0. A
Preview of Calculus (Chapter 0)
A. A Precalculus Review
1. The Real
Number Line and Order 
a) The Individual
Pieces
(1) Natural
Numbers;
N; {1,2,3,...}
(2) Whole
Numbers; W; {0,1,2,...}
(3) Integers;
IN; {...,-1,0,1,...}
(4) Rational
Numbers;
Q; 
| (a) Decimal Form:
repeater or terminator |
(5) Irrational
Numbers;
IR; 
| (a) Decimal Form:
non-repeating, non-terminating |
(6) Real
Numbers; R;  |
b) The Relationships
(1) Diagram of
Relationships
(2) Filling the Number Line
| (a) Line is not
filled until all of R is included |
|
c) Types of Intervals
d) Solving Inequalities |
2. Absolute
Value and Distance on the Real Number Line
a) Absolute
Value of a
Real Number
b) Distance on the Real Number Line
(1)  |
c) Intervals
Defined by Absolute
Value
(1) Double Inequalities
(AND) (
)
(2) Disjunction (OR) (
) |
d) Midpoint of an Interval
(1)  |
|
3. Exponents
and Radicals
a) Expressions
Involving Exponents
or Radicals
(1) Properties of
Exponents
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h)  |
(2) Cautions
(3) Evaluating Expressions With Integer Exponents
(4) Evaluating Expressions with Fractional Exponents
(5) Operations with Exponents
(6) Simplifying Expressions with Exponents
(7) Simplifying by Factoring
(8) Factors Involving Quotients |
b) Domain
of an Algebraic Expression
| (1) Finding the Domain
of an Algebraic
Expression |
|
4. Factoring
Polynomials
5. Fractions
and Rationalization
a) Operations on Fractions
b) Operations on Rational Algebraic Expressions
c) Expressions Involving Radicals
d) Rationalization Techniques
(1) Rationalize
Denominator
(a) Multiply top
& bottom by radical
(b) Multiply top & bottom by conjugate of bottom |
(2) Rationalize Numerator
| (a) Similar
Techniques to 5d1 above |
|
|
|
B. Functions, Graphs, and Limits
1. The
Cartesian Plane and the Distance Formula
a) The Cartesian
Plane
| (1)
Terms
(2) plotting points |
b) The Distance
Formula
(1) 
(2) From the Pythagorean Theorem
(3) Applications of the Distance
Formula
(4) Finding Points at a Specified Distance from a Given Point |
c) The Midpoint Formula
(1) 
(2) Ideal for finding "averages" |
|
2. Graphs
of Equations
a) The Graph of an
Equation
(1)
Terms
(a) table of values
(b) graph (set of all points satisfying equation)
(c) zero(s)
(d) intercepts |
|
b) Intercepts of a Graph
c) Circles
(1) Standard
Form  |
d) Points of Intersection
| (1) Finding the Equation
of a Circle
|
e) Mathematical Models |
3. Lines
in the Plane and
Slope
a) Linear Equations and Slope
(1) 
(2) slope intercept form of a line: 
(3) Meaning of positive, negative, zero, and undefined
slopes
(a) positive slope:
up hill (left to right)
(b) negative slope: down hill (left to right)
(c) zero slope: horizontal line
(d) undefined slope: vertical line |
|
b) Finding an Equation of a Line
(1) Point-Slope Form of a
Line: 
(2) slope intercept form of a line: 
(a) Find Slope
(Using Slope Formula)
(b) Substitute into Slope Intercept form of a line
(c) Solve for b (the y-intercept)
(d) Write the complete equation in slope intercept form |
(3) General Form of a line: 
(4) Vertical Line: 
; slope undefined; y-intercept undefined or none
(5) Horizontal Line: 
; slope:zero ; y-intercept: |
c) Parallel
Lines and Perpendicular
Lines
d) Intersection
of Lines
(1) To find Points of Intersection:
| (a) Solve Equations
Simultaneously using:
|
|
|
4. Functions
a) Terms
(1) Independent Variable
(2) Dependent Variable
(3) Definition
of a Function:
each value of the independent variable corresponds to one and only one
value for the dependent variable
(4) Domain:
the set of all possible
values of the independent variable
(5) Range:
the set of all possible
values of the dependent variable |
b) Determining Functional Relationships
(1) Using
Equation
(a) Isolate
Dependent Variable
(b) Determine if Function or not (Using Function Definition) |
(2) Using Graph
|
c) The Graph of a Function
d) A one to one Function
e) Function Notation
f) Combinations of Functions (Composite
Functions)
g) Inverse Functions |
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