A solid foundation in Pre-Calculus will prepare you to excel in Calculus by providing the background for concepts, problems, issues, and techniques you will encounter. Professor Tohoru M. will make sure you understand every concept and reinforce what you learned with many examples. Sample topics include everything from Polar Coordinates and Vectors, to Conic Sections, Trigonometry, and Probability. This course includes other Educator instructors specializing in Trigonometry, Algebra 2, and Statistics to make sure every aspect of Pre-Calculus is covered. Professor Tohoru M. received his B.S. from the Massachusetts Institute of Technology in Chemical Engineering and has over 15 years of experience teaching.

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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
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
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
Circle is 2 Pi Radians 2:31
Half-Circle and Right Angle 4:00
Converting Between Degrees and Radians 6:24
Formulas for Degrees and Radians 6:52
Coterminal, Complementary, Supplementary Angles 7:23
Coterminal Angles 7:30
Complementary Angles 9:40
Supplementary Angles 10:08
Example 1: Dividing a Circle 10:38
Example 2: Converting Between Degrees and Radians 11:56
Example 3: Quadrants and Coterminal Angles 14:18
Extra Example 1: Common Angle Conversions 8:02
Extra Example 2: Quadrants and Coterminal Angles 7:14
Sine and Cosine Functions 43:16
Intro 0:00
Sine and Cosine 0:15
Unit Circle 0:22
Coordinates on Unit Circle 1:03
Right Triangles 1:52
Master Right Triangle Formula: SOHCAHTOA 2:48
Odd Functions, Even Functions 4:40
Example: Odd Function 4:56
Example: Even Function 7:30
Example 1: Sine and Cosine 10:27
Example 2: Graphing Sine and Cosine Functions 14:39
Example 3: Right Triangle 21:40
Example 4: Odd, Even, or Neither 26:01
Extra Example 1: Right Triangle 4:05
Extra Example 2: Graphing Sine and Cosine Functions 5:23
Sine and Cosine Values of Special Angles 33:05
Intro 0:00
45-45-90 Triangle and 30-60-90 Triangle 0:08
45-45-90 Triangle 0:21
30-60-90 Triangle 2:06
Mnemonic: All Students Take Calculus (ASTC) 5:21
Using the Unit Circle 5:59
New Angles 6:21
Mnemonic: All Students Take Calculus 10:13
Example 1: Convert, Quadrant, Sine/Cosine 13:11
Example 2: Convert, Quadrant, Sine/Cosine 16:48
Example 3: All Angles and Quadrants 20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine 4:15
Extra Example 2: All Angles and Quadrants 4:03
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D 52:03
Intro 0:00
Amplitude and Period of a Sine Wave 0:38
Sine Wave Graph 0:58
Amplitude: Distance from Middle to Peak 1:18
Peak: Distance from Peak to Peak 2:41
Phase Shift and Vertical Shift 4:13
Phase Shift: Distance Shifted Horizontally 4:16
Vertical Shift: Distance Shifted Vertically 6:48
Example 1: Amplitude/Period/Phase and Vertical Shift 8:04
Example 2: Amplitude/Period/Phase and Vertical Shift 17:39
Example 3: Find Sine Wave Given Attributes 25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift 7:27
Extra Example 2: Find Cosine Wave Given Attributes 10:27
Tangent and Cotangent Functions 36:04
Intro 0:00
Tangent and Cotangent Definitions 0:21
Tangent Definition 0:25
Cotangent Definition 0:47
Master Formula: SOHCAHTOA 1:01
Mnemonic 1:16
Tangent and Cotangent Values 2:29
Remember Common Values of Sine and Cosine 2:46
90 Degrees Undefined 4:36
Slope and Menmonic: ASTC 5:47
Uses of Tangent 5:54
Example: Tangent of Angle is Slope 6:09
Sign of Tangent in Quadrants 7:49
Example 1: Graph Tangent and Cotangent Functions 10:42
Example 2: Tangent and Cotangent of Angles 16:09
Example 3: Odd, Even, or Neither 18:56
Extra Example 1: Tangent and Cotangent of Angles 2:27
Extra Example 2: Tangent and Cotangent of Angles 5:02
Secant and Cosecant Functions 27:18
Intro 0:00
Secant and Cosecant Definitions 0:17
Secant Definition 0:18
Cosecant Definition 0:33
Example 1: Graph Secant Function 0:48
Example 2: Values of Secant and Cosecant 6:49
Example 3: Odd, Even, or Neither 12:49
Extra Example 1: Graph of Cosecant Function 4:58
Extra Example 2: Values of Secant and Cosecant 5:19
Inverse Trigonometric Functions 32:58
Intro 0:00
Arcsine Function 0:24
Restrictions between -1 and 1 0:43
Arcsine Notation 1:26
Arccosine Function 3:07
Restrictions between -1 and 1 3:36
Cosine Notation 3:53
Arctangent Function 4:30
Between -Pi/2 and Pi/2 4:44
Tangent Notation 5:02
Example 1: Domain/Range/Graph of Arcsine 5:45
Example 2: Arcsin/Arccos/Arctan Values 10:46
Example 3: Domain/Range/Graph of Arctangent 17:14
Extra Example 1: Domain/Range/Graph of Arccosine 4:30
Extra Example 2: Arcsin/Arccos/Arctan Values 5:40
Computations of Inverse Trigonometric Functions 31:08
Intro 0:00
Inverse Trigonometric Function Domains and Ranges 0:31
Arcsine 0:41
Arccosine 1:14
Arctangent 1:41
Example 1: Arcsines of Common Values 2:44
Example 2: Odd, Even, or Neither 5:57
Example 3: Arccosines of Common Values 12:24
Extra Example 1: Arctangents of Common Values 5:50
Extra Example 2: Arcsin/Arccos/Arctan Values 8:51
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
Intro 0:00
How to Remember 0:48
Cofunction Identities 1:31
How to Remember Graphically 1:44
Where to Use Cofunction Identities 2:52
Example 1: Derive the Formula for cos(A-B) 3:08
Example 2: Use Addition and Subtraction Formulas 16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity 25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities 7:54
Extra Example 2: Convert to Radians and use Formulas 11:32
Double Angle Formulas 29:05
Intro 0:00
Main Formula 0:07
How to Remember from Addition Formula 0:18
Two Other Forms 1:35
Example 1: Find Sine and Cosine of Angle using Double Angle 3:16
Example 2: Prove Trigonometric Identity using Double Angle 9:37
Example 3: Use Addition and Subtraction Formulas 12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle 6:10
Extra Example 2: Prove Trigonometric Identity using Double Angle 3:18
Half-Angle Formulas 43:55
Intro 0:00
Main Formulas 0:09
Confusing Part 0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle 0:54
Example 2: Prove Trigonometric Identity using Half-Angle 11:51
Example 3: Prove the Half-Angle Formula for Tangents 18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle 7:16
Extra Example 2: Prove Trigonometric Identity using Half-Angle 3:34
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
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
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
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
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
Same Dimensions 0:25
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 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
Equation of a Circle 0:44
Example: Standard Form 1:11
Graphing Circles 1:47
Example: Circle 1:56
Center Not at Origin 3:07
Example: Completing the Square 3:51
Example 1: Equation of Circle 6:44
Example 2: Center and Radius 11:51
Example 4: Equation of Circle 16:57
Ellipses 46:51
Intro 0:00
What Are Ellipses? 0:11
Foci 0:23
Properties of Ellipses 1:43
Major Axis, Minor Axis 1:47
Center 1:54
Length of Major Axis and Minor Axis 3:21
Standard Form 5:33
Example: Standard Form of Ellipse 6:09
Vertical Major Axis 9:14
Example: Vertical Major Axis 9:46
Graphing Ellipses 12:51
Complete the Square and Symmetry 13:00
Example: Graphing Ellipse 13:16
Equation with Center at (h, k) 19:57
Horizontal and Vertical 20:14
Difference 20:27
Example: Center at (h, k) 20:55
Example 1: Equation of Ellipse 24:05
Example 2: Equation of Ellipse 27:57
Example 3: Equation of Ellipse 32:32
Example 4: Graph Ellipse 38:27
Hyperbolas 38:15
Intro 0:00
What are Hyperbolas? 0:12
Two Branches 0:18
Foci 0:38
Properties 2:00
Transverse Axis and Conjugate Axis 2:06
Vertices 2:46
Length of Transverse Axis 3:14
Distance Between Foci 3:31
Length of Conjugate Axis 3:38
Standard Form 5:45
Vertex Location 6:36
Known Points 6:52
Vertical Transverse Axis 7:26
Vertex Location 7:50
Asymptotes 8:36
Vertex Location 8:56
Rectangle 9:28
Diagonals 10:29
Graphing Hyperbolas 12:58
Example: Hyperbola 13:16
Equation with Center at (h, k) 16:32
Example: Center at (h, k) 17:21
Example 1: Equation of Hyperbola 19:20
Example 2: Equation of Hyperbola 22:48
Example 3: Graph Hyperbola 26:05
Example 4: Equation of Hyperbola 36:29
Conic Sections 18:43
Intro 0:00
Conic Sections 0:16
Double Cone Sections 0:24
Standard Form 1:27
General Form 1:37
Identify Conic Sections 2:16
B = 0 2:50
X and Y 3:22
Identify Conic Sections, Cont. 4:46
Parabola 5:17
Circle 5:51
Ellipse 6:31
Hyperbola 7:10
Example 1: Identify Conic Section 8:01
Example 2: Identify Conic Section 11:03
Example 3: Identify Conic Section 11:38
Example 4: Identify Conic Section 14:50
Intro 0:00
Solutions 2:49
Graphs of Possible Solutions 3:10
Example: Elimination 4:21
Solutions 11:39
Example: 0, 1, 2, 3, 4 Solutions 11:50
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
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