INSTRUCTORS Carleen Eaton Grant Fraser

Home » Mathematics » Algebra 2 » Proof and Mathematical Induction

Professor Fraser

Professor Fraser

Proof and Mathematical Induction

Slide Duration:

Table of Contents

Section 1: Equations and Inequalities
Expressions and Formulas

28m 56s

Intro
0:00
Order of Operations
0:51
Variables and Algebraic Expressions
0:57
Order of Operations
3:05
Monomials
5:25
Examples
5:37
Constant, Coefficient, Degree, Power
6:27
Polynomials
8:29
Examples
8:42
Terms, Like Terms, Binomial, Trinomial
8:59
Formulas
12:35
Examples: Area, Volume, Surface Area
12:50
Lecture Example 1
15:50
Lecture Example 2
21:31
Additional Example 3
-1
Additional Example 4
-2
Properties of Real Numbers

23m 45s

Intro
0:00
Real Numbers
0:15
Rational Numbers
0:40
Irrational Numbers
1:38
Venn Diagram of the Real Numbers
2:55
Properties of Real Numbers
6:49
Commutative Property
7:06
Associative Property
7:27
Identity Property
8:01
Inverse Property
8:42
Distributive Property
10:05
Lecture Example 1
10:43
Lecture Example 2
13:08
Additional Example 3
-1
Additional Example 4
-2
Solving Equations

24m 41s

Intro
0:00
Translations
0:11
Example: Verbal to Algebraic Expressions
0:44
Properties of Equality
2:51
Reflexive, Symmetric, Transitive Properties
2:58
Addition, Subtraction, Multiplication, Division
3:32
Solving Equations
6:09
Example
6:23
Solving for a Variable
8:49
Example: Surface Area of a Cone
8:58
Lecture Example 1
11:06
Lecture Example 2
12:39
Additional Example 3
-1
Additional Example 4
-2
Solving Absolute Value Equations

17m 36s

Intro
0:00
Absolute Value Expressions
0:10
Example: Positive Distance
0:15
Absolute Value Equations
1:07
Examples
1:18
No Solutions
2:54
Example: Empty Set
2:58
Number of Solutions
3:56
Examples
4:42
Lecture Example 1
6:42
Lecture Example 2
8:54
Additional Example 3
-1
Additional Example 4
-2
Solving Inequalities

19m 27s

Intro
0:00
Properties of Inequality
0:07
Addition Property
0:21
Subtraction Property
0:48
Example
1:02
Multiplication Properties
1:44
Multiplying by a Positive Number
1:48
Example: Positive
2:17
Multiplying by a Negative Number
2:25
Example: Negative
2:35
Division Properties
3:23
Example: Positive
3:32
Example: Negative
4:04
Describing the Solution Set
6:00
Set Builder Notation
6:15
Graphing
7:15
Lecture Example 1
8:04
Lecture Example 2
9:09
Additional Example 3
-1
Additional Example 4
-2
Solving Compound and Absolute Value Inequalities

24m 8s

Intro
0:00
Compound Inequalities
0:11
Example
0:33
'And' Inequality
3:41
Example: Set Intersection
4:00
'Or' Inequality
6:01
Example: Set Union
6:15
Absolute Value Inequalities
8:19
Examples
8:37
Lecture Example 1
11:43
Lecture Example 2
14:47
Additional Example 3
-1
Additional Example 4
-2
Section 2: Linear Relations and Functions
Relations and Functions

38m 15s

Intro
0:00
Coordinate Plane
0:38
Example: Origin and Quadrants
0:44
Relations
4:08
Example: Ordered Pairs
4:14
Domain and Range
5:05
Functions
5:57
Example: Mapping
6:11
One-to-One Functions
9:58
Example
10:05
Graphs of Relations
13:42
Example: Discrete and Continuous
13:55
Vertical Line Test
16:26
Examples
16:38
Equations, Relations, Functions
19:38
Example: Independent and Dependent Variables
19:45
Function Notation
21:51
Examples
22:27
Lecture Example 1
24:39
Lecture Example 2
28:29
Additional Example 3
-1
Additional Example 4
-2
Linear Equations

12m 50s

Intro
0:00
Linear Equations and Functions
0:12
Example: Linear Equation
0:21
Example: Linear Function
1:16
Standard Form
2:13
Examples
2:43
Graphing with Intercepts
3:26
Example: Intercepts
3:51
Lecture Example 1
6:25
Lecture Example 2
7:53
Additional Example 3
-1
Additional Example 4
-2
Slope

20m 7s

Intro
0:00
Definition of Slope
0:23
Interpretation of Slope
2:19
Example: 0 Slope and Undefined Slope
2:25
Example: Positive Slope
4:04
Example: Negative Slope
4:43
Parallel Lines
6:16
Perpendicular Lines
7:15
Lecture Example 1
8:20
Lecture Example 2
10:45
Additional Example 3
-1
Additional Example 4
-2
Writing Linear Functions

27m 36s

Intro
0:00
Slope Intercept Form
0:08
Origin of Form
0:21
Example
2:08
Point Slope Form
3:47
Origin of Form
4:01
Parallel and Perpendicular Lines
5:36
Example: Find Parallel Line
5:58
Lecture Example 1
8:27
Lecture Example 2
12:08
Additional Example 3
-1
Additional Example 4
-2
Special Functions

24m 28s

Intro
0:00
Step Functions
0:13
Graph
0:21
Example: Birthday Function
2:32
Absolute Value Functions
5:21
Graph
5:27
Piecewise Functions
7:34
Example
7:38
Lecture Example 1
10:20
Lecture Example 2
14:38
Additional Example 3
-1
Additional Example 4
-2
Graphing Inequalities

30m 37s

Intro
0:00
Graphing Linear Inequalities
0:11
Example: Linear Inequalities
0:20
Half Plane
2:04
Test Point
2:53
Graphing Absolute Value Inequalities
5:38
Example: Linear Inequalities
5:49
Example: Absolute Value
8:23
Lecture Example 1
11:39
Lecture Example 2
14:50
Additional Example 3
-1
Additional Example 4
-2
Section 3: Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

21m 27s

Intro
0:00
Systems of Equations
0:14
Solving by Graphing
0:34
Types of Systems
1:07
Independent (One Solution)
2:02
Dependent (Infinite Solutions)
2:30
Inconsistent (No Solutions, Parallel)
3:37
Lecture Example 1
4:52
Lecture Example 2
8:42
Additional Example 3
-1
Additional Example 4
-2
Solving Systems of Equations Algebraically

31m 26s

Intro
0:00
Solving by Substitution
0:15
Examples
0:50
Solving by Elimination
4:19
Examples
4:27
Solving by Multiplication
7:24
Examples
7:37
Inconsistent and Dependent Systems
11:42
Example: Spotting Differences
12:07
Lecture Example 1
15:00
Lecture Example 2
17:35
Additional Example 3
-1
Additional Example 4
-2
Solving Systems of Inequalities by Graphing

20m 43s

Intro
0:00
Solving by Graphing
0:10
Example: Single Inequality
0:14
No Solution
4:16
Example: No Solution
4:25
Lecture Example 1
6:25
Lecture Example 2
8:23
Additional Example 3
-1
Additional Example 4
-2
Solving Systems of Equations in 3 Variables

21m 27s

Intro
0:00
Solving Systems in Three Variables
0:15
Ordered Triple
0:36
Number of Solutions
1:32
Lecture Example 1
2:19
Lecture Example 2
6:14
Additional Example 3
-1
Additional Example 4
-2
Section 4: Matrices
Basic Matrix Concepts

14m 8s

Intro
0:00
What is a Matrix?
0:33
Example: Rectangular Array
0:41
Element
1:52
Examples: More Matrices
2:04
Dimensions
3:40
Examples
4:53
Special Matrices
6:31
(m x 1) Matrix
6:36
Square Matrix
7:01
Zero Matrix
7:38
Equal Matrices
8:23
Examples
8:32
Lecture Example 1
10:56
Lecture Example 2
11:28
Additional Example 3
-1
Additional Example 4
-2
Matrix Operations

16m 40s

Intro
0:00
Matrix Addition
0:10
Example
1:07
Matrix Subtraction
2:12
Example
2:31
Scalar Multiplication
3:23
Example
4:05
Properties of Matrix Operations
5:31
Commutative Property
5:48
Associative Property
5:59
Distributive Property
6:34
Lecture Example 1
7:03
Lecture Example 2
8:15
Additional Example 3
-1
Additional Example 4
-2
Matrix Multiplication

22m 47s

Intro
0:00
Dimension Requirement
0:19
Example
0:45
Matrix Multiplication
1:35
Example
2:21
Properties of Matrix Multiplication
6:46
Associative Property
6:59
Distributive Property
7:15
Commutative Property
7:39
Lecture Example 1
8:49
Lecture Example 2
11:43
Additional Example 3
-1
Additional Example 4
-2
Determinants

25m 47s

Intro
0:00
What is a Determinant
0:15
Determinant of a 2x2 Matrix
0:56
Difference from Matrices
1:16
Second Order Determinant
1:38
Example
2:06
Determinant of a 3x3 Matrix
3:20
Third Order Determinants
3:25
Origin of Equation (Minors)
3:38
Expansion by Minors
6:05
Example: 3x3 Matrix
8:55
Diagonal Method for 3x3 Matrix
12:45
Example
12:55
Lecture Example 1
17:03
Lecture Example 2
17:42
Additional Example 3
-1
Additional Example 4
-2
Cramer's Rule

25m 42s

Intro
0:00
System of 2 Equations in 2 Variables
0:27
Example
1:20
System of 3 Equations in 3 Variables
3:10
Example
3:51
Lecture Example 1
6:45
Lecture Example 2
10:22
Additional Example 3
-1
Additional Example 4
-2
Identity and Inverse Matrices

27m 1s

Intro
0:00
Identity Matrix
0:10
Example: 2x2 Matrix
2:18
Matrix Inverses
4:40
Example: Does Not Exist
6:04
Inverse of a 2x2 Matrix
8:17
Example
9:38
Lecture Example 1
13:19
Lecture Example 2
15:57
Additional Example 3
-1
Additional Example 4
-2
Solving Systems of Equations with Matrices

28m 40s

Intro
0:00
Matrix Equations
0:22
Example
0:40
Solving Systems of Equations
4:20
Example
5:58
Lecture Example 1
9:11
Lecture Example 2
15:09
Additional Example 3
-1
Additional Example 4
-2
Section 5: Quadratic Functions and Inequalities
Graphing Quadratic Equations

26m 36s

Intro
0:00
Quadratic Functions
0:10
Parabola
0:50
Example: Opens Upward
1:03
Example: Opens Downward
1:54
Properties of Parabolas
3:17
Axis of Symmetry
3:26
Vertex
4:05
Example
4:28
Maximum and Minimum Values
7:10
Example: Upwards/Minimum
7:32
Example: Downwards/Maximum
8:19
Lecture Example 1
9:09
Lecture Example 2
13:05
Additional Example 3
-1
Additional Example 4
-2
Solving Quadratic Equations by Graphing

19m 26s

Intro
0:00
Quadratic Equations
0:18
Example: Standard Form
0:55
Solving by Graphing
1:39
Roots
1:48
Example: 2 Solutions
1:56
Example: 1 Solution
2:39
Example: 0 Solutions
3:10
Estimating Solutions
3:55
Example
4:07
Lecture Example 1
5:16
Lecture Example 2
7:51
Additional Example 3
-1
Additional Example 4
-2
Solving Quadratic Equations by Factoring

17m 46s

Intro
0:00
Factoring Techniques
0:16
Greatest Common Factor (GCF)
0:29
Difference of Two Squares
1:45
Perfect Square Trinomials
2:07
General Trinomials
3:16
Zero Product Rule
4:50
Example
5:01
Lecture Example 1
6:19
Lecture Example 2
8:13
Additional Example 3
-1
Additional Example 4
-2
Imaginary and Complex Numbers

37m 41s

Intro
0:00
Properties of Square Roots
0:17
Example: Product and Quotient Rules
0:33
Imaginary Numbers
4:04
Powers of Imaginary Numbers
5:06
Example
6:27
Complex Numbers
7:21
Real and Complex Numbers
8:19
Equality
9:04
Example
9:17
Addition and Subtraction
9:43
Example
9:55
Complex Plane
11:38
Example
11:52
Multiplication
13:34
Example
13:43
Division
16:36
Complex Conjugates
16:45
Example
18:16
Lecture Example 1
23:40
Lecture Example 2
26:34
Additional Example 3
-1
Additional Example 4
-2
Completing the Square

16m 42s

Intro
0:00
Square Root Property
0:22
Examples
0:33
Completing the Square
1:48
Example: Making into Perfect Square
1:50
Solve Equations
3:43
Example
3:53
Equations Where 'a' Not Equal to 1
6:47
Example
6:57
Complex Solutions
10:14
Example
10:22
Lecture Example 1
11:30
Lecture Example 2
12:34
Additional Example 3
-1
Additional Example 4
-2
Quadratic Formula and the Discriminant

17m 44s

Intro
0:00
Quadratic Formula
0:37
Example
0:56
One Rational Root
3:10
Why It Works
3:26
Repeated/Double Root
3:49
Complex Solutions
4:31
Example
4:50
Discriminant
7:19
Discriminant Value and Root Type
8:50
Lecture Example 1
12:08
Lecture Example 2
14:15
Additional Example 3
-1
Additional Example 4
-2
Analyzing the Graphs of Quadratic Functions

23m

Intro
0:00
Vertex Form
0:24
Example
1:56
Significance of Coefficient 'a'
3:15
Example
3:39
Writing Quadratic Equations in Vertex Form
4:51
Examples
5:19
Lecture Example 1
8:14
Lecture Example 2
10:22
Additional Example 3
-1
Additional Example 4
-2
Graphing and Solving Quadratic Inequalities

34m 38s

Intro
0:00
Graphing Quadratic Inequalities
1:14
Example: Linear Inequality
1:29
Example: Quadratic Inequality
3:11
Solving Quadratic Inequalities
6:32
Example
6:38
Lecture Example 1
11:50
Lecture Example 2
15:09
Additional Example 3
-1
Additional Example 4
-2
Section 6: Polynomial Functions
Properties of Exponents

20m 28s

Intro
0:00
Simplifying Exponential Expressions
0:32
Negative Exponents
0:54
Example: Base 0
1:16
Examples
1:30
Properties of Exponents
2:22
Base and Exponent
2:52
Lecture Example 1
8:29
Lecture Example 2
10:58
Additional Example 3
-1
Additional Example 4
-2
Operations on Polynomials

16m 13s

Intro
0:00
Adding and Subtracting Polynomials
0:24
Example: Signs
0:34
Multiplying Polynomials
3:04
Example
3:12
Lecture Example 1
6:40
Lecture Example 2
7:21
Additional Example 3
-1
Additional Example 4
-2
Dividing Polynomials

29m 26s

Intro
0:00
Dividing by a Monomial
0:16
Example
0:28
Long Division
3:24
Example: Missing Terms, Remainder
3:49
Example: Long Division
6:51
Synthetic Division
10:13
Example
10:44
Divisor in Synthetic Division
13:18
Example: Coefficient Not 1
13:30
Lecture Example 1
16:41
Lecture Example 2
18:22
Additional Example 3
-1
Additional Example 4
-2
Polynomial Functions

29m 34s

Intro
0:00
Polynomial in One Variable
0:17
Degree n
0:30
Descending Order
0:43
Example: Leading Coefficient
1:04
Function Values
3:31
Example
3:42
Zeros of Polynomial Functions
5:45
Example: Zeros
6:04
End Behavior
9:51
Example: 4 Situations
10:51
Lecture Example 1
17:30
Lecture Example 2
19:11
Additional Example 3
-1
Additional Example 4
-2
Analyzing Graphs of Polynomials

34m 36s

Intro
0:00
Graphing Polynomial Functions
0:09
End Behavior
0:19
Examples: Degree and Sign of Polynomials
1:13
Location Principle
4:50
Example
6:19
Maximum and Minimum Points
7:34
Example: Relative Maximum and Relative Minimum
7:44
Lecture Example 1
10:17
Lecture Example 2
15:13
Additional Example 3
-1
Additional Example 4
-2
Solving Polynomial Equations

19m 23s

Intro
0:00
Factoring Polynomials
0:08
Example: Greatest Common Factor (GCF)
0:40
Example: Perfect Square Trinomials
1:30
Example: General Trinomials
2:48
Sum and Difference of Two Cubes
3:25
Example
4:18
Quadratic Form
6:20
Lecture Example 1
7:30
Lecture Example 2
10:43
Additional Example 3
-1
Additional Example 4
-2
Remainder and Factor Theorems

27m 52s

Intro
0:00
Remainder Theorem
0:04
Quotient and Remainder
0:30
Examples
1:34
Synthetic Substitution
5:04
Example
5:28
Factor Theorem
10:00
Factoring Polynomials
11:21
Example
11:51
Lecture Example 1
16:38
Lecture Example 2
18:41
Additional Example 3
-1
Additional Example 4
-2
Roots and Zeros

31m 4s

Intro
0:00
Numbers of Roots
0:10
Example: Real and Complex Roots
0:23
Descartes' Rule of Signs
3:43
Example: Positive Real Roots
4:58
Example: Negative Real Roots
8:00
Finding the Roots
12:11
Conjugate Roots
13:24
Lecture Example 1
15:41
Lecture Example 2
19:41
Additional Example 3
-1
Additional Example 4
-2
Rational Zero Theorem

29m 27s

Intro
0:00
Equation
0:14
Leading Coefficient and Constant Term
0:30
Example
2:15
Leading Coefficient Equal to 1
8:08
Example
9:20
Finding Rational Zeros
11:48
Lecture Example 1
12:10
Lecture Example 2
15:59
Additional Example 3
-1
Additional Example 4
-2
Section 7: Rational Equations and Inequalities
Operations on Functions

35m 12s

Intro
0:00
Arithmetic Operations
0:12
Example: Domain
0:25
Composition of Functions
7:35
Example
7:55
Composition is Not Commutative
17:13
Example
18:18
Lecture Example 1
21:51
Lecture Example 2
24:25
Additional Example 3
-1
Additional Example 4
-2
Inverse Functions and Relations

18m 12s

Intro
0:00
Inverse of a Relation
0:24
Example: Ordered Pairs
0:33
Inverse of a Function
2:15
Procedure to Construct an Inverse Function
4:28
Example: Inverse Function
4:58
Example: Inverse Function 2
7:31
Inverses and Compositions
8:41
Lecture Example 1
9:59
Lecture Example 2
10:45
Additional Example 3
-1
Additional Example 4
-2
Square Root Functions and Inequalities

26m 24s

Intro
0:00
Square Root Functions
0:06
Example: Not Square Root Function
0:23
Example: Square Root Function
1:17
Graphing Square Root Functions
3:11
Example: Radicand
3:21
Square Root Inequalities
6:51
Example
7:13
Lecture Example 1
11:27
Lecture Example 2
14:05
Additional Example 3
-1
Additional Example 4
-2
nth Roots

24m 6s

Intro
0:00
Definition of the nth Root
0:13
Example
0:36
Principal nth Root
2:18
Index
3:04
Examples
3:20
Using Absolute Values
6:25
Examples
6:52
Lecture Example 1
11:26
Lecture Example 2
13:17
Additional Example 3
-1
Additional Example 4
-2
Operations with Radical Expressions

34m 38s

Intro
0:00
Properties of Radicals
0:22
Example
1:37
Simplifying Radical Expressions
2:58
Examples
3:24
Rationalizing Denominators
4:08
Examples
4:18
Conjugate Radical Expressions
8:01
Example
8:09
Adding and Subtracting Radicals
11:23
Examples
11:44
Multiplying Radicals
12:57
Examples
13:03
Lecture Example 1
16:53
Lecture Example 2
20:11
Additional Example 3
-1
Additional Example 4
-2
Rational Exponents

24m 36s

Intro
0:00
Definition 1
0:24
nth Root
0:44
Example: Even
1:29
Definition 2
2:55
Simplifying Expressions
3:20
Examples
3:40
Simplified Form
7:07
Example
7:32
Lecture Example 1
8:18
Lecture Example 2
10:20
Additional Example 3
-1
Additional Example 4
-2
Solving Radical Equations and Inequalities

38m 46s

Intro
0:00
Radical Equations
0:23
Examples
0:34
Example: Radical Equation
4:47
Extraneous Roots
12:29
Eliminating nth Roots
14:28
Examples
14:54
Radical Inequalities
16:38
Example
17:18
Lecture Example 1
20:28
Lecture Example 2
22:57
Additional Example 3
-1
Additional Example 4
-2
Section 8: Radical Expressions and Equations
Multiplying and Dividing Rational Expressions

30m 11s

Intro
0:00
Simplifying Rational Expressions
0:12
Examples: Rational Expressions
0:31
Factoring -1
3:26
Example
3:33
Multiplying and Dividing Rational Expressions
4:50
Multiplying
5:08
Dividing
5:16
Example
6:10
Factoring
9:13
Example
9:33
Complex Fractions
13:15
Example
13:27
Lecture Example 1
15:36
Lecture Example 2
18:25
Additional Example 3
-1
Additional Example 4
-2
Adding and Subtracting Rational Exprsesions

51m 53s

Intro
0:00
Example: Fractions
0:22
Least Common Multiple (LCM)
1:36
Example
2:07
Adding and Subtracting
7:56
Least Common Denominator (LCD)
8:01
Example: Fractions
8:14
Example: Rational Expression
10:23
Equivalent Fractions
13:45
Example
14:20
Simplifying Complex Fractions
20:03
Example
20:28
Lecture Example 1
26:34
Lecture Example 2
31:06
Additional Example 3
-1
Additional Example 4
-2
Graphing Rational Functions

45m 13s

Intro
0:00
Rational Functions
0:35
Example
0:57
Breaks in Continuity
2:48
Discontinuities
3:19
Example: Excluded Values
3:52
Graphs and Discontinuities
4:36
Example: Hole Discontinuity
6:07
Example: Asymptote
8:53
Horizontal Asymptotes
13:34
Example
13:54
Lecture Example 1
17:58
Lecture Example 2
20:29
Additional Example 3
-1
Additional Example 4
-2
Direct, Joint, and Inverse Variation

21m 49s

Intro
0:00
Direct Variation
0:16
Constant of Variation
0:44
Graph of Direct Variation
1:28
Example: Straight Line
1:36
Joint Variation
2:55
Inverse Variation
4:17
Example
4:50
Graph of Inverse Variation
5:35
Example
6:00
Proportions
8:00
Example
9:28
Lecture Example 1
12:32
Lecture Example 2
14:26
Additional Example 3
-1
Additional Example 4
-2
Solving Rational Equations and Inequalities

53m 21s

Intro
0:00
Rational Equations
0:15
Example: Not Rational Equation
0:26
Example: X in Denominator
0:38
Example: LCD
1:08
Example: Rational Equations
5:19
Extraneous Solutions
12:08
Example
12:42
Rational Inequalities
15:31
Example
15:45
Example: Rational Inequalities
12:05
Lecture Example 1
32:06
Lecture Example 2
35:18
Additional Example 3
-1
Additional Example 4
-2
Section 9: Exponential and Logarithmic Relations
Exponential Functions

28m 22s

Intro
0:00
What is an Exponential Function?
0:11
Exponent and Base
0:38
Graphing Exponential Functions
1:31
Example
1:34
Properties
4:05
Growth and Decay
9:38
Equations
10:32
Example
11:05
Inequalities
13:00
Example
14:29
Lecture Example 1
16:48
Lecture Example 2
18:50
Additional Example 3
-1
Additional Example 4
-2
Logarithms and Logarithmic Functions

36m 31s

Intro
0:00
What are Logarithms?
0:17
Examples
1:30
Logarithmic Functions
4:09
Graph of the Logarithmic Function
4:52
Properties
9:08
Inverse Property
10:47
Equations
11:44
Example
12:11
Inequalities
14:45
Equations with Logarithms on Both Sides
17:00
Example
17:18
Inequalities with Logarithms on Both Sides
19:17
Example
19:32
Lecture Example 1
20:31
Lecture Example 2
22:38
Additional Example 3
-1
Additional Example 4
-2
Properties of Logarithms

29m 50s

Intro
0:00
Product Property
0:08
Example
0:26
Quotient Property
1:06
Example
1:12
Power Rule
3:29
Example
3:33
Equations
5:43
Example
6:19
Lecture Example 1
12:19
Lecture Example 2
16:13
Additional Example 3
-1
Additional Example 4
-2
Common Logarithms

27m 10s

Intro
0:00
What are Common Logarithms?
0:54
Base 10
0:58
Equations
2:00
Examples
2:22
Inequalities
5:35
Example
5:42
Change of Base
9:23
Example
10:09
Lecture Example 1
12:04
Lecture Example 2
15:16
Additional Example 3
-1
Additional Example 4
-2
Base 'e' and Natural Logarithms

19m 52s

Intro
0:00
The Number 'e'
0:32
Natural Base
0:44
Euler
1:12
Natural Exponential Function
1:38
Natural Log Function
2:44
Growth and Decay
2:55
Natural Logarithms
3:16
Graph (Inverse)
3:34
Equations and Inequalities
4:49
Lecture Example 1
7:21
Lecture Example 2
9:10
Additional Example 3
-1
Additional Example 4
-2
Exponential Growth and Decay

28m 10s

Intro
0:00
Decay
0:15
Fixed Percentage
0:24
Rate of Decay
2:35
Scientific Model of Decay (Exponential Decay)
4:17
Graph
5:19
Growth
6:19
Rate of Growth
6:36
Scientific Model of Growth (Exponential Growth)
6:41
Graph
6:48
Lecture Example 1
7:48
Additional Example 3
-1
Additional Example 4
-2
Section 10: Conic Sections
Midpoint and Distance Formulas

29m 35s

Intro
0:00
Midpoint Formula
0:35
Distance Formula
1:42
Example
2:52
Lecture Example 1
3:40
Lecture Example 2
6:37
Additional Example 3
-1
Additional Example 4
-2
Parabolas

26m 11s

Intro
0:00
What is a Parabola?
0:21
Focus and Directrix
0:33
Axis of Symmetry
1:41
Vertex
2:03
Example
2:15
Standard Form
3:11
Upward and Downward
4:07
Graphing Parabolas
5:24
Example
6:32
Latus Rectum
7:37
Horizontal Parabolas
9:10
Focus and Direction
12:31
Lecture Example 1
13:11
Lecture Example 2
16:46
Additional Example 3
-1
Additional Example 4
-2
Circles

17m 33s

Intro
0:00
What are Circles
0:17
Center, Radius
0:37
Equation (Standard Form)
0:46
Graphing
1:21
Center Not at Origin
1:53
Example
2:06
Lecture Example 1
4:16
Lecture Example 2
8:22
Additional Example 3
-1
Additional Example 4
-2
Ellipses

38m 57s

Intro
0:00
What are Ellipses?
0:59
Foci
1:04
Properties
3:47
Major Axis, Minor Axis
4:03
Standard Form
7:22
Example
8:05
Vertical Major Axis
10:12
Example
10:40
Graphing Ellipses
13:33
Example: Completing the Square
14:04
Equation with Center at (h,k)
17:25
Example
17:53
Lecture Example 1
19:36
Lecture Example 2
23:52
Additional Example 3
-1
Additional Example 4
-2
Hyperbolas

37m 59s

Intro
0:00
What are Hyperbolas?
1:09
Properties
2:35
Transverse Axis, Conjugate Axis
2:57
Center, Vertices
3:54
Standard Form
4:33
Vertical Transverse Axis
6:35
Asymptotes
10:17
Graphing Hyperbolas
13:44
Example
17:23
Equation with Center at (h,k)
18:20
Lecture Example 1
20:19
Lecture Example 2
23:25
Additional Example 3
-1
Additional Example 4
-2
Conic Sections

23m 10s

Intro
0:00
What are Conic Sections?
2:16
Standard Form
2:58
Example
5:29
Identifying Conic Sections
6:14
Example
6:55
Lecture Example 1
8:55
Lecture Example 2
11:18
Additional Example 3
-1
Additional Example 4
-2
Solving Quadratic Systems

28m 18s

Intro
0:00
Linear Quadratic Systems
0:04
Example
0:28
Solutions
3:13
Quadratic Quadratic System
3:36
Example: Elimination
3:45
Solutions
7:34
Systems of Quadratic Inequalities
7:55
Example
8:07
Lecture Example 1
11:10
Lecture Example 2
16:12
Additional Example 3
-1
Additional Example 4
-2
Section 11: Sequences and Series
Arithmetic Sequences

27m 44s

Intro
0:00
Sequences
0:27
Example: Term
0:36
Arithmetic Sequence
2:13
Common Difference
2:22
Example
2:35
Formula for nth Term
3:39
Example
4:29
Equation for nth Term
5:58
Example
6:10
Arithmetic Means
7:40
Example
8:13
Lecture Example 1
14:08
Lecture Example 2
15:35
Additional Example 3
-1
Additional Example 4
-2
Arithmetic Series

29m 12s

Intro
0:00
What are Arithmetic Series?
0:22
Example: Sequence
0:29
Example: Common Difference (d)
0:35
Sum of Arithmetic Series
2:52
Example
3:44
Sigma Notation
6:10
Example
6:48
Lecture Example 1
8:32
Lecture Example 2
12:39
Additional Example 3
-1
Additional Example 4
-2
Geometric Sequences

24m 52s

Intro
0:00
What are Geometric Sequences?
0:20
Common Ratio
1:03
Example
1:20
nth Term of a Geometric Sequence
3:39
Geometric Means
4:16
Example: Missing Term
5:06
Lecture Example 1
8:09
Lecture Example 2
11:42
Additional Example 3
-1
Additional Example 4
-2
Geometric Series

27m 2s

Intro
0:00
What are Geometric Series?
0:20
Example: Common Ratio
1:00
Sum of Geometric Series
2:27
Example
4:01
Sigma Notation
4:56
Example: Index
5:24
Example
6:20
Another Sum Formula
7:51
Specific Terms
9:19
Lecture Example 1
11:15
Lecture Example 2
14:30
Additional Example 3
-1
Additional Example 4
-2
Infinite Geometric Series

24m 1s

Intro
0:00
What are Infinite Geometric Series?
0:35
Partial Sums of the Infinite Series
1:17
Example
1:24
Sum of an Infinite Geometric Series
3:16
Convergent Series
3:25
Example
4:17
Sigma Notation
5:31
Example
5:43
Repeating Decimals
6:38
Example
6:48
Lecture Example 1
9:33
Lecture Example 2
11:20
Additional Example 3
-1
Additional Example 4
-2
Recursion and Special Sequences

17m 11s

Intro
0:00
Fibonacci Sequence
0:17
Example: Fibonacci Sequence
0:36
Example: Recursive Formula
2:38
Iteration
3:40
Example
4:57
Lecture Example 1
7:10
Lecture Example 2
9:03
Additional Example 3
-1
Additional Example 4
-2
Binomial Theorem

38m 25s

Intro
0:00
Pascal's Triangle
0:11
General Form
2:43
Properties
7:01
Binomial Theorem
9:20
Example
11:47
Finding a Specific Term
16:24
Example
16:32
Lecture Example 1
20:35
Lecture Example 2
23:30
Additional Example 3
-1
Additional Example 4
-2
Proof and Mathematical Induction

19m 53s

Intro
0:00
Math Induction Principle
0:19
Example
0:29
Counter Examples
5:00
Example
5:14
Lecture Example 1
7:16
Lecture Example 2
10:53
Additional Example 3
-1
Additional Example 4
-2
Loading...
This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of Algebra 2
Bookmark & Share Embed

Share this knowledge with your friends!

Copy & Paste this embed code into your website’s HTML

Please ensure that your website editor is in text mode when you paste the code.
(In Wordpress, the mode button is on the top right corner.)
  ×
  • - Allow users to view the embedded video in full-size.
Since this lesson is not free, only the preview will appear on your website.

  • Discussion

  • Answer Engine

  • Study Guides

  • Download Lecture Slides

  • Table of Contents

  • Related Books

Please login to ask a question and view discussion.

Start Learning Now

Our free lessons will get you started (Adobe Flash® required).
Get immediate access to our entire library.

Sign up for Educator.com

Membership Overview

  • Unlimited access to our entire library of courses.
  • Search and jump to exactly what you want to learn.
  • *Ask questions and get answers from the community and our teachers!
  • Practice questions with step-by-step solutions.
  • Download lesson files for programming and software training practice.
  • Track your course viewing progress.
  • Download lecture slides for taking notes.
  • Learn at your own pace... anytime, anywhere!

Proof and Mathematical Induction

  • Remember to do the case of n = 1 (the base case). This is usually very simple.

  • Always use the information in the case of n = k to prove the case of n = k + 1. Often factoring or a clever multiplication will work.

  • To find a counter-example, start with small values of n and increase by 1 until you find one.

Proof and Mathematical Induction

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Math Induction Principle 0:19
    • Example
  • Counter Examples 5:00
    • Example
  • Lecture Example 1 7:16
  • Lecture Example 2 10:53
  • Additional Example 3
  • Additional Example 4

Related Books

Educator®

Please sign in to participate in this lecture discussion.

Resetting Your Password?
or
Still having trouble? Contact us
OR

Start Learning Now

Our free lessons will get you started (Adobe Flash® required).
 Get immediate access to our entire library.

Membership Overview

  • Available 24/7. Unlimited Access to Our Entire Library.
  • Search and jump to exactly what you want to learn.
  • *Ask questions and get answers from the community and our teachers!
  • Practice questions with step-by-step solutions.
  • Download lecture slides for taking notes.
  • Track your course viewing progress.
  • Accessible anytime, anywhere with our Android and iOS apps.