  Vincent Selhorst-Jones

Understanding Exponents

Slide Duration:

Section 1: Introduction
Introduction to Precalculus

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
18:50
19:04
19:21
20:04
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
Section 2: Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
Section 3: Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
13:08
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
9:56
11:38
Derivation
11:43
Final Form
12:23
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
26:49
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
19:24
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
7:41
8:55
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
8:25
9:26
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
Section 5: Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
2:08
2:31
2:52
Half-Circle and Right Angle
4:00
6:24
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
9:43
Mnemonic: All Students Take Calculus
10:13
13:11
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Section 7: Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2

52m 52s

Intro
0:00
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
Section 8: Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
Section 9: Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
11:42
Example of Using a Point
12:41
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
Section 11: Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
Section 12: Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
Section 13: Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
Section 14: Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
11:51
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Section 15: Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
Section 16: Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
Section 17: Appendix: Graphing Calculators

10m 41s

Intro
0:00
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
9:17
9:19
9:35
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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### Understanding Exponents

• At heart, exponentiation is repeated multiplication. By definition, for any number x and any positive integer a,
 xa    = x·x ·x …x ·x .
We can expand on this fundamental idea to see how exponentiation can work with numbers that aren't just positive integers.
• Through multiplication, we can combine numbers that have the same exponent base:
 xa ·xb = xa+b.
• We can consider exponentiation acting upon exponentiation:
 (xa)b = xa·b.
• We can look at raising two numbers to the same exponent:
 xa ·ya = (xy)a.
• For any number at all, raising it to the 0 turns it into 1:
 x0 = 1.
• Raising a number to a negative will "flip" it to its reciprocal:
 x−a = 1 xa , ⎛⎝ x y ⎞⎠ = ⎛⎝ y x ⎞⎠ .
• Because of the above, we see that a denominator is effectively a negative exponent. This means if we have a fraction where the numerator and denominator have the same base, we can subtract the denominator's exponent from the numerator's exponent:
 xaxb = xa−b.
• Raising a number to a fraction is the equivalent of taking a root:
 x[1/2] = √x,                      x[1/3] = 3√ x ,                      x[1/n] = n√ x .
• If we want to find the value of raising something to an irrational number, we can find a decimal approximation of the true value by just using many decimals from our irrational number:
 8π    =   83.1415926…    ≈   83.14159.
The more accurate we need our approximation to be, the more decimals we can use from the irrational number.

### Understanding Exponents

Simplify each of the below expressions.
 x4·x9               t ·t3 ·t7 ·t8 ·t−5

• The important rule here is one of the most fundamental rules of exponents: xa ·xb = xa+b. [This rule is true for any variable or expression, we just use x for convenience.]
• For the first expression, x4 ·x9, the rule says that because they are directly multiplying each other and have the same base, we add the exponents.
 x4 ·x9     =     x4+9     =     x13
• Things work similarly for the second expression, we just need to repeat the process. We could add the first two together, then the next to the result, then the next, and so on. Alternatively, and more easily, since we're just going to wind up adding all the exponents together, we can add them all together in a single step:
 t ·t3 ·t7 ·t8 ·t−5     =     t1 + 3 + 7 + 8 −5     =     t14
[If you're not sure where the 1 in the sum comes from, it's because a t on its own can be seen as t1.]

x13               t14
Simplify each of the below expressions.
 (x3)2               ( ( ( t2 )5 )[1/4] )2

• The important rule here is that if you exponentiate something that's already been exponentiated, the exponents multiply: (xa)b = xa·b. [This rule is true for any variable or expression, we just use x for convenience.]
• For the first expression, (x3)2, the rule says that because we're raising it to the 2 after already raising it to the 3, we multiply the numbers:
 (x3)2     =     x3 ·2     =     x6
• Things work similarly for the second expression, we just need to repeat the process. We could start with the in-most part of the expression, multiply those exponents, then do that again with the next exponent, then again with the next. Alternatively, and more easily, since we're just going to wind up multiplying them all together, we can multiply them all together in a single step:
 ( ( ( t2 )5 )[1/4] )2     =     t2 ·5 ·[1/4] ·2     =     t5

x6               t5
Simplify each of the below expressions.
 x0               47190               ( a + b10 + 28)0

• The important rule here is that anything raised to the 0 becomes 1: x0 = 1. [This rule is true for any variable or expression, we just use x for convenience.]
• This is true for anything: if a number or a variable or an expression is raised to the 0, it becomes 1: 47190 = 1.
• This is true for the last expression as well: ( a + b10 + 28)0 = 1. It might be a little bit confusing because there are multiple things there, but think about it like this: no matter what (a + b10 + 28) comes out to be, it comes out to be a number. When you raise any number to the 0, it becomes 1. So, because the expression is in parentheses, we're raising the whole expression to the 0, so we get 1.

1               1               1
Simplify each of the below expressions.
 x−4 ⎛⎝ 4 7 ⎞⎠ −2 ⎛⎝ u−4v−6w2 ⎞⎠ −3

• The important rule here is that a negative exponent causes a fraction to "flip". If something isn't already written as a fraction, remember, you can always put it over 1, then flip the top and bottom. From the rules in the lesson, we have x−a = [1/(xa)]. [This rule is true for any variable or expression, we just use x for convenience.]
• Thus, for the first expression, the negative exponent causes it to "flip" into the bottom of a fraction.
 x−4 = 1 x4
• Things work similarly for the second expression as well. Because it has a negative exponent, the fraction flips, and then we apply the now-positive exponent to each part:
 ⎛⎝ 4 7 ⎞⎠ −2 = ⎛⎝ 7 4 ⎞⎠ 2 = 7242 = 49 16
• We continue this process with the third expression, the only difference is that we want to deal with the negative exponents inside the parentheses first. The negative exponent on the u causes it to flip to the bottom of the fraction, while the negative exponent on the v causes it to flip to the top:
 ⎛⎝ u−4v−6w2 ⎞⎠ −3 = ⎛⎝ v6u4w2 ⎞⎠ −3
Once that's taken care of, we apply the −3 exponent on the outside of the parentheses to the entire fraction:
 ⎛⎝ v6u4w2 ⎞⎠ −3 = ⎛⎝ u4w2v6 ⎞⎠ 3 = u4·3w2·3v6·3 = u12w6v18
[Alternatively, you could also use the rule of exponentiation on exponentiation causing exponents to multiply:
 ⎛⎝ u−4v−6w2 ⎞⎠ −3 = u(−4) (−3)v(−6)(−3)w2(−3) = u12v18w−6 = u12w6v18
Either way is acceptable and gets you to the same answer.]

[1/(x4)]               [49/16]               [(u12w6)/(v18)]
Simplify each of the below expressions.
 64[1/3]               ( 257 )[1/14]

• The important rule here is that fractional exponents are connected to roots: x[1/n] = n√{x}. [This rule is true for any variable or expression, we just use x for convenience.]
• Thus, for the first expression, we get the appropriate root:
 64[1/3]     = 3√ 64 .
Remember, 3√{64} is the cube root of 64: that is, the number that, when raised to the third power, will give us 64. After thinking about it for awhile, we realize that 4·4 ·4 = 64, and so 3√{64} = 4.
• For the next expression, we might be tempted to begin by figuring out what 257 is. Once we've used a calculator to find that out, we can attempt to take the 14 root of whatever giant number we get. The above would work, but there's a much easier way! Instead, begin by using one of our other rules: exponentiation on exponentiation causes the exponents to multiply.
 ( 257 )[1/14]     =     257 ·[1/14]     =     25[1/2]
At this point, it's much easier to apply the root rule:
 25[1/2]     = √ 25 =     5
[If you're wondering why it's written as √{25} and not 2√{25}, that's because the radical symbol (√{  }) is automatically assumed to be a square root (2√{  }) unless another number is put down to show that it is a different root.]

4              5
Approximate the below to the first two decimal places.
 5π

• It is impossible to write out the precise value of 5π with numbers because it will be an irrational number: one where the decimal expansion continues forever. However, we can get a good approximation by plugging in a decimal number for π, then using a calculator.
• What number should we use for π? Remember, π is also irrational, so whatever number we use for it, it will also be an approximation. While we could use the approximation of 3.14, the less accurate our π approximation is, the less accurate our end result will be. Therefore, let's use (at least) the first six digits of π: 3.14159. If you want even more accuracy, use even more digits of π.
• Now that we have a number that we can work with, we can plug it in to our calculator:
 5π≈ 53.14159 = 156.9918748…
The problem asked for the first two decimal places, so we cut it off to 156.99. [Notice that 5π≠ 156.9918748…. Using a much longer string of digits for π, we can find that
 5π = 156.9925….
Therefore, while 53.14159 = 156.9918748… is a good approximation, it is not perfect. In fact, no number can be perfect. Because we calculate the value using digits of π, we have to choose some specific number of digits. But since π goes on forever, we can only ever use an approximation of the true value of π, so our final result must also be an approximation. We can get very good approximations, but they can never be absolutely precise. The only way to represent what the number is absolutely precisely is by using what we started with: 5π.]

156.99 [If you got an answer that was close, but not quite the same as the above, you probably need to use more digits of π. Check out the steps for a more detailed discussion.]
Simplify the below expression.
 (x2 ·x−10 ·x7 )5(x80 ·x20 )−[1/5]

• To do this, we will need to use a combination of the rules we've gone over previously in these questions. These rules can be applied in many different orders, but will all give the same result. The steps below will give a method that is fairly direct and quick, but they are not the only way to find the answer.

•  (x2 ·x−10 ·x7 )5(x80 ·x20 )−[1/5] = (x2−10+7)5(x80+20 )−[1/5] = (x−1)5(x100 )−[1/5]

•  (x−1)5(x100 )−[1/5] = x−1 ·5x100 ·(− [1/5]) = x−5x−20
• Since both the numerator and denominator each have negative exponents on them, they both "flip" sides:
 x−5x−20 = x20x5
At this point, we can just cancel out like usual: x20 means multiplying 20 x's, while x5 means multiplying 5 x's, so the 5 on the bottom will cancel out 5 of those on top:
 x20x5 =     x15

x15
Simplify the below expression.
173 ·
2a

 3−6a ·38a36a ·176a

• To do this, we will need to use a combination of the rules we've gone over previously in these questions. These rules can be applied in many different orders, but will all give the same result. The steps below will give a method that is fairly direct and quick, but they are not the only way to find the answer.

• 173 ·
2a

 3−6a ·38a36a ·176a

=    173 ·
2a

 3−6a + 8a36a ·176a

=     173 ·
2a

 32a36a ·176a

• Notice that since 176a does not have the same base as 32a and 36a, it can't really interact with them directly: there is no way to combine them. For now, just set it aside from the fraction (although it must remain inside the radical).
173 ·
2a

 32a36a ·176a

=    173 ·
2a

 32a36a · 1 176a

=    173 ·
2a

 1 34a · 1 176a

• 173 ·
2a

 1 34a · 1 176a

=    173 ·
2a

3−4a ·17−6a

=     173 ·(3−4a ·17−6a )[1/2a]

•  173 ·(3−4a ·17−6a )[1/2a]     =    173 ·(3−4a ·[1/2a] ·17−6a ·[1/2a] )     =    173 ·( 3−2 ·17−3 )

•  173 ·17−3 ·3−2     =     170 · 1 32 =    1 · 1 9 = 1 9

[1/9]
Let f(x) = 17 − x[8/9] and g(x) = x[3/4]. Give (f °g ) (x) and simplify. Then find (f °g ) (27).

• To find (f °g ) (x), we compose the functions by plugging one into the other. Remember, it's almost always easier to know what to do by writing the above function composition in its equivalent form:
 ⎛⎝ f °g ⎞⎠ (x) = f ⎛⎝ g(x) ⎞⎠
• Plug g(x) in to f:
 f ⎛⎝ g(x) ⎞⎠ =     f( x[3/4])     =     17 − ( x[3/4] )[8/9]
• Simplify based on the rules of exponentiation:
 17 − ( x[3/4] )[8/9]     =    17 − x[3/4] ·[8/9]     =     17 − x[2/3]
Thus, in simplest form, we have (f °g ) (x) = 17 − x[2/3].
• To find the value (f °g ) (27), just plug x=27 in to what we just found:
 ⎛⎝ f °g ⎞⎠ (27)     =     17 − (27)[2/3]
From there, just simplify based on the rules of exponentiation:
 17 − (27)[2/3]     =     17 − ⎛⎝ 3√ 27 ⎞⎠ 2 =     17 − (3)2     =     17−9     =     8

(f °g ) (x) = 17 − x[2/3],     (f °g ) (27) = 8
Let t be a number such that 64t = [9/4]. What is 8t?

• Right now, we can't directly figure out what the value of t is. However, we can figure out how 8 connects to 64 in terms of exponentiation. [In a few lessons, once we learn about logarithms, we will be able to solve for t directly. But it would still be easier to do this problem the way shown below.]
• We can connect 8 and 64 through exponentiation by the following:
 √ 64 = 8     ⇒     64[1/2] = 8
Once we see that, we can replace 8 with 64[1/2].
• Plugging in to our expression 8t, we have
 8t     =     (64[1/2] )t     =     64[1/2] ·t     =     64t ·[1/2]     =     ( 64t )[1/2]
• We can now replace 64t with what we were given in the problem, then simplify:
( 64t )[1/2]     =
9

4

[1/2]

=      ⎛

 9 4

=     3

2

[3/2]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Understanding Exponents

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
• Introduction 0:05
• Fundamental Idea 1:46
• Expanding the Idea 2:28
• Multiplication of the Same Base
• Exponents acting on Exponents
• Different Bases with the Same Exponent
• To the Zero 5:35
• To the First
• Fundamental Rule with the Zero Power
• To the Negative 7:45
• Any Number to a Negative Power
• A Fraction to a Negative Power
• Division with Exponential Terms
• To the Fraction 11:33
• Square Root
• Any Root
• Summary of Rules 14:38
• To the Irrational 17:21
• Example 1 20:34
• Example 2 23:42
• Example 3 27:44
• Example 4 31:44
• Example 5 33:15

### Transcription: Understanding Exponents

Hi--welcome back to Educator.com.0000

Today, we are going to talk about understanding exponents.0002

At this point, we are quite used to using exponents; we have seen them a bunch, and we just did them a whole lot when we were working with polynomials.0005

We know that an exponent is just a shorthand way to express repeated multiplication.0012

For example, if we had 37, that would be a way of just saying 3 times 3 times 3 times 3 times 3 times 3 times 3.0016

That is 3, multiplied by itself 7 times; so the number of times it multiplies is what the exponent is; that is what it represents.0026

x2 is x times x; x3 is x times x times x (three x's); x4 would be four x's multiplied together; etc.0035

So, that is the idea of exponentiation.0044

It is clear how this process works when the exponents are positive integers; it is just multiplied by itself that many times.0047

But if we wanted to expand to any real number--what if we wanted to be able to exponentiate to any real number at all?0052

If we wanted to know things like 30, 37/8, 3-5, or 3√2, how can we deal with that?0058

This lesson will show us how to work with any kind of exponent--any real number.0066

You may have seen rules for this stuff in a previous math class; but we are also going to work through an understanding of why these rules are true.0070

Even if you remember what the rules are for what 30 is, for instance,0077

you might not have a good grasp of why 30 is what it is.0081

And so, this is the lesson that we are actually going to see how we build these rules, where they come from,0085

why we can trust in them, and also how we can make them ourselves, if we ever forget the rules.0089

If we forget the long summary list of all the rules for exponents, we will be able to just go back to our workshop0094

and work it out, even in the middle of an exam; it won't take that long for us to figure out, if we have a sense of how we get these things.0100

All right, at heart, exponentiation is this idea of repeated multiplication; that is the basic, fundamental idea.0107

By definition, for any number x and any positive integer a, xa is equal to x, multiplied by itself a times.0114

This is the key idea behind all of our coming rules for exponents.0123

If we start with this idea, and we just hold it as a fundamental truth and see all the places that it can take us,0126

we will be able to get all of these other cool rules that will explain how it will work for any real number.0132

We take this idea; we say that we believe in this; and then we move forward and try to figure out0137

what else has to be true if this idea right here is always going to work out.0141

Let's see: the first thing we can figure out: with this idea in mind, we can consider what happens when we multiply0147

(that should be "multiply," not "multiple") some xa and xb.0153

By definition, if we have xa times xb, then that means we have xa,0160

so we have x multiplied by itself a times; and xb is x multiplied by itself b times.0165

Now, that means we have some number of x's there and some number of x's there; we have a x's on one side and b x's on the other side.0172

So, how many is it together? Well, let's say, if I had a pile of 5 rocks over here, and I had a pile of 12 rocks over here, then in total, I would have 17 rocks.0179

If I had 3 rocks here and 4 rocks here, then I would have 7 rocks total.0190

So, if we have a rocks and b rocks here, then together that is going to be a + b rocks (or in this case, x's).0196

So, we put them together, and we see that we have a + b many x's showing up.0206

Thus, xa times xb equals xa + b.0211

This is another really fundamental property, and this is what is going to allow us to explore a lot of things in exponentiating with real numbers.0217

We can look at exponentiation acting on top of exponentiation.0226

If we have xa, and then raise that to the b, then that means we have xa multiplied by itself b times.0230

So, this will give us a total of a times b x's; why is that the case?0237

Well, imagine if, in each box, there are 5 rocks; well, if we have 3 of these boxes, then it is 3 times 5.0241

So, if we have a rocks in each box, a x's in each of these boxes (each of these xa's),0251

and we have b many of them, then it must be a times b of them in total.0258

Thus, (xa)b is equal to xab.0263

We just multiply the two exponents in that case.0268

We can also consider what happens if we have two different numbers, each raised to the same exponent.0272

If we have xa times ya, then we have a many x's and a many y's.0276

But we can also shuffle things around: xy is the same thing as yx; we are pretty confident in this.0283

That is one of the cool things about working with the real numbers--that they are commutative.0290

They are allowed to swap their spaces; 2 times 3 is the exact same thing as 3 times 2.0294

So, x times y is the same thing as y times x, which also means that, if we have xx times yy,0299

if we feel like it (because we are allowed to commute--we are allowed to swap the locations in multiplication),0306

then we could switch this to xyxy, which is exactly what we do here.0310

We take these x's and these y's, and we file them together: one x here, one y here, the next x here, and the next y here.0315

So, we will have xyxy; that will show up a total of a times, as well.0323

xa times ya...if we want, we can write it as (xy), that quantity, all raised to the a.0329

Now, let's try to figure out what happens when we raise a number to the 0--our first sort of difficult question.0337

x0 equals what? First, we can write x = x any time we want; just definitionally, that is the idea of equality; you are equal to yourself.0342

So, we can write x = x as x1 = x1.0354

We can put it to an exponent of 1, because that just means that it is itself, just multiplied once--0358

it is just there by itself, because there is only one of them.0363

So, there is nothing wrong with writing x = x as x1 = x1.0365

But not only that--we know that 1 is equal to 1 + 0; 1 + 0 is just 1.0369

So, if we want, we can take this 1 right here, and we will substitute it for 1 + 0; and we have x1 + 0.0376

And then, we will just sort of knock out this one, and leave it as x, just so it is easier to read what is going on.0383

So, we have x1 + 0 = x.0388

This means we have a 0 on the field that we can play with.0392

By our new property, xa + b equals xa times xb; it is our most basic property.0396

Then, we can break this up, and we can take the 1 and separate it from the 0.0402

So, we will have x1 and x0; and we will just write x1 as just x by itself, since that is what it is.0407

We have x(x0) = x; now, as long as x is not equal to 0 (if x is equal to 0, we can't really divide by it easily),0416

we divide by x; we cancel out the x's on both sides, since they show up on both sides.0428

They disappear, and we are left with x0 = 1.0433

There we go: thus, any number, as long as it isn't 0, raised to the 0, becomes 1.0437

So, we take any number at all; we raise it to the 0; it is going to become 1.0445

50 = 1; (-52)0 = 1; 47 million to the 0--you guessed it--equals 1.0449

So, whatever we take, if we raise it to the 0, it becomes 1.0461

Next, what happens when we raise something to a negative number?0466

For ease, let's just figure out what happens with x-1.0469

We begin similarly: x1 = x1; you can't stop me from saying that, just because it is the same thing on both sides.0472

And then, we can also say that 1 = 2 - 1; 2 minus 1 equals 1, so if we want, we can substitute in on this side here; and we have x2 - 1 = x.0479

Now, notice: we have a -1 on the field; so we use that property; we break it apart.0490

And so, by our property, xa + b becomes xa times xb, we have x2 times x-1.0496

So, x2 times x-1 = x; now, we can divide by x2, once again, as long as x is not equal to 0.0507

If we have that, then things get troublesome.0515

But if we divide by x2, the x2 goes from here over to here.0517

And so, we get x-1 = x/x2; and so, x/x2 cancels the x on top0525

and turns this to a 1, and we are left with = 1/x.0534

So, x-1 = 1/x; thus, any number, as long as it isn't 0, raised to a negative, flips to its reciprocal.0538

If we have some number, and we raise it to a negative, we will get that number, flipped into the reciprocal format.0547

5-1 = 1/5...so, what if we want to know what x-a is for any a at all?0555

Well, from the other work that we just did, we know that x-1 to the a is x to the -1 times a, so we can write it in that way, as well.0563

And so, we have 1/xa = 1/xa.0574

A negative in our exponent causes it to flip to its reciprocal format.0581

But it will still keep whatever that original exponent number was, as well.0585

The number will go with it, but the negative is a flip; so negatives flip, but this number that we are exponentiating to will still stay with it.0590

With this idea, we can consider if we had a fraction raised to the -1--not just a number, but a whole fraction.0599

(x/y)-1...well, you can't stop me from separating that into x times 1 over y.0604

And then, we can distribute that -1; remember, xy to the a is equal to xa times ya.0610

So, that -1 will go onto both the x and the 1/y; so we have x-1 times (1/y)-1.0615

x-1 flips to 1/x; (1/y) will flip to y/1, or just y; and so we have y/x.0623

So, negative exponents flip fractions; if we have a negative exponent, we flip whatever it is,0631

whether it is a fraction or a number--we flip to the reciprocal; great.0638

We can also look at if we have powers in the numerator and the denominator with the same base.0642

The base is just the thing that is having that exponentiation happening to it.0647

So, x is our base in almost all of these examples.0652

So, xa over xb...well, we can separate that into xa times (1/x)b.0656

We have xa on top, so we separate that: xa times 1/x, and that whole thing to the b.0662

That is equal to xa times x-b, because 1/x is equal to x-1.0668

So, 1/x to the b is equal to x-b; and now, xa times x-b0675

combines through addition, which, in this case, will become subtraction.0681

So, we have xa - b; thus, the denominator's power subtracts from the numerator's power.0684

That is another thing that we have gotten out of this.0691

Finally, what happens when we raise a number to a fraction?0694

For ease, let's look at just x1/2 first; that will make it easier to understand.0696

Once again, we start from the same place: x1 = x1.0701

And like usual, we want to bring 1/2 to bear; so we notice that 1 = 1/2 + 1/2--it is as simple as that.0704

So now, we can substitute it; we swap this in here, and we have x1/2 + 1/2 = x.0710

Now, notice: we can use our property xa + b = xa times xb,0719

the usual property we have been using, to separate this into x1/2 times x1/2 = x.0725

We already have a name for that--square root.0731

√x times √x = x; that is the idea behind square root.0736

The definition of square root is some number that, when you multiply it by itself, becomes the number you took the square root of.0742

So, √x times √x...√x is just some number that, when you multiply it by itself, becomes x.0749

So, if x1/2 times x1/2 is equal to x, then that must be the same thing as √x,0755

because it does the exact same property--itself times itself becomes x.0763

We already have a name for that: we call that square root.0768

With this property, we see that x1/2 = √x; they are equivalent.0773

We can expand this, by similar logic, to x1/n.0780

x1/n is the same thing as saying that x1/n times itself n times is equal to x,0784

because we have that n many 1/n's is equal to 1.0791

So, if we combine x1/n times x1/n, we will be adding 1/n to itself n times, which is equal to 1.0797

So, x1/n times itself n times is equal to our x, by the same logic that we split up x1/2.0803

By definition, the nth root of a number is something that, when it multiplies by itself n times...we get the original number.0810

The cube root of something, the third root of something, is a number such that, when it multiplies by itself 3 times, we get our original number.0819

The nth root of something is a number such that, when it multiplies by itself n times, we get the original number.0825

Well, look: we are multiplying it by itself n times; we are getting that original number out of it;0831

so it must be that x1/n is equal to the nth root of x.0835

With this idea in mind, we can use any rational number that we want at all.0840

We have xa/b; we can separate that into (xa)1/b.0843

And since we had just had this thing here, 1/n is the same thing as nth root, so we have 1/b becoming b√.0853

We have the b√xa, the bth root of xa.0860

We are just mixing the two properties.0865

The numerator is normal exponentiation, just multiplying by itself, like we normally would expect.0867

And then, the denominator takes a root; it takes that bth root, because it is 1/b.0872

At this point, we have a lot of different rules, and we can see a summary.0879

Our first, foremost, most fundamental rule of all is this idea right here, xa times xb = xa + b.0882

From there, we are able to figure out all of these other rules: (xa)b is equal to multiplying the two together.0891

So, if we have two different exponents raised on one thing, it multiplies them together.0898

If we have xa times ya, we can combine that to just having xy to one a.0904

x0 is always equal to 1, so if you are raising to the 0, you always come out to being 1,0910

as long as we are not dealing with x equal to 0, which we won't address.0917

x-a = 1/xa; we flip with negative exponents.0920

If you have a negative here, then you flip down to the bottom.0925

If we have a fraction with a negative, then the whole fraction flips to its reciprocal; we have y over x to the a.0929

And we also see that xa divided by xb becomes xa - b.0937

Finally, our nth root stuff: x1/n is equal to the nth root of x.0943

xa/b is equal to the bth root of xa,0949

which is the same thing as the bth root of x, to the a, because as opposed to splitting it...0953

we can split this as 1/b to the a, which we would see as b√x, all raised to the a.0958

And that is where we are getting that.0968

So, there are two different ways of looking at it.0969

Sometimes it will be more convenient to have the bth root of xa.0971

Other times, it will be more convenient to have the bth root of x, all raised to the a.0975

It will depend on the specific problem.0979

Remember: I really want you to take away this idea here.0980

If you forget any of these rules, you can figure them out from this fundamental, basic idea: xa times xb equals xa + b.0984

You just have to come up with some creative way to get the thing that you are trying to figure out,0997

whether it is fractions, whether it is 0, whether it is negative numbers--1001

you figure out some creative way to get 0, -1, 1/2, 1/n, something like that, to show up.1006

And then, you look at it, and you say, "Oh, I see--that is what it is!"1011

And so, even if you forget this in the middle of an exam--some place where you can't go and look it up in a book--1014

you can figure this out on your own; it is not that hard.1019

And having worked through it, and understanding how we are getting this, it is that much more likely to stick in your brain.1022

I know it seems like a lot of rules; but once you start using them, and you get used to using them, they will stick in your head.1028

And as long as you remember this one, you can ultimately get back anything that you have forgotten by accident.1034

All right, the final idea: what if we want to raise to an irrational?1039

So far, we have actually only discussed exponentiation using rational numbers.1043

That is the only thing that we have technically dealt with.1048

We have a/b for any a and b, but we haven't dealt with if it can't be expressed as a/b, like √2.1050

So, what if we want to raise something to an irrational number?1057

Let's say we want to look at 3√2; notice: if we want to, we can look at as many places of √2 as we want.1059

We can figure out that √2 is equal to 1.41421356...1067

and it will just keep marching on forever, because it is an irrational number,1072

so its decimal expansion goes on forever, never repeating, always changing, constantly going on forever.1075

But we can figure out what that is.1082

Furthermore, we know how to exponentiate to any rational number.1085

So, we can raise to any decimal, because any decimal is actually something that we can express as a rational.1089

For example, 1.4: if we want to, we can express that as 14 divided by 10.1094

1.414: if we wanted to, we could express that as 1414 divided by 1000.1100

All right, so we can do any of these based on all of the work that we just had.1110

We could do 31.4 as 14/10, 31.414 as 1414/1000...1114

we see that the work we have just done gives us a way to figure these things out.1122

Of course, it would be very difficult for us to do these by hand, but there are methods to do these things.1126

We could do it by hand, but we will leave it to the calculators, since they can do it so much faster.1130

We can use a calculator and get this done so much faster, because they have already been programmed with how these methods work.1134

So, we can take these various things and see: 31.4 becomes 4.6555.1140

31.414 becomes 4.7276; 31.41421 becomes 4.7287; 31.4142135 becomes 4.7288.1146

So, notice: as we use more and more of these decimals, we see the exponentiation, this 3√2, sort of stabilize to a single thing.1160

The 4 always gets used; the 7 always gets used; the 2 always gets used; the 8 always gets used.1169

We see that it is becoming more and more and more stable--that we are seeing more and more of these decimal places show up,1175

and they are not going to change--they are going to stay there forever.1181

So, while we can't get the whole number all at once (it is going to end up being irrational,1184

so it is going to also have a decimal expansion that continues forever, constantly changing), it is stabilizing to something.1189

So, we can get this idea that, while we can't write it down on paper (because it would require an infinite amount of paper), the number does exist.1197

And so, we can get as many decimals as we need for whatever our use is.1205

So, we won't formally define this: but we see that irrational exponents make sense, because we are stabilizing to some number.1209

As long as we use lots and lots of decimals when we calculate it out, 3 to the lots and lots of decimals,1215

from what we were originally trying to use as our irrational number,1221

we will be able to get something that is a very, very close approximation1224

to the exact number that we are trying to strive towards, but won't ever be able to perfectly reach.1228

All right, we are ready for some examples.1233

Evaluate 8/27, all raised to the -2/3.1236

With many of these examples, there are actually going to be multiple ways that we could approach it.1240

So, I will try to show you the various ways that you could go about it.1244

(8/27)-2/3: the first thing I would do is see that we have this negative sign.1247

So, I am going to flip and get rid of that negative: we have (27/8)2/3.1253

That equals...now we have 2 and thirds, so I would put the third into nth roots on both of them.1260

We have the third root, the cube root, of 27, and the cube root of 8.1268

And so, we now no longer have that dividing by 3 to worry about.1272

But we still have the squared, because we didn't get rid of it by putting anything out there.1277

So, the cube root of 27: 3 times 3 times 3 is 27, so we have 3; 2 times 2 times 2 is 8, so we have 2.1281

That is all raised to the 2, once again; so that is 32/22.1292

It gets distributed: 9/4, and there is our answer.1299

But there are also other ways that we could have done this.1304

We could have seen this as (8/27)-2/3, and we could have gone about this as flipping to (27/8)2/3.1306

And we could then put this as [(272)/(82)]1/3; and that is going to be kind of difficult for us to do.1318

So, we could do this with a calculator; and then we could take the cube root of 272 over 82.1327

And that would eventually simplify out to 9/4.1335

But that would be very difficult to do by hand.1338

But notice that we can do the cube root of 27, and we can do the cube root of 8.1340

So, this is probably the much easier way to do this.1343

Furthermore, we could even go about this by taking this as 8/27; we could put this as -2/3 on 8, and then 27-2/3.1346

And then, since they are both negative, they would flip into 272/3 over 82/3.1356

So, we would have 272/3 and 82/3; and then, we could, once again,1362

do either this method here or this method here.1368

At this point, I think pretty clearly that this here is our best bet--the easiest way to do it--1373

where we go through this method, because we see, "8...27...I am going to have to deal with cube roots."1380

8 and 27 are things I can easily take a cube root of, so I am going to do cube root first, then square.1389

And I will also get rid of that negative, as just a first step.1395

You can do these things in many different orders, because the rules all work together.1398

But you will want to get a sense, and as you work on more examples, you will get a sense,1402

of "Oh, the way that will make this problem easiest is for me to go through like this."1407

And so, you will develop an intuition about it.1411

And even if you end up going in a way that is not the easiest, it will still work out.1414

It just might require using a calculator or require a little extra effort.1418

All right, the next one: Simplify (x2/z)-2 times (x2y3z-1)3/y8.1421

All right, the first thing I would do is deal with the negatives, once again.1431

Usually that is easiest; so this will become z/x2, all raised to the now positive 2,1434

times...and let's distribute this 3; the 3 will go onto the x2, the y3, and the z-1.1441

So, it is x to the 2 times 3, because we have exponentiation on exponentiation; y to the 3 times 3, z to the -1 times 3, all divided by y8.1449

We can deal with this squared, and we get z2 over...that 2 also distributes onto the top and to the bottom,1463

so x2 squared is x2 times x2, or x4.1472

Also, it is x to the 2 times 2--another way of looking at it.1477

Times x6y9z-3, all over y8...1480

At this point, we see that we can cancel out the y8, and this becomes y1,1491

which we could also look at as y9 - 8, because we have one on top and one on bottom,1496

which would also become y1; so there are various ways to do this.1502

z2 times x6 times...I will move that over...let's put all of our variables so that they are near their similar ones...1508

We have x6 times y1 (which I will just leave as y), times z2, times z-3, all over x4.1520

And that is all we have on the bottom at this point.1533

So, we see that we have z2, and we see we have z-3; so that will cancel out,1536

and we will get -1, because -3 + 2...z2 times z-3 combine through addition,1539

because they are both just multiplying each other, so we have z2 - 3,1550

which becomes z-1, which is how we get what we have right here.1555

x6 divided by x4 will cancel out all but two of these, which we could also see as x6 - 4, which equals x2.1560

So, we have x2 times y times z-1.1572

And since z-1 is 1/z, we can write this as x2y, all over z.1577

Once again, like our first example, there are other things that we could have done at various points.1585

If we wanted to, we could have broken off here, and we probably could have written this as z2/x4,1589

on our next step, times x6y9, and then z-3/y8.1596

At this point, we see z-3, so we could move that over, and we could do x6/x4,1606

times y9/y8, times z2...1612

and since we had z-3, we could also write that as z3.1616

At this point, we have x6 - 4 times y9 - 8 times z2 - 3.1620

So, we have x2 times y1 times z-1, which also becomes x2y/z.1630

Or, if we wanted to, we could also just say, "We have 9 y's on top and 8 y's on the bottom;1639

so all of them will cancel on the bottom, and one will be left on the top," and similar things with the x's and the z's.1644

So, there are a variety of ways to look at these things, once you get into this.1650

And once again, it is about developing an intuition and just doing it a bunch of times.1653

And also, just be comfortable in the fact that whatever way you choose, as long as you follow the rules, they will all end up working out eventually.1657

The third example: Simplify n√(5n(53n))2n, divided by 56n2.1665

All right, this is a great one to show two different ways to approach this.1674

Let's leave that nth root intact in our first one.1676

nth root of 5n times 53n:1680

well, 5n times 53n, because they are multiplying, will go through addition: 5n + 3n.1683

So, we have 54n, all raised to the 2n, over, still, 56n2.1692

Let's expand that radical a bit, so we see the whole thing.1700

It equals...it still has that nth root, n√(54n(2n),1703

because it was exponentiation on exponentiation, 4n on 2n, over 56n2),1711

equals n√(58) (4 times 2 is 8; n times n is n2) over 56n2.1719

Now, at this point, you might be tempted to cancel out our n2's, but that would be improper,1730

because it is not like canceling 3/3, where we can cancel both of them, because they are dividing.1735

It is different, because it is about how many times they show up.1741

So, we have to use the rule that we have, which is that, when we have a fraction with something on top and something on the bottom,1743

it subtracts if they have the same thing on the bottom.1750

They are both 5's, so they can use this rule.1753

It is still the nth root, so it is going to be 58n2 - 6n2.1756

So, those are common terms, so that becomes...8n2 - 6n2...8 minus 6 is 2, but it still has n2, so it is 52n2.1765

nth root is just the same thing as saying 1/n; so it is 52n2(1/n) = 52n.1776

Great; an alternative way we could have done this, though, is that taking something to the nth root is 1/n, whenever we do it.1791

So, 5n times 53n...once again, that will be 54n, because they add together.1799

And that is all raised to the 2n, over 56n2.1805

Now, it was a radical of the whole thing, so it has to be that the whole thing is raised to the 1/n.1809

Now, we can distribute this, and we can say that that is 54n.1817

Let's leave that as 2n times 1/n, over 56n2 times 1/n,1823

because this 1/n will get applied to the top and to the bottom of our fraction.1833

54n...well, these n's cancel out; this 1/n cancels out the squared and leaves the n, so we have (54n)2/56n.1838

54n(2), because it was exponentiation on exponentiation, and still 56n,1854

equals 58n/56n, equals 58n - 6n, which equals 52n, just as well.1861

Great; so, these are two different ways of doing it, very different approaches--going through the inside,1875

or going from the outside in or inside out--but they both end up giving us the exact same answer.1879

One of the great things about all of these rules is that they all work together.1885

There is no preference of one rule versus the other.1888

So, sometimes it generates various different paths that we could go.1890

But you will develop an intuition; and once again, they all will end up working out.1893

Just make sure you practice these things on your own, and you will develop a sense for how this works.1898

And it will get faster and faster, the more you practice it.1902

All right, another example: f(x) = 7x2/3 - 2; g(x) = x6/5; give f composed with g(x), and simplify.1904

Now, remember: the first thing, when we talked about function composition:1913

f(g(x)) is almost always the way we want to switch to writing this, f acting on g acting on x.1917

What is g(x)? g(x) is x6/5, so it is f(x6/5).1925

So now, it is not about the x; it is about where the box of input goes into our formula for that function.1933

So, f(box) = 7(box)2/3 - 2.1941

In this case, our box is x6/5; we have 7 times box, x6/5;1946

and then, that box raised to the 2/3 - 2, because that is what our whole function said before.1956

So, it is 7 times x6/5; because it is exponentiation on exponentiation, it is multiplication;1962

2/3 - 2; 7 times x; 6/5 times 2/3; we notice that 6 can be broken into 3 times 2, so that knocks out this 3 and this 3.1969

And we are left with 2 times 2 on the top, so that is 4/5, minus 2.1980

So, 7 times x4/5 - 2 is what f composed with g comes out to be.1988

The final example: let x be a number such that 7x equals 3/5; what is 49x?1995

Now, at first, you might see a problem like this, and it is completely confusing, because you have no idea what to do.2002

But notice: we have 7 here; we have 49 here; so there is going to be some sort of clever trick2006

connecting the fact that 49 has something to do with 7.2012

How is 49 related to 7? 49 is equal to 72.2016

So, there is this connection between 49 and 7, so we can use that.2023

We can now apply that, and we can say, "49x...we know that 49 is equal to 72, so we can just substitute that out."2027

So, put 49x in parentheses; it is the same thing as (72)x.2038

There is nothing you can do to just stop substitution like that.2045

So, (72)x--that means 7 to the 2 times x, but we could also write that as 7 to the x times 2,2048

which we could then write as 7x, all raised to the 2, which would be...2057

we know what 7x is--it is 3/5!2062

So, we have 3/5, all raised to the 2, which means we have 9/25, because we square the 3, and we square the 5.2066

3/5 squared means that the square will go onto the 3 and go onto the 5.2078

All right, exponents are pretty cool stuff; they are really, really powerful.2083

It is important to get a good grasp of just working with them, though.2086

The only way that you will be able to get really comfortable with them is doing some practice.2089

So, just make sure that you do some practice with exponentiation, using exponents of various types.2092

But once you get in a bit of practice, you will get used to it; they are skills that stick with you.2097

And as long as you stay with this xa times xb = xa + b,2101

as long as you stay with that idea, you can figure out everything else if you get in a situation where you forget one of the rules.2109

All right, we will see you at Educator.com later--goodbye!2115