (ax+b)(cx+d)>0 graphically, where a>0 and c>0:

The solution lies in the interval where both lines are dashed and both lines are solid.

Hence, x>-d/c, e9780738670393_i0005.jpg is the solution to the above inequality.

1.3 ABSOLUTE VALUE

Definition: The absolute value of a real number x is defined as

For real numbers a and b:

|a| = |-a|

|ab| = |a|·|b|

-|a| ≤ a ≤ |a|

ab ≤ |a| |b|

|a+b| ² = (a+b)²

|a+b| ≤ |a| + |b| (Triangle Inequality)

|a-b| ≥ | |a| - |b | |

For positive values of b

|a| < b if and only if -b < a < b

|a| > b if and only if a > b or a < -b

|a| = b if and only if a = b or a = -b

1.4 SET NOTATION

A set is a collection of objects called elements. Let A and B be sets.

x ∈ A : x is an element of A

x ∉ B : x is not an element of B

A is a subset of B , (A ⊂ B) , means that A is contained in another set B and each element of A is also an element of B.