Vincent SelhorstJones
Area Under a Curve (Integrals)
Slide Duration:Table of Contents
10m 3s
 Intro0:00
 Title of the Course0:06
 Different Names for the Course0:07
 Precalculus0:12
 Math Analysis0:14
 Trigonometry0:16
 Algebra III0:20
 Geometry II0:24
 College Algebra0:30
 Same Concepts0:36
 How do the Lessons Work?0:54
 Introducing Concepts0:56
 Apply Concepts1:04
 Go through Examples1:25
 Who is this Course For?1:38
 Those Who Need eExtra Help with Class Work1:52
 Those Working on Material but not in Formal Class at School1:54
 Those Who Want a Refresher2:00
 Try to Watch the Whole Lesson2:20
 Understanding is So Important3:56
 What to Watch First5:26
 Lesson #2: Sets, Elements, and Numbers5:30
 Lesson #7: Idea of a Function5:33
 Lesson #6: Word Problems6:04
 What to Watch First, cont.6:46
 Lesson #2: Sets, Elements and Numbers6:56
 Lesson #3: Variables, Equations, and Algebra6:58
 Lesson #4: Coordinate Systems7:00
 Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope7:02
 Lesson #6: Word Problems7:10
 Lesson #7: Idea of a Function7:12
 Lesson #8: Graphs7:14
 Graphing Calculator Appendix7:40
 What to Watch Last8:46
 Let's get Started!9:48
45m 11s
 Intro0:00
 Introduction0:05
 Sets and Elements1:19
 Set1:20
 Element1:23
 Name a Set2:20
 Order The Elements Appear In Has No Effect on the Set2:55
 Describing/ Defining Sets3:28
 Directly Say All the Elements3:36
 Clearly Describing All the Members of the Set3:55
 Describing the Quality (or Qualities) Each member Of the Set Has In Common4:32
 Symbols: 'Element of' and 'Subset of'6:01
 Symbol is ∈6:03
 Subset Symbol is ⊂6:35
 Empty Set8:07
 Symbol is ∅8:20
 Since It's Empty, It is a Subset of All Sets8:44
 Union and Intersection9:54
 Union Symbol is ∪10:08
 Intersection Symbol is ∩10:18
 Sets Can Be Weird Stuff12:26
 Can Have Elements in a Set12:50
 We Can Have Infinite Sets13:09
 Example13:22
 Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times14:08
 This Set Has Infinitely Many Distinct Elements14:40
 Numbers as Sets16:03
 Natural Numbers ℕ16:16
 Including 0 and the Negatives ℤ18:13
 Rational Numbers ℚ19:27
 Can Express Rational Numbers with Decimal Expansions22:05
 Irrational Numbers23:37
 Real Numbers ℝ: Put the Rational and Irrational Numbers Together25:15
 Interval Notation and the Real Numbers26:45
 Include the End Numbers27:06
 Exclude the End Numbers27:33
 Example28:28
 Interval Notation: Infinity29:09
 Use ∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other29:14
 Always Use Parentheses29:50
 Examples30:27
 Example 131:23
 Example 235:26
 Example 338:02
 Example 442:21
35m 31s
 Intro0:00
 What is a Variable?0:05
 A Variable is a Placeholder for a Number0:11
 Affects the Output of a Function or a Dependent Variable0:24
 Naming Variables1:51
 Useful to Use Symbols2:21
 What is a Constant?4:14
 A Constant is a Fixed, Unchanging Number4:28
 We Might Refer to a Symbol Representing a Number as a Constant4:51
 What is a Coefficient?5:33
 A Coefficient is a Multiplicative Factor on a Variable5:37
 Not All Coefficients are Constants5:51
 Expressions and Equations6:42
 An Expression is a String of Mathematical Symbols That Make Sense Used Together7:05
 An Equation is a Statement That Two Expression Have the Same Value8:20
 The Idea of Algebra8:51
 Equality8:59
 If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same9:41
 Always Do The Exact Same Thing to Both Sides12:22
 Solving Equations13:23
 When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something13:33
 Look For What Values Makes the Equation True13:38
 Isolate the Variable by Doing Algebra14:37
 Order of Operations16:02
 Why Certain Operations are Grouped17:01
 When You Don't Have to Worry About Order17:39
 Distributive Property18:15
 It Allows Multiplication to Act Over Addition in Parentheses18:23
 We Can Use the Distributive Property in Reverse to Combine Like Terms19:05
 Substitution20:03
 Use Information From One Equation in Another Equation20:07
 Put Your Substitution in Parentheses20:44
 Example 123:17
 Example 225:49
 Example 328:11
 Example 430:02
35m 2s
 Intro0:00
 Inherent Order in ℝ0:05
 Real Numbers Come with an Inherent Order0:11
 Positive Numbers0:21
 Negative Numbers0:58
 'Less Than' and 'Greater Than'2:04
 Tip To Help You Remember the Signs2:56
 Inequality4:06
 Less Than or Equal and Greater Than or Equal4:51
 One Dimension: The Number Line5:36
 Graphically Represent ℝ on a Number Line5:43
 Note on Infinities5:57
 With the Number Line, We Can Directly See the Order We Put on ℝ6:35
 Ordered Pairs7:22
 Example7:34
 Allows Us to Talk About Two Numbers at the Same Time9:41
 Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ10:41
 Two Dimensions: The Plane13:13
 We Can Represent Ordered Pairs with the Plane13:24
 Intersection is known as the Origin14:31
 Plotting the Point14:32
 Plane = Coordinate Plane = Cartesian Plane = ℝ²17:46
 The Plane and Quadrants18:50
 Quadrant I19:04
 Quadrant II19:21
 Quadrant III20:04
 Quadrant IV20:20
 Three Dimensions: Space21:02
 Create Ordered Triplets21:09
 Visually Represent This21:19
 ThreeDimension = Space = ℝ³21:47
 Higher Dimensions22:24
 If We Have n Dimensions, We Call It nDimensional Space or ℝ to the nth Power22:31
 We Can Represent Places In This nDimensional Space As Ordered Groupings of n Numbers22:41
 Hard to Visualize Higher Dimensional Spaces23:18
 Example 125:07
 Example 226:10
 Example 328:58
 Example 431:05
48m 43s
 Intro0:00
 Introduction0:07
 Midpoint: One Dimension2:09
 Example of Something More Complex2:31
 Use the Idea of a Middle3:28
 Find the Midpoint of Arbitrary Values a and b4:17
 How They're Equivalent5:05
 Official Midpoint Formula5:46
 Midpoint: Two Dimensions6:19
 The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle6:38
 Arbitrary Pair of Points Example7:25
 Distance: One Dimension9:26
 Absolute Value10:54
 Idea of Forcing Positive11:06
 Distance: One Dimension, Formula11:47
 Distance Between Arbitrary a and b11:48
 Absolute Value Helps When the Distance is Negative12:41
 Distance Formula12:58
 The Pythagorean Theorem13:24
 a²+b²=c²13:50
 Distance: Two Dimensions14:59
 Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem15:16
 Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)16:21
 Slope19:30
 Slope is the Rate of Change19:41
 m = rise over run21:27
 Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)22:31
 Interpreting Slope24:12
 Positive Slope and Negative Slope25:40
 m=1, m=0, m=126:48
 Example 128:25
 Example 231:42
 Example 336:40
 Example 442:48
56m 31s
 Intro0:00
 Introduction0:05
 What is a Word Problem?0:45
 Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols0:48
 Requires Us to Think1:32
 Why Are They So Hard?2:11
 Reason 1: No Simple Formula to Solve Them2:16
 Reason 2: Harder to Teach Word Problems2:47
 You Can Learn How to Do Them!3:51
 Grades7:57
 'But I'm Never Going to Use This In Real Life'9:46
 Solving Word Problems12:58
 First: Understand the Problem13:37
 Second: What Are You Looking For?14:33
 Third: Set Up Relationships16:21
 Fourth: Solve It!17:48
 Summary of Method19:04
 Examples on Things Other Than Math20:21
 MathSpecific Method: What You Need Now25:30
 Understand What the Problem is Talking About25:37
 Set Up and Name Any Variables You Need to Know25:56
 Set Up Equations Connecting Those Variables to the Information in the Problem Statement26:02
 Use the Equations to Solve for an Answer26:14
 Tip26:58
 Draw Pictures27:22
 Breaking Into Pieces28:28
 Try Out Hypothetical Numbers29:52
 Student Logic31:27
 Jump In!32:40
 Example 134:03
 Example 239:15
 Example 344:22
 Example 450:24
39m 54s
 Intro0:00
 Introduction0:04
 What is a Function?1:06
 A Visual Example and NonExample1:30
 Function Notation3:47
 f(x)4:05
 Express What Sets the Function Acts On5:45
 Metaphors for a Function6:17
 Transformation6:28
 Map7:17
 Machine8:56
 Same Input Always Gives Same Output10:01
 If We Put the Same Input Into a Function, It Will Always Produce the Same Output10:11
 Example of Something That is Not a Function11:10
 A NonNumerical Example12:10
 The Functions We Will Use15:05
 Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers15:11
 Usually Told the Rule of a Given Function15:27
 How To Use a Function16:18
 Apply the Rule to Whatever Our Input Value Is16:28
 Make Sure to Wrap Your Substitutions in Parentheses17:09
 Functions and Tables17:36
 Table of Values, Sometimes Called a TTable17:46
 Example17:56
 Domain: What Goes In18:55
 The Domain is the Set of all Inputs That the Function Can Accept18:56
 Example19:40
 Range: What Comes Out21:27
 The Range is the Set of All Possible Outputs a Function Can Assign21:34
 Example21:49
 Another Example Would Be Our Initial Function From Earlier in This Lesson22:29
 Example 123:45
 Example 225:22
 Example 327:27
 Example 429:23
 Example 533:33
58m 26s
 Intro0:00
 Introduction0:04
 How to Interpret Graphs1:17
 Input / Independent Variable1:47
 Output / Dependent Variable2:00
 Graph as Input ⇒ Output2:23
 One Way to Think of a Graph: See What Happened to Various Inputs2:25
 Example2:47
 Graph as Location of Solution4:20
 A Way to See Solutions4:36
 Example5:20
 Which Way Should We Interpret?7:13
 Easiest to Think In Terms of How Inputs Are Mapped to Outputs7:20
 Sometimes It's Easier to Think In Terms of Solutions8:39
 Pay Attention to Axes9:50
 Axes Tell Where the Graph Is and What Scale It Has10:09
 Often, The Axes Will Be Square10:14
 Example12:06
 Arrows or No Arrows?16:07
 Will Not Use Arrows at the End of Our Graphs17:13
 Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops17:18
 How to Graph19:47
 Plot Points20:07
 Connect with Curves21:09
 If You Connect with Straight Lines21:44
 Graphs of Functions are Smooth22:21
 More Points ⇒ More Accurate23:38
 Vertical Line Test27:44
 If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function28:41
 Every Point on a Graph Tells Us Where the xValue Below is Mapped30:07
 Domain in Graphs31:37
 The Domain is the Set of All Inputs That a Function Can Accept31:44
 Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'33:19
 Range in Graphs33:53
 Graphing Calculators: Check the Appendix!36:55
 Example 138:37
 Example 245:19
 Example 350:41
 Example 453:28
 Example 555:50
48m 49s
 Intro0:00
 Introduction0:05
 Increasing Decreasing Constant0:43
 Looking at a Specific Graph1:15
 Increasing Interval2:39
 Constant Function4:15
 Decreasing Interval5:10
 Find Intervals by Looking at the Graph5:32
 Intervals Show xvalues; Write in Parentheses6:39
 Maximum and Minimums8:48
 Relative (Local) Max/Min10:20
 Formal Definition of Relative Maximum12:44
 Formal Definition of Relative Minimum13:05
 Max/Min, More Terms14:18
 Definition of Extrema15:01
 Average Rate of Change16:11
 Drawing a Line for the Average Rate16:48
 Using the Slope of the Secant Line17:36
 Slope in Function Notation18:45
 Zeros/Roots/xintercepts19:45
 What Zeros in a Function Mean20:25
 Even Functions22:30
 Odd Functions24:36
 Even/Odd Functions and Graphs26:28
 Example of an Even Function27:12
 Example of an Odd Function28:03
 Example 129:35
 Example 233:07
 Example 340:32
 Example 442:34
29m 20s
 Intro0:00
 Introduction0:04
 Don't Forget that Axes Matter!1:44
 The Constant Function2:40
 The Identity Function3:44
 The Square Function4:40
 The Cube Function5:44
 The Square Root Function6:51
 The Reciprocal Function8:11
 The Absolute Value Function10:19
 The Trigonometric Functions11:56
 f(x)=sin(x)12:12
 f(x)=cos(x)12:24
 Alternate Axes12:40
 The Exponential and Logarithmic Functions13:35
 Exponential Functions13:44
 Logarithmic Functions14:24
 Alternating Axes15:17
 Transformations and Compositions16:08
 Example 117:52
 Example 218:33
 Example 320:24
 Example 426:07
48m 35s
 Intro0:00
 Introduction0:04
 Vertical Shift1:12
 Graphical Example1:21
 A Further Explanation2:16
 Vertical Stretch/Shrink3:34
 Graph Shrinks3:46
 Graph Stretches3:51
 A Further Explanation5:07
 Horizontal Shift6:49
 Moving the Graph to the Right7:28
 Moving the Graph to the Left8:12
 A Further Explanation8:19
 Understanding Movement on the xaxis8:38
 Horizontal Stretch/Shrink12:59
 Shrinking the Graph13:40
 Stretching the Graph13:48
 A Further Explanation13:55
 Understanding Stretches from the xaxis14:12
 Vertical Flip (aka Mirror)16:55
 Example Graph17:07
 Multiplying the Vertical Component by 117:18
 Horizontal Flip (aka Mirror)18:43
 Example Graph19:01
 Multiplying the Horizontal Component by 119:54
 Summary of Transformations22:11
 Stacking Transformations24:46
 Order Matters25:20
 Transformation Example25:52
 Example 129:21
 Example 234:44
 Example 338:10
 Example 443:46
33m 24s
 Intro0:00
 Introduction0:04
 Arithmetic Combinations0:40
 Basic Operations1:20
 Definition of the Four Arithmetic Combinations1:40
 Composite Functions2:53
 The Function as a Machine3:32
 Function Compositions as Multiple Machines3:59
 Notation for Composite Functions4:46
 Two Formats6:02
 Another Visual Interpretation7:17
 How to Use Composite Functions8:21
 Example of on Function acting on Another9:17
 Example 111:03
 Example 215:27
 Example 321:11
 Example 427:06
51m 42s
 Intro0:00
 Introduction0:04
 Analogies to a Piecewise Function1:16
 Different Potatoes1:41
 Factory Production2:27
 Notations for Piecewise Functions3:39
 Notation Examples from Analogies6:11
 Example of a Piecewise (with Table)7:24
 Example of a NonNumerical Piecewise11:35
 Graphing Piecewise Functions14:15
 Graphing Piecewise Functions, Example16:26
 Continuous Functions16:57
 Statements of Continuity19:30
 Example of Continuous and NonContinuous Graphs20:05
 Interesting Functions: the Step Function22:00
 Notation for the Step Function22:40
 How the Step Function Works22:56
 Graph of the Step Function25:30
 Example 126:22
 Example 228:49
 Example 336:50
 Example 446:11
49m 37s
 Intro0:00
 Introduction0:04
 Analogy by picture1:10
 How to Denote the inverse1:40
 What Comes out of the Inverse1:52
 Requirement for Reversing2:02
 The Basketball Factory2:12
 The Importance of Information2:45
 OnetoOne4:04
 Requirement for Reversibility4:21
 When a Function has an Inverse4:43
 OnetoOne5:13
 Not OnetoOne5:50
 Not a Function6:19
 Horizontal Line Test7:01
 How to the test Works7:12
 OnetoOne8:12
 Not OnetoOne8:45
 Definition: Inverse Function9:12
 Formal Definition9:21
 Caution to Students10:02
 Domain and Range11:12
 Finding the Range of the Function Inverse11:56
 Finding the Domain of the Function Inverse12:11
 Inverse of an Inverse13:09
 Its just x!13:26
 Proof14:03
 Graphical Interpretation17:07
 Horizontal Line Test17:20
 Graph of the Inverse18:04
 Swapping Inputs and Outputs to Draw Inverses19:02
 How to Find the Inverse21:03
 What We Are Looking For21:21
 Reversing the Function21:38
 A Method to Find Inverses22:33
 Check Function is OnetoOne23:04
 Swap f(x) for y23:25
 Interchange x and y23:41
 Solve for y24:12
 Replace y with the inverse24:40
 Some Comments25:01
 Keeping Step 2 and 3 Straight25:44
 Switching to Inverse26:12
 Checking Inverses28:52
 How to Check an Inverse29:06
 Quick Example of How to Check29:56
 Example 131:48
 Example 234:56
 Example 339:29
 Example 446:19
28m 49s
 Intro0:00
 Introduction0:06
 Direct Variation1:14
 Same Direction1:21
 Common Example: Groceries1:56
 Different Ways to Say that Two Things Vary Directly2:28
 Basic Equation for Direct Variation2:55
 Inverse Variation3:40
 Opposite Direction3:50
 Common Example: Gravity4:53
 Different Ways to Say that Two Things Vary Indirectly5:48
 Basic Equation for Indirect Variation6:33
 Joint Variation7:27
 Equation for Joint Variation7:53
 Explanation of the Constant8:48
 Combined Variation9:35
 Gas Law as a Combination9:44
 Single Constant10:33
 Example 110:49
 Example 213:34
 Example 315:39
 Example 419:48
38m 41s
 Intro0:00
 Introduction0:04
 Definition of a Polynomial1:04
 Starting Integer2:06
 Structure of a Polynomial2:49
 The a Constants3:34
 Polynomial Function5:13
 Polynomial Equation5:23
 Polynomials with Different Variables5:36
 Degree6:23
 Informal Definition6:31
 Find the Largest Exponent Variable6:44
 Quick Examples7:36
 Special Names for Polynomials8:59
 Based on the Degree9:23
 Based on the Number of Terms10:12
 Distributive Property (aka 'FOIL')11:37
 Basic Distributive Property12:21
 Distributing Two Binomials12:55
 Longer Parentheses15:12
 Reverse: Factoring17:26
 LongTerm Behavior of Polynomials17:48
 Examples18:13
 Controlling TermTerm with the Largest Exponent19:33
 Positive and Negative Coefficients on the Controlling Term20:21
 Leading Coefficient Test22:07
 Even Degree, Positive Coefficient22:13
 Even Degree, Negative Coefficient22:39
 Odd Degree, Positive Coefficient23:09
 Odd Degree, Negative Coefficient23:27
 Example 125:11
 Example 227:16
 Example 331:16
 Example 434:41
41m 7s
 Intro0:00
 Introduction0:05
 Roots in Graphs1:17
 The xintercepts1:33
 How to Remember What 'Roots' Are1:50
 Naïve Attempts2:31
 Isolating Variables2:45
 Failures of Isolating Variables3:30
 Missing Solutions4:59
 Factoring: How to Find Roots6:28
 How Factoring Works6:36
 Why Factoring Works7:20
 Steps to Finding Polynomial Roots9:21
 Factoring: How to Find Roots CAUTION10:08
 Factoring is Not Easy11:32
 Factoring Quadratics13:08
 Quadratic Trinomials13:21
 Form of Factored Binomials13:38
 Factoring Examples14:40
 Factoring Quadratics, Check Your Work16:58
 Factoring Higher Degree Polynomials18:19
 Factoring a Cubic18:32
 Factoring a Quadratic19:04
 Factoring: Roots Imply Factors19:54
 Where a Root is, A Factor Is20:01
 How to Use Known Roots to Make Factoring Easier20:35
 Not all Polynomials Can be Factored22:30
 Irreducible Polynomials23:27
 Complex Numbers Help23:55
 Max Number of Roots/Factors24:57
 Limit to Number of Roots Equal to the Degree25:18
 Why there is a Limit25:25
 Max Number of Peaks/Valleys26:39
 Shape Information from Degree26:46
 Example Graph26:54
 Max, But Not Required28:00
 Example 128:37
 Example 231:21
 Example 336:12
 Example 438:40
39m 43s
 Intro0:00
 Introduction0:05
 Square Roots and Equations0:51
 Taking the Square Root to Find the Value of x0:55
 Getting the Positive and Negative Answers1:05
 Completing the Square: Motivation2:04
 Polynomials that are Easy to Solve2:20
 Making Complex Polynomials Easy to Solve3:03
 Steps to Completing the Square4:30
 Completing the Square: Method7:22
 Move C over7:35
 Divide by A7:44
 Find r7:59
 Add to Both Sides to Complete the Square8:49
 Solving Quadratics with Ease9:56
 The Quadratic Formula11:38
 Derivation11:43
 Final Form12:23
 Follow Format to Use Formula13:38
 How Many Roots?14:53
 The Discriminant15:47
 What the Discriminant Tells Us: How Many Roots15:58
 How the Discriminant Works16:30
 Example 1: Complete the Square18:24
 Example 2: Solve the Quadratic22:00
 Example 3: Solve for Zeroes25:28
 Example 4: Using the Quadratic Formula30:52
45m 34s
 Intro0:00
 Introduction0:05
 Parabolas0:35
 Examples of Different Parabolas1:06
 Axis of Symmetry and Vertex1:28
 Drawing an Axis of Symmetry1:51
 Placing the Vertex2:28
 Looking at the Axis of Symmetry and Vertex for other Parabolas3:09
 Transformations4:18
 Reviewing Transformation Rules6:28
 Note the Different Horizontal Shift Form7:45
 An Alternate Form to Quadratics8:54
 The Constants: k, h, a9:05
 Transformations Formed10:01
 Analyzing Different Parabolas10:10
 Switching Forms by Completing the Square11:43
 Vertex of a Parabola16:30
 Vertex at (h, k)16:47
 Vertex in Terms of a, b, and c Coefficients17:28
 Minimum/Maximum at Vertex18:19
 When a is Positive18:25
 When a is Negative18:52
 Axis of Symmetry19:54
 Incredibly Minor Note on Grammar20:52
 Example 121:48
 Example 226:35
 Example 328:55
 Example 431:40
46m 8s
 Intro0:00
 Introduction0:05
 Reminder: Roots Imply Factors1:32
 The Intermediate Value Theorem3:41
 The Basis: U between a and b4:11
 U is on the Function4:52
 Intermediate Value Theorem, Proof Sketch5:51
 If Not True, the Graph Would Have to Jump5:58
 But Graph is Defined as Continuous6:43
 Finding Roots with the Intermediate Value Theorem7:01
 Picking a and b to be of Different Signs7:10
 Must Be at Least One Root7:46
 Dividing a Polynomial8:16
 Using Roots and Division to Factor8:38
 Long Division Refresher9:08
 The Division Algorithm12:18
 How It Works to Divide Polynomials12:37
 The Parts of the Equation13:24
 Rewriting the Equation14:47
 Polynomial Long Division16:20
 Polynomial Long Division In Action16:29
 One Step at a Time20:51
 Synthetic Division22:46
 Setup23:11
 Synthetic Division, Example24:44
 Which Method Should We Use26:39
 Advantages of Synthetic Method26:49
 Advantages of Long Division27:13
 Example 129:24
 Example 231:27
 Example 336:22
 Example 440:55
45m 36s
 Intro0:00
 Introduction0:04
 A Wacky Idea1:02
 The Definition of the Imaginary Number1:22
 How it Helps Solve Equations2:20
 Square Roots and Imaginary Numbers3:15
 Complex Numbers5:00
 Real Part and Imaginary Part5:20
 When Two Complex Numbers are Equal6:10
 Addition and Subtraction6:40
 Deal with Real and Imaginary Parts Separately7:36
 Two Quick Examples7:54
 Multiplication9:07
 FOIL Expansion9:14
 Note What Happens to the Square of the Imaginary Number9:41
 Two Quick Examples10:22
 Division11:27
 Complex Conjugates13:37
 Getting Rid of i14:08
 How to Denote the Conjugate14:48
 Division through Complex Conjugates16:11
 Multiply by the Conjugate of the Denominator16:28
 Example17:46
 Factoring SoCalled 'Irreducible' Quadratics19:24
 Revisiting the Quadratic Formula20:12
 Conjugate Pairs20:37
 But Are the Complex Numbers 'Real'?21:27
 What Makes a Number Legitimate25:38
 Where Complex Numbers are Used27:20
 Still, We Won't See Much of C29:05
 Example 130:30
 Example 233:15
 Example 338:12
 Example 442:07
19m 9s
 Intro0:00
 Introduction0:05
 Idea: Hidden Roots1:16
 Roots in Complex Form1:42
 All Polynomials Have Roots2:08
 Fundamental Theorem of Algebra2:21
 Where Are All the Imaginary Roots, Then?3:17
 All Roots are Complex3:45
 Real Numbers are a Subset of Complex Numbers3:59
 The n Roots Theorem5:01
 For Any Polynomial, Its Degree is Equal to the Number of Roots5:11
 Equivalent Statement5:24
 Comments: Multiplicity6:29
 NonDistinct Roots6:59
 Denoting Multiplicity7:20
 Comments: Complex Numbers Necessary7:41
 Comments: Complex Coefficients Allowed8:55
 Comments: Existence Theorem9:59
 Proof Sketch of n Roots Theorem10:45
 First Root11:36
 Second Root13:23
 Continuation to Find all Roots16:00
33m 22s
 Intro0:00
 Introduction0:05
 Definition of a Rational Function1:20
 Examples of Rational Functions2:30
 Why They are Called 'Rational'2:47
 Domain of a Rational Function3:15
 Undefined at Denominator Zeros3:25
 Otherwise all Reals4:16
 Investigating a Fundamental Function4:50
 The Domain of the Function5:04
 What Occurs at the Zeroes of the Denominator5:20
 Idea of a Vertical Asymptote6:23
 What's Going On?6:58
 Approaching x=0 from the left7:32
 Approaching x=0 from the right8:34
 Dividing by Very Small Numbers Results in Very Large Numbers9:31
 Definition of a Vertical Asymptote10:05
 Vertical Asymptotes and Graphs11:15
 Drawing Asymptotes by Using a Dashed Line11:27
 The Graph Can Never Touch Its Undefined Point12:00
 Not All Zeros Give Asymptotes13:02
 Special Cases: When Numerator and Denominator Go to Zero at the Same Time14:58
 Cancel out Common Factors15:49
 How to Find Vertical Asymptotes16:10
 Figure out What Values Are Not in the Domain of x16:24
 Determine if the Numerator and Denominator Share Common Factors and Cancel16:45
 Find Denominator Roots17:33
 Note if Asymptote Approaches Negative or Positive Infinity18:06
 Example 118:57
 Example 221:26
 Example 323:04
 Example 430:01
34m 16s
 Intro0:00
 Introduction0:05
 Investigating a Fundamental Function0:53
 What Happens as x Grows Large1:00
 Different View1:12
 Idea of a Horizontal Asymptote1:36
 What's Going On?2:24
 What Happens as x Grows to a Large Negative Number2:49
 What Happens as x Grows to a Large Number3:30
 Dividing by Very Large Numbers Results in Very Small Numbers3:52
 Example Function4:41
 Definition of a Vertical Asymptote8:09
 Expanding the Idea9:03
 What's Going On?9:48
 What Happens to the Function in the Long Run?9:51
 Rewriting the Function10:13
 Definition of a Slant Asymptote12:09
 Symbolical Definition12:30
 Informal Definition12:45
 Beyond Slant Asymptotes13:03
 Not Going Beyond Slant Asymptotes14:39
 Horizontal/Slant Asymptotes and Graphs15:43
 How to Find Horizontal and Slant Asymptotes16:52
 How to Find Horizontal Asymptotes17:12
 Expand the Given Polynomials17:18
 Compare the Degrees of the Numerator and Denominator17:40
 How to Find Slant Asymptotes20:05
 Slant Asymptotes Exist When n+m=120:08
 Use Polynomial Division20:24
 Example 124:32
 Example 225:53
 Example 326:55
 Example 429:22
49m 7s
 Intro0:00
 Introduction0:05
 A Process for Graphing1:22
 1. Factor Numerator and Denominator1:50
 2. Find Domain2:53
 3. Simplifying the Function3:59
 4. Find Vertical Asymptotes4:59
 5. Find Horizontal/Slant Asymptotes5:24
 6. Find Intercepts7:35
 7. Draw Graph (Find Points as Necessary)9:21
 Draw Graph Example11:21
 Vertical Asymptote11:41
 Horizontal Asymptote11:50
 Other Graphing12:16
 Test Intervals15:08
 Example 117:57
 Example 223:01
 Example 329:02
 Example 433:37
44m 56s
 Intro0:00
 Introduction: Idea0:04
 Introduction: Prerequisites and Uses1:57
 Proper vs. Improper Polynomial Fractions3:11
 Possible Things in the Denominator4:38
 Linear Factors6:16
 Example of Linear Factors7:03
 Multiple Linear Factors7:48
 Irreducible Quadratic Factors8:25
 Example of Quadratic Factors9:26
 Multiple Quadratic Factors9:49
 Mixing Factor Types10:28
 Figuring Out the Numerator11:10
 How to Solve for the Constants11:30
 Quick Example11:40
 Example 114:29
 Example 218:35
 Example 320:33
 Example 428:51
35m 17s
 Intro0:00
 Introduction0:05
 Fundamental Idea1:46
 Expanding the Idea2:28
 Multiplication of the Same Base2:40
 Exponents acting on Exponents3:45
 Different Bases with the Same Exponent4:31
 To the Zero5:35
 To the First5:45
 Fundamental Rule with the Zero Power6:35
 To the Negative7:45
 Any Number to a Negative Power8:14
 A Fraction to a Negative Power9:58
 Division with Exponential Terms10:41
 To the Fraction11:33
 Square Root11:58
 Any Root12:59
 Summary of Rules14:38
 To the Irrational17:21
 Example 120:34
 Example 223:42
 Example 327:44
 Example 431:44
 Example 533:15
47m 4s
 Intro0:00
 Introduction0:05
 Definition of an Exponential Function0:48
 Definition of the Base1:02
 Restrictions on the Base1:16
 Computing Exponential Functions2:29
 Harder Computations3:10
 When to Use a Calculator3:21
 Graphing Exponential Functions: a>16:02
 Three Examples6:13
 What to Notice on the Graph7:44
 A Story8:27
 Story Diagram9:15
 Increasing Exponentials11:29
 Story Morals14:40
 Application: Compound Interest15:15
 Compounding Year after Year16:01
 Function for Compounding Interest16:51
 A Special Number: e20:55
 Expression for e21:28
 Where e stabilizes21:55
 Application: Continuously Compounded Interest24:07
 Equation for Continuous Compounding24:22
 Exponential Decay 0<a<125:50
 Three Examples26:11
 Why they 'lose' value26:54
 Example 127:47
 Example 233:11
 Example 336:34
 Example 441:28
40m 31s
 Intro0:00
 Introduction0:04
 Definition of a Logarithm, Base 20:51
 Log 2 Defined0:55
 Examples2:28
 Definition of a Logarithm, General3:23
 Examples of Logarithms5:15
 Problems with Unusual Bases7:38
 Shorthand Notation: ln and log9:44
 base e as ln10:01
 base 10 as log10:34
 Calculating Logarithms11:01
 using a calculator11:34
 issues with other bases11:58
 Graphs of Logarithms13:21
 Three Examples13:29
 Slow Growth15:19
 Logarithms as Inverse of Exponentiation16:02
 Using Base 216:05
 General Case17:10
 Looking More Closely at Logarithm Graphs19:16
 The Domain of Logarithms20:41
 Thinking about Logs like Inverses21:08
 The Alternate24:00
 Example 125:59
 Example 230:03
 Example 332:49
 Example 437:34
42m 33s
 Intro0:00
 Introduction0:04
 Basic Properties1:12
 Inverselog(exp)1:43
 A Key Idea2:44
 What We Get through Exponentiation3:18
 B Always Exists4:50
 Inverseexp(log)5:53
 Logarithm of a Power7:44
 Logarithm of a Product10:07
 Logarithm of a Quotient13:48
 Caution! There Is No Rule for loga(M+N)16:12
 Summary of Properties17:42
 Change of BaseMotivation20:17
 No Calculator Button20:59
 A Specific Example21:45
 Simplifying23:45
 Change of BaseFormula24:14
 Example 125:47
 Example 229:08
 Example 331:14
 Example 434:13
34m 10s
 Intro0:00
 Introduction0:05
 One to One Property1:09
 Exponential1:26
 Logarithmic1:44
 Specific Considerations2:02
 OnetoOne Property3:30
 Solving by OnetoOne4:11
 Inverse Property6:09
 Solving by Inverses7:25
 Dealing with Equations7:50
 Example of Taking an Exponent or Logarithm of an Equation9:07
 A Useful Property11:57
 Bring Down Exponents12:01
 Try to Simplify13:20
 Extraneous Solutions13:45
 Example 116:37
 Example 219:39
 Example 321:37
 Example 426:45
 Example 529:37
48m 46s
 Intro0:00
 Introduction0:06
 Applications of Exponential Functions1:07
 A Secret!2:17
 Natural Exponential Growth Model3:07
 Figure out r3:34
 A Secret!Why Does It Work?4:44
 e to the r Morphs4:57
 Example5:06
 Applications of Logarithmic Functions8:32
 Examples8:43
 What Logarithms are Useful For9:53
 Example 111:29
 Example 215:30
 Example 326:22
 Example 432:05
 Example 539:19
39m 5s
 Intro0:00
 Degrees0:22
 Circle is 360 Degrees0:48
 Splitting a Circle1:13
 Radians2:08
 Circle is 2 Pi Radians2:31
 One Radian2:52
 HalfCircle and Right Angle4:00
 Converting Between Degrees and Radians6:24
 Formulas for Degrees and Radians6:52
 Coterminal, Complementary, Supplementary Angles7:23
 Coterminal Angles7:30
 Complementary Angles9:40
 Supplementary Angles10:08
 Example 1: Dividing a Circle10:38
 Example 2: Converting Between Degrees and Radians11:56
 Example 3: Quadrants and Coterminal Angles14:18
 Extra Example 1: Common Angle Conversions1
 Extra Example 2: Quadrants and Coterminal Angles2
43m 16s
 Intro0:00
 Sine and Cosine0:15
 Unit Circle0:22
 Coordinates on Unit Circle1:03
 Right Triangles1:52
 Adjacent, Opposite, Hypotenuse2:25
 Master Right Triangle Formula: SOHCAHTOA2:48
 Odd Functions, Even Functions4:40
 Example: Odd Function4:56
 Example: Even Function7:30
 Example 1: Sine and Cosine10:27
 Example 2: Graphing Sine and Cosine Functions14:39
 Example 3: Right Triangle21:40
 Example 4: Odd, Even, or Neither26:01
 Extra Example 1: Right Triangle1
 Extra Example 2: Graphing Sine and Cosine Functions2
33m 5s
 Intro0:00
 454590 Triangle and 306090 Triangle0:08
 454590 Triangle0:21
 306090 Triangle2:06
 Mnemonic: All Students Take Calculus (ASTC)5:21
 Using the Unit Circle5:59
 New Angles6:21
 Other Quadrants9:43
 Mnemonic: All Students Take Calculus10:13
 Example 1: Convert, Quadrant, Sine/Cosine13:11
 Example 2: Convert, Quadrant, Sine/Cosine16:48
 Example 3: All Angles and Quadrants20:21
 Extra Example 1: Convert, Quadrant, Sine/Cosine1
 Extra Example 2: All Angles and Quadrants2
52m 3s
 Intro0:00
 Amplitude and Period of a Sine Wave0:38
 Sine Wave Graph0:58
 Amplitude: Distance from Middle to Peak1:18
 Peak: Distance from Peak to Peak2:41
 Phase Shift and Vertical Shift4:13
 Phase Shift: Distance Shifted Horizontally4:16
 Vertical Shift: Distance Shifted Vertically6:48
 Example 1: Amplitude/Period/Phase and Vertical Shift8:04
 Example 2: Amplitude/Period/Phase and Vertical Shift17:39
 Example 3: Find Sine Wave Given Attributes25:23
 Extra Example 1: Amplitude/Period/Phase and Vertical Shift1
 Extra Example 2: Find Cosine Wave Given Attributes2
36m 4s
 Intro0:00
 Tangent and Cotangent Definitions0:21
 Tangent Definition0:25
 Cotangent Definition0:47
 Master Formula: SOHCAHTOA1:01
 Mnemonic1:16
 Tangent and Cotangent Values2:29
 Remember Common Values of Sine and Cosine2:46
 90 Degrees Undefined4:36
 Slope and Menmonic: ASTC5:47
 Uses of Tangent5:54
 Example: Tangent of Angle is Slope6:09
 Sign of Tangent in Quadrants7:49
 Example 1: Graph Tangent and Cotangent Functions10:42
 Example 2: Tangent and Cotangent of Angles16:09
 Example 3: Odd, Even, or Neither18:56
 Extra Example 1: Tangent and Cotangent of Angles1
 Extra Example 2: Tangent and Cotangent of Angles2
27m 18s
 Intro0:00
 Secant and Cosecant Definitions0:17
 Secant Definition0:18
 Cosecant Definition0:33
 Example 1: Graph Secant Function0:48
 Example 2: Values of Secant and Cosecant6:49
 Example 3: Odd, Even, or Neither12:49
 Extra Example 1: Graph of Cosecant Function1
 Extra Example 2: Values of Secant and Cosecant2
32m 58s
 Intro0:00
 Arcsine Function0:24
 Restrictions between 1 and 10:43
 Arcsine Notation1:26
 Arccosine Function3:07
 Restrictions between 1 and 13:36
 Cosine Notation3:53
 Arctangent Function4:30
 Between Pi/2 and Pi/24:44
 Tangent Notation5:02
 Example 1: Domain/Range/Graph of Arcsine5:45
 Example 2: Arcsin/Arccos/Arctan Values10:46
 Example 3: Domain/Range/Graph of Arctangent17:14
 Extra Example 1: Domain/Range/Graph of Arccosine1
 Extra Example 2: Arcsin/Arccos/Arctan Values2
31m 8s
 Intro0:00
 Inverse Trigonometric Function Domains and Ranges0:31
 Arcsine0:41
 Arccosine1:14
 Arctangent1:41
 Example 1: Arcsines of Common Values2:44
 Example 2: Odd, Even, or Neither5:57
 Example 3: Arccosines of Common Values12:24
 Extra Example 1: Arctangents of Common Values1
 Extra Example 2: Arcsin/Arccos/Arctan Values2
19m 11s
 Intro0:00
 Pythagorean Identity0:17
 Pythagorean Triangle0:27
 Pythagorean Identity0:45
 Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity1:14
 Example 2: Find Angle Given Cosine and Quadrant4:18
 Example 3: Verify Trigonometric Identity8:00
 Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem1
 Extra Example 2: Find Angle Given Cosine and Quadrant2
23m 16s
 Intro0:00
 Main Formulas0:19
 Companion to Pythagorean Identity0:27
 For Cotangents and Cosecants0:52
 How to Remember0:58
 Example 1: Prove the Identity1:40
 Example 2: Given Tan Find Sec3:42
 Example 3: Prove the Identity7:45
 Extra Example 1: Prove the Identity1
 Extra Example 2: Given Sec Find Tan2
52m 52s
 Intro0:00
 Addition and Subtraction Formulas0:09
 How to Remember0:48
 Cofunction Identities1:31
 How to Remember Graphically1:44
 Where to Use Cofunction Identities2:52
 Example 1: Derive the Formula for cos(AB)3:08
 Example 2: Use Addition and Subtraction Formulas16:03
 Example 3: Use Addition and Subtraction Formulas to Prove Identity25:11
 Extra Example 1: Use cos(AB) and Cofunction Identities1
 Extra Example 2: Convert to Radians and use Formulas2
29m 5s
 Intro0:00
 Main Formula0:07
 How to Remember from Addition Formula0:18
 Two Other Forms1:35
 Example 1: Find Sine and Cosine of Angle using Double Angle3:16
 Example 2: Prove Trigonometric Identity using Double Angle9:37
 Example 3: Use Addition and Subtraction Formulas12:38
 Extra Example 1: Find Sine and Cosine of Angle using Double Angle1
 Extra Example 2: Prove Trigonometric Identity using Double Angle2
43m 55s
 Intro0:00
 Main Formulas0:09
 Confusing Part0:34
 Example 1: Find Sine and Cosine of Angle using HalfAngle0:54
 Example 2: Prove Trigonometric Identity using HalfAngle11:51
 Example 3: Prove the HalfAngle Formula for Tangents18:39
 Extra Example 1: Find Sine and Cosine of Angle using HalfAngle1
 Extra Example 2: Prove Trigonometric Identity using HalfAngle2
25m 43s
 Intro0:00
 Master Formula for Right Angles0:11
 SOHCAHTOA0:15
 Only for Right Triangles1:26
 Example 1: Find All Angles in a Triangle2:19
 Example 2: Find Lengths of All Sides of Triangle7:39
 Example 3: Find All Angles in a Triangle11:00
 Extra Example 1: Find All Angles in a Triangle1
 Extra Example 2: Find Lengths of All Sides of Triangle2
56m 40s
 Intro0:00
 Law of Sines Formula0:18
 SOHCAHTOA0:27
 Any Triangle0:59
 Graphical Representation1:25
 Solving Triangle Completely2:37
 When to Use Law of Sines2:55
 ASA, SAA, SSA, AAA2:59
 SAS, SSS for Law of Cosines7:11
 Example 1: How Many Triangles Satisfy Conditions, Solve Completely8:44
 Example 2: How Many Triangles Satisfy Conditions, Solve Completely15:30
 Example 3: How Many Triangles Satisfy Conditions, Solve Completely28:32
 Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely1
 Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely2
49m 5s
 Intro0:00
 Law of Cosines Formula0:23
 Graphical Representation0:34
 Relates Sides to Angles1:00
 Any Triangle1:20
 Generalization of Pythagorean Theorem1:32
 When to Use Law of Cosines2:26
 SAS, SSS2:30
 Heron's Formula4:49
 Semiperimeter S5:11
 Example 1: How Many Triangles Satisfy Conditions, Solve Completely5:53
 Example 2: How Many Triangles Satisfy Conditions, Solve Completely15:19
 Example 3: Find Area of a Triangle Given All Side Lengths26:33
 Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely1
 Extra Example 2: Length of Third Side and Area of Triangle2
27m 37s
 Intro0:00
 Master Right Triangle Formula and Law of Cosines0:19
 SOHCAHTOA0:27
 Law of Cosines1:23
 Heron's Formula2:22
 Semiperimeter S2:37
 Example 1: Area of Triangle with Two Sides and One Angle3:12
 Example 2: Area of Triangle with Three Sides6:11
 Example 3: Area of Triangle with Three Sides, No Heron's Formula8:50
 Extra Example 1: Area of Triangle with Two Sides and One Angle1
 Extra Example 2: Area of Triangle with Two Sides and One Angle2
34m 25s
 Intro0:00
 Formulas to Remember0:11
 SOHCAHTOA0:15
 Law of Sines0:55
 Law of Cosines1:48
 Heron's Formula2:46
 Example 1: Telephone Pole Height4:01
 Example 2: Bridge Length7:48
 Example 3: Area of Triangular Field14:20
 Extra Example 1: Kite Height1
 Extra Example 2: Roads to a Town2
55m 40s
 Intro0:00
 Introduction0:04
 Graphs as Location of 'True'1:49
 All Locations that Make the Function True2:25
 Understand the Relationship Between Solutions and the Graph3:43
 Systems as Graphs4:07
 Equations as Lines4:20
 Intersection Point5:19
 Three Possibilities for Solutions6:17
 Independent6:24
 Inconsistent6:36
 Dependent7:06
 Solving by Substitution8:37
 Solve for One Variable9:07
 Substitute into the Second Equation9:34
 Solve for Both Variables10:12
 What If a System is Inconsistent or Dependent?11:08
 No Solutions11:25
 Infinite Solutions12:30
 Solving by Elimination13:56
 Example14:22
 Determining the Number of Solutions16:30
 Why Elimination Makes Sense17:25
 Solving by Graphing Calculator19:59
 Systems with More than Two Variables23:22
 Example 125:49
 Example 230:22
 Example 334:11
 Example 438:55
 Example 546:01
 (Non) Example 653:37
1h 13s
 Intro0:00
 Introduction0:04
 Inequality RefresherSolutions0:46
 Equation Solutions vs. Inequality Solutions1:02
 Essentially a Wide Variety of Answers1:35
 RefresherNegative Multiplication Flips1:43
 RefresherNegative Flips: Why?3:19
 Multiplication by a Negative3:43
 The Relationship Flips3:55
 RefresherStick to Basic Operations4:34
 Linear Equations in Two Variables6:50
 Graphing Linear Inequalities8:28
 Why It Includes a Whole Section8:43
 How to Show The Difference Between Strict and Not Strict Inequalities10:08
 Dashed LineNot Solutions11:10
 Solid LineAre Solutions11:24
 Test Points for Shading11:42
 Example of Using a Point12:41
 Drawing Shading from the Point13:14
 Graphing a System14:53
 Set of Solutions is the Overlap15:17
 Example15:22
 Solutions are Best Found Through Graphing18:05
 Linear ProgrammingIdea19:52
 Use a Linear Objective Function20:15
 Variables in Objective Function have Constraints21:24
 Linear ProgrammingMethod22:09
 Rearrange Equations22:21
 Graph22:49
 Critical Solution is at the Vertex of the Overlap23:40
 Try Each Vertice24:35
 Example 124:58
 Example 228:57
 Example 333:48
 Example 443:10
41m 1s
 Intro0:00
 Introduction0:06
 Substitution1:12
 Example1:22
 Elimination3:46
 Example3:56
 Elimination is Less Useful for Nonlinear Systems4:56
 Graphing5:56
 Using a Graphing Calculator6:44
 Number of Solutions8:44
 Systems of Nonlinear Inequalities10:02
 Graph Each Inequality10:06
 Dashed and/or Solid10:18
 Shade Appropriately11:14
 Example 113:24
 Example 215:50
 Example 322:02
 Example 429:06
 Example 4, cont.33:40
1h 9m 31s
 Intro0:00
 Introduction0:10
 Magnitude of the Force0:22
 Direction of the Force0:48
 Vector0:52
 Idea of a Vector1:30
 How Vectors are Denoted2:00
 Component Form3:20
 Angle Brackets and Parentheses3:50
 Magnitude/Length4:26
 Denoting the Magnitude of a Vector5:16
 Direction/Angle7:52
 Always Draw a Picture8:50
 Component Form from Magnitude & Angle10:10
 Scaling by Scalars14:06
 Unit Vectors16:26
 Combining Vectors  Algebraically18:10
 Combining Vectors  Geometrically19:54
 Resultant Vector20:46
 Alternate Component Form: i, j21:16
 The Zero Vector23:18
 Properties of Vectors24:20
 No Multiplication (Between Vectors)28:30
 Dot Product29:40
 Motion in a Medium30:10
 Fish in an Aquarium Example31:38
 More Than Two Dimensions33:12
 More Than Two Dimensions  Magnitude34:18
 Example 135:26
 Example 238:10
 Example 345:48
 Example 450:40
 Example 4, cont.56:07
 Example 501:32
35m 20s
 Intro0:00
 Introduction0:08
 Dot Product  Definition0:42
 Dot Product Results in a Scalar, Not a Vector2:10
 Example in Two Dimensions2:34
 Angle and the Dot Product2:58
 The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors2:59
 Proof of Dot Product Formula4:14
 Won't Directly Help Us Better Understand Vectors4:18
 Dot Product  Geometric Interpretation4:58
 We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are7:26
 Dot Product  Perpendicular Vectors8:24
 If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other8:54
 Cross Product  Definition11:08
 Cross Product Only Works in Three Dimensions11:09
 Cross Product  A Mnemonic12:16
 The Determinant of a 3 x 3 Matrix and Standard Unit Vectors12:17
 Cross Product  Geometric Interpretations14:30
 The RightHand Rule15:17
 Cross Product  Geometric Interpretations Cont.17:00
 Example 118:40
 Example 222:50
 Example 324:04
 Example 426:20
 Bonus Round29:18
 Proof: Dot Product Formula29:24
 Proof: Dot Product Formula, cont.30:38
54m 7s
 Intro0:00
 Introduction0:08
 Definition of a Matrix3:02
 Size or Dimension3:58
 Square Matrix4:42
 Denoted by Capital Letters4:56
 When are Two Matrices Equal?5:04
 Examples of Matrices6:44
 Rows x Columns6:46
 Talking About Specific Entries7:48
 We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries8:32
 Using Entries to Talk About Matrices10:08
 Scalar Multiplication11:26
 Scalar = Real Number11:34
 Example12:36
 Matrix Addition13:08
 Example14:22
 Matrix Multiplication15:00
 Example18:52
 Matrix Multiplication, cont.19:58
 Matrix Multiplication and Order (Size)25:26
 Make Sure Their Orders are Compatible25:27
 Matrix Multiplication is NOT Commutative28:20
 Example30:08
 Special Matrices  Zero Matrix (0)32:48
 Zero Matrix Has 0 for All of its Entries32:49
 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 Entries34:15
 Example 136:16
 Example 240:00
 Example 344:54
 Example 450:08
47m 12s
 Intro0:00
 Introduction0:06
 Not All Matrices Are Invertible1:30
 What Must a Matrix Have to Be Invertible?2:08
 Determinant2:32
 The Determinant is a Real Number Associated With a Square Matrix2:38
 If the Determinant of a Matrix is Nonzero, the Matrix is Invertible3:40
 Determinant of a 2 x 2 Matrix4:34
 Think in Terms of Diagonals5:12
 Minors and Cofactors  Minors6:24
 Example6:46
 Minors and Cofactors  Cofactors8:00
 Cofactor is Closely Based on the Minor8:01
 Alternating Sign Pattern9:04
 Determinant of Larger Matrices10:56
 Example13:00
 Alternative Method for 3x3 Matrices16:46
 Not Recommended16:48
 Inverse of a 2 x 2 Matrix19:02
 Inverse of Larger Matrices20:00
 Using Inverse Matrices21:06
 When Multiplied Together, They Create the Identity Matrix21:24
 Example 123:45
 Example 227:21
 Example 332:49
 Example 436:27
 Finding the Inverse of Larger Matrices41:59
 General Inverse Method  Step 143:25
 General Inverse Method  Step 243:27
 General Inverse Method  Step 2, cont.43:27
 General Inverse Method  Step 345:15
58m 34s
 Intro0:00
 Introduction0:12
 Augmented Matrix1:44
 We Can Represent the Entire Linear System With an Augmented Matrix1:50
 Row Operations3:22
 Interchange the Locations of Two Rows3:50
 Multiply (or Divide) a Row by a Nonzero Number3:58
 Add (or Subtract) a Multiple of One Row to Another4:12
 Row Operations  Keep Notes!5:50
 Suggested Symbols7:08
 GaussJordan Elimination  Idea8:04
 GaussJordan Elimination  Idea, cont.9:16
 Reduced RowEchelon Form9:18
 GaussJordan Elimination  Method11:36
 Begin by Writing the System As An Augmented Matrix11:38
 GaussJordan Elimination  Method, cont.13:48
 Cramer's Rule  2 x 2 Matrices17:08
 Cramer's Rule  n x n Matrices19:24
 Solving with Inverse Matrices21:10
 Solving Inverse Matrices, cont.25:28
 The Mighty (Graphing) Calculator26:38
 Example 129:56
 Example 233:56
 Example 337:00
 Example 3, cont.45:04
 Example 451:28
53m 33s
 Intro0:00
 Introduction0:06
 Definition1:10
 Plane Curve1:24
 The Key Idea2:00
 Graphing with Parametric Equations2:52
 Same Graph, Different Equations5:04
 How Is That Possible?5:36
 Same Graph, Different Equations, cont.5:42
 Here's Another to Consider7:56
 Same Plane Curve, But Still Different8:10
 A Metaphor for Parametric Equations9:36
 Think of Parametric Equations As a Way to Describe the Motion of An Object9:38
 Graph Shows Where It Went, But Not Speed10:32
 Eliminating Parameters12:14
 Rectangular Equation12:16
 Caution13:52
 Creating Parametric Equations14:30
 Interesting Graphs16:38
 Graphing Calculators, Yay!19:18
 Example 122:36
 Example 228:26
 Example 337:36
 Example 441:00
 Projectile Motion44:26
 Example 547:00
48m 7s
 Intro0:00
 Introduction0:04
 Polar Coordinates Give Us a Way To Describe the Location of a Point0:26
 Polar Equations and Functions0:50
 Plotting Points with Polar Coordinates1:06
 The Distance of the Point from the Origin1:09
 The Angle of the Point1:33
 Give Points as the Ordered Pair (r,θ)2:03
 Visualizing Plotting in Polar Coordinates2:32
 First Way We Can Plot2:39
 Second Way We Can Plot2:50
 First, We'll Look at Visualizing r, Then θ3:09
 Rotate the Length CounterClockwise by θ3:38
 Alternatively, We Can Visualize θ, Then r4:06
 'Polar Graph Paper'6:17
 Horizontal and Vertical Tick Marks Are Not Useful for Polar6:42
 Use Concentric Circles to Helps Up See Distance From the Pole7:08
 Can Use Arc Sectors to See Angles7:57
 Multiple Ways to Name a Point9:17
 Examples9:30
 For Any Angle θ, We Can Make an Equivalent Angle10:44
 Negative Values for r11:58
 If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out12:22
 Another Way to Name the Same Point: Add π to θ and Make r Negative13:44
 Converting Between Rectangular and Polar14:37
 Rectangular Way to Name14:43
 Polar Way to Name14:52
 The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle15:08
 Connect Both Systems Through Basic Trigonometry15:38
 Equation to Convert From Polar to Rectangular Coordinate Systems16:55
 Equation to Convert From Rectangular to Polar Coordinate Systems17:13
 Converting to Rectangular is Easy17:20
 Converting to Polar is a Bit Trickier17:21
 Draw Pictures18:55
 Example 119:50
 Example 225:17
 Example 331:05
 Example 435:56
 Example 541:49
38m 16s
 Intro0:00
 Introduction0:04
 Equations and Functions1:16
 Independent Variable1:21
 Dependent Variable1:30
 Examples1:46
 Always Assume That θ Is In Radians2:44
 Graphing in Polar Coordinates3:29
 Graph is the Same Way We Graph 'Normal' Stuff3:32
 Example3:52
 Graphing in Polar  Example, Cont.6:45
 Tips for Graphing9:23
 Notice Patterns10:19
 Repetition13:39
 Graphing Equations of One Variable14:39
 Converting Coordinate Types16:16
 Use the Same Conversion Formulas From the Previous Lesson16:23
 Interesting Graphs17:48
 Example 118:03
 Example 218:34
 Graphing Calculators, Yay!19:07
 Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works19:11
 Check Out the Appendix19:26
 Example 121:36
 Example 228:13
 Example 334:24
 Example 435:52
40m 43s
 Intro0:00
 Polar Coordinates0:49
 Rectangular Form0:52
 Polar Form1:25
 R and Theta1:51
 Polar Form Conversion2:27
 R and Theta2:35
 Optimal Values4:05
 Euler's Formula4:25
 Multiplying Two Complex Numbers in Polar Form6:10
 Multiply r's Together and Add Exponents6:32
 Example 1: Convert Rectangular to Polar Form7:17
 Example 2: Convert Polar to Rectangular Form13:49
 Example 3: Multiply Two Complex Numbers17:28
 Extra Example 1: Convert Between Rectangular and Polar Forms1
 Extra Example 2: Simplify Expression to Polar Form2
57m 37s
 Intro0:00
 Introduction to DeMoivre's Theorem0:10
 n nth Roots3:06
 DeMoivre's Theorem: Finding nth Roots3:52
 Relation to Unit Circle6:29
 One nth Root for Each Value of k7:11
 Example 1: Convert to Polar Form and Use DeMoivre's Theorem8:24
 Example 2: Find Complex Eighth Roots15:27
 Example 3: Find Complex Roots27:49
 Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem1
 Extra Example 2: Find Complex Fourth Roots2
31m 36s
 Intro0:00
 Introduction0:08
 Combinatorics0:56
 Definition: Event1:24
 Example1:50
 Visualizing an Event3:02
 Branching line diagram3:06
 Addition Principle3:40
 Example4:18
 Multiplication Principle5:42
 Example6:24
 Pigeonhole Principle8:06
 Example10:26
 Draw Pictures11:06
 Example 112:02
 Example 214:16
 Example 317:34
 Example 421:26
 Example 525:14
44m 3s
 Intro0:00
 Introduction0:08
 Permutation0:42
 Combination1:10
 Towards a Permutation Formula2:38
 How Many Ways Can We Arrange the Letters A, B, C, D, and E?3:02
 Towards a Permutation Formula, cont.3:34
 Factorial Notation6:56
 Symbol Is '!'6:58
 Examples7:32
 Permutation of n Objects8:44
 Permutation of r Objects out of n9:04
 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 Permutations14:46
 What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?14:48
 Distinguishable Permutations, cont.17:04
 Combinations19:04
 Combinations, cont.20:56
 Example 123:10
 Example 226:16
 Example 328:28
 Example 431:52
 Example 533:58
 Example 636:34
36m 58s
 Intro0:00
 Introduction0:06
 Definition: Sample Space1:18
 Event = Something Happening1:20
 Sample Space1:36
 Probability of an Event2:12
 Let E Be An Event and S Be The Corresponding Sample Space2:14
 'Equally Likely' Is Important3:52
 Fair and Random5:26
 Interpreting Probability6:34
 How Can We Interpret This Value?7:24
 We Can Represent Probability As a Fraction, a Decimal, Or a Percentage8:04
 One of Multiple Events Occurring9:52
 Mutually Exclusive Events10:38
 What If The Events Are Not Mutually Exclusive?12:20
 Taking the Possibility of Overlap Into Account13:24
 An Event Not Occurring17:14
 Complement of E17:22
 Independent Events19:36
 Independent19:48
 Conditional Events21:28
 What Is The Events Are Not Independent Though?21:30
 Conditional Probability22:16
 Conditional Events, cont.23:51
 Example 125:27
 Example 227:09
 Example 328:57
 Example 430:51
 Example 534:15
41m 27s
 Intro0:00
 What is a Parabola?0:20
 Definition of a Parabola0:29
 Focus0:59
 Directrix1:15
 Axis of Symmetry3:08
 Vertex3:33
 Minimum or Maximum3:44
 Standard Form4:59
 Horizontal Parabolas5:08
 Vertex Form5:19
 Upward or Downward5:41
 Example: Standard Form6:06
 Graphing Parabolas8:31
 Shifting8:51
 Example: Completing the Square9:22
 Symmetry and Translation12:18
 Example: Graph Parabola12:40
 Latus Rectum17:13
 Length18:15
 Example: Latus Rectum18:35
 Horizontal Parabolas18:57
 Not Functions20:08
 Example: Horizontal Parabola21:21
 Focus and Directrix24:11
 Horizontal24:48
 Example 1: Parabola Standard Form25:12
 Example 2: Graph Parabola30:00
 Example 3: Graph Parabola33:13
 Example 4: Parabola Equation37:28
21m 3s
 Intro0:00
 What are Circles?0:08
 Example: Equidistant0:17
 Radius0:32
 Equation of a Circle0:44
 Example: Standard Form1:11
 Graphing Circles1:47
 Example: Circle1:56
 Center Not at Origin3:07
 Example: Completing the Square3:51
 Example 1: Equation of Circle6:44
 Example 2: Center and Radius11:51
 Example 3: Radius15:08
 Example 4: Equation of Circle16:57
46m 51s
 Intro0:00
 What Are Ellipses?0:11
 Foci0:23
 Properties of Ellipses1:43
 Major Axis, Minor Axis1:47
 Center1:54
 Length of Major Axis and Minor Axis3:21
 Standard Form5:33
 Example: Standard Form of Ellipse6:09
 Vertical Major Axis9:14
 Example: Vertical Major Axis9:46
 Graphing Ellipses12:51
 Complete the Square and Symmetry13:00
 Example: Graphing Ellipse13:16
 Equation with Center at (h, k)19:57
 Horizontal and Vertical20:14
 Difference20:27
 Example: Center at (h, k)20:55
 Example 1: Equation of Ellipse24:05
 Example 2: Equation of Ellipse27:57
 Example 3: Equation of Ellipse32:32
 Example 4: Graph Ellipse38:27
38m 15s
 Intro0:00
 What are Hyperbolas?0:12
 Two Branches0:18
 Foci0:38
 Properties2:00
 Transverse Axis and Conjugate Axis2:06
 Vertices2:46
 Length of Transverse Axis3:14
 Distance Between Foci3:31
 Length of Conjugate Axis3:38
 Standard Form5:45
 Vertex Location6:36
 Known Points6:52
 Vertical Transverse Axis7:26
 Vertex Location7:50
 Asymptotes8:36
 Vertex Location8:56
 Rectangle9:28
 Diagonals10:29
 Graphing Hyperbolas12:58
 Example: Hyperbola13:16
 Equation with Center at (h, k)16:32
 Example: Center at (h, k)17:21
 Example 1: Equation of Hyperbola19:20
 Example 2: Equation of Hyperbola22:48
 Example 3: Graph Hyperbola26:05
 Example 4: Equation of Hyperbola36:29
18m 43s
 Intro0:00
 Conic Sections0:16
 Double Cone Sections0:24
 Standard Form1:27
 General Form1:37
 Identify Conic Sections2:16
 B = 02:50
 X and Y3:22
 Identify Conic Sections, Cont.4:46
 Parabola5:17
 Circle5:51
 Ellipse6:31
 Hyperbola7:10
 Example 1: Identify Conic Section8:01
 Example 2: Identify Conic Section11:03
 Example 3: Identify Conic Section11:38
 Example 4: Identify Conic Section14:50
57m 45s
 Intro0:00
 Introduction0:06
 Definition: Sequence0:28
 Infinite Sequence2:08
 Finite Sequence2:22
 Length2:58
 Formula for the nth Term3:22
 Defining a Sequence Recursively5:54
 Initial Term7:58
 Sequences and Patterns10:40
 First, Identify a Pattern12:52
 How to Get From One Term to the Next17:38
 Tips for Finding Patterns19:52
 More Tips for Finding Patterns24:14
 Even More Tips26:50
 Example 130:32
 Example 234:54
 Fibonacci Sequence34:55
 Example 338:40
 Example 445:02
 Example 549:26
 Example 651:54
40m 27s
 Intro0:00
 Introduction0:06
 Definition: Series1:20
 Why We Need Notation2:48
 Simga Notation (AKA Summation Notation)4:44
 Thing Being Summed5:42
 Index of Summation6:21
 Lower Limit of Summation7:09
 Upper Limit of Summation7:23
 Sigma Notation, Example7:36
 Sigma Notation for Infinite Series9:08
 How to Reindex10:58
 How to Reindex, Expanding12:56
 How to Reindex, Substitution16:46
 Properties of Sums19:42
 Example 123:46
 Example 225:34
 Example 327:12
 Example 429:54
 Example 532:06
 Example 637:16
31m 36s
 Intro0:00
 Introduction0:05
 Definition: Arithmetic Sequence0:47
 Common Difference1:13
 Two Examples1:19
 Form for the nth Term2:14
 Recursive Relation2:33
 Towards an Arithmetic Series Formula5:12
 Creating a General Formula10:09
 General Formula for Arithmetic Series14:23
 Example 115:46
 Example 217:37
 Example 322:21
 Example 424:09
 Example 527:14
39m 27s
 Intro0:00
 Introduction0:06
 Definition0:48
 Form for the nth Term2:42
 Formula for Geometric Series5:16
 Infinite Geometric Series11:48
 Diverges13:04
 Converges14:48
 Formula for Infinite Geometric Series16:32
 Example 120:32
 Example 222:02
 Example 326:00
 Example 430:48
 Example 534:28
49m 53s
 Intro0:00
 Introduction0:06
 Belief Vs. Proof1:22
 A Metaphor for Induction6:14
 The Principle of Mathematical Induction11:38
 Base Case13:24
 Inductive Step13:30
 Inductive Hypothesis13:52
 A Remark on Statements14:18
 Using Mathematical Induction16:58
 Working Example19:58
 Finding Patterns28:46
 Example 130:17
 Example 237:50
 Example 342:38
1h 13m 13s
 Intro0:00
 Introduction0:06
 We've Learned That a Binomial Is An Expression That Has Two Terms0:07
 Understanding Binomial Coefficients1:20
 Things We Notice2:24
 What Goes In the Blanks?5:52
 Each Blank is Called a Binomial Coefficient6:18
 The Binomial Theorem6:38
 Example8:10
 The Binomial Theorem, cont.10:46
 We Can Also Write This Expression Compactly Using Sigma Notation12:06
 Proof of the Binomial Theorem13:22
 Proving the Binomial Theorem Is Within Our Reach13:24
 Pascal's Triangle15:12
 Pascal's Triangle, cont.16:12
 Diagonal Addition of Terms16:24
 Zeroth Row18:04
 First Row18:12
 Why Do We Care About Pascal's Triangle?18:50
 Pascal's Triangle, Example19:26
 Example 121:26
 Example 224:34
 Example 328:34
 Example 432:28
 Example 537:12
 Time for the Fireworks!43:38
 Proof of the Binomial Theorem43:44
 We'll Prove This By Induction44:04
 Proof (By Induction)46:36
 Proof, Base Case47:00
 Proof, Inductive Step  Notation Discussion49:22
 Induction Step49:24
 Proof, Inductive Step  Setting Up52:26
 Induction Hypothesis52:34
 What We What To Show52:44
 Proof, Inductive Step  Start54:18
 Proof, Inductive Step  Middle55:38
 Expand Sigma Notations55:48
 Proof, Inductive Step  Middle, cont.58:40
 Proof, Inductive Step  Checking In01:08
 Let's Check In With Our Original Goal01:12
 Want to Show01:18
 Lemma  A Mini Theorem02:18
 Proof, Inductive Step  Lemma02:52
 Proof of Lemma: Let's Investigate the Left Side03:08
 Proof, Inductive Step  Nearly There07:54
 Proof, Inductive Step  End!09:18
 Proof, Inductive Step  End!, cont.11:01
40m 22s
 Intro0:00
 Introduction0:05
 Motivating Example1:26
 Fuzzy Notion of a Limit3:38
 Limit is the Vertical Location a Function is Headed Towards3:44
 Limit is What the Function Output is Going to Be4:15
 Limit Notation4:33
 Exploring Limits  'Ordinary' Function5:26
 Test Out5:27
 Graphing, We See The Answer Is What We Would Expect5:44
 Exploring Limits  Piecewise Function6:45
 If We Modify the Function a Bit6:49
 Exploring Limits  A Visual Conception10:08
 Definition of a Limit12:07
 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=c12:49
 We Are Considering x Approaching From All Directions, Not Just One Side13:10
 Limits Do Not Always Exist15:47
 Finding Limits19:49
 Graphs19:52
 Tables21:48
 Precise Methods24:53
 Example 126:06
 Example 227:39
 Example 330:51
 Example 433:11
 Example 537:07
57m 11s
 Intro0:00
 Introduction0:06
 New Greek Letters2:42
 Delta3:14
 Epsilon3:46
 Sometimes Called the EpsilonDelta Definition of a Limit3:56
 Formal Definition of a Limit4:22
 What does it MEAN!?!?5:00
 The Groundwork5:38
 Set Up the Limit5:39
 The Function is Defined Over Some Portion of the Reals5:58
 The Horizontal Location is the Value the Limit Will Approach6:28
 The Vertical Location L is Where the Limit Goes To7:00
 The EpsilonDelta Part7:26
 The Hard Part is the Second Part of the Definition7:30
 Second Half of Definition10:04
 Restrictions on the Allowed x Values10:28
 The EpsilonDelta Part, cont.13:34
 Sherlock Holmes and Dr. Watson15:08
 The Adventure of the DeltaEpsilon Limit15:16
 Setting15:18
 We Begin By Setting Up the Game As Follows15:52
 The Adventure of the DeltaEpsilon, cont.17:24
 This Game is About Limits17:46
 What If I Try Larger?19:39
 Technically, You Haven't Proven the Limit20:53
 Here is the Method21:18
 What We Should Concern Ourselves With22:20
 Investigate the Left Sides of the Expressions25:24
 We Can Create the Following Inequalities28:08
 Finally…28:50
 Nothing Like a Good Proof to Develop the Appetite30:42
 Example 131:02
 Example 1, cont.36:26
 Example 241:46
 Example 2, cont.47:50
32m 40s
 Intro0:00
 Introduction0:08
 Method  'Normal' Functions2:04
 The Easiest Limits to Find2:06
 It Does Not 'Break'2:18
 It Is Not Piecewise2:26
 Method  'Normal' Functions, Example3:38
 Method  'Normal' Functions, cont.4:54
 The Functions We're Used to Working With Go Where We Expect Them To Go5:22
 A Limit is About Figuring Out Where a Function is 'Headed'5:42
 Method  Canceling Factors7:18
 One Weird Thing That Often Happens is Dividing By 07:26
 Method  Canceling Factors, cont.8:16
 Notice That The Two Functions Are Identical With the Exception of x=08:20
 Method  Canceling Factors, cont.10:00
 Example10:52
 Method  Rationalization12:04
 Rationalizing a Portion of Some Fraction12:05
 Conjugate12:26
 Method  Rationalization, cont.13:14
 Example13:50
 Method  Piecewise16:28
 The Limits of Piecewise Functions16:30
 Example 117:42
 Example 218:44
 Example 320:20
 Example 422:24
 Example 524:24
 Example 627:12
32m 43s
 Intro0:00
 Introduction0:06
 Motivating Example0:56
 Continuity  Idea2:14
 Continuous Function2:18
 All Parts of Function Are Connected2:28
 Function's Graph Can Be Drawn Without Lifting Pencil2:36
 There Are No Breaks or Holes in Graph2:56
 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  Definition5:16
 A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does5:18
 Discontinuous6:02
 Discontinuity6:10
 Continuity and 'Normal' Functions6:48
 Return of the Motivating Example8:14
 OneSided Limit8:48
 OneSided Limit  Definition9:16
 Only Considers One Side9:20
 Be Careful to Keep Track of Which Symbol Goes With Which Side10:06
 OneSided Limit  Example10:50
 There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits11:16
 Normal Limits and OneSided Limits12:08
 Limits of Piecewise Functions14:12
 'Breakover' Points14:22
 We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other15:34
 Example 116:40
 Example 218:54
 Example 322:00
 Example 426:36
32m 49s
 Intro0:00
 Introduction0:06
 Definition: Limit of a Function at Infinity1:44
 A Limit at Infinity Works Very Similarly to How a Normal Limit Works2:38
 Evaluating Limits at Infinity4:08
 Rational Functions4:17
 Examples4:30
 For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator5:22
 There are Three Possibilities6:36
 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 This10:26
 Limit of a Sequence12:20
 What Value Does the Sequence Tend to Do in the LongRun?12:41
 The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity12:52
 Numerical Evaluation14:16
 Numerically: Plug in Numbers and See What Comes Out14:24
 Example 116:42
 Example 221:00
 Example 322:08
 Example 426:14
 Example 528:10
 Example 631:06
51m 13s
 Intro0:00
 Introduction0:08
 The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing0:16
 Instantaneous Slop0:22
 Instantaneous Rate of Change0:28
 Slope1:24
 The Vertical Change Divided by the Horizontal1:40
 Idea of Instantaneous Slope2:10
 What If We Wanted to Apply the Idea of Slope to a NonLine?2:14
 Tangent to a Circle3:52
 What is the Tangent Line for a Circle?4:42
 Tangent to a Curve5:20
 Towards a Derivative  Average Slope6:36
 Towards a Derivative  Average Slope, cont.8:20
 An Approximation11:24
 Towards a Derivative  General Form13:18
 Towards a Derivative  General Form, cont.16:46
 An h Grows Smaller, Our Slope Approximation Becomes Better18:44
 Towards a Derivative  Limits!20:04
 Towards a Derivative  Limits!, cont.22:08
 We Want to Show the Slope at x=122:34
 Towards a Derivative  Checking Our Slope23:12
 Definition of the Derivative23:54
 Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point23:58
 Differentiation24:54
 Notation for the Derivative25:58
 The Derivative is a Very Important Idea In Calculus26:04
 The Important Idea27:34
 Why Did We Learn the Formal Definition to Find a Derivative?28:18
 Example 130:50
 Example 236:06
 Example 340:24
 The Power Rule44:16
 Makes It Easier to Find the Derivative of a Function44:24
 Examples45:04
 n Is Any Constant Number45:46
 Example 446:26
45m 26s
 Intro0:00
 Introduction0:06
 Integral0:12
 Idea of Area Under a Curve1:18
 Approximation by Rectangles2:12
 The Easiest Way to Find Area is With a Rectangle2:18
 Various Methods for Choosing Rectangles4:30
 Rectangle Method  LeftMost Point5:12
 The LeftMost Point5:16
 Rectangle Method  RightMost Point5:58
 The RightMost Point6:00
 Rectangle Method  MidPoint6:42
 Horizontal MidPoint6:48
 Rectangle Method  Maximum (Upper Sum)7:34
 Maximum Height7:40
 Rectangle Method  Minimum8:54
 Minimum Height9:02
 Evaluating the Area Approximation10:08
 Split the Interval Into n SubIntervals10:30
 More Rectangles, Better Approximation12:14
 The More We Us , the Better Our Approximation Becomes12:16
 Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity12:44
 Finding Area with a Limit13:08
 If This Limit Exists, It Is Called the Integral From a to b14:08
 The Process of Finding Integrals is Called Integration14:22
 The Big Reveal14:40
 The Integral is Based on the Antiderivative14:46
 The Big Reveal  Wait, Why?16:28
 The Rate of Change for the Area is Based on the Height of the Function16:50
 Height is the Derivative of Area, So Area is Based on the Antiderivative of Height17:50
 Example 119:06
 Example 222:48
 Example 329:06
 Example 3, cont.35:14
 Example 440:14
10m 41s
 Intro0:00
 Should You Buy?0:06
 Should I Get a Graphing Utility?0:20
 Free Graphing Utilities  Web Based0:38
 Personal Favorite: Desmos0:58
 Free Graphing Utilities  Offline Programs1:18
 GeoGebra1:31
 Microsoft Mathematics1:50
 Grapher2:18
 Other Graphing Utilities  Tablet/Phone2:48
 Should You Buy a Graphing Calculator?3:22
 The Only Real Downside4:10
 Deciding on Buying4:20
 If You Plan on Continuing in Math and/or Science4:26
 If Money is Not Particularly Tight for You4:32
 If You Don't Plan to Continue in Math and Science5:02
 If You Do Plan to Continue and Money Is Tight5:28
 Which to Buy5:44
 Which Graphing Calculator is Best?5:46
 Too Many Factors5:54
 Do Your Research6:12
 The Old Standby7:10
 TI83 (Plus)7:16
 TI84 (Plus)7:18
 Tips for Purchasing9:17
 Buy Online9:19
 Buy Used9:35
 Ask Around10:09
10m 51s
 Intro0:00
 Read the Manual0:06
 Skim It0:20
 Play Around and Experiment0:34
 Syntax0:40
 Definition of Syntax in English and Math0:46
 Pay Careful Attention to Your Syntax When Working With a Calculator2:08
 Make Sure You Use Parentheses to Indicate the Proper Order of Operations2:16
 Think About the Results3:54
 Settings4:58
 You'll Almost Never Need to Change the Settings on Your Calculator5:00
 Tell Calculator In Settings Whether the Angles Are In Radians or Degrees5:26
 Graphing Mode6:32
 Error Messages7:10
 Don't Panic7:11
 Internet Search7:32
 So Many Things8:14
 More Powerful Than You Realize8:18
 Other Things Your Graphing Calculator Can Do8:24
 Playing Around9:16
10m 38s
 Intro0:00
 Graphing Functions0:18
 Graphing Calculator Expects the Variable to Be x0:28
 Syntax0:58
 The Syntax We Choose Will Affect How the Function Graphs1:00
 Use Parentheses1:26
 The Viewing Window2:00
 One of the Most Important Ideas When Graphing Is To Think About The Viewing Window2:01
 For Example2:30
 The Viewing Window, cont.2:36
 Window Settings3:24
 Manually Choose Window Settings4:20
 x Min4:40
 x Max4:42
 y Min4:44
 y Max4:46
 Changing the x Scale or y Scale5:08
 Window Settings, cont.5:44
 Table of Values7:38
 Allows You to Quickly Churn Out Values for Various Inputs7:42
 For example7:44
 Changing the Independent Variable From 'Automatic' to 'Ask'8:50
9m 45s
 Intro0:00
 Points of Interest0:06
 Interesting Points on the Graph0:11
 Roots/Zeros (Zero)0:18
 Relative Minimums (Min)0:26
 Relative Maximums (Max)0:32
 Intersections (Intersection)0:38
 Finding Points of Interest  Process1:48
 Graph the Function1:49
 Adjust Viewing Window2:12
 Choose Point of Interest Type2:54
 Identify Where Search Should Occur3:04
 Give a Guess3:36
 Get Result4:06
 Advanced Technique: Arbitrary Solving5:10
 Find Out What Input Value Causes a Certain Output5:12
 For Example5:24
 Advanced Technique: Calculus7:18
 Derivative7:22
 Integral7:30
 But How Do You Show Work?8:20
7m 8s
 Intro0:00
 Change Graph Type0:08
 Located in General 'Settings'0:16
 Graphing in Parametric1:06
 Set Up Both Horizontal Function and Vertical Function1:08
 For Example2:04
 Graphing in Polar4:00
 For Example4:28
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For more information, please see full course syllabus of Pre Calculus
1 answer
Last reply by: Professor SelhorstJones
Sun Mar 8, 2015 9:10 PM
Post by Jamal Tischler on March 8, 2015
If we try to evaluate the integral of 1/x from a=1 to b=2 with the summation we get the limit of n*(1/(n+1)+1/(n+2)+...+1/(n+n)). How do we evaluate this ?
1 answer
Last reply by: Professor SelhorstJones
Mon Aug 26, 2013 11:09 PM
Post by Richard Gregory on August 22, 2013
Thank you very much Vincent. I really enjoyed this course and will be moving on to further maths 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.