I. Introduction 

Introduction to Math Analysis 
10:03 
 
Intro 
0:00  
 
Title of the Course 
0:06  
 
 Different Names for the Course 
0:07  
 
 Precalculus 
0:12  
 
 Math Analysis 
0:14  
 
 Trigonometry 
0:16  
 
 Algebra III 
0:20  
 
 Geometry II 
0:24  
 
 College Algebra 
0:30  
 
 Same Concepts 
0:36  
 
How do the Lessons Work? 
0:54  
 
 Introducing Concepts 
0:56  
 
 Apply Concepts 
1:04  
 
 Go through Examples 
1:25  
 
Who is this Course For? 
1:38  
 
 Those Who Need eExtra Help with Class Work 
1:52  
 
 Those Working on Material but not in Formal Class at School 
1:54  
 
 Those Who Want a Refresher 
2:00  
 
Try to Watch the Whole Lesson 
2:20  
 
Understanding is So Important 
3:56  
 
What to Watch First 
5:26  
 
 Lesson #2: Sets, Elements, and Numbers 
5:30  
 
 Lesson #7: Idea of a Function 
5:33  
 
 Lesson #6: Word Problems 
6:04  
 
What to Watch First, cont. 
6:46  
 
 Lesson #2: Sets, Elements and Numbers 
6:56  
 
 Lesson #3: Variables, Equations, and Algebra 
6:58  
 
 Lesson #4: Coordinate Systems 
7:00  
 
 Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope 
7:02  
 
 Lesson #6: Word Problems 
7:10  
 
 Lesson #7: Idea of a Function 
7:12  
 
 Lesson #8: Graphs 
7:14  
 
Graphing Calculator Appendix 
7:40  
 
What to Watch Last 
8:46  
 
Let's get Started! 
9:48  

Sets, Elements, & Numbers 
45:11 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Sets and Elements 
1:19  
 
 Set 
1:20  
 
 Element 
1:23  
 
 Name a Set 
2:20  
 
 Order The Elements Appear In Has No Effect on the Set 
2:55  
 
Describing/ Defining Sets 
3:28  
 
 Directly Say All the Elements 
3:36  
 
 Clearly Describing All the Members of the Set 
3:55  
 
 Describing the Quality (or Qualities) Each member Of the Set Has In Common 
4:32  
 
Symbols: 'Element of' and 'Subset of' 
6:01  
 
 Symbol is ∈ 
6:03  
 
 Subset Symbol is ⊂ 
6:35  
 
Empty Set 
8:07  
 
 Symbol is ∅ 
8:20  
 
 Since It's Empty, It is a Subset of All Sets 
8:44  
 
Union and Intersection 
9:54  
 
 Union Symbol is ∪ 
10:08  
 
 Intersection Symbol is ∩ 
10:18  
 
Sets Can Be Weird Stuff 
12:26  
 
 Can Have Elements in a Set 
12:50  
 
 We Can Have Infinite Sets 
13:09  
 
 Example 
13:22  
 
 Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times 
14:08  
 
 This Set Has Infinitely Many Distinct Elements 
14:40  
 
Numbers as Sets 
16:03  
 
 Natural Numbers ℕ 
16:16  
 
 Including 0 and the Negatives ℤ 
18:13  
 
 Rational Numbers ℚ 
19:27  
 
 Can Express Rational Numbers with Decimal Expansions 
22:05  
 
 Irrational Numbers 
23:37  
 
 Real Numbers ℝ: Put the Rational and Irrational Numbers Together 
25:15  
 
Interval Notation and the Real Numbers 
26:45  
 
 Include the End Numbers 
27:06  
 
 Exclude the End Numbers 
27:33  
 
 Example 
28:28  
 
Interval Notation: Infinity 
29:09  
 
 Use ∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other 
29:14  
 
 Always Use Parentheses 
29:50  
 
 Examples 
30:27  
 
Example 1 
31:23  
 
Example 2 
35:26  
 
Example 3 
38:02  
 
Example 4 
42:21  

Variables, Equations, & Algebra 
35:31 
 
Intro 
0:00  
 
What is a Variable? 
0:05  
 
 A Variable is a Placeholder for a Number 
0:11  
 
 Affects the Output of a Function or a Dependent Variable 
0:24  
 
Naming Variables 
1:51  
 
 Useful to Use Symbols 
2:21  
 
What is a Constant? 
4:14  
 
 A Constant is a Fixed, Unchanging Number 
4:28  
 
 We Might Refer to a Symbol Representing a Number as a Constant 
4:51  
 
What is a Coefficient? 
5:33  
 
 A Coefficient is a Multiplicative Factor on a Variable 
5:37  
 
 Not All Coefficients are Constants 
5:51  
 
Expressions and Equations 
6:42  
 
 An Expression is a String of Mathematical Symbols That Make Sense Used Together 
7:05  
 
 An Equation is a Statement That Two Expression Have the Same Value 
8:20  
 
The Idea of Algebra 
8:51  
 
 Equality 
8:59  
 
 If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same 
9:41  
 
 Always Do The Exact Same Thing to Both Sides 
12:22  
 
Solving Equations 
13:23  
 
 When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something 
13:33  
 
 Look For What Values Makes the Equation True 
13:38  
 
 Isolate the Variable by Doing Algebra 
14:37  
 
Order of Operations 
16:02  
 
 Why Certain Operations are Grouped 
17:01  
 
 When You Don't Have to Worry About Order 
17:39  
 
Distributive Property 
18:15  
 
 It Allows Multiplication to Act Over Addition in Parentheses 
18:23  
 
 We Can Use the Distributive Property in Reverse to Combine Like Terms 
19:05  
 
Substitution 
20:03  
 
 Use Information From One Equation in Another Equation 
20:07  
 
 Put Your Substitution in Parentheses 
20:44  
 
Example 1 
23:17  
 
Example 2 
25:49  
 
Example 3 
28:11  
 
Example 4 
30:02  

Coordinate Systems 
35:02 
 
Intro 
0:00  
 
Inherent Order in ℝ 
0:05  
 
 Real Numbers Come with an Inherent Order 
0:11  
 
 Positive Numbers 
0:21  
 
 Negative Numbers 
0:58  
 
'Less Than' and 'Greater Than' 
2:04  
 
 Tip To Help You Remember the Signs 
2:56  
 
 Inequality 
4:06  
 
 Less Than or Equal and Greater Than or Equal 
4:51  
 
One Dimension: The Number Line 
5:36  
 
 Graphically Represent ℝ on a Number Line 
5:43  
 
 Note on Infinities 
5:57  
 
 With the Number Line, We Can Directly See the Order We Put on ℝ 
6:35  
 
Ordered Pairs 
7:22  
 
 Example 
7:34  
 
 Allows Us to Talk About Two Numbers at the Same Time 
9:41  
 
 Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ 
10:41  
 
Two Dimensions: The Plane 
13:13  
 
 We Can Represent Ordered Pairs with the Plane 
13:24  
 
 Intersection is known as the Origin 
14:31  
 
 Plotting the Point 
14:32  
 
 Plane = Coordinate Plane = Cartesian Plane = ℝ² 
17:46  
 
The Plane and Quadrants 
18:50  
 
 Quadrant I 
19:04  
 
 Quadrant II 
19:21  
 
 Quadrant III 
20:04  
 
 Quadrant IV 
20:20  
 
Three Dimensions: Space 
21:02  
 
 Create Ordered Triplets 
21:09  
 
 Visually Represent This 
21:19  
 
 ThreeDimension = Space = ℝ³ 
21:47  
 
Higher Dimensions 
22:24  
 
 If We Have n Dimensions, We Call It nDimensional Space or ℝ to the nth Power 
22:31  
 
 We Can Represent Places In This nDimensional Space As Ordered Groupings of n Numbers 
22:41  
 
 Hard to Visualize Higher Dimensional Spaces 
23:18  
 
Example 1 
25:07  
 
Example 2 
26:10  
 
Example 3 
28:58  
 
Example 4 
31:05  

Midpoints, Distance, the Pythagorean Theorem, & Slope 
48:43 
 
Intro 
0:00  
 
Introduction 
0:07  
 
Midpoint: One Dimension 
2:09  
 
 Example of Something More Complex 
2:31  
 
 Use the Idea of a Middle 
3:28  
 
 Find the Midpoint of Arbitrary Values a and b 
4:17  
 
 How They're Equivalent 
5:05  
 
 Official Midpoint Formula 
5:46  
 
Midpoint: Two Dimensions 
6:19  
 
 The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle 
6:38  
 
 Arbitrary Pair of Points Example 
7:25  
 
Distance: One Dimension 
9:26  
 
Absolute Value 
10:54  
 
 Idea of Forcing Positive 
11:06  
 
Distance: One Dimension, Formula 
11:47  
 
 Distance Between Arbitrary a and b 
11:48  
 
 Absolute Value Helps When the Distance is Negative 
12:41  
 
 Distance Formula 
12:58  
 
The Pythagorean Theorem 
13:24  
 
 a²+b²=c² 
13:50  
 
Distance: Two Dimensions 
14:59  
 
 Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem 
15:16  
 
 Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂) 
16:21  
 
Slope 
19:30  
 
 Slope is the Rate of Change 
19:41  
 
 m = rise over run 
21:27  
 
 Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂) 
22:31  
 
Interpreting Slope 
24:12  
 
 Positive Slope and Negative Slope 
25:40  
 
 m=1, m=0, m=1 
26:48  
 
Example 1 
28:25  
 
Example 2 
31:42  
 
Example 3 
36:40  
 
Example 4 
42:48  

Word Problems 
56:31 
 
Intro 
0:00  
 
Introduction 
0:05  
 
What is a Word Problem? 
0:45  
 
 Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols 
0:48  
 
 Requires Us to Think 
1:32  
 
Why Are They So Hard? 
2:11  
 
 Reason 1: No Simple Formula to Solve Them 
2:16  
 
 Reason 2: Harder to Teach Word Problems 
2:47  
 
 You Can Learn How to Do Them! 
3:51  
 
 Grades 
7:57  
 
 'But I'm Never Going to Use This In Real Life' 
9:46  
 
Solving Word Problems 
12:58  
 
 First: Understand the Problem 
13:37  
 
 Second: What Are You Looking For? 
14:33  
 
 Third: Set Up Relationships 
16:21  
 
 Fourth: Solve It! 
17:48  
 
Summary of Method 
19:04  
 
Examples on Things Other Than Math 
20:21  
 
MathSpecific Method: What You Need Now 
25:30  
 
 Understand What the Problem is Talking About 
25:37  
 
 Set Up and Name Any Variables You Need to Know 
25:56  
 
 Set Up Equations Connecting Those Variables to the Information in the Problem Statement 
26:02  
 
 Use the Equations to Solve for an Answer 
26:14  
 
Tip 
26:58  
 
 Draw Pictures 
27:22  
 
 Breaking Into Pieces 
28:28  
 
 Try Out Hypothetical Numbers 
29:52  
 
 Student Logic 
31:27  
 
 Jump In! 
32:40  
 
Example 1 
34:03  
 
Example 2 
39:15  
 
Example 3 
44:22  
 
Example 4 
50:24  
II. Functions 

Idea of a Function 
39:54 
 
Intro 
0:00  
 
Introduction 
0:04  
 
What is a Function? 
1:06  
 
A Visual Example and NonExample 
1:30  
 
Function Notation 
3:47  
 
 f(x) 
4:05  
 
 Express What Sets the Function Acts On 
5:45  
 
Metaphors for a Function 
6:17  
 
 Transformation 
6:28  
 
 Map 
7:17  
 
 Machine 
8:56  
 
Same Input Always Gives Same Output 
10:01  
 
 If We Put the Same Input Into a Function, It Will Always Produce the Same Output 
10:11  
 
 Example of Something That is Not a Function 
11:10  
 
A NonNumerical Example 
12:10  
 
The Functions We Will Use 
15:05  
 
 Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers 
15:11  
 
 Usually Told the Rule of a Given Function 
15:27  
 
How To Use a Function 
16:18  
 
 Apply the Rule to Whatever Our Input Value Is 
16:28  
 
 Make Sure to Wrap Your Substitutions in Parentheses 
17:09  
 
Functions and Tables 
17:36  
 
 Table of Values, Sometimes Called a TTable 
17:46  
 
 Example 
17:56  
 
Domain: What Goes In 
18:55  
 
 The Domain is the Set of all Inputs That the Function Can Accept 
18:56  
 
 Example 
19:40  
 
Range: What Comes Out 
21:27  
 
 The Range is the Set of All Possible Outputs a Function Can Assign 
21:34  
 
 Example 
21:49  
 
 Another Example Would Be Our Initial Function From Earlier in This Lesson 
22:29  
 
Example 1 
23:45  
 
Example 2 
25:22  
 
Example 3 
27:27  
 
Example 4 
29:23  
 
Example 5 
33:33  

Graphs 
58:26 
 
Intro 
0:00  
 
Introduction 
0:04  
 
How to Interpret Graphs 
1:17  
 
 Input / Independent Variable 
1:47  
 
 Output / Dependent Variable 
2:00  
 
Graph as Input ⇒ Output 
2:23  
 
 One Way to Think of a Graph: See What Happened to Various Inputs 
2:25  
 
 Example 
2:47  
 
Graph as Location of Solution 
4:20  
 
 A Way to See Solutions 
4:36  
 
 Example 
5:20  
 
Which Way Should We Interpret? 
7:13  
 
 Easiest to Think In Terms of How Inputs Are Mapped to Outputs 
7:20  
 
 Sometimes It's Easier to Think In Terms of Solutions 
8:39  
 
Pay Attention to Axes 
9:50  
 
 Axes Tell Where the Graph Is and What Scale It Has 
10:09  
 
 Often, The Axes Will Be Square 
10:14  
 
 Example 
12:06  
 
Arrows or No Arrows? 
16:07  
 
 Will Not Use Arrows at the End of Our Graphs 
17:13  
 
 Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops 
17:18  
 
How to Graph 
19:47  
 
 Plot Points 
20:07  
 
 Connect with Curves 
21:09  
 
 If You Connect with Straight Lines 
21:44  
 
 Graphs of Functions are Smooth 
22:21  
 
 More Points ⇒ More Accurate 
23:38  
 
Vertical Line Test 
27:44  
 
 If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function 
28:41  
 
 Every Point on a Graph Tells Us Where the xValue Below is Mapped 
30:07  
 
Domain in Graphs 
31:37  
 
 The Domain is the Set of All Inputs That a Function Can Accept 
31:44  
 
 Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window' 
33:19  
 
Range in Graphs 
33:53  
 
Graphing Calculators: Check the Appendix! 
36:55  
 
Example 1 
38:37  
 
Example 2 
45:19  
 
Example 3 
50:41  
 
Example 4 
53:28  
 
Example 5 
55:50  

Properties of Functions 
48:49 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Increasing Decreasing Constant 
0:43  
 
 Looking at a Specific Graph 
1:15  
 
 Increasing Interval 
2:39  
 
 Constant Function 
4:15  
 
 Decreasing Interval 
5:10  
 
Find Intervals by Looking at the Graph 
5:32  
 
Intervals Show xvalues; Write in Parentheses 
6:39  
 
Maximum and Minimums 
8:48  
 
Relative (Local) Max/Min 
10:20  
 
 Formal Definition of Relative Maximum 
12:44  
 
 Formal Definition of Relative Minimum 
13:05  
 
Max/Min, More Terms 
14:18  
 
 Definition of Extrema 
15:01  
 
Average Rate of Change 
16:11  
 
 Drawing a Line for the Average Rate 
16:48  
 
 Using the Slope of the Secant Line 
17:36  
 
 Slope in Function Notation 
18:45  
 
Zeros/Roots/xintercepts 
19:45  
 
 What Zeros in a Function Mean 
20:25  
 
Even Functions 
22:30  
 
Odd Functions 
24:36  
 
Even/Odd Functions and Graphs 
26:28  
 
 Example of an Even Function 
27:12  
 
 Example of an Odd Function 
28:03  
 
Example 1 
29:35  
 
Example 2 
33:07  
 
Example 3 
40:32  
 
Example 4 
42:34  

Function Petting Zoo 
29:20 
 
Intro 
0:00  
 
Introduction 
0:04  
 
Don't Forget that Axes Matter! 
1:44  
 
The Constant Function 
2:40  
 
The Identity Function 
3:44  
 
The Square Function 
4:40  
 
The Cube Function 
5:44  
 
The Square Root Function 
6:51  
 
The Reciprocal Function 
8:11  
 
The Absolute Value Function 
10:19  
 
The Trigonometric Functions 
11:56  
 
 f(x)=sin(x) 
12:12  
 
 f(x)=cos(x) 
12:24  
 
 Alternate Axes 
12:40  
 
The Exponential and Logarithmic Functions 
13:35  
 
 Exponential Functions 
13:44  
 
 Logarithmic Functions 
14:24  
 
 Alternating Axes 
15:17  
 
Transformations and Compositions 
16:08  
 
Example 1 
17:52  
 
Example 2 
18:33  
 
Example 3 
20:24  
 
Example 4 
26:07  

Transformation of Functions 
48:35 
 
Intro 
0:00  
 
Introduction 
0:04  
 
Vertical Shift 
1:12  
 
 Graphical Example 
1:21  
 
 A Further Explanation 
2:16  
 
Vertical Stretch/Shrink 
3:34  
 
 Graph Shrinks 
3:46  
 
 Graph Stretches 
3:51  
 
 A Further Explanation 
5:07  
 
Horizontal Shift 
6:49  
 
 Moving the Graph to the Right 
7:28  
 
 Moving the Graph to the Left 
8:12  
 
 A Further Explanation 
8:19  
 
 Understanding Movement on the xaxis 
8:38  
 
Horizontal Stretch/Shrink 
12:59  
 
 Shrinking the Graph 
13:40  
 
 Stretching the Graph 
13:48  
 
 A Further Explanation 
13:55  
 
 Understanding Stretches from the xaxis 
14:12  
 
Vertical Flip (aka Mirror) 
16:55  
 
 Example Graph 
17:07  
 
 Multiplying the Vertical Component by 1 
17:18  
 
Horizontal Flip (aka Mirror) 
18:43  
 
 Example Graph 
19:01  
 
 Multiplying the Horizontal Component by 1 
19:54  
 
Summary of Transformations 
22:11  
 
Stacking Transformations 
24:46  
 
 Order Matters 
25:20  
 
 Transformation Example 
25:52  
 
Example 1 
29:21  
 
Example 2 
34:44  
 
Example 3 
38:10  
 
Example 4 
43:46  

Composite Functions 
33:24 
 
Intro 
0:00  
 
Introduction 
0:04  
 
Arithmetic Combinations 
0:40  
 
 Basic Operations 
1:20  
 
 Definition of the Four Arithmetic Combinations 
1:40  
 
Composite Functions 
2:53  
 
The Function as a Machine 
3:32  
 
Function Compositions as Multiple Machines 
3:59  
 
Notation for Composite Functions 
4:46  
 
 Two Formats 
6:02  
 
Another Visual Interpretation 
7:17  
 
How to Use Composite Functions 
8:21  
 
 Example of on Function acting on Another 
9:17  
 
Example 1 
11:03  
 
Example 2 
15:27  
 
Example 3 
21:11  
 
Example 4 
27:06  

Piecewise Functions 
51:42 
 
Intro 
0:00  
 
Introduction 
0:04  
 
Analogies to a Piecewise Function 
1:16  
 
 Different Potatoes 
1:41  
 
 Factory Production 
2:27  
 
Notations for Piecewise Functions 
3:39  
 
 Notation Examples from Analogies 
6:11  
 
Example of a Piecewise (with Table) 
7:24  
 
Example of a NonNumerical Piecewise 
11:35  
 
Graphing Piecewise Functions 
14:15  
 
Graphing Piecewise Functions, Example 
16:26  
 
Continuous Functions 
16:57  
 
 Statements of Continuity 
19:30  
 
 Example of Continuous and NonContinuous Graphs 
20:05  
 
Interesting Functions: the Step Function 
22:00  
 
 Notation for the Step Function 
22:40  
 
 How the Step Function Works 
22:56  
 
 Graph of the Step Function 
25:30  
 
Example 1 
26:22  
 
Example 2 
28:49  
 
Example 3 
36:50  
 
Example 4 
46:11  

Inverse Functions 
49:37 
 
Intro 
0:00  
 
Introduction 
0:04  
 
Analogy by picture 
1:10  
 
 How to Denote the inverse 
1:40  
 
 What Comes out of the Inverse 
1:52  
 
Requirement for Reversing 
2:02  
 
 The Basketball Factory 
2:12  
 
 The Importance of Information 
2:45  
 
OnetoOne 
4:04  
 
 Requirement for Reversibility 
4:21  
 
 When a Function has an Inverse 
4:43  
 
 OnetoOne 
5:13  
 
 Not OnetoOne 
5:50  
 
 Not a Function 
6:19  
 
Horizontal Line Test 
7:01  
 
 How to the test Works 
7:12  
 
 OnetoOne 
8:12  
 
 Not OnetoOne 
8:45  
 
Definition: Inverse Function 
9:12  
 
 Formal Definition 
9:21  
 
 Caution to Students 
10:02  
 
Domain and Range 
11:12  
 
 Finding the Range of the Function Inverse 
11:56  
 
 Finding the Domain of the Function Inverse 
12:11  
 
Inverse of an Inverse 
13:09  
 
 Its just x! 
13:26  
 
 Proof 
14:03  
 
Graphical Interpretation 
17:07  
 
 Horizontal Line Test 
17:20  
 
 Graph of the Inverse 
18:04  
 
 Swapping Inputs and Outputs to Draw Inverses 
19:02  
 
How to Find the Inverse 
21:03  
 
 What We Are Looking For 
21:21  
 
 Reversing the Function 
21:38  
 
A Method to Find Inverses 
22:33  
 
 Check Function is OnetoOne 
23:04  
 
 Swap f(x) for y 
23:25  
 
 Interchange x and y 
23:41  
 
 Solve for y 
24:12  
 
 Replace y with the inverse 
24:40  
 
Some Comments 
25:01  
 
 Keeping Step 2 and 3 Straight 
25:44  
 
 Switching to Inverse 
26:12  
 
Checking Inverses 
28:52  
 
 How to Check an Inverse 
29:06  
 
 Quick Example of How to Check 
29:56  
 
Example 1 
31:48  
 
Example 2 
34:56  
 
Example 3 
39:29  
 
Example 4 
46:19  

Variation Direct and Inverse 
28:49 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Direct Variation 
1:14  
 
 Same Direction 
1:21  
 
 Common Example: Groceries 
1:56  
 
 Different Ways to Say that Two Things Vary Directly 
2:28  
 
 Basic Equation for Direct Variation 
2:55  
 
Inverse Variation 
3:40  
 
 Opposite Direction 
3:50  
 
 Common Example: Gravity 
4:53  
 
 Different Ways to Say that Two Things Vary Indirectly 
5:48  
 
 Basic Equation for Indirect Variation 
6:33  
 
Joint Variation 
7:27  
 
 Equation for Joint Variation 
7:53  
 
 Explanation of the Constant 
8:48  
 
Combined Variation 
9:35  
 
 Gas Law as a Combination 
9:44  
 
 Single Constant 
10:33  
 
Example 1 
10:49  
 
Example 2 
13:34  
 
Example 3 
15:39  
 
Example 4 
19:48  
III. Polynomials 

Intro to Polynomials 
38:41 
 
Intro 
0:00  
 
Introduction 
0:04  
 
Definition of a Polynomial 
1:04  
 
 Starting Integer 
2:06  
 
 Structure of a Polynomial 
2:49  
 
 The a Constants 
3:34  
 
 Polynomial Function 
5:13  
 
 Polynomial Equation 
5:23  
 
 Polynomials with Different Variables 
5:36  
 
Degree 
6:23  
 
 Informal Definition 
6:31  
 
 Find the Largest Exponent Variable 
6:44  
 
 Quick Examples 
7:36  
 
Special Names for Polynomials 
8:59  
 
 Based on the Degree 
9:23  
 
 Based on the Number of Terms 
10:12  
 
Distributive Property (aka 'FOIL') 
11:37  
 
 Basic Distributive Property 
12:21  
 
 Distributing Two Binomials 
12:55  
 
 Longer Parentheses 
15:12  
 
 Reverse: Factoring 
17:26  
 
LongTerm Behavior of Polynomials 
17:48  
 
 Examples 
18:13  
 
 Controlling TermTerm with the Largest Exponent 
19:33  
 
 Positive and Negative Coefficients on the Controlling Term 
20:21  
 
Leading Coefficient Test 
22:07  
 
 Even Degree, Positive Coefficient 
22:13  
 
 Even Degree, Negative Coefficient 
22:39  
 
 Odd Degree, Positive Coefficient 
23:09  
 
 Odd Degree, Negative Coefficient 
23:27  
 
Example 1 
25:11  
 
Example 2 
27:16  
 
Example 3 
31:16  
 
Example 4 
34:41  

Roots (Zeros) of Polynomials 
41:07 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Roots in Graphs 
1:17  
 
 The xintercepts 
1:33  
 
 How to Remember What 'Roots' Are 
1:50  
 
Naïve Attempts 
2:31  
 
 Isolating Variables 
2:45  
 
 Failures of Isolating Variables 
3:30  
 
 Missing Solutions 
4:59  
 
Factoring: How to Find Roots 
6:28  
 
 How Factoring Works 
6:36  
 
 Why Factoring Works 
7:20  
 
 Steps to Finding Polynomial Roots 
9:21  
 
Factoring: How to Find Roots CAUTION 
10:08  
 
Factoring is Not Easy 
11:32  
 
Factoring Quadratics 
13:08  
 
 Quadratic Trinomials 
13:21  
 
 Form of Factored Binomials 
13:38  
 
 Factoring Examples 
14:40  
 
Factoring Quadratics, Check Your Work 
16:58  
 
Factoring Higher Degree Polynomials 
18:19  
 
 Factoring a Cubic 
18:32  
 
 Factoring a Quadratic 
19:04  
 
Factoring: Roots Imply Factors 
19:54  
 
 Where a Root is, A Factor Is 
20:01  
 
 How to Use Known Roots to Make Factoring Easier 
20:35  
 
Not all Polynomials Can be Factored 
22:30  
 
 Irreducible Polynomials 
23:27  
 
 Complex Numbers Help 
23:55  
 
Max Number of Roots/Factors 
24:57  
 
 Limit to Number of Roots Equal to the Degree 
25:18  
 
 Why there is a Limit 
25:25  
 
Max Number of Peaks/Valleys 
26:39  
 
 Shape Information from Degree 
26:46  
 
 Example Graph 
26:54  
 
Max, But Not Required 
28:00  
 
Example 1 
28:37  
 
Example 2 
31:21  
 
Example 3 
36:12  
 
Example 4 
38:40  

Completing the Square and the Quadratic Formula 
39:43 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Square Roots and Equations 
0:51  
 
 Taking the Square Root to Find the Value of x 
0:55  
 
 Getting the Positive and Negative Answers 
1:05  
 
Completing the Square: Motivation 
2:04  
 
 Polynomials that are Easy to Solve 
2:20  
 
 Making Complex Polynomials Easy to Solve 
3:03  
 
 Steps to Completing the Square 
4:30  
 
Completing the Square: Method 
7:22  
 
 Move C over 
7:35  
 
 Divide by A 
7:44  
 
 Find r 
7:59  
 
 Add to Both Sides to Complete the Square 
8:49  
 
Solving Quadratics with Ease 
9:56  
 
The Quadratic Formula 
11:38  
 
 Derivation 
11:43  
 
 Final Form 
12:23  
 
Follow Format to Use Formula 
13:38  
 
How Many Roots? 
14:53  
 
The Discriminant 
15:47  
 
 What the Discriminant Tells Us: How Many Roots 
15:58  
 
 How the Discriminant Works 
16:30  
 
Example 1: Complete the Square 
18:24  
 
Example 2: Solve the Quadratic 
22:00  
 
Example 3: Solve for Zeroes 
25:28  
 
Example 4: Using the Quadratic Formula 
30:52  

Properties of Quadratic Functions 
45:34 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Parabolas 
0:35  
 
 Examples of Different Parabolas 
1:06  
 
Axis of Symmetry and Vertex 
1:28  
 
 Drawing an Axis of Symmetry 
1:51  
 
 Placing the Vertex 
2:28  
 
 Looking at the Axis of Symmetry and Vertex for other Parabolas 
3:09  
 
Transformations 
4:18  
 
 Reviewing Transformation Rules 
6:28  
 
 Note the Different Horizontal Shift Form 
7:45  
 
An Alternate Form to Quadratics 
8:54  
 
 The Constants: k, h, a 
9:05  
 
 Transformations Formed 
10:01  
 
 Analyzing Different Parabolas 
10:10  
 
Switching Forms by Completing the Square 
11:43  
 
Vertex of a Parabola 
16:30  
 
 Vertex at (h, k) 
16:47  
 
 Vertex in Terms of a, b, and c Coefficients 
17:28  
 
Minimum/Maximum at Vertex 
18:19  
 
 When a is Positive 
18:25  
 
 When a is Negative 
18:52  
 
Axis of Symmetry 
19:54  
 
Incredibly Minor Note on Grammar 
20:52  
 
Example 1 
21:48  
 
Example 2 
26:35  
 
Example 3 
28:55  
 
Example 4 
31:40  

Intermediate Value Theorem and Polynomial Division 
46:08 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Reminder: Roots Imply Factors 
1:32  
 
The Intermediate Value Theorem 
3:41  
 
 The Basis: U between a and b 
4:11  
 
 U is on the Function 
4:52  
 
Intermediate Value Theorem, Proof Sketch 
5:51  
 
 If Not True, the Graph Would Have to Jump 
5:58  
 
 But Graph is Defined as Continuous 
6:43  
 
Finding Roots with the Intermediate Value Theorem 
7:01  
 
 Picking a and b to be of Different Signs 
7:10  
 
 Must Be at Least One Root 
7:46  
 
Dividing a Polynomial 
8:16  
 
 Using Roots and Division to Factor 
8:38  
 
Long Division Refresher 
9:08  
 
The Division Algorithm 
12:18  
 
 How It Works to Divide Polynomials 
12:37  
 
 The Parts of the Equation 
13:24  
 
 Rewriting the Equation 
14:47  
 
Polynomial Long Division 
16:20  
 
 Polynomial Long Division In Action 
16:29  
 
 One Step at a Time 
20:51  
 
Synthetic Division 
22:46  
 
 Setup 
23:11  
 
Synthetic Division, Example 
24:44  
 
Which Method Should We Use 
26:39  
 
 Advantages of Synthetic Method 
26:49  
 
 Advantages of Long Division 
27:13  
 
Example 1 
29:24  
 
Example 2 
31:27  
 
Example 3 
36:22  
 
Example 4 
40:55  

Complex Numbers 
45:36 
 
Intro 
0:00  
 
Introduction 
0:04  
 
A Wacky Idea 
1:02  
 
 The Definition of the Imaginary Number 
1:22  
 
 How it Helps Solve Equations 
2:20  
 
Square Roots and Imaginary Numbers 
3:15  
 
Complex Numbers 
5:00  
 
 Real Part and Imaginary Part 
5:20  
 
 When Two Complex Numbers are Equal 
6:10  
 
Addition and Subtraction 
6:40  
 
 Deal with Real and Imaginary Parts Separately 
7:36  
 
 Two Quick Examples 
7:54  
 
Multiplication 
9:07  
 
 FOIL Expansion 
9:14  
 
 Note What Happens to the Square of the Imaginary Number 
9:41  
 
 Two Quick Examples 
10:22  
 
Division 
11:27  
 
Complex Conjugates 
13:37  
 
 Getting Rid of i 
14:08  
 
 How to Denote the Conjugate 
14:48  
 
Division through Complex Conjugates 
16:11  
 
 Multiply by the Conjugate of the Denominator 
16:28  
 
 Example 
17:46  
 
Factoring SoCalled 'Irreducible' Quadratics 
19:24  
 
 Revisiting the Quadratic Formula 
20:12  
 
 Conjugate Pairs 
20:37  
 
But Are the Complex Numbers 'Real'? 
21:27  
 
 What Makes a Number Legitimate 
25:38  
 
 Where Complex Numbers are Used 
27:20  
 
Still, We Won't See Much of C 
29:05  
 
Example 1 
30:30  
 
Example 2 
33:15  
 
Example 3 
38:12  
 
Example 4 
42:07  

Fundamental Theorem of Algebra 
19:09 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Idea: Hidden Roots 
1:16  
 
 Roots in Complex Form 
1:42  
 
 All Polynomials Have Roots 
2:08  
 
Fundamental Theorem of Algebra 
2:21  
 
Where Are All the Imaginary Roots, Then? 
3:17  
 
 All Roots are Complex 
3:45  
 
 Real Numbers are a Subset of Complex Numbers 
3:59  
 
The n Roots Theorem 
5:01  
 
 For Any Polynomial, Its Degree is Equal to the Number of Roots 
5:11  
 
 Equivalent Statement 
5:24  
 
Comments: Multiplicity 
6:29  
 
 NonDistinct Roots 
6:59  
 
 Denoting Multiplicity 
7:20  
 
Comments: Complex Numbers Necessary 
7:41  
 
Comments: Complex Coefficients Allowed 
8:55  
 
Comments: Existence Theorem 
9:59  
 
Proof Sketch of n Roots Theorem 
10:45  
 
 First Root 
11:36  
 
 Second Root 
13:23  
 
 Continuation to Find all Roots 
16:00  
IV. Rational Functions 

Rational Functions and Vertical Asymptotes 
33:22 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Definition of a Rational Function 
1:20  
 
 Examples of Rational Functions 
2:30  
 
 Why They are Called 'Rational' 
2:47  
 
Domain of a Rational Function 
3:15  
 
 Undefined at Denominator Zeros 
3:25  
 
 Otherwise all Reals 
4:16  
 
Investigating a Fundamental Function 
4:50  
 
 The Domain of the Function 
5:04  
 
 What Occurs at the Zeroes of the Denominator 
5:20  
 
Idea of a Vertical Asymptote 
6:23  
 
What's Going On? 
6:58  
 
 Approaching x=0 from the left 
7:32  
 
 Approaching x=0 from the right 
8:34  
 
 Dividing by Very Small Numbers Results in Very Large Numbers 
9:31  
 
Definition of a Vertical Asymptote 
10:05  
 
Vertical Asymptotes and Graphs 
11:15  
 
 Drawing Asymptotes by Using a Dashed Line 
11:27  
 
 The Graph Can Never Touch Its Undefined Point 
12:00  
 
Not All Zeros Give Asymptotes 
13:02  
 
 Special Cases: When Numerator and Denominator Go to Zero at the Same Time 
14:58  
 
 Cancel out Common Factors 
15:49  
 
How to Find Vertical Asymptotes 
16:10  
 
 Figure out What Values Are Not in the Domain of x 
16:24  
 
 Determine if the Numerator and Denominator Share Common Factors and Cancel 
16:45  
 
 Find Denominator Roots 
17:33  
 
 Note if Asymptote Approaches Negative or Positive Infinity 
18:06  
 
Example 1 
18:57  
 
Example 2 
21:26  
 
Example 3 
23:04  
 
Example 4 
30:01  

Horizontal Asymptotes 
34:16 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Investigating a Fundamental Function 
0:53  
 
 What Happens as x Grows Large 
1:00  
 
 Different View 
1:12  
 
Idea of a Horizontal Asymptote 
1:36  
 
What's Going On? 
2:24  
 
 What Happens as x Grows to a Large Negative Number 
2:49  
 
 What Happens as x Grows to a Large Number 
3:30  
 
 Dividing by Very Large Numbers Results in Very Small Numbers 
3:52  
 
 Example Function 
4:41  
 
Definition of a Vertical Asymptote 
8:09  
 
Expanding the Idea 
9:03  
 
What's Going On? 
9:48  
 
 What Happens to the Function in the Long Run? 
9:51  
 
 Rewriting the Function 
10:13  
 
Definition of a Slant Asymptote 
12:09  
 
 Symbolical Definition 
12:30  
 
 Informal Definition 
12:45  
 
Beyond Slant Asymptotes 
13:03  
 
Not Going Beyond Slant Asymptotes 
14:39  
 
Horizontal/Slant Asymptotes and Graphs 
15:43  
 
How to Find Horizontal and Slant Asymptotes 
16:52  
 
How to Find Horizontal Asymptotes 
17:12  
 
 Expand the Given Polynomials 
17:18  
 
 Compare the Degrees of the Numerator and Denominator 
17:40  
 
How to Find Slant Asymptotes 
20:05  
 
 Slant Asymptotes Exist When n+m=1 
20:08  
 
 Use Polynomial Division 
20:24  
 
Example 1 
24:32  
 
Example 2 
25:53  
 
Example 3 
26:55  
 
Example 4 
29:22  

Graphing Asymptotes in a Nutshell 
49:07 
 
Intro 
0:00  
 
Introduction 
0:05  
 
A Process for Graphing 
1:22  
 
1. Factor Numerator and Denominator 
1:50  
 
2. Find Domain 
2:53  
 
3. Simplifying the Function 
3:59  
 
4. Find Vertical Asymptotes 
4:59  
 
5. Find Horizontal/Slant Asymptotes 
5:24  
 
6. Find Intercepts 
7:35  
 
7. Draw Graph (Find Points as Necessary) 
9:21  
 
Draw Graph Example 
11:21  
 
 Vertical Asymptote 
11:41  
 
 Horizontal Asymptote 
11:50  
 
 Other Graphing 
12:16  
 
Test Intervals 
15:08  
 
Example 1 
17:57  
 
Example 2 
23:01  
 
Example 3 
29:02  
 
Example 4 
33:37  

Partial Fractions 
44:56 
 
Intro 
0:00  
 
Introduction: Idea 
0:04  
 
Introduction: Prerequisites and Uses 
1:57  
 
Proper vs. Improper Polynomial Fractions 
3:11  
 
Possible Things in the Denominator 
4:38  
 
Linear Factors 
6:16  
 
 Example of Linear Factors 
7:03  
 
 Multiple Linear Factors 
7:48  
 
Irreducible Quadratic Factors 
8:25  
 
 Example of Quadratic Factors 
9:26  
 
 Multiple Quadratic Factors 
9:49  
 
Mixing Factor Types 
10:28  
 
Figuring Out the Numerator 
11:10  
 
 How to Solve for the Constants 
11:30  
 
 Quick Example 
11:40  
 
Example 1 
14:29  
 
Example 2 
18:35  
 
Example 3 
20:33  
 
Example 4 
28:51  
V. Exponential & Logarithmic Functions 

Understanding Exponents 
35:17 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Fundamental Idea 
1:46  
 
Expanding the Idea 
2:28  
 
 Multiplication of the Same Base 
2:40  
 
 Exponents acting on Exponents 
3:45  
 
 Different Bases with the Same Exponent 
4:31  
 
To the Zero 
5:35  
 
 To the First 
5:45  
 
 Fundamental Rule with the Zero Power 
6:35  
 
To the Negative 
7:45  
 
 Any Number to a Negative Power 
8:14  
 
 A Fraction to a Negative Power 
9:58  
 
 Division with Exponential Terms 
10:41  
 
To the Fraction 
11:33  
 
 Square Root 
11:58  
 
 Any Root 
12:59  
 
Summary of Rules 
14:38  
 
To the Irrational 
17:21  
 
Example 1 
20:34  
 
Example 2 
23:42  
 
Example 3 
27:44  
 
Example 4 
31:44  
 
Example 5 
33:15  

Exponential Functions 
47:04 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Definition of an Exponential Function 
0:48  
 
 Definition of the Base 
1:02  
 
 Restrictions on the Base 
1:16  
 
Computing Exponential Functions 
2:29  
 
 Harder Computations 
3:10  
 
 When to Use a Calculator 
3:21  
 
Graphing Exponential Functions: a>1 
6:02  
 
 Three Examples 
6:13  
 
 What to Notice on the Graph 
7:44  
 
A Story 
8:27  
 
 Story Diagram 
9:15  
 
 Increasing Exponentials 
11:29  
 
 Story Morals 
14:40  
 
Application: Compound Interest 
15:15  
 
 Compounding Year after Year 
16:01  
 
 Function for Compounding Interest 
16:51  
 
A Special Number: e 
20:55  
 
 Expression for e 
21:28  
 
 Where e stabilizes 
21:55  
 
Application: Continuously Compounded Interest 
24:07  
 
 Equation for Continuous Compounding 
24:22  
 
Exponential Decay 0<a<1 
25:50  
 
 Three Examples 
26:11  
 
 Why they 'lose' value 
26:54  
 
Example 1 
27:47  
 
Example 2 
33:11  
 
Example 3 
36:34  
 
Example 4 
41:28  

Introduction to Logarithms 
40:31 
 
Intro 
0:00  
 
Introduction 
0:04  
 
Definition of a Logarithm, Base 2 
0:51  
 
 Log 2 Defined 
0:55  
 
 Examples 
2:28  
 
Definition of a Logarithm, General 
3:23  
 
Examples of Logarithms 
5:15  
 
 Problems with Unusual Bases 
7:38  
 
Shorthand Notation: ln and log 
9:44  
 
 base e as ln 
10:01  
 
 base 10 as log 
10:34  
 
Calculating Logarithms 
11:01  
 
 using a calculator 
11:34  
 
 issues with other bases 
11:58  
 
Graphs of Logarithms 
13:21  
 
 Three Examples 
13:29  
 
 Slow Growth 
15:19  
 
Logarithms as Inverse of Exponentiation 
16:02  
 
 Using Base 2 
16:05  
 
 General Case 
17:10  
 
 Looking More Closely at Logarithm Graphs 
19:16  
 
The Domain of Logarithms 
20:41  
 
 Thinking about Logs like Inverses 
21:08  
 
 The Alternate 
24:00  
 
Example 1 
25:59  
 
Example 2 
30:03  
 
Example 3 
32:49  
 
Example 4 
37:34  

Properties of Logarithms 
42:33 
 
Intro 
0:00  
 
Introduction 
0:04  
 
Basic Properties 
1:12  
 
Inverselog(exp) 
1:43  
 
A Key Idea 
2:44  
 
 What We Get through Exponentiation 
3:18  
 
 B Always Exists 
4:50  
 
Inverseexp(log) 
5:53  
 
Logarithm of a Power 
7:44  
 
Logarithm of a Product 
10:07  
 
Logarithm of a Quotient 
13:48  
 
Caution! There Is No Rule for loga(M+N) 
16:12  
 
Summary of Properties 
17:42  
 
Change of BaseMotivation 
20:17  
 
 No Calculator Button 
20:59  
 
 A Specific Example 
21:45  
 
 Simplifying 
23:45  
 
Change of BaseFormula 
24:14  
 
Example 1 
25:47  
 
Example 2 
29:08  
 
Example 3 
31:14  
 
Example 4 
34:13  

Solving Exponential and Logarithmic Equations 
34:10 
 
Intro 
0:00  
 
Introduction 
0:05  
 
One to One Property 
1:09  
 
 Exponential 
1:26  
 
 Logarithmic 
1:44  
 
 Specific Considerations 
2:02  
 
 OnetoOne Property 
3:30  
 
Solving by OnetoOne 
4:11  
 
Inverse Property 
6:09  
 
Solving by Inverses 
7:25  
 
 Dealing with Equations 
7:50  
 
 Example of Taking an Exponent or Logarithm of an Equation 
9:07  
 
A Useful Property 
11:57  
 
 Bring Down Exponents 
12:01  
 
 Try to Simplify 
13:20  
 
Extraneous Solutions 
13:45  
 
Example 1 
16:37  
 
Example 2 
19:39  
 
Example 3 
21:37  
 
Example 4 
26:45  
 
Example 5 
29:37  

Application of Exponential and Logarithmic Functions 
48:46 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Applications of Exponential Functions 
1:07  
 
A Secret! 
2:17  
 
 Natural Exponential Growth Model 
3:07  
 
 Figure out r 
3:34  
 
A Secret!Why Does It Work? 
4:44  
 
 e to the r Morphs 
4:57  
 
 Example 
5:06  
 
Applications of Logarithmic Functions 
8:32  
 
 Examples 
8:43  
 
 What Logarithms are Useful For 
9:53  
 
Example 1 
11:29  
 
Example 2 
15:30  
 
Example 3 
26:22  
 
Example 4 
32:05  
 
Example 5 
39:19  
VI. Trigonometric Functions 

Angles 
39:05 
 
Intro 
0:00  
 
Degrees 
0:22  
 
 Circle is 360 Degrees 
0:48  
 
 Splitting a Circle 
1:13  
 
Radians 
2:08  
 
 Circle is 2 Pi Radians 
2:31  
 
 One Radian 
2:52  
 
 HalfCircle and Right Angle 
4:00  
 
Converting Between Degrees and Radians 
6:24  
 
 Formulas for Degrees and Radians 
6:52  
 
Coterminal, Complementary, Supplementary Angles 
7:23  
 
 Coterminal Angles 
7:30  
 
 Complementary Angles 
9:40  
 
 Supplementary Angles 
10:08  
 
Example 1: Dividing a Circle 
10:38  
 
Example 2: Converting Between Degrees and Radians 
11:56  
 
Example 3: Quadrants and Coterminal Angles 
14:18  
 
Extra Example 1: Common Angle Conversions 
8:02  
 
Extra Example 2: Quadrants and Coterminal Angles 
7:14  

Sine and Cosine Functions 
43:16 
 
Intro 
0:00  
 
Sine and Cosine 
0:15  
 
 Unit Circle 
0:22  
 
 Coordinates on Unit Circle 
1:03  
 
 Right Triangles 
1:52  
 
 Adjacent, Opposite, Hypotenuse 
2:25  
 
 Master Right Triangle Formula: SOHCAHTOA 
2:48  
 
Odd Functions, Even Functions 
4:40  
 
 Example: Odd Function 
4:56  
 
 Example: Even Function 
7:30  
 
Example 1: Sine and Cosine 
10:27  
 
Example 2: Graphing Sine and Cosine Functions 
14:39  
 
Example 3: Right Triangle 
21:40  
 
Example 4: Odd, Even, or Neither 
26:01  
 
Extra Example 1: Right Triangle 
4:05  
 
Extra Example 2: Graphing Sine and Cosine Functions 
5:23  

Sine and Cosine Values of Special Angles 
33:05 
 
Intro 
0:00  
 
454590 Triangle and 306090 Triangle 
0:08  
 
 454590 Triangle 
0:21  
 
 306090 Triangle 
2:06  
 
Mnemonic: All Students Take Calculus (ASTC) 
5:21  
 
 Using the Unit Circle 
5:59  
 
 New Angles 
6:21  
 
 Other Quadrants 
9:43  
 
 Mnemonic: All Students Take Calculus 
10:13  
 
Example 1: Convert, Quadrant, Sine/Cosine 
13:11  
 
Example 2: Convert, Quadrant, Sine/Cosine 
16:48  
 
Example 3: All Angles and Quadrants 
20:21  
 
Extra Example 1: Convert, Quadrant, Sine/Cosine 
4:15  
 
Extra Example 2: All Angles and Quadrants 
4:03  

Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D 
52:03 
 
Intro 
0:00  
 
Amplitude and Period of a Sine Wave 
0:38  
 
 Sine Wave Graph 
0:58  
 
 Amplitude: Distance from Middle to Peak 
1:18  
 
 Peak: Distance from Peak to Peak 
2:41  
 
Phase Shift and Vertical Shift 
4:13  
 
 Phase Shift: Distance Shifted Horizontally 
4:16  
 
 Vertical Shift: Distance Shifted Vertically 
6:48  
 
Example 1: Amplitude/Period/Phase and Vertical Shift 
8:04  
 
Example 2: Amplitude/Period/Phase and Vertical Shift 
17:39  
 
Example 3: Find Sine Wave Given Attributes 
25:23  
 
Extra Example 1: Amplitude/Period/Phase and Vertical Shift 
7:27  
 
Extra Example 2: Find Cosine Wave Given Attributes 
10:27  

Tangent and Cotangent Functions 
36:04 
 
Intro 
0:00  
 
Tangent and Cotangent Definitions 
0:21  
 
 Tangent Definition 
0:25  
 
 Cotangent Definition 
0:47  
 
Master Formula: SOHCAHTOA 
1:01  
 
 Mnemonic 
1:16  
 
Tangent and Cotangent Values 
2:29  
 
 Remember Common Values of Sine and Cosine 
2:46  
 
 90 Degrees Undefined 
4:36  
 
Slope and Menmonic: ASTC 
5:47  
 
 Uses of Tangent 
5:54  
 
 Example: Tangent of Angle is Slope 
6:09  
 
 Sign of Tangent in Quadrants 
7:49  
 
Example 1: Graph Tangent and Cotangent Functions 
10:42  
 
Example 2: Tangent and Cotangent of Angles 
16:09  
 
Example 3: Odd, Even, or Neither 
18:56  
 
Extra Example 1: Tangent and Cotangent of Angles 
2:27  
 
Extra Example 2: Tangent and Cotangent of Angles 
5:02  

Secant and Cosecant Functions 
27:18 
 
Intro 
0:00  
 
Secant and Cosecant Definitions 
0:17  
 
 Secant Definition 
0:18  
 
 Cosecant Definition 
0:33  
 
Example 1: Graph Secant Function 
0:48  
 
Example 2: Values of Secant and Cosecant 
6:49  
 
Example 3: Odd, Even, or Neither 
12:49  
 
Extra Example 1: Graph of Cosecant Function 
4:58  
 
Extra Example 2: Values of Secant and Cosecant 
5:19  

Inverse Trigonometric Functions 
32:58 
 
Intro 
0:00  
 
Arcsine Function 
0:24  
 
 Restrictions between 1 and 1 
0:43  
 
 Arcsine Notation 
1:26  
 
Arccosine Function 
3:07  
 
 Restrictions between 1 and 1 
3:36  
 
 Cosine Notation 
3:53  
 
Arctangent Function 
4:30  
 
 Between Pi/2 and Pi/2 
4:44  
 
 Tangent Notation 
5:02  
 
Example 1: Domain/Range/Graph of Arcsine 
5:45  
 
Example 2: Arcsin/Arccos/Arctan Values 
10:46  
 
Example 3: Domain/Range/Graph of Arctangent 
17:14  
 
Extra Example 1: Domain/Range/Graph of Arccosine 
4:30  
 
Extra Example 2: Arcsin/Arccos/Arctan Values 
5:40  

Computations of Inverse Trigonometric Functions 
31:08 
 
Intro 
0:00  
 
Inverse Trigonometric Function Domains and Ranges 
0:31  
 
 Arcsine 
0:41  
 
 Arccosine 
1:14  
 
 Arctangent 
1:41  
 
Example 1: Arcsines of Common Values 
2:44  
 
Example 2: Odd, Even, or Neither 
5:57  
 
Example 3: Arccosines of Common Values 
12:24  
 
Extra Example 1: Arctangents of Common Values 
5:50  
 
Extra Example 2: Arcsin/Arccos/Arctan Values 
8:51  
VII. Trigonometric Identities 

Pythagorean Identity 
19:11 
 
Intro 
0:00  
 
Pythagorean Identity 
0:17  
 
 Pythagorean Triangle 
0:27  
 
 Pythagorean Identity 
0:45  
 
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity 
1:14  
 
Example 2: Find Angle Given Cosine and Quadrant 
4:18  
 
Example 3: Verify Trigonometric Identity 
8:00  
 
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem 
3:32  
 
Extra Example 2: Find Angle Given Cosine and Quadrant 
3:55  

Identity Tan(squared)x+1=Sec(squared)x 
23:16 
 
Intro 
0:00  
 
Main Formulas 
0:19  
 
 Companion to Pythagorean Identity 
0:27  
 
 For Cotangents and Cosecants 
0:52  
 
 How to Remember 
0:58  
 
Example 1: Prove the Identity 
1:40  
 
Example 2: Given Tan Find Sec 
3:42  
 
Example 3: Prove the Identity 
7:45  
 
Extra Example 1: Prove the Identity 
2:22  
 
Extra Example 2: Given Sec Find Tan 
4:34  

Addition and Subtraction Formulas 
52:52 
 
Intro 
0:00  
 
Addition and Subtraction Formulas 
0:09  
 
 How to Remember 
0:48  
 
Cofunction Identities 
1:31  
 
 How to Remember Graphically 
1:44  
 
 Where to Use Cofunction Identities 
2:52  
 
Example 1: Derive the Formula for cos(AB) 
3:08  
 
Example 2: Use Addition and Subtraction Formulas 
16:03  
 
Example 3: Use Addition and Subtraction Formulas to Prove Identity 
25:11  
 
Extra Example 1: Use cos(AB) and Cofunction Identities 
7:54  
 
Extra Example 2: Convert to Radians and use Formulas 
11:32  

Double Angle Formulas 
29:05 
 
Intro 
0:00  
 
Main Formula 
0:07  
 
 How to Remember from Addition Formula 
0:18  
 
 Two Other Forms 
1:35  
 
Example 1: Find Sine and Cosine of Angle using Double Angle 
3:16  
 
Example 2: Prove Trigonometric Identity using Double Angle 
9:37  
 
Example 3: Use Addition and Subtraction Formulas 
12:38  
 
Extra Example 1: Find Sine and Cosine of Angle using Double Angle 
6:10  
 
Extra Example 2: Prove Trigonometric Identity using Double Angle 
3:18  

HalfAngle Formulas 
43:55 
 
Intro 
0:00  
 
Main Formulas 
0:09  
 
 Confusing Part 
0:34  
 
Example 1: Find Sine and Cosine of Angle using HalfAngle 
0:54  
 
Example 2: Prove Trigonometric Identity using HalfAngle 
11:51  
 
Example 3: Prove the HalfAngle Formula for Tangents 
18:39  
 
Extra Example 1: Find Sine and Cosine of Angle using HalfAngle 
7:16  
 
Extra Example 2: Prove Trigonometric Identity using HalfAngle 
3:34  
VIII. Applications of Trigonometry 

Trigonometry in Right Angles 
25:43 
 
Intro 
0:00  
 
Master Formula for Right Angles 
0:11  
 
 SOHCAHTOA 
0:15  
 
 Only for Right Triangles 
1:26  
 
Example 1: Find All Angles in a Triangle 
2:19  
 
Example 2: Find Lengths of All Sides of Triangle 
7:39  
 
Example 3: Find All Angles in a Triangle 
11:00  
 
Extra Example 1: Find All Angles in a Triangle 
5:10  
 
Extra Example 2: Find Lengths of All Sides of Triangle 
4:18  

Law of Sines 
56:40 
 
Intro 
0:00  
 
Law of Sines Formula 
0:18  
 
 SOHCAHTOA 
0:27  
 
 Any Triangle 
0:59  
 
 Graphical Representation 
1:25  
 
 Solving Triangle Completely 
2:37  
 
When to Use Law of Sines 
2:55  
 
 ASA, SAA, SSA, AAA 
2:59  
 
 SAS, SSS for Law of Cosines 
7:11  
 
Example 1: How Many Triangles Satisfy Conditions, Solve Completely 
8:44  
 
Example 2: How Many Triangles Satisfy Conditions, Solve Completely 
15:30  
 
Example 3: How Many Triangles Satisfy Conditions, Solve Completely 
28:32  
 
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely 
8:01  
 
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely 
15:11  

Law of Cosines 
49:05 
 
Intro 
0:00  
 
Law of Cosines Formula 
0:23  
 
 Graphical Representation 
0:34  
 
 Relates Sides to Angles 
1:00  
 
 Any Triangle 
1:20  
 
 Generalization of Pythagorean Theorem 
1:32  
 
When to Use Law of Cosines 
2:26  
 
 SAS, SSS 
2:30  
 
Heron's Formula 
4:49  
 
 Semiperimeter S 
5:11  
 
Example 1: How Many Triangles Satisfy Conditions, Solve Completely 
5:53  
 
Example 2: How Many Triangles Satisfy Conditions, Solve Completely 
15:19  
 
Example 3: Find Area of a Triangle Given All Side Lengths 
26:33  
 
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely 
11:05  
 
Extra Example 2: Length of Third Side and Area of Triangle 
9:17  

Finding the Area of a Triangle 
27:37 
 
Intro 
0:00  
 
Master Right Triangle Formula and Law of Cosines 
0:19  
 
 SOHCAHTOA 
0:27  
 
 Law of Cosines 
1:23  
 
Heron's Formula 
2:22  
 
 Semiperimeter S 
2:37  
 
Example 1: Area of Triangle with Two Sides and One Angle 
3:12  
 
Example 2: Area of Triangle with Three Sides 
6:11  
 
Example 3: Area of Triangle with Three Sides, No Heron's Formula 
8:50  
 
Extra Example 1: Area of Triangle with Two Sides and One Angle 
2:54  
 
Extra Example 2: Area of Triangle with Two Sides and One Angle 
6:48  

Word Problems and Applications of Trigonometry 
34:25 
 
Intro 
0:00  
 
Formulas to Remember 
0:11  
 
 SOHCAHTOA 
0:15  
 
 Law of Sines 
0:55  
 
 Law of Cosines 
1:48  
 
 Heron's Formula 
2:46  
 
Example 1: Telephone Pole Height 
4:01  
 
Example 2: Bridge Length 
7:48  
 
Example 3: Area of Triangular Field 
14:20  
 
Extra Example 1: Kite Height 
4:36  
 
Extra Example 2: Roads to a Town 
10:34  
IX. Systems of Equations and Inequalities 

Systems of Linear Equations 
55:40 
 
Intro 
0:00  
 
Introduction 
0:04  
 
Graphs as Location of 'True' 
1:49  
 
 All Locations that Make the Function True 
2:25  
 
 Understand the Relationship Between Solutions and the Graph 
3:43  
 
Systems as Graphs 
4:07  
 
 Equations as Lines 
4:20  
 
 Intersection Point 
5:19  
 
Three Possibilities for Solutions 
6:17  
 
 Independent 
6:24  
 
 Inconsistent 
6:36  
 
 Dependent 
7:06  
 
Solving by Substitution 
8:37  
 
 Solve for One Variable 
8:67  
 
 Substitute into the Second Equation 
9:34  
 
 Solve for Both Variables 
10:12  
 
 What If a System is Inconsistent or Dependent? 
11:08  
 
 No Solutions 
11:25  
 
 Infinite Solutions 
12:30  
 
Solving by Elimination 
13:56  
 
 Example 
14:22  
 
 Determining the Number of Solutions 
16:30  
 
Why Elimination Makes Sense 
17:25  
 
Solving by Graphing Calculator 
19:59  
 
Systems with More than Two Variables 
23:22  
 
Example 1 
25:49  
 
Example 2 
30:22  
 
Example 3 
34:11  
 
Example 4 
38:55  
 
Example 5 
46:01  
 
(Non) Example 6 
53:37  

Systems of Linear Inequalities 
1:00:13 
 
Intro 
0:00  
 
Introduction 
0:04  
 
Inequality RefresherSolutions 
0:46  
 
 Equation Solutions vs. Inequality Solutions 
1:02  
 
 Essentially a Wide Variety of Answers 
1:35  
 
RefresherNegative Multiplication Flips 
1:43  
 
RefresherNegative Flips: Why? 
3:19  
 
 Multiplication by a Negative 
3:43  
 
 The Relationship Flips 
3:55  
 
RefresherStick to Basic Operations 
4:34  
 
Linear Equations in Two Variables 
6:50  
 
Graphing Linear Inequalities 
8:28  
 
 Why It Includes a Whole Section 
8:43  
 
 How to Show The Difference Between Strict and Not Strict Inequalities 
10:08  
 
 Dashed LineNot Solutions 
11:10  
 
 Solid LineAre Solutions 
11:24  
 
Test Points for Shading 
11:42  
 
 Example of Using a Point 
12:41  
 
 Drawing Shading from the Point 
13:14  
 
Graphing a System 
14:53  
 
 Set of Solutions is the Overlap 
15:17  
 
 Example 
15:22  
 
Solutions are Best Found Through Graphing 
18:05  
 
Linear ProgrammingIdea 
19:52  
 
 Use a Linear Objective Function 
20:15  
 
 Variables in Objective Function have Constraints 
21:24  
 
Linear ProgrammingMethod 
22:09  
 
 Rearrange Equations 
22:21  
 
 Graph 
22:49  
 
 Critical Solution is at the Vertex of the Overlap 
23:40  
 
 Try Each Vertice 
24:35  
 
Example 1 
24:58  
 
Example 2 
28:57  
 
Example 3 
33:48  
 
Example 4 
43:10  

Nonlinear Systems 
41:01 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Substitution 
1:12  
 
 Example 
1:22  
 
Elimination 
3:46  
 
 Example 
3:56  
 
 Elimination is Less Useful for Nonlinear Systems 
4:56  
 
Graphing 
5:56  
 
 Using a Graphing Calculator 
6:44  
 
Number of Solutions 
8:44  
 
Systems of Nonlinear Inequalities 
10:02  
 
 Graph Each Inequality 
10:06  
 
 Dashed and/or Solid 
10:18  
 
 Shade Appropriately 
11:14  
 
Example 1 
13:24  
 
Example 2 
15:50  
 
Example 3 
22:02  
 
Example 4 
29:06  
 
 Example 4, cont. 
33:40  
X. Vectors and Matrices 

Vectors 
1:09:31 
 
Intro 
0:00  
 
Introduction 
0:10  
 
 Magnitude of the Force 
0:22  
 
 Direction of the Force 
0:48  
 
 Vector 
0:52  
 
Idea of a Vector 
1:30  
 
 How Vectors are Denoted 
2:00  
 
Component Form 
3:20  
 
 Angle Brackets and Parentheses 
3:50  
 
Magnitude/Length 
4:26  
 
 Denoting the Magnitude of a Vector 
5:16  
 
Direction/Angle 
7:52  
 
 Always Draw a Picture 
8:50  
 
Component Form from Magnitude & Angle 
10:10  
 
Scaling by Scalars 
14:06  
 
Unit Vectors 
16:26  
 
Combining Vectors  Algebraically 
18:10  
 
Combining Vectors  Geometrically 
19:54  
 
 Resultant Vector 
20:46  
 
Alternate Component Form: i, j 
21:16  
 
The Zero Vector 
23:18  
 
Properties of Vectors 
24:20  
 
No Multiplication (Between Vectors) 
28:30  
 
 Dot Product 
29:40  
 
Motion in a Medium 
30:10  
 
 Fish in an Aquarium Example 
31:38  
 
More Than Two Dimensions 
33:12  
 
More Than Two Dimensions  Magnitude 
34:18  
 
Example 1 
35:26  
 
Example 2 
38:10  
 
Example 3 
45:48  
 
Example 4 
50:40  
 
 Example 4, cont. 
56:07  
 
Example 5 
61:32  

Dot Product & Cross Product 
35:20 
 
Intro 
0:00  
 
Introduction 
0:08  
 
Dot Product  Definition 
0:42  
 
 Dot Product Results in a Scalar, Not a Vector 
2:10  
 
 Example in Two Dimensions 
2:34  
 
Angle and the Dot Product 
2:58  
 
 The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors 
2:59  
 
Proof of Dot Product Formula 
4:14  
 
 Won't Directly Help Us Better Understand Vectors 
4:18  
 
Dot Product  Geometric Interpretation 
4:58  
 
 We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are 
7:26  
 
Dot Product  Perpendicular Vectors 
8:24  
 
 If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other 
8:54  
 
Cross Product  Definition 
11:08  
 
 Cross Product Only Works in Three Dimensions 
11:09  
 
Cross Product  A Mnemonic 
12:16  
 
 The Determinant of a 3 x 3 Matrix and Standard Unit Vectors 
12:17  
 
Cross Product  Geometric Interpretations 
14:30  
 
 The RightHand Rule 
15:17  
 
 Cross Product  Geometric Interpretations Cont. 
17:00  
 
Example 1 
18:40  
 
Example 2 
22:50  
 
Example 3 
24:04  
 
Example 4 
26:20  
 
Bonus Round 
29:18  
 
Proof: Dot Product Formula 
29:24  
 
 Proof: Dot Product Formula, cont. 
30:38  

Matrices 
54:07 
 
Intro 
0:00  
 
Introduction 
0:08  
 
Definition of a Matrix 
3:02  
 
 Size or Dimension 
3:58  
 
 Square Matrix 
4:42  
 
 Denoted by Capital Letters 
4:56  
 
 When are Two Matrices Equal? 
5:04  
 
Examples of Matrices 
6:44  
 
 Rows x Columns 
6:46  
 
Talking About Specific Entries 
7:48  
 
 We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries 
8:32  
 
Using Entries to Talk About Matrices 
10:08  
 
Scalar Multiplication 
11:26  
 
 Scalar = Real Number 
11:34  
 
 Example 
12:36  
 
Matrix Addition 
13:08  
 
 Example 
14:22  
 
Matrix Multiplication 
15:00  
 
 Example 
18:52  
 
 Matrix Multiplication, cont. 
19:58  
 
Matrix Multiplication and Order (Size) 
25:26  
 
 Make Sure Their Orders are Compatible 
25:27  
 
Matrix Multiplication is NOT Commutative 
28:20  
 
 Example 
30:08  
 
Special Matrices  Zero Matrix (0) 
32:48  
 
 Zero Matrix Has 0 for All of its Entries 
32:49  
 
Special Matrices  Identity Matrix (I) 
34:14  
 
 Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries 
34:15  
 
Example 1 
36:16  
 
Example 2 
40:00  
 
Example 3 
44:54  
 
Example 4 
50:08  

Determinants & Inverses of Matrices 
47:12 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Not All Matrices Are Invertible 
1:30  
 
 What Must a Matrix Have to Be Invertible? 
2:08  
 
Determinant 
2:32  
 
 The Determinant is a Real Number Associated With a Square Matrix 
2:38  
 
 If the Determinant of a Matrix is Nonzero, the Matrix is Invertible 
3:40  
 
Determinant of a 2 x 2 Matrix 
4:34  
 
 Think in Terms of Diagonals 
5:12  
 
Minors and Cofactors  Minors 
6:24  
 
 Example 
6:46  
 
Minors and Cofactors  Cofactors 
8:00  
 
 Cofactor is Closely Based on the Minor 
8:01  
 
 Alternating Sign Pattern 
9:04  
 
Determinant of Larger Matrices 
10:56  
 
 Example 
13:00  
 
Alternative Method for 3x3 Matrices 
16:46  
 
 Not Recommended 
16:48  
 
Inverse of a 2 x 2 Matrix 
19:02  
 
Inverse of Larger Matrices 
20:00  
 
Using Inverse Matrices 
21:06  
 
 When Multiplied Together, They Create the Identity Matrix 
21:24  
 
Example 1 
23:45  
 
Example 2 
27:21  
 
Example 3 
32:49  
 
Example 4 
36:27  
 
Finding the Inverse of Larger Matrices 
41:59  
 
General Inverse Method  Step 1 
43:25  
 
General Inverse Method  Step 2 
43:27  
 
 General Inverse Method  Step 2, cont. 
43:27  
 
General Inverse Method  Step 3 
45:15  

Using Matrices to Solve Systems of Linear Equations 
58:34 
 
Intro 
0:00  
 
Introduction 
0:12  
 
Augmented Matrix 
1:44  
 
 We Can Represent the Entire Linear System With an Augmented Matrix 
1:50  
 
Row Operations 
3:22  
 
 Interchange the Locations of Two Rows 
3:50  
 
 Multiply (or Divide) a Row by a Nonzero Number 
3:58  
 
 Add (or Subtract) a Multiple of One Row to Another 
4:12  
 
Row Operations  Keep Notes! 
5:50  
 
 Suggested Symbols 
7:08  
 
GaussJordan Elimination  Idea 
8:04  
 
 GaussJordan Elimination  Idea, cont. 
9:16  
 
 Reduced RowEchelon Form 
9:18  
 
GaussJordan Elimination  Method 
11:36  
 
 Begin by Writing the System As An Augmented Matrix 
11:38  
 
 GaussJordan Elimination  Method, cont. 
13:48  
 
Cramer's Rule  2 x 2 Matrices 
17:08  
 
Cramer's Rule  n x n Matrices 
19:24  
 
Solving with Inverse Matrices 
21:10  
 
 Solving Inverse Matrices, cont. 
25:28  
 
The Mighty (Graphing) Calculator 
26:38  
 
Example 1 
29:56  
 
Example 2 
33:56  
 
Example 3 
37:00  
 
 Example 3, cont. 
45:04  
 
Example 4 
51:28  
XI. Alternate Ways to Graph 

Parametric Equations 
53:33 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Definition 
1:10  
 
 Plane Curve 
1:24  
 
 The Key Idea 
2:00  
 
Graphing with Parametric Equations 
2:52  
 
Same Graph, Different Equations 
5:04  
 
 How Is That Possible? 
5:36  
 
 Same Graph, Different Equations, cont. 
5:42  
 
 Here's Another to Consider 
7:56  
 
 Same Plane Curve, But Still Different 
8:10  
 
A Metaphor for Parametric Equations 
9:36  
 
 Think of Parametric Equations As a Way to Describe the Motion of An Object 
9:38  
 
 Graph Shows Where It Went, But Not Speed 
10:32  
 
Eliminating Parameters 
12:14  
 
 Rectangular Equation 
12:16  
 
 Caution 
13:52  
 
Creating Parametric Equations 
14:30  
 
Interesting Graphs 
16:38  
 
Graphing Calculators, Yay! 
19:18  
 
Example 1 
22:36  
 
Example 2 
28:26  
 
Example 3 
37:36  
 
Example 4 
41:00  
 
Projectile Motion 
44:26  
 
Example 5 
47:00  

Polar Coordinates 
48:07 
 
Intro 
0:00  
 
Introduction 
0:04  
 
 Polar Coordinates Give Us a Way To Describe the Location of a Point 
0:26  
 
 Polar Equations and Functions 
0:50  
 
Plotting Points with Polar Coordinates 
1:06  
 
 The Distance of the Point from the Origin 
1:09  
 
 The Angle of the Point 
1:33  
 
 Give Points as the Ordered Pair (r,θ) 
2:03  
 
Visualizing Plotting in Polar Coordinates 
2:32  
 
 First Way We Can Plot 
2:39  
 
 Second Way We Can Plot 
2:50  
 
 First, We'll Look at Visualizing r, Then θ 
3:09  
 
 Rotate the Length CounterClockwise by θ 
3:38  
 
 Alternatively, We Can Visualize θ, Then r 
4:06  
 
'Polar Graph Paper' 
6:17  
 
 Horizontal and Vertical Tick Marks Are Not Useful for Polar 
6:42  
 
 Use Concentric Circles to Helps Up See Distance From the Pole 
7:08  
 
 Can Use Arc Sectors to See Angles 
7:57  
 
Multiple Ways to Name a Point 
9:17  
 
 Examples 
9:30  
 
 For Any Angle θ, We Can Make an Equivalent Angle 
10:44  
 
Negative Values for r 
11:58  
 
 If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out 
12:22  
 
 Another Way to Name the Same Point: Add π to θ and Make r Negative 
13:44  
 
Converting Between Rectangular and Polar 
14:37  
 
 Rectangular Way to Name 
14:43  
 
 Polar Way to Name 
14:52  
 
 The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle 
15:08  
 
 Connect Both Systems Through Basic Trigonometry 
15:38  
 
 Equation to Convert From Polar to Rectangular Coordinate Systems 
16:55  
 
 Equation to Convert From Rectangular to Polar Coordinate Systems 
17:13  
 
 Converting to Rectangular is Easy 
17:20  
 
 Converting to Polar is a Bit Trickier 
17:21  
 
Draw Pictures 
18:55  
 
Example 1 
19:50  
 
Example 2 
25:17  
 
Example 3 
31:05  
 
Example 4 
35:56  
 
Example 5 
41:49  

Polar Equations & Functions 
38:16 
 
Intro 
0:00  
 
Introduction 
0:04  
 
Equations and Functions 
1:16  
 
 Independent Variable 
1:21  
 
 Dependent Variable 
1:30  
 
 Examples 
1:46  
 
 Always Assume That θ Is In Radians 
2:44  
 
Graphing in Polar Coordinates 
3:29  
 
 Graph is the Same Way We Graph 'Normal' Stuff 
3:32  
 
 Example 
3:52  
 
Graphing in Polar  Example, Cont. 
6:45  
 
Tips for Graphing 
9:23  
 
 Notice Patterns 
10:19  
 
 Repetition 
13:39  
 
Graphing Equations of One Variable 
14:39  
 
Converting Coordinate Types 
16:16  
 
 Use the Same Conversion Formulas From the Previous Lesson 
16:23  
 
Interesting Graphs 
17:48  
 
 Example 1 
18:03  
 
 Example 2 
18:34  
 
Graphing Calculators, Yay! 
19:07  
 
 Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works 
19:11  
 
 Check Out the Appendix 
19:26  
 
Example 1 
21:36  
 
Example 2 
28:13  
 
Example 3 
34:24  
 
Example 4 
35:52  
XII. Complex Numbers and Polar Coordinates 

Polar Form of Complex Numbers 
40:43 
 
Intro 
0:00  
 
Polar Coordinates 
0:49  
 
 Rectangular Form 
0:52  
 
 Polar Form 
1:25  
 
 R and Theta 
1:51  
 
Polar Form Conversion 
2:27  
 
 R and Theta 
2:35  
 
 Optimal Values 
4:05  
 
 Euler's Formula 
4:25  
 
Multiplying Two Complex Numbers in Polar Form 
6:10  
 
 Multiply r's Together and Add Exponents 
6:32  
 
Example 1: Convert Rectangular to Polar Form 
7:17  
 
Example 2: Convert Polar to Rectangular Form 
13:49  
 
Example 3: Multiply Two Complex Numbers 
17:28  
 
Extra Example 1: Convert Between Rectangular and Polar Forms 
6:48  
 
Extra Example 2: Simplify Expression to Polar Form 
7:48  

DeMoivre's Theorem 
57:37 
 
Intro 
0:00  
 
Introduction to DeMoivre's Theorem 
0:10  
 
 n nth Roots 
3:06  
 
DeMoivre's Theorem: Finding nth Roots 
3:52  
 
 Relation to Unit Circle 
6:29  
 
 One nth Root for Each Value of k 
7:11  
 
Example 1: Convert to Polar Form and Use DeMoivre's Theorem 
8:24  
 
Example 2: Find Complex Eighth Roots 
15:27  
 
Example 3: Find Complex Roots 
27:49  
 
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem 
7:41  
 
Extra Example 2: Find Complex Fourth Roots 
14:36  
XIII. Counting & Probability 

Counting 
31:36 
 
Intro 
0:00  
 
Introduction 
0:08  
 
 Combinatorics 
0:56  
 
Definition: Event 
1:24  
 
 Example 
1:50  
 
Visualizing an Event 
3:02  
 
 Branching line diagram 
3:06  
 
Addition Principle 
3:40  
 
 Example 
4:18  
 
Multiplication Principle 
5:42  
 
 Example 
6:24  
 
Pigeonhole Principle 
8:06  
 
 Example 
10:26  
 
Draw Pictures 
11:06  
 
Example 1 
12:02  
 
Example 2 
14:16  
 
Example 3 
17:34  
 
Example 4 
21:26  
 
Example 5 
25:14  

Permutations & Combinations 
44:03 
 
Intro 
0:00  
 
Introduction 
0:08  
 
 Permutation 
0:42  
 
 Combination 
1:10  
 
Towards a Permutation Formula 
2:38  
 
 How Many Ways Can We Arrange the Letters A, B, C, D, and E? 
3:02  
 
 Towards a Permutation Formula, cont. 
3:34  
 
Factorial Notation 
6:56  
 
 Symbol Is '!' 
6:58  
 
 Examples 
7:32  
 
Permutation of n Objects 
8:44  
 
Permutation of r Objects out of n 
9:04  
 
 What If We Have More Objects Than We Have Slots to Fit Them Into? 
9:46  
 
 Permutation of r Objects Out of n, cont. 
10:28  
 
Distinguishable Permutations 
14:46  
 
 What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other? 
14:48  
 
 Distinguishable Permutations, cont. 
17:04  
 
Combinations 
19:04  
 
 Combinations, cont. 
20:56  
 
Example 1 
23:10  
 
Example 2 
26:16  
 
Example 3 
28:28  
 
Example 4 
31:52  
 
Example 5 
33:58  
 
Example 6 
36:34  

Probability 
36:58 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Definition: Sample Space 
1:18  
 
 Event = Something Happening 
1:20  
 
 Sample Space 
1:36  
 
Probability of an Event 
2:12  
 
 Let E Be An Event and S Be The Corresponding Sample Space 
2:14  
 
'Equally Likely' Is Important 
3:52  
 
 Fair and Random 
5:26  
 
Interpreting Probability 
6:34  
 
 How Can We Interpret This Value? 
7:24  
 
 We Can Represent Probability As a Fraction, a Decimal, Or a Percentage 
8:04  
 
One of Multiple Events Occurring 
9:52  
 
 Mutually Exclusive Events 
10:38  
 
 What If The Events Are Not Mutually Exclusive? 
12:20  
 
 Taking the Possibility of Overlap Into Account 
13:24  
 
An Event Not Occurring 
17:14  
 
 Complement of E 
17:22  
 
Independent Events 
19:36  
 
 Independent 
19:48  
 
Conditional Events 
21:28  
 
 What Is The Events Are Not Independent Though? 
21:30  
 
 Conditional Probability 
22:16  
 
 Conditional Events, cont. 
23:51  
 
Example 1 
25:27  
 
Example 2 
27:09  
 
Example 3 
28:57  
 
Example 4 
30:51  
 
Example 5 
34:15  
XIV. Conic Sections 

Parabolas 
41:27 
 
Intro 
0:00  
 
What is a Parabola? 
0:20  
 
 Definition of a Parabola 
0:29  
 
 Focus 
0:59  
 
 Directrix 
1:15  
 
 Axis of Symmetry 
3:08  
 
Vertex 
3:33  
 
 Minimum or Maximum 
3:44  
 
Standard Form 
4:59  
 
 Horizontal Parabolas 
5:08  
 
 Vertex Form 
5:19  
 
 Upward or Downward 
5:41  
 
 Example: Standard Form 
6:06  
 
Graphing Parabolas 
8:31  
 
 Shifting 
8:51  
 
 Example: Completing the Square 
9:22  
 
 Symmetry and Translation 
12:18  
 
 Example: Graph Parabola 
12:40  
 
Latus Rectum 
17:13  
 
 Length 
18:15  
 
 Example: Latus Rectum 
18:35  
 
Horizontal Parabolas 
18:57  
 
 Not Functions 
20:08  
 
 Example: Horizontal Parabola 
21:21  
 
Focus and Directrix 
24:11  
 
 Horizontal 
24:48  
 
Example 1: Parabola Standard Form 
25:12  
 
Example 2: Graph Parabola 
30:00  
 
Example 3: Graph Parabola 
33:13  
 
Example 4: Parabola Equation 
37:28  

Circles 
21:03 
 
Intro 
0:00  
 
What are Circles? 
0:08  
 
 Example: Equidistant 
0:17  
 
 Radius 
0:32  
 
Equation of a Circle 
0:44  
 
 Example: Standard Form 
1:11  
 
Graphing Circles 
1:47  
 
 Example: Circle 
1:56  
 
Center Not at Origin 
3:07  
 
 Example: Completing the Square 
3:51  
 
Example 1: Equation of Circle 
6:44  
 
Example 2: Center and Radius 
11:51  
 
Example 3: Radius 
15:08  
 
Example 4: Equation of Circle 
16:57  

Ellipses 
46:51 
 
Intro 
0:00  
 
What Are Ellipses? 
0:11  
 
 Foci 
0:23  
 
Properties of Ellipses 
1:43  
 
 Major Axis, Minor Axis 
1:47  
 
 Center 
1:54  
 
 Length of Major Axis and Minor Axis 
3:21  
 
Standard Form 
5:33  
 
 Example: Standard Form of Ellipse 
6:09  
 
Vertical Major Axis 
9:14  
 
 Example: Vertical Major Axis 
9:46  
 
Graphing Ellipses 
12:51  
 
 Complete the Square and Symmetry 
13:00  
 
 Example: Graphing Ellipse 
13:16  
 
Equation with Center at (h, k) 
19:57  
 
 Horizontal and Vertical 
20:14  
 
 Difference 
20:27  
 
 Example: Center at (h, k) 
20:55  
 
Example 1: Equation of Ellipse 
24:05  
 
Example 2: Equation of Ellipse 
27:57  
 
Example 3: Equation of Ellipse 
32:32  
 
Example 4: Graph Ellipse 
38:27  

Hyperbolas 
38:15 
 
Intro 
0:00  
 
What are Hyperbolas? 
0:12  
 
 Two Branches 
0:18  
 
 Foci 
0:38  
 
Properties 
2:00  
 
 Transverse Axis and Conjugate Axis 
2:06  
 
 Vertices 
2:46  
 
 Length of Transverse Axis 
3:14  
 
 Distance Between Foci 
3:31  
 
 Length of Conjugate Axis 
3:38  
 
Standard Form 
5:45  
 
 Vertex Location 
6:36  
 
 Known Points 
6:52  
 
Vertical Transverse Axis 
7:26  
 
 Vertex Location 
7:50  
 
Asymptotes 
8:36  
 
 Vertex Location 
8:56  
 
 Rectangle 
9:28  
 
 Diagonals 
10:29  
 
Graphing Hyperbolas 
12:58  
 
 Example: Hyperbola 
13:16  
 
Equation with Center at (h, k) 
16:32  
 
 Example: Center at (h, k) 
17:21  
 
Example 1: Equation of Hyperbola 
19:20  
 
Example 2: Equation of Hyperbola 
22:48  
 
Example 3: Graph Hyperbola 
26:05  
 
Example 4: Equation of Hyperbola 
36:29  

Conic Sections 
18:43 
 
Intro 
0:00  
 
Conic Sections 
0:16  
 
 Double Cone Sections 
0:24  
 
Standard Form 
1:27  
 
 General Form 
1:37  
 
Identify Conic Sections 
2:16  
 
 B = 0 
2:50  
 
 X and Y 
3:22  
 
Identify Conic Sections, Cont. 
4:46  
 
 Parabola 
5:17  
 
 Circle 
5:51  
 
 Ellipse 
6:31  
 
 Hyperbola 
7:10  
 
Example 1: Identify Conic Section 
8:01  
 
Example 2: Identify Conic Section 
11:03  
 
Example 3: Identify Conic Section 
11:38  
 
Example 4: Identify Conic Section 
14:50  
XV. Sequences, Series, & Induction 

Introduction to Sequences 
57:45 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Definition: Sequence 
0:28  
 
 Infinite Sequence 
2:08  
 
 Finite Sequence 
2:22  
 
 Length 
2:58  
 
Formula for the nth Term 
3:22  
 
Defining a Sequence Recursively 
5:54  
 
 Initial Term 
7:58  
 
Sequences and Patterns 
10:40  
 
 First, Identify a Pattern 
12:52  
 
 How to Get From One Term to the Next 
17:38  
 
Tips for Finding Patterns 
19:52  
 
 More Tips for Finding Patterns 
24:14  
 
 Even More Tips 
26:50  
 
Example 1 
30:32  
 
Example 2 
34:54  
 
 Fibonacci Sequence 
34:55  
 
Example 3 
38:40  
 
Example 4 
45:02  
 
Example 5 
49:26  
 
Example 6 
51:54  

Introduction to Series 
40:27 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Definition: Series 
1:20  
 
Why We Need Notation 
2:48  
 
Simga Notation (AKA Summation Notation) 
4:44  
 
 Thing Being Summed 
5:42  
 
 Index of Summation 
6:21  
 
 Lower Limit of Summation 
7:09  
 
 Upper Limit of Summation 
7:23  
 
Sigma Notation, Example 
7:36  
 
Sigma Notation for Infinite Series 
9:08  
 
How to Reindex 
10:58  
 
 How to Reindex, Expanding 
12:56  
 
 How to Reindex, Substitution 
16:46  
 
Properties of Sums 
19:42  
 
Example 1 
23:46  
 
Example 2 
25:34  
 
Example 3 
27:12  
 
Example 4 
29:54  
 
Example 5 
32:06  
 
Example 6 
37:16  

Arithmetic Sequences & Series 
31:36 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Definition: Arithmetic Sequence 
0:47  
 
 Common Difference 
1:13  
 
 Two Examples 
1:19  
 
Form for the nth Term 
2:14  
 
 Recursive Relation 
2:33  
 
Towards an Arithmetic Series Formula 
5:12  
 
Creating a General Formula 
10:09  
 
General Formula for Arithmetic Series 
14:23  
 
Example 1 
15:46  
 
Example 2 
17:37  
 
Example 3 
22:21  
 
Example 4 
24:09  
 
Example 5 
27:14  

Geometric Sequences & Series 
39:27 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Definition 
0:48  
 
Form for the nth Term 
2:42  
 
Formula for Geometric Series 
5:16  
 
Infinite Geometric Series 
11:48  
 
 Diverges 
13:04  
 
 Converges 
14:48  
 
Formula for Infinite Geometric Series 
16:32  
 
Example 1 
20:32  
 
Example 2 
22:02  
 
Example 3 
26:00  
 
Example 4 
30:48  
 
Example 5 
34:28  

Mathematical Induction 
49:53 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Belief Vs. Proof 
1:22  
 
A Metaphor for Induction 
6:14  
 
The Principle of Mathematical Induction 
11:38  
 
 Base Case 
13:24  
 
 Inductive Step 
13:30  
 
 Inductive Hypothesis 
13:52  
 
A Remark on Statements 
14:18  
 
Using Mathematical Induction 
16:58  
 
Working Example 
19:58  
 
 Finding Patterns 
28:46  
 
Example 1 
30:17  
 
Example 2 
37:50  
 
Example 3 
42:38  

The Binomial Theorem 
1:13:13 
 
Intro 
0:00  
 
Introduction 
0:06  
 
 We've Learned That a Binomial Is An Expression That Has Two Terms 
0:07  
 
Understanding Binomial Coefficients 
1:20  
 
 Things We Notice 
2:24  
 
 What Goes In the Blanks? 
5:52  
 
 Each Blank is Called a Binomial Coefficient 
6:18  
 
The Binomial Theorem 
6:38  
 
 Example 
8:10  
 
 The Binomial Theorem, cont. 
10:46  
 
 We Can Also Write This Expression Compactly Using Sigma Notation 
12:06  
 
Proof of the Binomial Theorem 
13:22  
 
 Proving the Binomial Theorem Is Within Our Reach 
13:24  
 
Pascal's Triangle 
15:12  
 
 Pascal's Triangle, cont. 
16:12  
 
 Diagonal Addition of Terms 
16:24  
 
 Zeroth Row 
18:04  
 
 First Row 
18:12  
 
 Why Do We Care About Pascal's Triangle? 
18:50  
 
 Pascal's Triangle, Example 
19:26  
 
Example 1 
21:26  
 
Example 2 
24:34  
 
Example 3 
28:34  
 
Example 4 
32:28  
 
Example 5 
37:12  
 
Time for the Fireworks! 
43:38  
 
Proof of the Binomial Theorem 
43:44  
 
 We'll Prove This By Induction 
44:04  
 
 Proof (By Induction) 
46:36  
 
Proof, Base Case 
47:00  
 
Proof, Inductive Step  Notation Discussion 
49:22  
 
 Induction Step 
49:24  
 
Proof, Inductive Step  Setting Up 
52:26  
 
 Induction Hypothesis 
52:34  
 
 What We What To Show 
52:44  
 
Proof, Inductive Step  Start 
54:18  
 
Proof, Inductive Step  Middle 
55:38  
 
 Expand Sigma Notations 
55:48  
 
 Proof, Inductive Step  Middle, cont. 
58:40  
 
Proof, Inductive Step  Checking In 
1:01:08  
 
 Let's Check In With Our Original Goal 
1:01:12  
 
 Want to Show 
1:01:18  
 
 Lemma  A Mini Theorem 
1:02:18  
 
Proof, Inductive Step  Lemma 
1:02:52  
 
 Proof of Lemma: Let's Investigate the Left Side 
1:03:08  
 
Proof, Inductive Step  Nearly There 
1:07:54  
 
Proof, Inductive Step  End! 
1:09:18  
 
 Proof, Inductive Step  End!, cont. 
1:11:01  
XVI. Preview of Calculus 

Idea of a Limit 
40:22 
 
Intro 
0:00  
 
Introduction 
0:05  
 
Motivating Example 
1:26  
 
Fuzzy Notion of a Limit 
3:38  
 
 Limit is the Vertical Location a Function is Headed Towards 
3:44  
 
 Limit is What the Function Output is Going to Be 
4:15  
 
 Limit Notation 
4:33  
 
Exploring Limits  'Ordinary' Function 
5:26  
 
 Test Out 
5:27  
 
 Graphing, We See The Answer Is What We Would Expect 
5:44  
 
Exploring Limits  Piecewise Function 
6:45  
 
 If We Modify the Function a Bit 
6:49  
 
Exploring Limits  A Visual Conception 
10:08  
 
Definition of a Limit 
12:07  
 
 If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L. 
12:09  
 
 We Are Not Concerned with f(x) at x=c 
12:49  
 
 We Are Considering x Approaching From All Directions, Not Just One Side 
13:10  
 
Limits Do Not Always Exist 
15:47  
 
Finding Limits 
19:49  
 
 Graphs 
19:52  
 
 Tables 
21:48  
 
 Precise Methods 
24:53  
 
Example 1 
26:06  
 
Example 2 
27:39  
 
Example 3 
30:51  
 
Example 4 
33:11  
 
Example 5 
37:07  

Formal Definition of a Limit 
57:11 
 
Intro 
0:00  
 
Introduction 
0:06  
 
New Greek Letters 
2:42  
 
 Delta 
3:14  
 
 Epsilon 
3:46  
 
 Sometimes Called the EpsilonDelta Definition of a Limit 
3:56  
 
Formal Definition of a Limit 
4:22  
 
What does it MEAN!?!? 
5:00  
 
The Groundwork 
5:38  
 
 Set Up the Limit 
5:39  
 
 The Function is Defined Over Some Portion of the Reals 
5:58  
 
 The Horizontal Location is the Value the Limit Will Approach 
6:28  
 
 The Vertical Location L is Where the Limit Goes To 
7:00  
 
The EpsilonDelta Part 
7:26  
 
 The Hard Part is the Second Part of the Definition 
7:30  
 
 Second Half of Definition 
10:04  
 
 Restrictions on the Allowed x Values 
10:28  
 
The EpsilonDelta Part, cont. 
13:34  
 
 Sherlock Holmes and Dr. Watson 
15:08  
 
The Adventure of the DeltaEpsilon Limit 
15:16  
 
 Setting 
15:18  
 
 We Begin By Setting Up the Game As Follows 
15:52  
 
The Adventure of the DeltaEpsilon, cont. 
17:24  
 
 This Game is About Limits 
17:46  
 
 What If I Try Larger? 
19:39  
 
 Technically, You Haven't Proven the Limit 
20:53  
 
 Here is the Method 
21:18  
 
 What We Should Concern Ourselves With 
22:20  
 
 Investigate the Left Sides of the Expressions 
25:24  
 
 We Can Create the Following Inequalities 
28:08  
 
 Finally… 
28:50  
 
 Nothing Like a Good Proof to Develop the Appetite 
30:42  
 
Example 1 
31:02  
 
 Example 1, cont. 
36:26  
 
Example 2 
41:46  
 
 Example 2, cont. 
47:50  

Finding Limits 
32:40 
 
Intro 
0:00  
 
Introduction 
0:08  
 
Method  'Normal' Functions 
2:04  
 
 The Easiest Limits to Find 
2:06  
 
 It Does Not 'Break' 
2:18  
 
 It Is Not Piecewise 
2:26  
 
Method  'Normal' Functions, Example 
3:38  
 
Method  'Normal' Functions, cont. 
4:54  
 
 The Functions We're Used to Working With Go Where We Expect Them To Go 
5:22  
 
 A Limit is About Figuring Out Where a Function is 'Headed' 
5:42  
 
Method  Canceling Factors 
7:18  
 
 One Weird Thing That Often Happens is Dividing By 0 
7:26  
 
 Method  Canceling Factors, cont. 
8:16  
 
 Notice That The Two Functions Are Identical With the Exception of x=0 
8:20  
 
 Method  Canceling Factors, cont. 
10:00  
 
 Example 
10:52  
 
Method  Rationalization 
12:04  
 
 Rationalizing a Portion of Some Fraction 
12:05  
 
 Conjugate 
12:26  
 
 Method  Rationalization, cont. 
13:14  
 
 Example 
13:50  
 
Method  Piecewise 
16:28  
 
 The Limits of Piecewise Functions 
16:30  
 
Example 1 
17:42  
 
Example 2 
18:44  
 
Example 3 
20:20  
 
Example 4 
22:24  
 
Example 5 
24:24  
 
Example 6 
27:12  

Continuity & OneSided Limits 
32:43 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Motivating Example 
0:56  
 
Continuity  Idea 
2:14  
 
 Continuous Function 
2:18  
 
 All Parts of Function Are Connected 
2:28  
 
 Function's Graph Can Be Drawn Without Lifting Pencil 
2:36  
 
 There Are No Breaks or Holes in Graph 
2:56  
 
 Continuity  Idea, cont. 
3:38  
 
 We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping' 
3:52  
 
Continuity  Definition 
5:16  
 
 A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does 
5:18  
 
 Discontinuous 
6:02  
 
 Discontinuity 
6:10  
 
Continuity and 'Normal' Functions 
6:48  
 
Return of the Motivating Example 
8:14  
 
 OneSided Limit 
8:48  
 
OneSided Limit  Definition 
9:16  
 
 Only Considers One Side 
9:20  
 
 Be Careful to Keep Track of Which Symbol Goes With Which Side 
10:06  
 
OneSided Limit  Example 
10:50  
 
 There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits 
11:16  
 
Normal Limits and OneSided Limits 
12:08  
 
Limits of Piecewise Functions 
14:12  
 
 'Breakover' Points 
14:22  
 
 We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other 
15:34  
 
Example 1 
16:40  
 
Example 2 
18:54  
 
Example 3 
22:00  
 
Example 4 
26:36  

Limits at Infinity & Limits of Sequences 
32:49 
 
Intro 
0:00  
 
Introduction 
0:06  
 
Definition: Limit of a Function at Infinity 
1:44  
 
 A Limit at Infinity Works Very Similarly to How a Normal Limit Works 
2:38  
 
Evaluating Limits at Infinity 
4:08  
 
 Rational Functions 
4:17  
 
 Examples 
4:30  
 
 For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator 
5:22  
 
 There are Three Possibilities 
6:36  
 
 Evaluating Limits at Infinity, cont. 
8:08  
 
 Does the Function Grow Without Bound? Will It 'Settle Down' Over Time? 
10:06  
 
 Two Good Ways to Think About This 
10:26  
 
Limit of a Sequence 
12:20  
 
 What Value Does the Sequence Tend to Do in the LongRun? 
12:41  
 
 The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity 
12:52  
 
Numerical Evaluation 
14:16  
 
 Numerically: Plug in Numbers and See What Comes Out 
14:24  
 
Example 1 
16:42  
 
Example 2 
21:00  
 
Example 3 
22:08  
 
Example 4 
26:14  
 
Example 5 
28:10  
 
Example 6 
31:06  

Instantaneous Slope & Tangents (Derivatives) 
51:13 
 
Intro 
0:00  
 
Introduction 
0:08  
 
 The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing 
0:16  
 
 Instantaneous Slop 
0:22  
 
 Instantaneous Rate of Change 
0:28  
 
Slope 
1:24  
 
 The Vertical Change Divided by the Horizontal 
1:40  
 
Idea of Instantaneous Slope 
2:10  
 
 What If We Wanted to Apply the Idea of Slope to a NonLine? 
2:14  
 
Tangent to a Circle 
3:52  
 
 What is the Tangent Line for a Circle? 
4:42  
 
Tangent to a Curve 
5:20  
 
Towards a Derivative  Average Slope 
6:36  
 
 Towards a Derivative  Average Slope, cont. 
8:20  
 
 An Approximation 
11:24  
 
Towards a Derivative  General Form 
13:18  
 
 Towards a Derivative  General Form, cont. 
16:46  
 
 An h Grows Smaller, Our Slope Approximation Becomes Better 
18:44  
 
Towards a Derivative  Limits! 
20:04  
 
 Towards a Derivative  Limits!, cont. 
22:08  
 
 We Want to Show the Slope at x=1 
22:34  
 
Towards a Derivative  Checking Our Slope 
23:12  
 
Definition of the Derivative 
23:54  
 
 Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point 
23:58  
 
 Differentiation 
24:54  
 
Notation for the Derivative 
25:58  
 
 The Derivative is a Very Important Idea In Calculus 
26:04  
 
The Important Idea 
27:34  
 
 Why Did We Learn the Formal Definition to Find a Derivative? 
28:18  
 
Example 1 
30:50  
 
Example 2 
36:06  
 
Example 3 
40:24  
 
The Power Rule 
44:16  
 
 Makes It Easier to Find the Derivative of a Function 
44:24  
 
 Examples 
45:04  
 
 n Is Any Constant Number 
45:46  
 
Example 4 
46:26  

Area Under a Curve (Integrals) 
45:26 
 
Intro 
0:00  
 
Introduction 
0:06  
 
 Integral 
0:12  
 
Idea of Area Under a Curve 
1:18  
 
Approximation by Rectangles 
2:12  
 
 The Easiest Way to Find Area is With a Rectangle 
2:18  
 
Various Methods for Choosing Rectangles 
4:30  
 
Rectangle Method  LeftMost Point 
5:12  
 
 The LeftMost Point 
5:16  
 
Rectangle Method  RightMost Point 
5:58  
 
 The RightMost Point 
6:00  
 
Rectangle Method  MidPoint 
6:42  
 
 Horizontal MidPoint 
6:48  
 
Rectangle Method  Maximum (Upper Sum) 
7:34  
 
 Maximum Height 
7:40  
 
Rectangle Method  Minimum 
8:54  
 
 Minimum Height 
9:02  
 
Evaluating the Area Approximation 
10:08  
 
 Split the Interval Into n SubIntervals 
10:30  
 
More Rectangles, Better Approximation 
12:14  
 
 The More We Us , the Better Our Approximation Becomes 
12:16  
 
 Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity 
12:44  
 
Finding Area with a Limit 
13:08  
 
 If This Limit Exists, It Is Called the Integral From a to b 
14:08  
 
 The Process of Finding Integrals is Called Integration 
14:22  
 
The Big Reveal 
14:40  
 
 The Integral is Based on the Antiderivative 
14:46  
 
The Big Reveal  Wait, Why? 
16:28  
 
 The Rate of Change for the Area is Based on the Height of the Function 
16:50  
 
 Height is the Derivative of Area, So Area is Based on the Antiderivative of Height 
17:50  
 
Example 1 
19:06  
 
Example 2 
22:48  
 
Example 3 
29:06  
 
 Example 3, cont. 
35:14  
 
Example 4 
40:14  
XVII. Appendix: Graphing Calculators 

Buying a Graphing Calculator 
10:41 
 
Intro 
0:00  
 
Should You Buy? 
0:06  
 
 Should I Get a Graphing Utility? 
0:20  
 
Free Graphing Utilities  Web Based 
0:38  
 
 Personal Favorite: Desmos 
0:58  
 
Free Graphing Utilities  Offline Programs 
1:18  
 
 GeoGebra 
1:31  
 
 Microsoft Mathematics 
1:50  
 
 Grapher 
2:18  
 
Other Graphing Utilities  Tablet/Phone 
2:48  
 
Should You Buy a Graphing Calculator? 
3:22  
 
 The Only Real Downside 
4:10  
 
Deciding on Buying 
4:20  
 
 If You Plan on Continuing in Math and/or Science 
4:26  
 
 If Money is Not Particularly Tight for You 
4:32  
 
 If You Don't Plan to Continue in Math and Science 
5:02  
 
 If You Do Plan to Continue and Money Is Tight 
5:28  
 
Which to Buy 
5:44  
 
 Which Graphing Calculator is Best? 
5:46  
 
 Too Many Factors 
5:54  
 
 Do Your Research 
6:12  
 
The Old Standby 
7:10  
 
 TI83 (Plus) 
7:16  
 
 TI84 (Plus) 
7:18  
 
Tips for Purchasing 
9:17  
 
 Buy Online 
9:19  
 
 Buy Used 
9:35  
 
 Ask Around 
10:09  

Graphing Calculator Basics 
10:51 
 
Intro 
0:00  
 
Read the Manual 
0:06  
 
 Skim It 
0:20  
 
 Play Around and Experiment 
0:34  
 
Syntax 
0:40  
 
 Definition of Syntax in English and Math 
0:46  
 
 Pay Careful Attention to Your Syntax When Working With a Calculator 
2:08  
 
 Make Sure You Use Parentheses to Indicate the Proper Order of Operations 
2:16  
 
 Think About the Results 
3:54  
 
Settings 
4:58  
 
 You'll Almost Never Need to Change the Settings on Your Calculator 
5:00  
 
 Tell Calculator In Settings Whether the Angles Are In Radians or Degrees 
5:26  
 
 Graphing Mode 
6:32  
 
Error Messages 
7:10  
 
 Don't Panic 
7:11  
 
 Internet Search 
7:32  
 
So Many Things 
8:14  
 
 More Powerful Than You Realize 
8:18  
 
 Other Things Your Graphing Calculator Can Do 
8:24  
 
Playing Around 
9:16  

Graphing Functions, Window Settings, & Table of Values 
10:38 
 
Intro 
0:00  
 
Graphing Functions 
0:18  
 
 Graphing Calculator Expects the Variable to Be x 
0:28  
 
Syntax 
0:58  
 
 The Syntax We Choose Will Affect How the Function Graphs 
1:00  
 
 Use Parentheses 
1:26  
 
The Viewing Window 
2:00  
 
 One of the Most Important Ideas When Graphing Is To Think About The Viewing Window 
2:01  
 
 For Example 
2:30  
 
 The Viewing Window, cont. 
2:36  
 
Window Settings 
3:24  
 
 Manually Choose Window Settings 
4:20  
 
 x Min 
4:40  
 
 x Max 
4:42  
 
 y Min 
4:44  
 
 y Max 
4:46  
 
 Changing the x Scale or y Scale 
5:08  
 
 Window Settings, cont. 
5:44  
 
Table of Values 
7:38  
 
 Allows You to Quickly Churn Out Values for Various Inputs 
7:42  
 
 For example 
7:44  
 
 Changing the Independent Variable From 'Automatic' to 'Ask' 
8:50  

Finding Points of Interest 
9:45 
 
Intro 
0:00  
 
Points of Interest 
0:06  
 
 Interesting Points on the Graph 
0:11  
 
 Roots/Zeros (Zero) 
0:18  
 
 Relative Minimums (Min) 
0:26  
 
 Relative Maximums (Max) 
0:32  
 
 Intersections (Intersection) 
0:38  
 
Finding Points of Interest  Process 
1:48  
 
 Graph the Function 
1:49  
 
 Adjust Viewing Window 
2:12  
 
 Choose Point of Interest Type 
2:54  
 
 Identify Where Search Should Occur 
3:04  
 
 Give a Guess 
3:36  
 
 Get Result 
4:06  
 
Advanced Technique: Arbitrary Solving 
5:10  
 
 Find Out What Input Value Causes a Certain Output 
5:12  
 
 For Example 
5:24  
 
Advanced Technique: Calculus 
7:18  
 
 Derivative 
7:22  
 
 Integral 
7:30  
 
But How Do You Show Work? 
8:20  

Parametric & Polar Graphs 
7:08 
 
Intro 
0:00  
 
Change Graph Type 
0:08  
 
 Located in General 'Settings' 
0:16  
 
Graphing in Parametric 
1:06  
 
 Set Up Both Horizontal Function and Vertical Function 
1:08  
 
 For Example 
2:04  
 
Graphing in Polar 
4:00  
 
 For Example 
4:28  