Vincent Selhorst-Jones

Midpoints, Distance, the Pythagorean Theorem, & Slope

Slide Duration:

Section 1: Introduction
Introduction to Math Analysis

10m 3s

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

45m 11s

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

35m 31s

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
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

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
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
18:50
19:04
19:21
20:04
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional 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

48m 43s

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

56m 31s

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
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
Math-Specific 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
Section 2: Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
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 Non-Numerical 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 T-Table
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

58m 26s

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 x-Value 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

48m 49s

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 x-values; 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/x-intercepts
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

29m 20s

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

48m 35s

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 x-axis
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 x-axis
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

33m 24s

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

51m 42s

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 Non-Numerical 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 Non-Continuous 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

49m 37s

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
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
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 One-to-One
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
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

28m 49s

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
Section 3: Polynomials
Intro to Polynomials

38m 41s

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
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
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

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
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
13:08
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
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

39m 43s

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
9:56
11:38
Derivation
11:43
Final Form
12:23
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
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52

45m 34s

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
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

46m 8s

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
26:49
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

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
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
19:24
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

19m 9s

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
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
7:41
8:55
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
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

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

34m 16s

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

49m 7s

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

44m 56s

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
8:25
9:26
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
Section 5: Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

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

47m 4s

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

40m 31s

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
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

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(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 Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
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

48m 46s

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
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
2:08
2:31
2:52
Half-Circle and Right Angle
4:00
6:24
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
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
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
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
9:43
Mnemonic: All Students Take Calculus
10:13
13:11
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

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
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

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
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
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

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
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

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
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

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
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Section 7: Trigonometric Identities
Pythagorean Identity

19m 11s

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
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

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

52m 52s

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

29m 5s

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

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
Section 8: Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

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

56m 40s

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

49m 5s

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
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

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
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

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
-1
Extra Example 2: Roads to a Town
-2
Section 9: Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

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
9:07
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

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick 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 Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
11:42
Example of Using a Point
12:41
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 Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
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

41m 1s

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
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: Vectors and Matrices
Vectors

1h 9m 31s

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
1:01:32
Dot Product & Cross Product

35m 20s

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 Right-Hand 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

54m 7s

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
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
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

47m 12s

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

58m 34s

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
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan 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
Section 11: Alternate Ways to Graph
Parametric Equations

53m 33s

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

48m 7s

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 Counter-Clockwise 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

38m 16s

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
Section 12: Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

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
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

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
-1
Extra Example 2: Find Complex Fourth Roots
-2
Section 13: Counting & Probability
Counting

31m 36s

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
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

44m 3s

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

36m 58s

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
Section 14: Conic Sections
Parabolas

41m 27s

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

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
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
11:51
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

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

38m 15s

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

18m 43s

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
Section 15: Sequences, Series, & Induction
Introduction to Sequences

57m 45s

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

40m 27s

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

31m 36s

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

39m 27s

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

49m 53s

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

1h 13m 13s

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
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
Section 16: Preview of Calculus
Idea of a Limit

40m 22s

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

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta 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 Epsilon-Delta 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 Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
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

32m 40s

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
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 & One-Sided Limits

32m 43s

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
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided 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

32m 49s

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
10:26
Limit of a Sequence
12:20
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 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)

51m 13s

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 Non-Line?
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)

45m 26s

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 - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
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 Sub-Intervals
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
Section 17: Appendix: Graphing Calculators

10m 41s

Intro
0:00
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
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
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
9:17
9:19
9:35
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
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
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

10m 38s

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

9m 45s

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
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
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

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
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• ## Related Books

### Midpoints, Distance, the Pythagorean Theorem, & Slope

• To find the midpoint in one dimension, we take the average of the two numbers involved:
 a+b 2 .
• To find the midpoint in two dimensions (in the plane), we take the average location for each dimension on its own:
 ⎛⎝ x1 + x22 , y1 + y22 ⎞⎠ .
• To find the distance between two points in one dimension, we subtract one from the other. However, that could potentially cause a negative to pop up, so we deal with that by taking the absolute value of the result. Thus, the distance between any two numbers is
 |a−b|.
• The Pythagorean theorem says, "On a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two legs." In other words, if the two legs (the shorter sides) are a and b, while the hypotenuse (the longest side) is c, then we have
 a2 + b2 = c2.
• The Pythagorean theorem allows us to find the distance between two points in the plane. We can plot the points, draw in a triangle, then figure out the lengths of the two legs. From there, we use the theorem to find the hypotenuse, which is the distance between them. This gives the distance formula
 d = √ (x2 − x1)2 + (y2 − y1)2
• Slope is a way to discuss how "steep" a line is. Another way to interpret it is the rate of change: the rate the line increases (or decreases) for every "step" to the right. We symbolize slope with m, and it is defined as any of the following equivalent things:
 m = riserun = vertical changehorizontal change = y2 − y1x2 − x1 .
• Slope tells us how much the value of a line will change for every "step" to the right. A slope of m=−3 means that if we go 1 unit right, the line will drop down by 3 units. It is the line's rate of change.
• The idea of slope is very important in math (especially in calculus), so it's useful to have an intuitive sense of how slope works. Keep these facts in mind when thinking about slope:
• Positive (+) slope ⇒ line rises (when going right),
• Negative (−) slope ⇒ line falls (when going right),
• Bigger number (+ or −) ⇒ steeper line,
• m=1  ⇒ line rises at 45° angle,
• m=0  ⇒ line is horizontal,
• m=−1  ⇒ line falls at 45° angle.

### Midpoints, Distance, the Pythagorean Theorem, & Slope

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

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

### Transcription: Midpoints, Distance, the Pythagorean Theorem, & Slope

Hi; welcome back to Educator.com.0000

Today we are going to talk about midpoints, distance, the Pythagorean theorem, and slope.0002

We have a bunch of things to talk about.0006

The concepts in this lesson, like all the other introductory lessons, are all ideas you have seen in previous math classes.0008

None of this should be totally new to you, but we definitely want to review them.0013

There are some important concepts for the course in here.0017

We are going to be talking about these things later on; we are not going to directly talk about these other ideas.0020

We are not going to be teaching on them directly (other than this lesson, where we will be doing that directly).0025

But it is going to be assumed that you understand them all.0030

They are all going to repeatedly show up as we work on more complex things, so we really want to make sure0034

that all of these things are totally understood now, and that we really know what we are doing.0038

Not only that, but we really want to understand what we are doing.0042

We don't just want to be able to do these things; we want to understand how it works--how these formulas are operating:0045

not just how to use the formulas, but how they work--why they work--what they mean--what is causing them to be the way they are.0053

As we get into more advanced math, like this course right here, it is going to become more and more important0061

for you to understand the big picture--not just how you can do this one problem, but why doing the problem this way makes sense.0066

As we see more and more complex ideas, it is absolutely necessary for you to be able to make sense of why we are doing the things we are.0073

If you are just doing it because that is what you are told to, and that was the step that has to come next,0080

eventually things are going to fall apart, and you are not going to be able to see what the next step is going to be.0085

As you get older and older, you take more responsibility; as you get into more and more advanced subjects,0089

you are expected to understand what is going on and be able to take things on yourself.0093

Back in algebra, you were able to just take step-by-step formulas and apply them.0097

But now, you have to understand why those step-by-step formulas work,0101

because you have to understand why that works, so you can now tackle more complex ideas.0104

So, don't just understand how you can use these things, but understand what is going on on a deeper level.0109

That is what I really want you to get here, and what I want you to get out of the entire course at large.0113

That should be the goal of your education at this point: being able to understand what it is doing--0118

why it works--not just going through it so you can get the next grade.0123

All right, let's get started.0127

Let's say we want to find the point that is halfway between 0 and 4.0129

It seems pretty easy, right? Halfway between 0 and 4...well, there is 0; click over; click over; look, it is 2.0133

It is just half of 4: 4 over 2 equals 2.0140

Well, what if I want something a little bit more complex--like I want to figure out the midpoint between -5 and 17?0144

What point is halfway between those two?0149

There are two ways we could approach this idea.0151

Say we want to find the midpoint between -5 and 17: there are two ways to look at this question.0153

First, we can look at it through the idea of distance.0158

The distance between -5 and 17: how far would we have to travel to get from -5 to 17?0161

We would have to travel 22: - 5 to 0--we travel 5; 0 on to 17--we travel another 17.0166

We can also look at it from the point of view of 17 minus the one before it (-5); we get 22 either way we do it.0173

Technically, we haven't formally defined what distance means; we will in just a little bit.0180

But this makes sense: we can see that that should be 22.0184

So, to find the midpoint, what we can do is start at -5, and then we will work halfway up.0187

We will go up 22/2: we start at -5, and then we add 22/2.0194

22/2 is 11, so -5 + 11...we get 6: 6 is our answer; 6 is our midpoint.0199

Let's look at another way to do it, though: what if, instead, we wanted to find it through the idea of a middle?0206

We are looking for the midpoint, so it makes sense that the midpoint has to be halfway between them.0212

What is going to be halfway between them? Well, it would be the average of the two numbers.0217

What would be (if we could combine those two and figure it out) the most common place you have between the two?0221

What is going to be the middle between the two? It would be the average.0226

So, we take 17; we take -5; and we don't have to worry about the distance; we just realize, "Look, the midpoint is going to be halfway."0229

To be halfway, you have to be at the average of the two values.0236

So, we take -5 + 17, and we divide it by 2; that gets us 12/2, which is 6, the exact same thing we figured out last time.0239

That is great: it agrees with our previous work, and that is definitely what we want.0247

Since both of these things made logical sense to us, they had better both work; otherwise there is going to be a mistake somewhere in there.0250

All right, at this point, we could maybe look at this in general.0256

Let's say somebody hands us two points, A and B, and guarantees that A comes before B--that A is less than or equal to B.0260

We have this order that A will come before B, or maybe on top of B.0268

We don't know what they are, but we still want to be able to talk about the midpoint.0271

From our previous work, we have two ways to find this.0275

The distance from A to B is going to be B - A, so if we move up half of that distance, it is going to be (B - A)/2.0277

So, we start at A; and then we would add (B - A)/2; that is our half of the distance idea.0288

But we can also think, "Look, I know I am going to be here, and I know I am here; so I am looking for the place that is halfway between them."0293

So, we get the middle: that is (A + B)/2.0300

Now, from what we saw before...we saw both of these ways worked; as we had hoped, they give the same value.0304

We can go with either A + (B - A)/2 or (A + B)/2; they both are the same thing.0311

Let's prove that they are actually equivalent: if we start with our half-distance formula right here,0318

and we have A + (B - A)/2, well, let's try to put them on the same fraction.0323

We make it 2A/2; we can now combine fractions, and we get (2A + B - A)/2.0327

At this point, 2A - A...we get (A + B)/2; so sure enough, our half distance is equal to our middle; they are just two different ways of saying the same thing.0333

Since we have two ways to find the midpoint, and they are really just equivalent ways to get the same answer,0347

we will just, out of laziness, make one of them the official one.0352

A good motivator to do anything is because it is the easier way, as opposed to having to memorize two different things,0356

or go with the slightly longer one, let's go with the slightly shorter one; so we make our midpoint formula (A + B)/2; there we are.0361

Whatever the two points are that we are trying to find the middle between them, it is just (A + B)/2,0369

because we are just looking for the middle place, and the middle place must be the average of our two locations.0374

What if we wanted to do this in two dimensions, though?0380

What if we wanted to find the midpoint between (0,0) and (6,2).0382

Now, notice: we could look at this, as opposed to trying to figure out what is the midpoint on the line that connects the two of them0385

(I wish I had made that slightly more perfect)--instead of trying to figure out, "What is the middle going to be here?"0393

we can say, "Well, we know that there is going to be some distance vertical and some distance horizontal."0397

It must be that it splits our distance halfway horizontally, and it splits our distance halfway vertically.0406

So, the two of them come together, and that is our midpoint.0413

It is going to have to be the horizontal middle: our horizontal distance was 6, so 6/2 is where we are going.0419

Our vertical distance was 2, so it is 2/2, so our location (that is going to be halfway between them)0425

will be half of our horizontal distance and half of our vertical distance: (6/2,2/2), or (3,1).0431

The midpoint is going to occur at the horizontal middle and the vertical middle, put together into a single point.0439

What if we are doing this for some arbitrary pair of points, (x1,y1) and (x2,y2)?0446

Well, the same basic idea: we can think, "What is the vertical, and what is the horizontal?"0450

What are going to be the midpoints of those two things?0457

We think with that idea, and we are able to come up with the same logic that is going to occur at the horizontal and vertical middles.0459

This point is going to be the same vertical height, because we never changed height as we went along.0466

So, horizontally, we are going to have changed to a new horizontal location; but vertically, this thing right here is going to be the same thing here.0473

The same sort of idea here: y2 is going to change when we switch down, as we go down.0484

But horizontally, we didn't change there; so we have fixed things here, so the point that we are meeting the two at--0488

if we drop a perpendicular and throw out a horizontal, we are going to meet up at (x2,y1).0498

So, the midpoint horizontally is going to be (x2 + x1)/2,0505

which will get us the middle location, because it is the average of our two horizontal locations.0516

The average of our two vertical locations is going to be (y2 + y1)/2.0521

We bring these two things together, and we get where our middle is--we get our midpoint that way.0528

So, from the midpoint in one dimension, we can figure out what it is horizontally.0538

The horizontal motion was x1 + x2, so its midpoint is (x1 + x2)/2.0542

And vertically, our locations were y1 and y2, so the middle of our vertical locations will be (y1 + y2)/2; great.0548

Our midpoint formula is just (x1 + x2)/2, and (y1 + y2)/2; awesome.0559

The next idea: distance--what if we want to find the distance between 2 and 7.0567

That is easy: 7 minus 2 equals 5--done, right?0571

Well, we could make a mistake, though; we are not perfect; what if we accidentally put it in in the wrong order.0575

We put in 2 - 7 = -5: well, that doesn't really make sense, because distance has to be a positive length.0581

There is no such thing as a negative length, if we are measuring something; you can't say, "Oh, that man is -2 meters tall."0589

It doesn't make sense; we can't talk about his distance, his length, as being -2.0595

So, -5 doesn't really work; but notice, 5 and -5...they are very different in one way; but in another way, they are very similar.0599

One of them is the same thing, just with a negative sign; the other one is the same thing, but with a positive sign.0610

So, they are the same number, but with different signs on them.0616

In one way, we can think of 5 and -5 as being very different numbers; they are opposites, after all.0620

But in another way, we can think of them being the same number, but with different signs.0624

They are the same distance from 0; so what we really want is some way of being able to force "positive-ness."0628

5 and -5 are pretty close to both being the same thing; it is just that one of them is the wrong sign.0637

So, if we could force it to be positive, it wouldn't matter if we did 7 - 2 or 2 - 7,0643

because, since we are forcing positive, it will always give us the same thing.0648

We would always find that distance, even if we put it in the wrong way.0651

This is where the idea of absolute value comes in.0654

We call this idea of forcing a positive, making something always come out as positive, absolute value.0657

It is represented by vertical bars on either side.0664

So, whatever we want to take the absolute value of, we just put inside of two vertical bars.0666

We could have |x - 5|, and whatever comes out after we plug in x, we would take its absolute value--we would force positive-ness.0670

It is going to take negatives, and it will make them positive; and it will take positives and not do anything; also with 0, it won't do anything.0677

If you are positive, you stay positive; if you are 0, you stay 0; if you are negative, you flip to being the positive version.0683

You hit it with another negative: so -5 would become 5, but positive 5 would just become positive 5, as it already started.0689

-47 would become 47; 47 would just stay as 47; great.0697

With this idea of absolute value, we can now tackle how we talk about distance in one dimension.0703

So, if we want to talk about it just arbitrarily, if we have two points, A and B, and we don't know which comes first--0707

we don't know if it is going to be A then B, or B then A--we have no idea which comes first (if it is A first or B first)--0714

but we still want to be able to talk about what the distance is between them,0726

well, our previous logic can tell us that one of these two is going to be right: A - B or B - A.0729

But the other one is going to be wrong, although almost correct, because it will be the negative version.0735

So, we have A - B versus B - A: we want some way of being able to say, "Let's just get rid of the negative signs," right?0741

Let's force everything to be positive; then, it doesn't matter what order we put it in, because it is going to be the same distance,0750

because it is just a negative version or a positive version;0756

it doesn't matter, because we will flip everything to positive; we will always get the distance.0758

So, we toss some absolute values on there: absolute value to the rescue!0761

We wrap them in absolute values, and they both become the same positive, correct distance.0764

So, the absolute value of A - B is the same thing as the absolute value of B - A,0770

because the only difference would be whether it is negative or positive.0774

And now they are both forced to be positive, so |A - B| is equal to |B - A|,0776

which is just going to be the distance between A and B, which is the distance between B and A.0782

For ease, we will just make the first one official: so the absolute value of A - B is the distance between those two locations.0787

We just take the absolute value of the difference, and that gives us how far the two things are apart0794

when it is in one dimension--when we are just on the number line.0799

What if we are in more dimensions, though? Let's take a look at the Pythagorean theorem, because we will need that to discuss two dimensions.0803

To discuss distance in two dimensions, we need to understand the Pythagorean theorem.0809

You have probably learned this before; if it isn't really something that you know well, you are going to want to go back and relearn it.0813

Make sure you have this idea, because it is going to show up all sorts of places in precalculus and in calculus.0817

And it is definitely going to show up a whole bunch in the trigonometry portion of this course.0823

So, definitely make sure that you go and re-study it if you don't remember it.0826

What it was: we have a right triangle (a right angle in the corner): the square of the hypotenuse0829

(that is the long side, the side that is opposite our right angle) is equal to the sum of the squares of the other two legs.0835

So, we square each of the other two, smaller legs.0846

And when we add them together, a2 + b2...each of the smaller legs squared,0849

then added together...that is going to be equal to our hypotenuse, squared.0854

a2 + b2 = c2: leg 1 squared, plus leg 2 squared, equals hypotenuse squared.0859

That is the idea of the Pythagorean theorem.0866

So, any time we see a right triangle showing up, anything we have showing up with perpendiculars--0868

it is a good idea to think, "Oh, I wonder if I could use the Pythagorean theorem here."0873

It will be very, very useful in a whole bunch of situations.0877

If you are not really comfortable with using it at this point, definitely go back and review this idea.0880

Either search for it on the Internet, or just try to do a couple of exercises and make sure you have practiced on it.0883

Or go review it on Educator.com: listen to the lecture, and then practice some exercises.0888

But you want to make sure that you are definitely comfortable with the Pythagorean theorem,0891

because it is going to show up a whole lot for the rest of the time you are doing math.0894

All right, on to distance in two dimensions: what if we wanted to find the distance between (0,0) and (6,8)?0898

Well, we can't just subtract and take absolute values, because we have two dimensions that we are running in.0904

We have to deal with both of these things at once.0910

What we do is say, "Let's turn this into a triangle."0912

We drop a perpendicular from (6,8); we now have this right angle in the corner.0916

And with this right angle in the corner, we can use the Pythagorean theorem.0921

When we did midpoint, we dropped down perpendiculars; we drew out horizontals and perpendiculars.0926

And we were able to get a right triangle going on, which will help us to find middle locations for horizontal and vertical.0931

Now, we are allowing us to find horizontal lengths and vertical lengths.0937

We break it into horizontal and vertical parts.0940

So, if we are at (6,8) up here, then the distance that we traveled horizontally is 6.0943

The distance that we traveled vertically is 8: remember, 6 is because that is the horizontal portion; 8 is because that is the vertical portion.0949

So, we use the Pythagorean theorem: we know that d2, the diagonal, the hypotenuse,0957

is going to be equal to 62 + 82 = 36 + 64 = 100.0961

So, for our diagonal, our distance, d, equals 10.0968

So, we can figure out that this has to be 10, up here on that side, because we can turn it into a right triangle, which allows us to apply the Pythagorean theorem.0972

What if we look at this in a more general way, where we just get two arbitrary points,0982

where we don't know what they are--(x1,y1), our first point, and (x2,y2).0985

Now, I didn't talk about this explicitly the first time, but when I say x1, I am not saying x times 1; I am just saying our first x.0990

x the first, y the first, x the second, y the second--(x1,y1),(x2,y2)--0996

that is what you should interpret when you see those little subscripts, those little numbers on the bottom right.1003

So, (x1,y1), (x2,y2): they are just two arbitrary points, sitting out in a plane.1008

We can continue with this idea: we will make a triangle.1013

We will toss out a horizontal from this one; we will go straight with a horizontal out.1016

And we will drop directly down with a vertical, like this; and that will guarantee us that we have a right triangle that we can now work with.1023

And now, we have a way of being able to talk about the distance of that.1034

So, we draw that in, and we can say, "Oh, what is the horizontal length?"1037

Well, since we ended up at x2 (because it is going to have the same horizontal location as our second point),1040

we went from x1 to x2; our distance is the absolute value of (x2 - x1).1047

The horizontal length is going to be the absolute value of (x2 - x1).1053

What is the vertical length--what is the vertical leg of our triangle?1056

Well, we end up at y2; and what is the location that we are starting on this triangle?1059

It is going to be y1, because it is going to be the same as over here.1064

So, we take y2 - y1; the absolute value of that is going to be our vertical length.1067

The length of the vertical leg of the triangle is the absolute value of (y2 - y1).1074

So, if we want to know what the distance of the diagonal is, it is going to be d2 = (|x2 - x1|)2 + (|y2 - y1|)2.1078

Now, there is a little thing that we can notice at this point.1096

OK, if I have |x2 - x1|, and then I square it, well, if I just take1099

x2 - x1, and I square that, that is going to be the same thing.1104

Remember, if I have (-7) squared, that comes out to be 49, which is the same thing as 7 squared.1108

So, we don't have to take an absolute value to begin with, because, when it is the number times itself, if it has a negative,1116

if it multiplies by itself with another negative, those negatives are going to cancel each other out.1123

But if we start on a positive, we are going to have no negatives anyway.1127

So, the absolute value of x2 - x1, squared, is equal to the quantity1130

(x2 - x1), squared, because they are both going to come out to be positive, in any case.1134

So, this is also the same for y; so we can actually drop our absolute values--we don't have to worry about absolute values when we are doing this.1139

And the distance will be equal to the square root, because it is d2 equals this thing squared, plus this thing squared1144

(horizontal length squared, plus vertical length squared); so d equals the square root--take the square root of both sides, so we get just d.1151

It is the square root of ((x2 - x1)2 + (y2 - y1)2).1157

So, it is the difference in our two horizontal locations, squared, plus the difference in our two vertical locations, squared; great.1163

That is distance in two dimensions.1168

All right, slope: slope is a way to discuss how steep a line is--how quickly it is going up--how much it is changing one way or the other.1171

Another way to interpret is the rate of change--the rate that the line increases or decreases for every step to the right.1181

So, if we take one step over to the right, it tells us how much up we should go or how much down we should go,1190

depending on if it is a positive slope or a negative slope.1195

So, it is one step over, and we change by the slope.1198

Either way we look at slope--whether we look at it as how steep it is (the angle we are working at)1202

or if we look at it as the rate of change (how much we are changing for every step we take on the line), we define it the same way.1206

We take some arbitrary portion of the line--any chunk of the line--and we see how much it "rises" and how much it "runs."1213

So, rise is the vertical amount of change, and run is the horizontal amount of change.1219

So, if we had some chunk of line, like this, then what we would do is just set two arbitrary points, here and here.1224

And then, we would say, "OK, how much did we move vertically?" (that is our run) and "How much did we move horizontally?"...1233

Oops, sorry, not our run--I said the wrong thing there: our vertical change is our rise--you rise vertically.1243

And our horizontal change is our run, because you run along the ground; you run along (generally) horizontal things.1251

So, our rise, compared to our run--we divide the rise by the run.1258

We symbolize it with m: why do we symbolize it with m? Because, clearly, the first letter of slope starts with m, right?--it makes sense.1262

I am kidding; there is actually no good reason, and nobody knows why we use m.1270

Anyway, m equals rise over run; that is how we symbolize it--the amount that we rise by, divided by the amount that we run.1274

We can also talk about this, since rise and run are just other words for vertical change and horizontal change.1281

The slope is equal to the amount of vertical change in our line, divided by the amount of horizontal change in our line.1286

Now, keep in mind: vertical change could go down, at which point we would have a negative rise (I keep accidentally swapping them).1292

We would have a negative rise if we ended up dropping down.1300

All right, let's go back really quickly and address this asterisk.1305

Why is it that we can look at any arbitrary portion of the line?1308

Why doesn't it matter which section of the line we look at?1312

Shouldn't it matter if we look at a big section or a small section, or if we look at a high section or a low section?1315

No, because the line never changes slope: that is what it means to be a line.1320

Every section of it is going along at the same steepness; every section of it is going along at the same rate of change.1324

If we put a bunch of sections together, they will all agree on their slope.1331

If we look at just one tiny section in a very different place, it is still going to have the same slope.1334

Whatever part of the line we look at, it will always have the same slope.1338

So, we get the same value for slope, no matter where we look on the line.1341

So, that is why we don't have to worry about what portion of the line we are considering.1345

We just look somewhere, and that is our slope; great.1349

So, if we have two points, they can define a line; so we want to find the slope between two arbitrary points, (x1,y1) and (x2,y2).1353

Well, what we have to do is say, "All right, how much did I rise in that chunk?"1361

And I compare it to how much I ran in that chunk.1368

We figure out both of those, and we will be able to get what our slope has to be.1372

So, we build a right triangle to help us find these distances.1376

Since this here is going to be the same as the horizontal of our second point, that matches up there.1379

And it is going to be the same as the vertical, since vertical doesn't change as we go horizontally.1385

Those match up there, so the amount of run that we have is (x2 - x1).1388

That is how much we changed as we went from left to right.1395

And the amount vertically that we changed is (y2 - y1), because we went up from y1 to y2).1398

We went up from y1 to some y2; so it is y2 - y1.1405

So, our rise is y2 - y1, and our run is x2 - x1.1411

Our slope is equal to the rise divided by the run, which is our vertical change divided by our horizontal change.1417

That gets us (y2 - y1)/(x2 - x1).1426

We have been using this formula for years, but now...hopefully, you actually understood it before...1429

but even if you didn't understand it before, why this formula was slope,1434

hopefully now you are thinking, "Oh, now I see why slope is what it is!"1437

It is because it is just coming from rise over run; that is how we defined it; so we get (y2 - y1)/(x2 - x1).1440

because (y2 - y1 is how much we rose, and (x2 - x1) is how much we ran.1446

Being able to understand what we are doing with slope, though, requires being able to interpret it on an intuitive basis.1452

We want to know what to immediately imagine when we are talking about something that has a slope of 50.1457

So, slope tells us how much the value of a line changes for every step to the right.1464

If we have a slope of m = 2, then that means, if we take one step to the right, then we will go up 2 steps.1469

Our line will end up looking like this.1480

If we had a slope of, say, -3, then it would be that for one step to the right, we take 3 steps down, so our line would look like that.1487

What we have here is a way of being able to talk about the line's rate of change--how much you change for one click.1508

You click over, and you change by your slope.1514

So, we can think of slope as how steep it is (bigger numbers will make it steeper, because it means more steps to be made in our rate of change).1517

All right, there is one step, but how many times we go down or how many times we go up is the number of our slope.1524

So, a line's rate of change is its slope: it is a way of talking about how fast this line is changing as we slide along it.1530

You want to keep these facts in mind as we think about slope.1539

If we have a positive slope, it means that the line is rising; we are always thinking about it as we go from left to right.1543

That is how we are always reading how our slope works: it is always what happens as we go from left to right.1552

So, positive slope means we rise as we go to the right; a negative slope means that we fall when we go to the right.1558

We either go up by positive, or we go down because it is negative.1565

A bigger number, whether it is positive or it is negative, means a steeper line.1568

The steeper the line is, the bigger the slope has to be; the bigger the slope is, the steeper the line is.1573

A big slope, like, say, m = 50, is going to be really, really super steep.1581

It is going to go up really, really fast, because for every step it takes to the right, it has to take 50 steps up.1587

Similarly, in m = -50, it is going to be very similar; but for every step it takes to the right, it takes 50 steps down; so it is super steep going down.1592

A big number, whether it is a big positive number or a big negative number--that is going to imply a steep line.1603

Some specific locations to keep in mind: if m is equal to 1, then that means our line rises at a 45-degree angle,1609

because for every step to the right, we take one step up.1615

So, it means that we have a nice, even-sided triangle: 45...these two have to be the same.1618

If we have m = -1, then for every step we take over, we take a step down.1626

So, we have the same idea; but instead, we are going down now; so these two angles have to be the same.1633

We have a nice 45-degree angle in that triangle as well--what makes up the line.1639

So, we are either rising at 45 degrees (if we have a positive one) or we are falling at 45 degrees (if we have a negative one).1644

And if m = 0, then we take one step over; we take no steps up; we just continue taking steps over forever and ever.1650

So, m = 0 means our line is horizontal; m = 1 means our line rises at 45 degrees; m = -1 means the line falls at 45 degrees.1657

This also means that everything between positive 45 and -45 is all going to happen in fractions--things that are between -1 and 1.1664

If we want to get really steep lines, that is as we approach either positive infinity, or as we approach negative infinity.1676

We can never be perfectly vertical with a slope, because that will require either positive infinity or negative infinity.1681

And we are not able to actually call those out, because they are not really numbers.1689

But as we go from 1 and click up more and more and more, and approach infinity more and more and more,1692

we will need larger and larger numbers to become steeper and steeper and steeper.1696

All right, there are lots of ideas that we have covered here; now we are ready to start talking about some examples.1700

First, the idea of midpoints: if we have a midpoint, and we are looking between -3 and 37, then remember,1706

our formula for midpoints was just (A + B)/2; it is the average of the two.1712

So, the average of -3 and 37...put those two together: we get 34/2, which equals 17; so, 17 is our answer.1718

If we want to find (6,2) to (1,-12), then we do it on each of the components, because we look at the horizontal average, and we look at the vertical average.1731

So, (6 + 1)/2, the average of our horizontal components, and (2 + -12), the average of our vertical components, will get us 7/2 and -10/2.1743

7/2...we can't simplify that anymore, so that will lock in; but -10/2...we can simplify that, so we get -5.1761

(7/2,-5): that is our midpoint for this one right here.1769

And our last one: what if somebody handed us things that weren't numbers--they hand us 2a, 3b, 6k, -7b?1775

They are numbers, in that a represents some number--it is a placeholder--it is a variable.1782

b represents a number; k represents a number; they all represent numbers.1786

But we can't actually solve and get numbers, like we did with these previous two ideas, these previous two questions.1789

But we can still use the numbers--we can still use the variables.1796

We just put them into the formula, just the same: we are still looking for what is the average of our horizontals--what is the average of 2a and 6k.1799

What is the average of our horizontal locations?1808

And what is the average of our vertical locations, 3b + -7b?1811

We are still looking for the same sort of average ideas, horizontal and vertical.1817

It just is that we can't combine 2a and 6k, because a and k are speaking totally different languages.1822

And b and b, we can combine, because they are speaking the same language.1828

So, 2a + 6k--we can't combine that, but we can have our fraction, the denominator,1831

go onto both of them: 2a/2 + 6k/2; we have the denominator split onto both of them.1835

And 3b + -7b; that begins -4b/2; so 2a/2 becomes just a; 6k/2 becomes 3k; no comma--they are combined together through addition.1842

But they can't do anything more: a and k don't speak the same language, so they can't combine.1856

But we have a + 3k; we know that is what our horizontal location is.1861

So, if we were given a and k later, we can easily get what the midpoint is, in terms of actual numbers.1864

-4b/2: that is -2b; there we are--we don't have numbers in the terms of 53 or something,1869

but we have answers that are still pretty good.1879

If we get what these variables are later--if we somehow get them because we solve for them,1882

or somebody hands them to us--we will be able to immediately find out what actual numbers would be.1886

And this gives us a great idea of where the midpoint is, based on variables.1890

We don't have to be working with numbers to be able to solve for these things; we can also just put in variables,1894

and just follow the exact same rules that the numbers would follow.1898

The next one: let's talk about distance--what is the distance between -7 and 8?1903

Remember, we do this based off of the difference between the two numbers, its absolute value.1907

So, we could take |-7 - 8| or |8 - -7|; either way we do this, -7 - 8 will be -15; 8 - -7 will turn that into positive 15.1914

Either way, they both equal 15; so the answer is 15.1929

The distance between -7 and 8 is 15, which makes sense, because -7 clicks up to 0 by going 7, and 0 clicks up to 8 by going 8; so 7 + 8 is 15.1933

Great, that makes a lot of sense.1943

What if we want to figure out what the distance is between (3,7) and (9,-1)?1945

Well, remember: now we are working off of what we figured out before, with the Pythagorean theorem and how that applied to distance.1949

So, it is going to be the square root of the difference between our horizontals, squared, plus our difference between our verticals, squared.1955

It is the square root of all of those things.1965

So, formulaically, it is d = √[(x2 - x1)2 + (y2 - y1)2]--the square root of all of that.1967

So, in this case, let's arbitrarily set this as our second one; and we will set this as our first one.1981

The distance is equal to the square root of (x2 - x1)2...that would be...1987

the second x is 9, minus...the first x is 3, squared; plus our y portion: the second y is -1, minus the first y (is 7), squared.1995

So, distance is equal to the square root of 6 squared plus -8 squared; distance equals √(36 + 64),2009

which means distance is equal to √100, which equals 10.2024

So, the distance between those two points is 10.2028

The final one: what if we get some things that don't turn out nicely?2031

We have these ugly decimal numbers: we still follow the exact same idea.2035

The distance is equal to the square root of...arbitrarily, we will just make this one the first one, and this one the second one...2039

the answer would turn out the same, because of all of the things that we talked about before.2045

It doesn't matter which gets turned into the second and which gets turned into the first; the distance is going to be the same between them.2048

If that doesn't really make sense, go back to when we talked about how we figured out that this is the formula.2053

And notice that (x2 - x1)2 is the same as (x1 - x2)2.2057

It comes up with the ideas where we were talking about absolute value before.2063

So, the first x is -0.2, minus the second x (is 2.5), squared, plus...the first y is 1, minus the second y (is 1.7), squared.2066

So, distance equals the square root of...-0.2 - 2.5 becomes -2.7, squared; plus...1 - 1.7 becomes -0.7, squared.2084

We use a calculator to figure this out; or we could do it by hand, but I used the calculator when I figured it out.2100

And I figured out before that (-2.7)2 becomes 7.29, plus...(-0.7)2 becomes 0.49,2105

because those negatives just end up canceling with each other when they square against themselves.2115

So, this is 7.29 + 0.49; so our distance is equal to the square root of 7.78 (what we get when we combine these two numbers--7.78).2119

That is the distance between those two things.2129

Of course, the square root of 7.78 is kind of hard to actually use, if we had to measure something and cut something,2132

if we were making, say, a bookshelf (who knows?); so we could approximate this.2139

We could take the square root of 7.78 with our calculator, and that would be approximately equal to 2.79.2144

In reality, the decimals actually keep going, and they keep shifting around forever, because 7.78 is not a perfect square.2149

So, we can't just come up with a nice, easy number; but 2.79...we can just cut it off, and we will say, "Yes, 2.79 is pretty good."2155

So, we get an approximate value; the true answer, though, is √7.78; that is what the answer really is.2164

But if we need to be able to work with something that we can actually know what the number is pretty close to,2172

because we are more concerned with having a sense of where it is than knowing precisely the right answer, then we could also get 2.79.2176

Most teachers would probably accept both; technically, √7.78 is a better answer than 2.79,2183

because we have lost a tiny bit of accuracy when we take the square root, and then we round it.2190

But we will be able to use both; they are both very good answers.2195

All right, the next one: what would be the distance between the origin and (3,12,4)?2199

So, we talked about things in two dimensions before: what if we had to deal with three dimensions, though?2206

This problem is going to happen in three dimensions.2211

We talked briefly about three-dimensional things before, where we have three different axes.2213

They are each perpendicular: so here is our x-axis; here is our y-axis; here is our z-axis; so it goes x, y, z, like this.2221

If we were to plot this point, just to try to get a sense of what is going on, then we would count out 3 on x;2234

we would count out 12 on y; and we would count out 4 on z.2242

So, our point would be out 3; from 3 out to a distance of 12; and then, from there, up 4.2248

I am starting to accidentally encroach into my words.2258

So, our line is going to look like this.2262

Now, notice, though: we could also think about a cross-section here.2266

This is getting a little complex to see: let me see if I can get it a little bit more sensible with my hands.2269

Imagine that here is our x; here is our y; here is our z.2275

So, our number is go x forward; it is going to go y 12, and it is going to go up 4 (forward by 3).2279

We go out like this: we are going to start here, and we are going to go in all of these, all at once.2286

We go out by 3; we go over by 12; and we go up by 4.2292

We could also think of this as being a cross-section: we could take a cross-section, and we could make a triangle in here.2298

We can say, "What happens in the x,y plane?": here is x; here is y.2307

We go over 3; we go up 12; and we can talk about how we got here--we can do that portion of our trip.2316

We go and travel x and y; then we go up.2327

We travel x and y; and we could figure out, "Look, the x and y plane is perpendicular to the z portion of our axis."2330

So, that is going to be perpendicular there, as well; so we can figure out what this length here is.2337

And then, we already know what this is; it has to be 4, because our height was 4.2342

If we could figure out what this portion right here is, what this length for our cross-section, the base of our triangle, is, we would be good to go.2347

We look at the x,y plane--we look at this portion here--and we can turn it into a nice, flat object that we can see--a nice planar object.2356

So, we do the same things that we have been doing before.2365

Here, 3 is our horizontal; 12 is our vertical; so our distance is going to be the square root of 12 squared, plus 3 squared.2368

Well, let's swap that around; that was exactly correct, but just to keep doing the exact same way we have been doing it precisely before.2379

We will have 32, because we had horizontal before first; and then 122, because we are always following with vertical.2387

The other way was just as right: 32 + 122 is the same thing as 122 + 32, after all.2393

But that way, we are just following our nice pattern from before.2398

So, we can figure out what the length of this portion right here is.2402

We have it right here: we now have that--now we just bring in this thing, and we just do another one.2408

For d of our triangle here, let's use a different color; we will go with green for the distance in our three-dimensional object.2414

In our three-dimensional object, it is going to be the square root of what it was in the x,y plane--that distance,2424

squared (what was it in the x,y plane? It was the square root of 3 squared plus 12 squared; that was its distance before;2430

but we have to square it), plus...what was the jump that it had up--what was its vertical leap in the z?--that was 4, so plus 4 squared.2438

Now, notice: when we take a square root and square it, like we have in here, d =...the square root of 32 plus2447

122, squared, is just going to crack it open, and we will get 32 + 122, and then plus 42.2453

We simplify this, and we get d = √(9 + 144 + 16); simplify that some more; we get √169, which equals 13; so our distance is 13.2460

What we have done is: we are able to look at how it changed on the first plane.2476

We sort of take a cut, so that we can look at how it changed in the x,y plane; and then, we put on the z.2481

Now, you might be getting a sense of "Oh, maybe there is something we could do in general."2489

And I didn't tell you this before, because it is not really going to come up much in this course.2494

But we can actually get a distance for three dimensions, as well.2497

It is going to be the distance of the square root of a whole bunch of stuff, now...of (x2 - x1)2,2500

the square in our horizontal, plus (y2 - y1)2, the square in our vertical,2510

plus (z2 - z1)2, the square in our coming out of the x,y plane.2517

That comes out, because what we do is clear out the x,y plane first.2525

It is going to have a square root around it; but then, when we put in that z--when we toss out that coming out perpendicularly--2528

we are going to square root again, because now we are doing another right triangle.2536

And so, it will simplify to just each one of these differences, squared.2539

This might be a slightly complex idea for you, so don't worry if this didn't make sense.2542

Just take it out of your head; throw it away; it is not really going to come up.2546

It is just a really cool thing that...if you are thinking, "Oh, there is something interesting going on here"...you are right!2549

This is the interesting thing that is going on; we can actually generalize this to three dimensions.2553

And if we wanted, we could even keep going to 4, 5, 6...any number of dimensions we want.2558

And you might have some idea of what is going to happen as we go on to four dimensions.2562

See if you can figure out what goes on in four dimensions--it is kind of cool.2565

All right, Example 4: What is the slope between (-1,8) and (1,14)?2568

So remember, we figured out that slope is rise over run; the amount that we rise is our change in our vertical,2573

(y2 - y1)--our two vertical locations--the change--and our run is our two horizontal locations--their change.2583

So, in this case, arbitrarily, let's make this one the second one, and this one the first one.2589

There is no particular reason; it is just because that one came first, and that one came second.2595

So, let's look at it that way: it also makes sense, because if we were to draw a picture of it, we would have something like this:2600

(-1,8), and then (1,14); so it makes sense that they give this one as the second one and this one as the first one.2606

But as we will see in a little bit, it actually doesn't matter which one we choose first.2612

So, we want to find out the slope between (-1,8) and (1,14).2616

So then, m = (y2 - y1)/(x2 - x1).2621

Our second y is 14; so 14 minus our first y (is 8), divided by our second x (is 1); our first x (is -1), so minus -1; that equals 6...2632

1 minus -1...those negatives cancel; we get 6/2 = 3, so our slope is 3.2648

What is the slope going to be if we switch our first and second points?2655

Instead, we make this one 1, and we make this one 2; well, we do the same thing: m =...2659

it is going to be...our new second one is 8, minus our new first one (is 14), divided by x2,2667

our new second one (is -1), minus our new first one (is 1).2681

8 - 14 is -6: -1 minus 1 is -2; will you look at that--these cancel, and we get the exact same thing.2688

If we switch our first and second points, which we arbitrarily decided to make second and first, does it affect what the slope comes out to be? No.2699

It doesn't--why? Because of the negatives: it introduces negatives on both the top and the bottom; they cancel out.2708

So, if we have negatives showing up because of the switch, they are going to show up on both the top and the bottom, so we will always see cancellation.2713

So, it doesn't matter if we plug in our one as the first one, or if we plug in that one as the second one.2719

It doesn't matter which one gets to be called first and second, as long as we match up our seconds and our firsts.2725

They have to match up vertically: if we have one point be the second point on the top, the second y-coordinate,2731

then it has to also be the second x-coordinate; it has to come first on the bottom, as well.2737

So, we have to make sure that the points match up vertically.2741

8 and -1; (-1,8); 14 and 1; (1,14); they match up there.2745

Let's prove this, though: if we want to prove that this always comes out to be the case,2753

to prove this, what we want to show is that it doesn't matter if it is (y2 - y1)/(x2 - x1),2759

versus (y1 - y2)/(x1 - x2); if we swap the location of which gets to come first,2771

which gets to be more on the left in the fraction, it doesn't matter which gets to be more on the left and which is more on the right.2781

That is what we want to show; so how do we prove it?2787

Well, let's start with this one here: we will have (y2 - y1)/(x2 - x1).2789

Now, we want to be able to get that to start looking like this thing.2800

And we say, "Well, (y2 - y1)...that is pretty much the same thing, but it has a negative introduced to it."2804

So, how could we introduce some negatives here?2811

Well, let's write it again: (y2 - y1)/(x2 - x1).2814

We could multiply it by 1, right...wait, wait, what?--yes, 1, right?--I can multiply anything by 1, any time I want.2818

You can't stop me from multiplying by 1; I can take any number and multiply it by 1, and it has no effect.2826

So, everything is equal to just itself times 1.2832

Now, the cool thing about math is that there are a lot of ways to say the number 1.2835

I can say 1 as 1, but I can also say it as 1/1; or I could say it as 5/5, or I could say it as -1/-1.2840

And that is how we introduce our negatives; and this is also the idea that is coming along when we change denominators.2849

We introduce by multiplying the same thing on the top and the bottom; we multiply by -1 on the top and -1 on the bottom.2855

Now, notice: since we are multiplying the top, we are not just multiplying the first part of the top.2861

We are multiplying the whole top; because it is multiplication, it is going to apply to this fraction as if it started in parentheses.2865

So, times -1, over -1; (y2 - y1) times -1 becomes -y2 + y1,2872

over -x2 + x1; and this thing right here is just the exact same thing as this thing right here.2881

We have just swapped the location; instead of -y2 + y1,2891

it becomes y1 - y2, what we are a little more used to seeing.2895

So, we have managed to prove that it doesn't matter what order we put it into, using this (y2 - y1)/(x2 - x1) formula.2899

It doesn't matter, because it is going to end up giving out the same answers.2905

But the really key idea to think about, when we are talking about slope, is that it is the rise over the run.2908

It is the rate of change--how quickly the line is changing.2912

All right, I hope you learned a bunch here; I hope it has been a great refresher, and everything is really understandable,2915

because we will be using these things a whole bunch, later on.2919

All right, see you at Educator.com later--goodbye!2922