CHAPTER 1

CHAPTER 1

1. 1.1 INTRODUCTION TO CALCULUS

   • REAL NUMBERS AND THE REAL LINE
     • Rational numbers
       • Integers,Z
         • Whole,W
           • Natural,N
     • Irrational numbers
     • Notations
       • Set
       • Real line
       • Interval
       • Inequalities
         • Rules
           • 1)a<b=>a+c<b+c
           • 2)a<b=>a-c<b-c
           • 3)a<b and c>0=>ac<bcI
           • 4)a<b and c<0=>bc<ac
           • 5)Special case:a<b=>-b<-a
           • 6)a>0=>(1/a)>0
           • 7)a and b both +ve or -ve, a<b=>(1/b)<(1/a)
     • Absolute Values
       • IxI is the distance between point x and the origin on the real line
       • important:
       • Properties
         • I-aI=IaI
         • IabI=IaI.IbI
         • Ia/bI=Ia/bI
         • Ia+bI<=IaI+IbI

2. 1.2 FUNCTIONS AND THEIR GRAPH

   • A function is a rule that relates how are quantity depends on other quantities
   • Vertical Line Test
     • to test whether the graph is a function or not
   • Types of function
     • Constant
     • Linear
     • Quadratic
     • Polinomial
     • Power
     • Rational
     • Root
     • Absolute
     • Greatest integer

3. 1.3 COMBINATIONS OF FUNCTIONS AND INVERSES

   • If f and g are functions then for every x that belongs to the domains of both f and g, we define: (f+g)(x)=f(x)+g(x),(f-g)(x)=f(x)-g(x),fg(x)=f(x) g(x)
   • Composite function
     • (g o f)(x)=g(f(x))
     • (f o g)(x)=fg(x)
   • THE INVERSE FUNCTION AND EVEN/ODD FUNCTIONS
     • One-to-one function
       • f(x)=f(x)=>x=x
     • Inverse function
       • (f o f)(x)=x
       • Domain f (x) is range of f(x)
     • Even & odd function
       • even function: f(-x)=f(x)
       • odd function:f(-x)=-f(x)
       • Properties
         • Even (+/-) Even=Even, Odd (+/-) Odd=Odd
         • Even(Odd ) (+/-) Odd(Even)=Not Even nor Odd
         • Even x Even = Even, Odd x Odd = Even
         • Product even & odd functions is odd

4. 1.4 TRIGONOMETRIC FUNCTIONS

   • Identities
     • tan x=sin x/cos x
     • cot x=cos x/sin x
     • csc x=i/sin x
     • sec x=1/cos x
     • cos x+sin x=1
     • 1 + tan x=sec x
     • cot x + 1=csc x
   • Reduction
     • sin(-x)=-sin x
     • cos(-x)=cos x
     • sin(x+ )=cos x
     • cos(x+ )-sin x
     • sin( -x )=cos x
     • cos( -x)=sin x
   • Addition & Substraction
     • sin(x y)=sin x cos y cos x sin y
     • cos(x y)=cos x cos y sin x sin y
     • tan(x y)=(tan x tan y)/(1 tan x tan y)
   • Double - Angle
     • sin 2x=2sinx cosx
     • cos 2x=cos x-sin x=1-2sin x=2cos x-1
     • tan 2x =(2 tan x)/(1-tan x)
   • Half-Angle
     • tan x=(1-cos 2x)/(sin 2x)=(sin 2x)/(1+cos 2x)
     • sin x=(1/2)(1-cos 2x)
     • cos x=(1/2)(1+cos 2x)
   • Product-to-sum
     • sin x sin y=(1/2)[cos(x-y)-cos(x+y)]
     • sin x cos y=(1/2)[sin(x+y)+sin(x-y)]
     • cos x cos y=(1/2)[cos(x+y)-cos(x-y)]