Hyperbolas, Pre Calculus

Hyperbolas

IX. Analytic Geometry (Conic Sections): Lecture 5 | 38:15 min

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Hyperbolas

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I. Polar Coordinates and Complex Numbers
	Polar Coordinates	29:32
	Graphing Polar Coordinates	64:42
	Introduction to Complex Numbers	34:10
	Simplifying Complex Numbers	38:44
	Polar Forms of Complex Numbers	33:50
	Products and Quotients of Complex Numbers in Polar Form	26:12
	Powers and Roots of Complex Numbers and de Moivre's Formula	63:15
II. Mathematical Analysis
	Mathematical Induction	30:39
	Fundamental Theorem of Algebra	28:55
	Rational Functions	45:48
III. Trigonometry (Functions)
	Angles	39:05
	Sine and Cosine Functions	43:16
	Sine and Cosine Values of Special Angles	33:05
	Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D	52:03
	Tangent and Cotangent Functions	36:04
	Secant and Cosecant Functions	27:18
	Inverse Trigonometric Functions	32:58
	Computations of Inverse Trigonometric Functions	31:08
IV. Trigonometry (Identities)
	Pythagorean Identity	19:11
	Identity Tan(squared)x+1=Sec(squared)x	23:16
	Addition and Subtraction Formulas	52:52
	Double Angle Formulas	29:05
	Half-Angle Formulas	43:55
V. Trigonometry (Applications)
	Trigonometry in Right Angles	25:43
	Law of Sines	56:40
	Law of Cosines	49:05
	Finding the Area of a Triangle	27:37
	Word Problems and Applications of Trigonometry	34:25
	DeMoivre's Theorem	57:37
VI. Vectors and Parametric Equations
	Geometric Vectors	40:10
	Algebraic Representation of Vectors	54:36
	Vectors in 3D Space	50:08
	Vectors and Parametric Equations	34:10
	Using Parametric Equations to Model Motion	31:13
VII. Linear Algebra (Matrices)
	Basic Matrix Concepts	11:34
	Matrix Operations	21:36
	Matrix Multiplication	29:36
	Determinants	33:13
	Cramer's Rule	28:25
	Identity and Inverse Matrices	22:25
	Gauss Jordan Method	58:48
VIII. Sequences and Series
	Arithmetic Sequences	21:16
	Arithmetic Series	21:36
	Geometric Sequences	23:03
	Geometric Series	22:43
	Infinite Geometric Series	18:32
	Recursion and Special Sequences	14:34
	Binomial Theorem	48:30
IX. Analytic Geometry (Conic Sections)
	Midpoint and Distance Formulas	32:42
	Parabolas	41:27
	Circles	21:03
	Ellipses	46:51
	Hyperbolas	38:15
	Conic Sections	18:43
	Solving Quadratic Systems	47:04
X. Probability
	Experiment, Outcomes, and Sample Space	14:54
	Calculating Probability	14:13
	Probability and Events	22:08
	Intersection of Events and the Multiplication Rule	19:58
	Union of Events and the Addition Rule	18:28
	Bayes' Rule	16:59
	Probability II	77:39

Mathematics: Pre Calculus

I. Polar Coordinates and Complex Numbers
	Polar Coordinates			29:32
		Intro		0:00
		Rectangular vs. Polar Coordinates		0:04
			Describing a Location of a Point on a Plane: Rectangular	0:07
			Describing a Location of a Point on a Plane: Polar Coordinate	3:04
		Coordinate Conversion		4:23
			Using SOHCAHTOA	4:24
		Rectangular to Polar		6:03
			Rectangular Coordinate to Polar Coordinates	6:04
		Polar to Rectangular		9:16
			Polar Coordinates to Rectangular Coordinates	9:19
		Equation Conversion		11:25
			Example: Circle with r = 2	11:40
		Linear Equation with Slope		14:57
			Example: Describing Line	14:58
		Vertical or Horizontal Line		18:16
				19:22
		Extra Example 1: Plot the Polar Coordinates		21:16
		Extra Example 2: plot the polar Coordinates		23:44
		Extra Example 3: Convert to Linear Equation		24:56
		Extra Example 4: Convert to Linear Equation		26:09
	Graphing Polar Coordinates			64:42
		Intro		0:00
		Six Types of Polar Graphs		0:07
			Line	0:31
			Circle	1:08
			Limacon	1:46
			Rose	2:16
			Lemniscate	2:50
			Spirals	3:15
		Step in Graphing		3:50
			Determine Type of Graph	4:27
			Use Information Given by a, b, and n	4:39
			Plot Critical Points	4:52
		Circle		6:43
			Circle: r = ncosθ	6:44
		Circle Example		8:10
			Example: r = 6sinθ	8:13
		Limacon		12:58
			Inner Loop	13:06
			Cardioid	20:18
		Limacon, cont.		26:00
			Dimpled	26:05
			Convex	31:40
		Rose		33:30
			r = a sin nθ	33:31
			Length of Petals = a	33:39
		Rose: Even and Odd		34:06
			Even n (2n Petals)	34:07
			Odd n (n Petals)	39:28
		Lemniscate		41:08
			'Bowtie'	41:09
			Length of Petals	41:25
			Number of Petals	41:29
		Lemniscate Example		41:46
			Example: r² = 9sin2θ	41:47
		Spiral		44:13
			Spiral: r = a + bθ	44:14
			Example: r = 2θ	44:34
		Other Curves		46:31
			Conchoid	47:20
			Hyperbolic Spiral	47:35
			Strophoid	47:39
			Logarithmic Spiral	48:00
		Extra Example 1: Identify Graph		49:20
		Extra Example 2: Identify Graph		51:51
		Extra Example 3: Identify Graph		54:22
		Extra Example 4: Identify Graph		58:42
	Introduction to Complex Numbers			34:10
		Intro		0:00
		Imaginary Numbers		0:06
			n-th Degree Polynomials	0:40
			Multiplicities and Complex Roots	1:04
			What is an Imaginary Number?	2:00
		Complex Numbers		7:27
			Complex Numbers: a + bi	7:55
			Electrical Engineering	9:00
		Operations with Complex Numbers		9:53
			Cartesian, Rectangular, or Algebraic Form	10:13
			1) Addition/ Subtraction	10:38
			2) Multiplication	12:30
			3) Rationalizing the Denominator (Using the Complex Conjugate)	15:05
		Powers of i		20:53
			i = √-1	20:58
		Extra Example 1: Add/ Subtract the Complex Numbers		25:03
		Extra Example 2: Multiply the Complex Numbers		26:09
		Extra Example 3: Powers of i		28:39
		Extra Example 4: Rationalize the Denominator Using Complex Conjugate		30:32
	Simplifying Complex Numbers			38:44
		Intro		0:00
		Negative Sign Under Radical		1:01
			Example: Simplifying Negative sign Under Radical	1:02
		Like Terms		3:54
			Example: Simplifying Like Terms	3:55
		Rationalizing the Denominator		5:36
			Example: Simplify 3/i	5:52
			Example: Simplify (3+i)/5i	7:47
		Distribution and FOIL		10:25
			Example: 6i(3 - 2i)	10:49
			Example: (2 + 4i)(2 - 6i)	12:02
		Complex Conjugates		13:54
			Example: Complex Conjugates	13:55
		Powers of i		19:17
			Example: Find Powers of i	19:18
		Extra Example 1: Simplify the Expression		24:28
		Extra Example 2: Simplify the Expression		28:03
		Extra Example 3: Simplify the Expression		32:42
		Extra Example 4: Powers of i		35:10
	Polar Forms of Complex Numbers			33:50
		Intro		0:00
		Complex Plane		0:25
			Definition and Example of Complex Plane	0:26
		Representing Complex Numbers		2:51
			Example: Representing Complex Numbers	2:55
		Absolute Value of a Complex Number		4:25
			Definition of the Absolute Value of a + bi	4:26
		Trigonometric or Polar Form of a Complex Number		6:38
			Trigonometric Form of the Complex Number z = a + bi	7:00
			Modulus of z	8:14
			Argument of z	8:27
		Euler's Formula		9:36
			Example: Euler's Formula	9:38
		Complex to Trigonometric		11:11
			Example: Write Complex Number in Trigonometric Form	11:12
		Complex to Standard Form		16:35
			Example: Write Complex Number in Standard Form	16:36
		Extra Example 1: Give the Polar Coordinates		21:26
		Extra Example 2: Give the Polar Coordinates		24:11
		Extra Example 3: Find the Rectangular Coordinates		26:03
		Extra Example 4: Find the Rectangular Coordinates		30:01
	Products and Quotients of Complex Numbers in Polar Form			26:12
		Intro		0:00
		Sum and Difference		0:14
			Example of Sum and Difference Formulas	0:16
		Multiplication		1:00
			Multiplication Formulas	1:24
		Multiplications Example		5:53
			Example: Find the Product of 2 Complex Numbers	5:54
		Division		8:25
			Methods for Division	8:26
			Euler's Equation	8:58
		Division Example		10:43
			Example: Find the Quotient of the Complex Numbers	10:45
		Extra Example 1: Find the Product		14:46
		Extra Example 2: Simplify		17:08
		Extra Example 3: Find the Quotient		20:18
		Extra Example 4: Find the Quotient		22:59
	Powers and Roots of Complex Numbers and de Moivre's Formula			63:15
		Intro		0:00
		Powers of Complex Numbers		0:19
			Methods of Finding Powers of Complex Numbers	0:47
		De Moivre's Formula		3:07
			De Moivre's Formula	3:40
			De Moivre's Theorem	3:51
			De Moivre's Formula Example	4:56
			Example: Using De Moivre's Theorem	4:57
		De Moivre's Formula Example		9:08
			Example: Using De Moivre's Theorem	9:09
		Roots of Complex Numbers		11:43
			Equation for Finding the n-th Roots of a Complex Number	11:44
		Origin of Formula		13:47
			Fundamental Theorem of Algebra	13:48
		Origin of Formula, cont.		17:37
			Origin of Formula	17:38
		n-th Root Example		22:53
			Example: Calculate the 6-th Roots	22:54
		Extra Example 1: De Moivre's Theorem		29:59
		Extra Example 2: Cube Roots of 8i		36:50
		Extra Example 3: Find the Four Fourth Roots of -16		44:43
		Extra Example 4: Find the 3 Cube Root of -2 + 2i		52:08
II. Mathematical Analysis
	Mathematical Induction			30:39
		Intro		0:00
		What is Induction?		0:07
			Example of Mathematical Induction	0:09
		Historical Background		1:36
			French Mathematician Pierre de Fermat	1:42
		Euler		2:55
			Leonhard Euler	2:57
		Principle of Mathematical Induction		4:05
			Example: Statement Involving Positive Integer	4:08
		Extra Example 1: Mathematical Induction		6:35
		Extra Example 2: Mathematical Induction		12:18
		Extra Example 3: Mathematical Induction		16:08
		Extra Example 4: Mathematical Induction		21:33
	Fundamental Theorem of Algebra			28:55
		Intro		0:00
		Fundamental Theorem of Algebra		0:08
			Fundamental Theorem of Algebra	0:09
			Example of the Fundamental Theorem of Algebra	1:34
		Rational Zero Test		2:40
			Rational Zero Test	2:41
			Example of the Rational Zero Test	3:30
		Binomial Factors		5:47
			Binomial Factors	5:48
			Example of the Binomial Factors	6:18
		Descarte Rule of Signs		9:41
			Descarte Rule of Signs	9:42
		Descarte Rule of Signs Example		11:23
			Example: Describing Polynomial	11:24
		Extra Example 1: Find Zeroes of a Polynomial Function		13:09
		Extra Example 2: Find all the Zeroes		17:37
		Extra Example 3: Find the Polynomial		21:28
		Extra Example 4: Descarte Rule of Signs		25:37
	Rational Functions			45:48
		Intro		0:00
		Graph of Rational Functions		0:06
			Rational Functions	0:07
		Vertical Asymptote		0:53
			Example: y=1/x	0:55
		Horizontal Asymptote		2:52
			Example: Horizontal Asymptote	2:53
		Horizontal Asymptote Cases		4:27
			1st Case	4:40
			2nd Case	7:15
			3rd Case	9:00
			Special Case	10:31
		Intercepts		13:06
			x-intercepts	13:15
			y-intercepts	14:04
		Discontinuities		15:29
			Example: Discontinuities	15:30
		Examples		17:08
			H(x)= 5/(x + 3)	17:12
		Extra Example 1: Graph and Identify the Function		22:02
		Extra Example 2: Graph and Identify the Function		29:04
		Extra Example 3: Graph and Identify the Function		34:27
		Extra Example 4: Graph and Identify the Function		41:29
III. Trigonometry (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
IV. Trigonometry (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
V. Trigonometry (Applications)
	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
	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
VI. Vectors and Parametric Equations
	Geometric Vectors			40:10
		Intro		0:00
		Magnitude and Direction		0:07
			Describing Quantities	0:12
			William Rowan Hamilton	1:42
			James Maxwell	2:04
		Vector Representation		2:43
			Scalar	3:50
			2 Vectors may not be Collinear	4:30
		Algebraically or Geometrically		5:44
			Representing Vectors	5:45
		Adding and Subtracting Vectors		7:58
			Addition: u + v	8:15
			Subtraction: u - v	10:16
		Multiplying Vectors		12:57
			Scalar Multiplication: 2u + (1/3)v	12:58
		Unit Vectors		15:28
			Standard Vectors Along x and y Axis	15:29
		Extra Example 1: Sketch 3u - (1/2)v		17:58
		Extra Example 2: Sketch the Vector		20:31
		Extra Example 3: Resultant Velocity of a Plane		22:57
		Extra Example 4: Add the Scalar Multiples		35:33
	Algebraic Representation of Vectors			54:36
		Intro		0:00
		Component Form		0:08
			Component Form	0:29
			Using Standard Unit Vector i, j	1:31
		Vector Operations		3:21
			Addition	3:30
			Subtraction	4:24
			Scalar Multiplication	5:40
		Unit Vectors		6:17
			Example: Find the Unit Vector	6:36
			Example: Find the Angle	9:42
		Dot Product		12:27
			Example: Dot Product	12:28
		Determining Orthogonality		22:28
			Example: Orthogonal	22:29
			Parallel	26:07
		Projections		28:52
			Finding Vector Components (Projection)	28:54
		Extra Example 1: Calculate u - v		35:40
		Extra Example 2: Find the Unit Vector		36:41
		Extra Example 3: Find the Angle		39:32
		Extra Example 4: Find the Direction and Magnitude of the Resultant Force		42:25
	Vectors in 3D Space			50:08
		Intro		0:00
		Algebraic Vector Operations in 3D		0:07
			Addition	1:11
			Subtraction	3:10
			Scalar Multiplication	4:30
			Magnitude (Distance Formula)	4:55
		Algebraic Vector Operations in 3D		7:54
			Equation of a Sphere	8:07
			Midpoint Formula	8:56
			Dot Product	9:55
			Angle Between 2 Vectors	11:00
			Quadrants/Octants	14:24
		Cross Product		16:13
			Cross Product/ Determinant	16:17
		Vector and Planes		21:01
			Example: x +3y + 6z = 12	21:02
		Cartesian Equation		24:39
			Cartesian Equation	24:41
		Equation of the Plane		31:16
			Example: Find the Equation of the Plane	31:17
		Extra Example 1: Find u - v		33:34
		Extra Example 2: Are the Vectors Perpendicular?		35:20
		Extra Example 3: Find the Equation of the Plane		37:32
		Extra Example 4: Non-zero Vector, Equation of a Plane, and Area		40:16
	Vectors and Parametric Equations			34:10
		Intro		0:00
		Rectangular and Parametric Equations		0:08
			Introduction to Rectangular and Parametric Equations	0:18
		Converting Parametric to Rectangular		2:35
			Example: Converting Parametric to Rectangular	2:36
		Trigonometric Functions		7:12
			Example: Trigonometric Functions	7:13
		Converting Rectangular to Parametric		10:08
			Example: Converting Rectangular to Parametric	10:09
		Extra Example 1: Parametric Equations		11:43
		Extra Example 2: Rectangular Equations		15:05
		Extra Example 3: Rectangular Equation		18:24
		Extra Example 4: Describe the Cycloid		22:17
	Using Parametric Equations to Model Motion			31:13
		Intro		0:00
		Vector Equations		0:23
			Linear Motion	1:06
			Parametric Equations	3:56
		Vector Equations		5:17
			Parametric Equations	5:18
		Parametric Equations on Curves		7:41
			Example: Parametric Equations on Curves	7:42
		Lissajous Curve		12:17
			Examples: Lissajous Curve	12:18
		Extra Example 1: Vector Equation		16:55
		Extra Example 2: Velocity, Vector Equation, and Parametric Equations		18:34
		Extra Example 3: Velocity and Speed		22:27
		Extra Example 4: Hyperbola		22:52
VII. Linear Algebra (Matrices)
	Basic Matrix Concepts			11:34
		Intro		0:00
		What is a Matrix		0:26
			Brackets	0:46
			Designation	1:21
			Element	1:47
			Matrix Equations	1:59
		Dimensions		2:27
			Rows (m) and Columns (n)	2:37
			Examples: Dimensions	2:43
		Special Matrices		4:22
			Row Matrix	4:32
			Column Matrix	5:00
			Zero Matrix	6:00
		Equal Matrices		6:30
			Example: Corresponding Elements	6:36
		Example 1: Matrix Dimension		8:12
		Example 2: Matrix Dimension		9:03
		Example 3: Zero Matrix		9:38
		Example 4: Row and Column Matrix		10:26
	Matrix Operations			21:36
		Intro		0:00
		Matrix Addition		0:18
			Same Dimensions	0:25
			Example: Adding Matrices	1:04
		Matrix Subtraction		3:42
			Same Dimensions	3:48
			Example: Subtracting Matrices	4:04
		Scalar Multiplication		6:08
			Scalar Constant	6:24
			Example: Multiplying Matrices	6:32
		Properties of Matrix Operations		8:23
			Commutative Property	8:41
			Associative Property	9:08
			Distributive Property	9:44
		Example 1: Matrix Addition		10:24
		Example 2: Matrix Subtraction		11:58
		Example 3: Scalar Multiplication		14:23
		Example 4: Matrix Properties		16:09
	Matrix Multiplication			29:36
		Intro		0:00
		Dimension Requirement		0:17
			n = p	0:24
			Resulting Product Matrix (m x q)	1:21
			Example: Multiplication	1:54
		Matrix Multiplication		3:38
			Example: Matrix Multiplication	4:07
		Properties of Matrix Multiplication		10:46
			Associative Property	11:00
			Associative Property (Scalar)	11:28
			Distributive Property	12:06
			Distributive Property (Scalar)	12:30
		Example 1: Possible Matrices		13:31
		Example 2: Multiplying Matrices		17:08
		Example 3: Multiplying Matrices		20:41
		Example 4: Matrix Properties		24:41
	Determinants			33:13
		Intro		0:00
		What is a Determinant		0:13
			Square Matrices	0:23
			Vertical Bars	0:41
		Determinant of a 2x2 Matrix		1:21
			Second Order Determinant	1:37
			Formula	1:45
			Example: 2x2 Determinant	1:58
		Determinant of a 3x3 Matrix		2:50
			Expansion by Minors	3:08
			Third Order Determinant	3:19
			Expanding Row One	4:06
			Example: 3x3 Determinant	6:40
		Diagonal Method for 3x3 Matrices		13:24
			Example: Diagonal Method	13:36
		Example 1: Determinant of 2x2		18:59
		Example 2: Determinant of 3x3		20:03
		Example 3: Determinant of 3x3		25:35
		Example 4: Determinant of 3x3		29:22
	Cramer's Rule			28:25
		Intro		0:00
		System of Two Equations in Two Variables		0:16
			One Variable	0:50
			Determinant of Denominator	1:14
			Determinants of Numerators	2:23
			Example: System of Equations	3:34
		System of Three Equations in Three Variables		7:06
			Determinant of Denominator	7:17
			Determinants of Numerators	7:52
		Example 1: Two Equations		8:57
		Example 2: Two Equations		13:21
		Example 3: Three Equations		17:11
		Example 4: Three Equations		23:43
	Identity and Inverse Matrices			22:25
		Intro		0:00
		Identity Matrix		0:13
			Example: 2x2 Identity Matrix	0:30
			Example: 4x4 Identity Matrix	0:50
			Properties of Identity Matrices	1:24
			Example: Multiplying Identity Matrix	2:52
		Matrix Inverses		5:30
			Writing Matrix Inverse	6:07
		Inverse of a 2x2 Matrix		6:39
			Example: 2x2 Matrix	7:31
		Example 1: Inverse Matrix		10:18
		Example 2: Find the Inverse Matrix		13:04
		Example 3: Find the Inverse Matrix		17:53
		Example 4: Find the Inverse Matrix		20:44
	Gauss Jordan Method			58:48
		Intro		0:00
		Two Equations		0:13
			Example: Using Substitution/ Elimination	0:14
		Three Equations or Higher		2:39
			History	2:40
		Row Echelon Form and Back Substitution		3:58
			Row-Echelon Form	5:01
			Back Substitution	10:49
		Gaussian Elimination		12:32
			Steps in Gaussian Elimination	12:33
		Augmented Matrix		15:17
			Example: Augmented Matrix	15:18
			More Example: Augmented Matrix	17:14
		Gauss-Jordan Elimination		26:19
			Example: Gauss-Jordan Elimination	26:20
		Extra Example 1: Augmented Matrix		34:21
		Extra Example 2: Augmented Matrix and Row-Echelon Form		40:58
		Extra Example 3: Gaussian Elimination and Back Substitution		49:21
		Extra Example 4: Gauss-Jordan Elimination		52:34
VIII. Sequences and Series
	Arithmetic Sequences			21:16
		Intro		0:00
		Sequences		0:10
			General Form of Sequence	0:16
			Example: Finite/Infinite Sequences	0:33
		Arithmetic Sequences		0:28
			Common Difference	2:41
			Example: Arithmetic Sequence	2:50
		Formula for the nth Term		3:51
			Example: nth Term	4:32
		Equation for the nth Term		6:37
			Example: Using Formula	6:56
		Arithmetic Means		9:47
			Example: Arithmetic Means	10:16
		Example 1: nth Term		12:38
		Example 2: Arithmetic Means		13:49
		Example 3: Arithmetic Means		16:12
		Example 4: nth Term		18:26
	Arithmetic Series			21:36
		Intro		0:00
		What are Arithmetic Series?		0:11
			Common Difference	0:28
			Example: Arithmetic Sequence	0:43
			Example: Arithmetic Series	1:09
			Finite/Infinite Series	1:36
		Sum of Arithmetic Series		2:27
			Example: Sum	3:21
		Sigma Notation		5:53
			Index	6:14
			Example: Sigma Notation	7:14
		Example 1: First Term		9:00
		Example 2: Three Terms		10:52
		Example 3: Sum of Series		14:14
		Example 4: Sum of Series		18:13
	Geometric Sequences			23:03
		Intro		0:00
		Geometric Sequences		0:11
			Common Difference	0:38
			Common Ratio	1:08
			Example: Geometric Sequence	2:38
		nth Term of a Geometric Sequence		4:41
			Example: nth Term	4:56
		Geometric Means		6:51
			Example: Geometric Mean	7:09
		Example 1: 9th Term		12:04
		Example 2: Geometric Means		15:18
		Example 3: nth Term		18:32
		Example 4: Three Terms		20:59
	Geometric Series			22:43
		Intro		0:00
		What are Geometric Series?		0:11
			List of Numbers	0:24
			Example: Geometric Series	1:12
		Sum of Geometric Series		2:16
			Example: Sum of Geometric Series	2:41
		Sigma Notation		4:21
			Lower Index, Upper Index	4:38
			Example: Sigma Notation	4:57
		Another Sum Formula		6:08
			Example: n Unknown	6:28
		Specific Terms		7:41
			Sum Formula	7:56
			Example: Specific Term	8:11
		Example 1: Sum of Geometric Series		10:02
		Example 2: Sum of 8 Terms		14:15
		Example 3: Sum of Geometric Series		18:23
		Example 4: First Term		20:16
	Infinite Geometric Series			18:32
		Intro		0:00
		What are Infinite Geometric Series		0:10
			Example: Finite	0:29
			Example: Infinite	0:51
			Partial Sums	1:09
			Formula	1:37
		Sum of an Infinite Geometric Series		2:39
			Convergent Series	2:58
			Example: Sum of Convergent Series	3:28
		Sigma Notation		7:31
			Example: Sigma	8:17
		Repeating Decimals		8:42
			Example: Repeating Decimal	8:53
		Example 1: Sum of Infinite Geometric Series		12:15
		Example 2: Repeating Decimal		13:24
		Example 3: Sum of Infinite Geometric Series		15:14
		Example 4: Repeating Decimal		16:48
	Recursion and Special Sequences			14:34
		Intro		0:00
		Fibonacci Sequence		0:05
			Background of Fibonacci	0:23
			Recursive Formula	0:37
			Fibonacci Sequence	0:52
			Example: Recursive Formula	2:18
		Iteration		3:49
			Example: Iteration	4:30
		Example 1: Five Terms		7:08
		Example 2: Three Terms		9:00
		Example 3: Five Terms		10:38
		Example 4: Three Iterates		12:41
	Binomial Theorem			48:30
		Intro		0:00
		Pascal's Triangle		0:06
			Expand Binomial	0:13
			Pascal's Triangle	4:26
		Properties		6:52
			Example: Properties of Binomials	6:58
		Factorials		9:11
			Product	9:28
			Example: Factorial	9:45
		Binomial Theorem		11:08
			Example: Binomial Theorem	13:48
		Finding a Specific Term		18:36
			Example: Specific Term	19:26
		Example 1: Expand		24:39
		Example 2: Fourth Term		30:26
		Example 3: Five Terms		36:13
		Example 4: Three Iterates		45:07
IX. Analytic Geometry (Conic Sections)
	Midpoint and Distance Formulas			32:42
		Intro		0:00
		Midpoint Formula		0:15
			Example: Midpoint	0:30
		Distance Formula		2:30
			Example: Distance	2:52
		Example 1: Midpoint and Distance		4:58
		Example 2: Midpoint and Distance		8:07
		Example 3: Median Length		18:51
		Example 4: Perimeter and Area		23:36
	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
	Solving Quadratic Systems			47:04
		Intro		0:00
		Linear Quadratic Systems		0:22
			Example: Linear Quadratic System	0:45
		Solutions		2:49
			Graphs of Possible Solutions	3:10
		Quadratic Quadratic System		4:10
			Example: Elimination	4:21
		Solutions		11:39
			Example: 0, 1, 2, 3, 4 Solutions	11:50
		Systems of Quadratic Inequalities		12:48
			Example: Quadratic Inequality	13:09
		Example 1: Solve Quadratic System		21:42
		Example 2: Solve Quadratic System		29:13
		Example 3: Solve Quadratic System		35:02
		Example 4: Solve Quadratic Inequality		40:29
X. Probability
	Experiment, Outcomes, and Sample Space			14:54
		Intro		0:00
		Basic Definitions		0:29
			Experiment	0:35
			Outcomes	0:55
			Sample Space	1:04
		Examples		1:34
			Roll a Die	1:39
			Flip a Coin	2:33
		Simple and Compound Events		3:30
			Event	3:43
			Simple Event	3:58
			Compound Event	4:27
		Example		5:14
		Extra Example 1		0:59
		Extra Example 2		4:21
	Calculating Probability			14:13
		Intro		0:00
		What is Probability		0:27
			Range	0:53
			Sum of Probabilities	1:26
			Example: Football Game	2:05
		Classical Probability		2:53
			Equally Likely Outcomes	3:05
			Example: Coin Flip	4:08
			Example: Die Roll	5:12
		Relative Frequency		6:44
			Example	7:22
		Subjective Probability		9:38
			Example	10:06
		Extra Example 1		1:04
		Extra Example 2		1:33
	Probability and Events			22:08
		Intro		0:00
		Mutually Exclusive Events		0:17
			Example: Coin Flip	0:27
			Example: Die Roll	3:03
		Independent Events		5:13
			Notation	3:31
			Example: Coin	6:01
		Independent Events, cont.		9:19
			Example: Coffee Drinkers	9:23
		Mutually Exclusive vs Independent		13:03
		Complementary Events		14:08
			Example: Coffee Drinkers	15:37
		Extra Example 1		1:16
		Extra Example 2		3:32
	Intersection of Events and the Multiplication Rule			19:58
		Intro		0:00
		Intersection of Events		0:08
			Venn Diagram	1:20
		Multiplication Rule		2:22
			Joint Probability	2:23
			Example	3:23
		Example		6:30
		Multiplication Rule for Independent Events		10:30
			Example	11:39
		Joint Probability of Mutually Exclusive Events		15:24
		Extra Example 1		1:24
		Extra Example 2		2:09
	Union of Events and the Addition Rule			18:28
		Intro		0:00
		Union of Events		0:06
			Venn Diagram	0:52
		Addition Rule		2:01
			Example: Coffee Drinkers	3:25
		Example		6:26
		Addition Rule for Mutually Exclusive Events		9:11
		Example		10:27
		Extra Example 1		2:41
		Extra Example 2		1:15
	Bayes' Rule			16:59
		Intro		0:00
		Partition of Events		0:07
			Venn Diagram	0:17
		Law of Total Probability		3:12
		Bayes' Rule		6:11
		Example		9:09
		Extra Example 1		4:07
	Probability II			77:39
		Intro		0:00
		What is Probability		0:06
			Foundation of Probability	0:31
			What is Probability?	1:34
		Terms		5:12
			Common Probability Terms	5:13
		Types of Probability Problems		13:41
			Probability of Either of 2 Events	13:44
		Types of Probability Problems		16:53
			Probability of Either of 2 Mutually Exclusive Events	17:08
		'At Least' Problems		18:29
			Example: 'At Least' Problem	18:30
		Likely Scenarios		24:16
			Cards, Dice, Flipping Coins, and Colored Marbles	24:17
			Example: 'and ' Scenario	24:54
			Example: 'or' Scenario	25:57
		Events Occurring Together		30:29
			Example: Conditional Probability	30:30
		Conditional Probability		37:10
			Problems Involving Conditional Probability	37:11
		Conditional Probability		44:48
			Example: Binomial Probability Theorem	44:57
		Combinations		51:00
			Examples: Combinations	51:14
		Expected Value		54:08
			Definition of Expected Value	54:09
			Example: Expected Value	54:27
		Extra Example 1: Conditional Probability		58:33
		Extra Example 2: Expected Value		63:39
		Extra Example 3: Marbles		72:52

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Mathematics: Pre Calculus

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