Vincent Selhorst-Jones

Rational Functions and Vertical Asymptotes

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
- Three-Dimension = Space = ℝ³21:47
- Higher Dimensions22:24
- If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power22:31
- We Can Represent Places In This n-Dimensional 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
- Math-Specific 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 Non-Example1:30
- 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 Non-Numerical 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 T-Table17:46
- 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 x-Value 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 x-values; 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/x-intercepts19:45
- 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 x-axis8:38
- Horizontal Stretch/Shrink12:59
- Shrinking the Graph13:40
- Stretching the Graph13:48
- A Further Explanation13:55
- Understanding Stretches from the x-axis14:12
- 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 Non-Numerical Piecewise11:35
- Graphing Piecewise Functions14:15
- Graphing Piecewise Functions, Example16:26
- Continuous Functions16:57
- Statements of Continuity19:30
- Example of Continuous and Non-Continuous 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
- One-to-One4:04
- Requirement for Reversibility4:21
- When a Function has an Inverse4:43
- One-to-One5:13
- Not One-to-One5:50
- Not a Function6:19
- Horizontal Line Test7:01
- How to the test Works7:12
- One-to-One8:12
- Not One-to-One8:45
- 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 One-to-One23:04
- 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
- Long-Term Behavior of Polynomials17:48
- Examples18:13
- Controlling Term--Term 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 x-intercepts1:33
- 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 So-Called '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
- Non-Distinct 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
- Inverse--log(exp)1:43
- A Key Idea2:44
- What We Get through Exponentiation3:18
- B Always Exists4:50
- Inverse--exp(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 Base--Motivation20:17
- No Calculator Button20:59
- A Specific Example21:45
- Simplifying23:45
- Change of Base--Formula24:14
- 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
- One-to-One Property3:30
- Solving by One-to-One4:11
- 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
- Half-Circle 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 Conversions-1
- Extra Example 2: Quadrants and Coterminal Angles-2

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 Triangle-1
- Extra Example 2: Graphing Sine and Cosine Functions-2

33m 5s

- Intro0:00
- 45-45-90 Triangle and 30-60-90 Triangle0:08
- 45-45-90 Triangle0:21
- 30-60-90 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/Cosine-1
- Extra Example 2: All Angles and Quadrants-2

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 Shift-1
- Extra Example 2: Find Cosine Wave Given Attributes-2

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 Angles-1
- Extra Example 2: Tangent and Cotangent of Angles-2

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 Function-1
- Extra Example 2: Values of Secant and Cosecant-2

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 Arccosine-1
- Extra Example 2: Arcsin/Arccos/Arctan Values-2

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 Values-1
- Extra Example 2: Arcsin/Arccos/Arctan Values-2

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 Theorem-1
- Extra Example 2: Find Angle Given Cosine and Quadrant-2

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 Identity-1
- Extra Example 2: Given Sec Find Tan-2

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(A-B)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(A-B) and Cofunction Identities-1
- Extra Example 2: Convert to Radians and use Formulas-2

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 Angle-1
- Extra Example 2: Prove Trigonometric Identity using Double Angle-2

43m 55s

- Intro0:00
- Main Formulas0:09
- Confusing Part0:34
- Example 1: Find Sine and Cosine of Angle using Half-Angle0:54
- Example 2: Prove Trigonometric Identity using Half-Angle11:51
- Example 3: Prove the Half-Angle Formula for Tangents18:39
- Extra Example 1: Find Sine and Cosine of Angle using Half-Angle-1
- Extra Example 2: Prove Trigonometric Identity using Half-Angle-2

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 Triangle-1
- Extra Example 2: Find Lengths of All Sides of Triangle-2

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 Completely-1
- Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely-2

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 Completely-1
- Extra Example 2: Length of Third Side and Area of Triangle-2

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 Angle-1
- Extra Example 2: Area of Triangle with Two Sides and One Angle-2

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 Height-1
- Extra Example 2: Roads to a Town-2

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 Refresher-Solutions0:46
- Equation Solutions vs. Inequality Solutions1:02
- Essentially a Wide Variety of Answers1:35
- Refresher--Negative Multiplication Flips1:43
- Refresher--Negative Flips: Why?3:19
- Multiplication by a Negative3:43
- The Relationship Flips3:55
- Refresher--Stick 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 Line--Not Solutions11:10
- Solid Line--Are 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 Programming-Idea19:52
- Use a Linear Objective Function20:15
- Variables in Objective Function have Constraints21:24
- Linear Programming-Method22:09
- 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 51:01:32

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 Right-Hand 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
- Gauss-Jordan Elimination - Idea8:04
- Gauss-Jordan Elimination - Idea, cont.9:16
- Reduced Row-Echelon Form9:18
- Gauss-Jordan Elimination - Method11:36
- Begin by Writing the System As An Augmented Matrix11:38
- Gauss-Jordan 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 Counter-Clockwise 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 Forms-1
- Extra Example 2: Simplify Expression to Polar Form-2

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 Theorem-1
- Extra Example 2: Find Complex Fourth Roots-2

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 In1:01:08
- Let's Check In With Our Original Goal1:01:12
- Want to Show1:01:18
- Lemma - A Mini Theorem1:02:18
- Proof, Inductive Step - Lemma1:02:52
- Proof of Lemma: Let's Investigate the Left Side1:03:08
- Proof, Inductive Step - Nearly There1:07:54
- Proof, Inductive Step - End!1:09:18
- Proof, Inductive Step - End!, cont.1:11:01

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 Epsilon-Delta 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 Epsilon-Delta 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 Epsilon-Delta Part, cont.13:34
- Sherlock Holmes and Dr. Watson15:08
- The Adventure of the Delta-Epsilon Limit15:16
- Setting15:18
- We Begin By Setting Up the Game As Follows15:52
- The Adventure of the Delta-Epsilon, 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
- One-Sided Limit8:48
- One-Sided Limit - Definition9:16
- Only Considers One Side9:20
- Be Careful to Keep Track of Which Symbol Goes With Which Side10:06
- One-Sided Limit - Example10:50
- There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits11:16
- Normal Limits and One-Sided 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 Long-Run?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 Non-Line?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 - Left-Most Point5:12
- The Left-Most Point5:16
- Rectangle Method - Right-Most Point5:58
- The Right-Most Point6:00
- Rectangle Method - Mid-Point6:42
- Horizontal Mid-Point6:48
- 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 Sub-Intervals10:30
- 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
- TI-83 (Plus)7:16
- TI-84 (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

For more information, please see full course syllabus of Math Analysis

# Math Analysis Rational Functions and Vertical Asymptotes

In this lesson, our instructor Vincent Selhorst-Jones will teach you about Rational Functions and Vertical Asymptotes. Starting with the definition of a rational function, Vincent will then explain the domain of a rational function and how you can go about investigating a fundamental function. You’ll explore the idea of a vertical asymptote, as well as, learn how to find them and graph them. The lesson ends with four worked-out examples.

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0 answers

Post by Collin Yang on December 19, 2022

For example 1, why is the domain x ?-5, 1 but when I graphed it, it showed that it could be x = 1? Shouldn't the factors cancel before doing looking for the domain?

1 answer

Last reply by: Professor Selhorst-Jones

Tue Sep 8, 2020 7:53 PM

Post by Cynthia Chen on August 30, 2020

In 15:30, isn't 0.00001 divided by 0.0001 equal to 0.1?

1 answer

Last reply by: Professor Selhorst-Jones

Thu May 14, 2020 11:46 AM

Post by Leonardo Luo on November 27, 2018

Is 1/0 infinity or undefined?

1 answer

Last reply by: Professor Selhorst-Jones

Mon Sep 30, 2013 9:05 AM

Post by Mike Olyer on September 29, 2013

Why is 1 not a vertical asymptote for the function in example one? Why does canceling out allow this?