CHAPTER 1: BASIC ALGEBRA TECHNIQUES
1.1 SETS
Set Notation:
Set notation is used to describe membership in a class, collection or group.
A set is any collection of objects with a rule to determine if a
Translations of graphs
Shifting a graph vertically up or down.
Example 1:
The graph of the equation y = x2 is a parabola opening upwards with the y axis as an axis of symmetry and having its lowest po
Sets, Real Numbers and Intervals
Sets and set operations
A set is any well-defined collection of (mathematical) objects.
A set can be specified by listing its elements, or members, within curly bracke
Reflections of graphs in the x and y axes
Introductory example.
Suppose that we have already constructed a table of values for the graph of the equation y = x + 4 and that we wish to draw the graph of
Arithmetic Operations on Real Numbers
Arithmetic operations performed on rational numbers.
Two fractions (rational numbers)
a
b
and
c
d
can be added by the means of the formula:
a
c
+
b
ad+bc
=
d
bd
.
Rationalizing denominators and numerators
Introductory example.
Suppose that you are given the problem of obtaining an approximate decimal value for the expression
1
given that
2 1
rounded to 6 figure
Rational expressions
Preliminaries.
Recall that a division of real numbers
a
1
( 1 )
=
a
a
b
, where b 0, can be performed by multiplying the real number a by the multiplicative inverse
of a. Thus
a
=
Rational exponents
Summary of formulas involving general integer exponents:
Before considering rational exponents it is appropriate to recall definitions and properties connected with integer exponent
Solving quadratic inequalities
A sign chart of an expression is a number line that shows where the expression is positive, negative or 0.
For example, the following picture shows a sign chart for 2 x
Quadratic equations and quadratic functions
Introduction.
A quadratic function is a function f of the form
f( x ) = a x2 + b x + c - (i),
where a 0.
A quadratic equation is an equation of the form
a x
Polynomials
Basic concepts connected with polynomials
A polynomial in a single variable x is an expression of the form:
(n 1)
an x n + an 1 x
+ . . . . + a2 x2 + a1 x + a0,
where the coefficients a0,
CHAPTER 1: BASIC ALGEBRA TECHNIQUES
1.2 FUNDAMENTALS OF ARITHMETIC
Fractions:
A fraction a is made up of a numerator (the expression in the top) and a denominator (the
b
expression in the bottom). Not
CHAPTER 1: BASIC ALGEBRA TECHNIQUES
1.3 ALGEBRAIC EXPRESSIONS
Terminology:
Algebra uses symbols [ usually letters ] to represent quantities which are unknown or which can vary.
A term is made up of a
CHAPTER 1: BASIC ALGEBRA TECHNIQUES
1.4 SOLVING EQUATIONS AND INEQUALITIES
Solving Polynomial Equations:
An equation is a statement of equality between two expressions, and this may or may not be true
CHAPTER 5: INTRODUCTORY CALCULUS
5.3 Applications of Derivatives
Curve Sketching:
The value of the derivative of a function represents the slope of the tangent line to the curve of the
function at a p
CHAPTER 5: INTRODUCTORY CALCULUS
5.2 Differentiation and Derivatives
Determining an instantaneous rate of change is a common application. The expression discussed in
f ah f a
5.1, h , is often referre
CHAPTER 5: INTRODUCTORY CALCULUS
5.1 Limits
The Definition of a Limit at a Number:
One of the most fundamental concepts and tools in calculus is the limit, which is defined as follows.
Let f be a func
CHAPTER 4: EXPONENTIAL AND LOGARITHMIC FUNCTIONS
4.3 Applications of Exponential and Logarithm Functions
Typical applications of these functions are to problems involving the rate of change of a quant
CHAPTER 4: EXPONENTIAL AND LOGARITHM FUNCTIONS
4.2 Logarithm Functions
Because an exponential function is always monotone, it has an inverse which is also a function. We
define the inverse function of
CHAPTER 4: EXPONENTIAL AND LOGARITHMIC FUNCTIONS
4.1 Exponential Functions
We defined the expression a x in 1.2. The value a is called the base and x is referred to as the
exponent. Evaluating such ex
CHAPTER 3: ANALYTIC GEOMETRY
3.3 Trigonometric Functions
In order to describe trigonometric functions and derive some of their properties, we introduce the
unit circle. In the xy-plane, the unit circl
CHAPTER 3: ANALYTIC GEOMETRY
3.2 CONICS
There are special figures in the xy-plane called conic sections. These can be described as the
intersection of a right circular cone and a plane, and are called
CHAPTER 3: ANALYTIC GEOMETRY
3.1 LINES
Two Points in a Plane:
We can compare two real numbers, a and b, by using the law of trichotomy: either a b, a b, or
a b. We can also compare the relative size o
CHAPTER 2: FUNCTIONS
Some final notes on inverses:
S The graphs of f and f 1 are reflections of each other through the line y x.
S A relation/function is symmetric about the line y x if and only if th
CHAPTER 2: FUNCTIONS
2.1 THE BASICS
Cartesian Coordinates in the Plane:
In order to gain a better understanding of numbers and equations, we often use geometric
representations. We have already used t
Non-linear Systems of Equations in two variables
Example 1:
Question: Solve the following system of equations and interpret the solution graphically.
cfw_
y=x+1
x2 + y2 = 25
See [9.4, #7]
.
Solution:
Multiplication of polynomials
Some terminology.
Consider the expression
a2 b + 2 a c 5 a3 d +
acd
3
+3
7,
where a, b, c and d are (unknown) real numbers. a, b, c and d are the variables in the express
Even and odd functions
A function f is called even provided that
f( x ) = f( x )
for all numbers x in the domain of f.
If a is a positive number, an even function has the same value at a negative numb
Solving Equations
Introductory example.
Solve the equation
x ( x + 2 ) = 15 - (i).
We need to find all the real numbers x such the statement x ( x + 2 ) = 15 is true.
Putting the problem into words, w