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Math Analysis Online Prof. Vincent Selhorst-Jones
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Level Intermediate
89 Lessons (58hr : 54min)
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Audio: English
English

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

Table of Contents
Course Details
About Prof. Selhorst-Jones

Section 1: 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
			Tip To Help You Remember the Signs	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
		The Plane and Quadrants		18:50
			Quadrant I	19:04
			Quadrant II	19:21
			Quadrant III	20:04
			Quadrant IV	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			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
			Grades	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			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 Basketball Factory	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
		Some Comments		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			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
Section 3: 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
		Leading Coefficient Test		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			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
		Factoring Quadratics		13:08
			Quadratic Trinomials	13:21
			Form of Factored Binomials	13:38
			Factoring Examples	14:40
		Factoring Quadratics, Check Your Work		16:58
		Factoring Higher Degree Polynomials		18:19
			Factoring a Cubic	18:32
			Factoring a Quadratic	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			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
		Solving Quadratics with Ease		9:56
		The Quadratic Formula		11:38
			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
	Properties of Quadratic Functions			45:34
		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
			Advantages of Synthetic Method	26:49
			Advantages of Long Division	27:13
		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
		Addition and Subtraction		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
		Factoring So-Called 'Irreducible' Quadratics		19:24
			Revisiting the Quadratic Formula	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			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
		Comments: Multiplicity		6:29
			Non-Distinct Roots	6:59
			Denoting Multiplicity	7:20
		Comments: Complex Numbers Necessary		7:41
		Comments: Complex Coefficients Allowed		8:55
		Comments: Existence Theorem		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			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
		Irreducible Quadratic Factors		8:25
			Example of Quadratic Factors	9:26
			Multiple Quadratic Factors	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			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
Section 6: Trigonometric Functions
	Angles			39:05
		Intro		0:00
		Degrees		0:22
			Circle is 360 Degrees	0:48
			Splitting a Circle	1:13
		Radians		2:08
			Circle is 2 Pi Radians	2:31
			One Radian	2:52
			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
			Adjacent, Opposite, Hypotenuse	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		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
			Other Quadrants	9:43
			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
Section 7: 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
	Addition and Subtraction Formulas			52:52
		Intro		0:00
		Addition and Subtraction Formulas		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		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
Section 8: 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
Section 9: 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
		Test Points for Shading		11:42
			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
			Shade Appropriately	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			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
		Talking About Specific Entries		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
		Matrix Addition		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			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
Section 11: 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
Section 12: 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
Section 13: 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
		Addition Principle		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			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
Section 14: 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
			Radius	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
		Example 2: Center and Radius		11:51
		Example 3: Radius		15:08
		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
Section 15: 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
			Diagonal Addition of Terms	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			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
			Two Good Ways to Think About This	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)			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
Section 17: Appendix: Graphing Calculators
	Buying a Graphing Calculator			10:41
		Intro		0:00
		Should You Buy?		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
		Deciding on Buying		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
		Which to Buy		5:44
			Which Graphing Calculator is Best?	5:46
			Too Many Factors	5:54
			Do Your Research	6:12
		The Old Standby		7:10
			TI-83 (Plus)	7:16
			TI-84 (Plus)	7:18
		Tips for Purchasing		9:17
			Buy Online	9:19
			Buy Used	9:35
			Ask Around	10:09
	Graphing Calculator Basics			10:51
		Intro		0:00
		Read the Manual		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
			Think About the Results	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			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
			Adjust Viewing Window	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
		Advanced Technique: Arbitrary Solving		5:10
			Find Out What Input Value Causes a Certain Output	5:12
			For Example	5:24
		Advanced Technique: Calculus		7:18
			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

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.

Additional Features:

Free Sample Lessons
Closed Captioning (CC)
Downloadable Lecture Slides
Study Guides
Instructor Comments

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

Professor Selhorst-Jones has been teaching 10+ years and double-majored in Mathematics and Theater at Pomona College, as well as received an M.F.A. in Acting from Harvard University.

Student Testimonials:

"Thank you very much Vincent. I really enjoyed this course and will be moving on to further math classes. Your teaching style is excellent and I admire your enthusiasm for the topic. I always struggled to understand the meaning of calculus in school and I believe the meaning is more important than the technical application. Your lectures have filled in all the holes school created." — Richard G.

“Thank for the video! It is very well explained and easy to understand.” — Akilah M.

“Awesome Video!!! It grabbed my intuition and slammed it into synchrony with this concept. Thank you! The dialogue made things really enjoyable.” — Jason T.

"What a perfect way to recap old material. I love the way you appeal to intuition and then explain it in math terms. Outstanding!" — Jonathan T.

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Course Overview:

Level Intermediate
89 Lessons (58hr : 54min)
27 already enrolled!
Audio: English
English

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Table of Contents

Section 1: Introduction

Section 2: Functions

Section 3: Polynomials

Section 4: Rational Functions

Section 5: Exponential & Logarithmic Functions

Section 6: Trigonometric Functions

Section 7: Trigonometric Identities

Section 8: Applications of Trigonometry

Section 9: Systems of Equations and Inequalities

Section 10: Vectors and Matrices

Section 11: Alternate Ways to Graph

Section 12: Complex Numbers and Polar Coordinates

Section 13: Counting & Probability

Section 14: Conic Sections

Section 15: Sequences, Series, & Induction

Section 16: Preview of Calculus

Section 17: Appendix: Graphing Calculators

Student Testimonials:

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Start Learning Now

Membership Overview

Math Analysis Online Prof. Vincent Selhorst-Jones Facebook Twitter More

Table of Contents

Section 1: Introduction

Section 2: Functions

Section 3: Polynomials

Section 4: Rational Functions

Section 5: Exponential & Logarithmic Functions

Section 6: Trigonometric Functions

Section 7: Trigonometric Identities

Section 8: Applications of Trigonometry

Section 9: Systems of Equations and Inequalities

Section 10: Vectors and Matrices

Section 11: Alternate Ways to Graph

Section 12: Complex Numbers and Polar Coordinates

Section 13: Counting & Probability

Section 14: Conic Sections

Section 15: Sequences, Series, & Induction

Section 16: Preview of Calculus

Section 17: Appendix: Graphing Calculators

Student Testimonials:

Related Courses:

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Membership Overview

Math Analysis Online Prof. Vincent Selhorst-Jones
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