Join Professor Vincent Selhorst-Jones in your preparation to grasp a solid foundation in Math Analysis so you can excel in Calculus and beyond.

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## I. Introduction

Introduction to Math Analysis 10:03
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 45:11
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 35:31
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
Put Your Substitution in Parentheses 20:44
Example 1 23:17
Example 2 25:49
Example 3 28:11
Example 4 30:02
Coordinate Systems 35:02
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
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
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 48:43
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 56:31
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
'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

## II. Functions

Idea of a Function 39:54
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 58:26
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 48:49
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 29:20
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 48:35
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 33:24
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 51:42
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 49:37
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
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
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 28:49
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

## III. Polynomials

Intro to Polynomials 38:41
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
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 41:07
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
Form of Factored Binomials 13:38
Factoring Examples 14:40
Factoring Higher Degree Polynomials 18:19
Factoring a Cubic 18:32
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 39:43
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
Derivation 11:43
Final Form 12:23
Follow Format to Use Formula 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
Example 2: Solve the Quadratic 22:00
Example 3: Solve for Zeroes 25:28
Example 4: Using the Quadratic Formula 30:52
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
An Alternate Form to Quadratics 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 46:08
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
Example 1 29:24
Example 2 31:27
Example 3 36:22
Example 4 40:55
Complex Numbers 45:36
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
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
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 19:09
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
Non-Distinct Roots 6:59
Denoting Multiplicity 7:20
Proof Sketch of n Roots Theorem 10:45
First Root 11:36
Second Root 13:23
Continuation to Find all Roots 16:00

## IV. Rational Functions

Rational Functions and Vertical Asymptotes 33:22
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 34:16
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 49:07
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 44:56
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
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

## V. Exponential & Logarithmic Functions

Understanding Exponents 35:17
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 47:04
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 40:31
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
Thinking about Logs like Inverses 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 42:33
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 34:10
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 48:46
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

## VI. Trigonometric Functions

Angles 39:05
Intro 0:00
Degrees 0:22
Circle is 360 Degrees 0:48
Splitting a Circle 1:13
Circle is 2 Pi Radians 2:31
Half-Circle and Right Angle 4:00
Converting Between Degrees and Radians 6:24
Formulas for Degrees and Radians 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 8:02
Extra Example 2: Quadrants and Coterminal Angles 7:14
Sine and Cosine Functions 43:16
Intro 0:00
Sine and Cosine 0:15
Unit Circle 0:22
Coordinates on Unit Circle 1:03
Right Triangles 1:52
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 4:05
Extra Example 2: Graphing Sine and Cosine Functions 5:23
Sine and Cosine Values of Special Angles 33:05
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
Mnemonic: All Students Take Calculus 10:13
Example 1: Convert, Quadrant, Sine/Cosine 13:11
Example 2: Convert, Quadrant, Sine/Cosine 16:48
Example 3: All Angles and Quadrants 20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine 4:15
Extra Example 2: All Angles and Quadrants 4:03
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D 52:03
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 7:27
Extra Example 2: Find Cosine Wave Given Attributes 10:27
Tangent and Cotangent Functions 36:04
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
Sign of Tangent in Quadrants 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 2:27
Extra Example 2: Tangent and Cotangent of Angles 5:02
Secant and Cosecant Functions 27:18
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 4:58
Extra Example 2: Values of Secant and Cosecant 5:19
Inverse Trigonometric Functions 32:58
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 4:30
Extra Example 2: Arcsin/Arccos/Arctan Values 5:40
Computations of Inverse Trigonometric Functions 31:08
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 5:50
Extra Example 2: Arcsin/Arccos/Arctan Values 8:51

## VII. Trigonometric Identities

Pythagorean Identity 19:11
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 3:32
Extra Example 2: Find Angle Given Cosine and Quadrant 3:55
Identity Tan(squared)x+1=Sec(squared)x 23:16
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 2:22
Extra Example 2: Given Sec Find Tan 4:34
Intro 0:00
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 7:54
Extra Example 2: Convert to Radians and use Formulas 11:32
Double Angle Formulas 29:05
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 6:10
Extra Example 2: Prove Trigonometric Identity using Double Angle 3:18
Half-Angle Formulas 43:55
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 7:16
Extra Example 2: Prove Trigonometric Identity using Half-Angle 3:34

## VIII. Applications of Trigonometry

Trigonometry in Right Angles 25:43
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 5:10
Extra Example 2: Find Lengths of All Sides of Triangle 4:18
Law of Sines 56:40
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 8:01
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely 15:11
Law of Cosines 49:05
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 11:05
Extra Example 2: Length of Third Side and Area of Triangle 9:17
Finding the Area of a Triangle 27:37
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 2:54
Extra Example 2: Area of Triangle with Two Sides and One Angle 6:48
Word Problems and Applications of Trigonometry 34:25
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 4:36
Extra Example 2: Roads to a Town 10:34

## IX. Systems of Equations and Inequalities

Systems of Linear Equations 55:40
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 8:67
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 1:00:13
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
Example of Using a Point 12:41
Drawing Shading from the Point 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 41:01
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
Example 1 13:24
Example 2 15:50
Example 3 22:02
Example 4 29:06
Example 4, cont. 33:40

## X. Vectors and Matrices

Vectors 1:09:31
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 61:32
Dot Product & Cross Product 35:20
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 54:07
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
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
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 47:12
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 58:34
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

## XI. Alternate Ways to Graph

Parametric Equations 53:33
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 48:07
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 38:16
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

## XII. Complex Numbers and Polar Coordinates

Polar Form of Complex Numbers 40:43
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 6:48
Extra Example 2: Simplify Expression to Polar Form 7:48
DeMoivre's Theorem 57:37
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 7:41
Extra Example 2: Find Complex Fourth Roots 14:36

## XIII. Counting & Probability

Counting 31:36
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
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 44:03
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 36:58
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

## XIV. Conic Sections

Parabolas 41:27
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 21:03
Intro 0:00
What are Circles? 0:08
Example: Equidistant 0:17
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
Example 2: Center and Radius 11:51
Example 4: Equation of Circle 16:57
Ellipses 46:51
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 38:15
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 18:43
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

## XV. Sequences, Series, & Induction

Introduction to Sequences 57:45
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 40:27
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 31:36
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 39:27
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 49:53
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 1:13:13
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
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

## XVI. Preview of Calculus

Idea of a Limit 40:22
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 57:11
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
This Game is About Limits 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 32:40
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
A Limit is About Figuring Out Where a Function is 'Headed' 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 32:43
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 32:49
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
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) 51:13
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) 45:26
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

## XVII. Appendix: Graphing Calculators

Intro 0:00
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
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
Which Graphing Calculator is Best? 5:46
Too Many Factors 5:54
The Old Standby 7:10
TI-83 (Plus) 7:16
TI-84 (Plus) 7:18
Graphing Calculator Basics 10:51
Intro 0:00
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
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 10:38
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 9:45
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
Choose Point of Interest Type 2:54
Identify Where Search Should Occur 3:04
Give a Guess 3:36
Get Result 4:06
Find Out What Input Value Causes a Certain Output 5:12
For Example 5:24
Derivative 7:22
Integral 7:30
But How Do You Show Work? 8:20
Parametric & Polar Graphs 7:08
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

## Course Details:

Duration: 58 hours, 54 minutes

Number of Lessons: 89

Professor Vincent Selhorst-Jones will make sure you understand every concept and reinforce what you learned through many examples. Schools have different names for this course such as Precalculus and Algebra 3, however, the concepts are the same and will provide you the background for anything you might encounter afterwards.

• Free Sample Lessons
• Closed Captioning (CC)
• Study Guides

Topics Include:

• Variables, Equations & Algebra
• Properties of Functions
• Roots of Polynomials
• Rational Functions
• Exponential & Logarithmic Functions
• Trigonometric Functions
• Law of Sines & Cosines
• Vectors & Matrices
• Parametric Equations
• Probability
• Conic Sections
• Calculus Preview