Vincent Selhorst-Jones

Systems of Linear Inequalities

Slide Duration:

Section 1: Introduction
Introduction to Precalculus

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

 3 answersLast reply by: Professor Selhorst-JonesSun Jul 24, 2016 12:59 PMPost by Joshua Jacob on July 23, 2014Professor Vincent,Could you explain the second equation in blue for example 2?I graphed it out and even graphed it in my calculator and the shaded area is under the line, not above it.Thanks for the amazing lectures!Josh

### Systems of Linear Inequalities

• Remember, we solve an inequality in much the same way we solve an equation. However, instead of giving just a single answer, an inequality has an infinite variety of solutions. Also, remember, when doing algebra with an inequality, multiplying or dividing both sides by a negative number will flip the direction of the inequality sign.
• When we graph the answers to a linear inequality, we shade in the entire region that satisfies the inequality. Furthermore, the boundary line changes depending on if the inequality is strict ( < or > ) or not strict ( ≤ or ≥ ). If it is strict, the line is dashed, while if it is not strict, it is a solid line.
• To help figure out where to shade, begin by just graphing the boundary of the inequality (as if it were an equation). Then, once you've drawn the boundary (with the appropriate solid or dashed line), test some point on one side of the boundary. If the point satisfies the inequality, shade that side. If not, shade the other side.
• When finding the solutions to a system of linear inequalities, it is best to graph each inequality in the system. When we worked with linear equalities (=), we could use substitution or elimination, but that was because we were solving for a single answer. With a system of inequalities, though, we aren't going to get just one answer-we're going to have infinitely many (or none, if the shadings don't overlap). Thus, graph each inequality, shade each appropriately as above, and the solutions to the system are where all the shadings overlap.
• An excellent application of linear inequalities is linear programming. It helps us optimize systems and make the best choice given certain requirements. We start with a linear objective function that we are trying to maximize or minimize (such as profit or cost). The objective function is based off some number of variables. The variables in the objective function also have various constraints (given as a system of linear inequalities). This gives a region of feasible solutions that are allowed by the constraints. Our objective is to find the maximum (or minimum) of our objective function within those feasible solutions.
• The theory of linear programming says that if there is a max/min for the objective function within the constraints, it occurs at a vertex ("corner") of the feasible solutions. Thus, to find the max/min of an objective function, we begin by finding the locations of the vertices by solving for intersection points. Once we know all the vertices, we can try each of them in our objective function, and whichever is highest/lowest is our max/min.

### Systems of Linear Inequalities

Graph the solutions to the system.
 y ≤ 2x − 3               y >− 1 2 x + 5
• It helps to begin by putting the inequalities into slope-intercept form (y=mx+b, where m is the slope and b is the y-intercept). Of course, since we're dealing with inequalities, we won't be able to precisely fit the form (since it has an =' sign), but we can use the same structure. In this problem, both inequalities are already in that structure, so we're ready to start plotting them.
• Plot out a line based on each inequality. While plotting the line on the graph, treat it as if it were a normal equation that you were graphing. The only thing to keep in mind is that ≤ and ≥ get solid lines, while the strict inequalities of < and > get dashed lines.
• Once the lines are plotted on the graph (and solid or dashed, according to the inequality type), shade each one according to its inequality. This can be done in one of two ways. First, if it's in the slope-intercept structure (like ours are), you can look at the "direction" of the inequality sign. If the y is greater (y ≥ or y > ), then we shade above the line. If the y is lesser (y ≤ or y < ), then we shade below the line. Alternatively, if you're not sure of the above or the inequality is not in slope-intercept, choose a "test point" on the graph. Plug it in to the inequality. If it works, shade the side that the test point is on. If it does not satisfy the inequality (it "breaks"), shade the opposite side.
• Once you have shaded each of the inequality lines appropriately, look for where the shadings overlap. The overlap is the set of solutions.
Graph the solutions to the system.
 3x+y < 5               2x+2y <−4
• It helps to begin by putting the inequalities into slope-intercept form (y=mx+b, where m is the slope and b is the y-intercept). Of course, since we're dealing with inequalities, we won't be able to precisely fit the form (since it has an =' sign), but we can use the same structure.
 3x+y < 5     ⇒     y <−3x + 5

 2x+2y <−4     ⇒     2y <−2x−4     ⇒     y <−x−2
• Plot out a line based on each inequality. While plotting the line on the graph, treat it as if it were a normal equation that you were graphing. The only thing to keep in mind is that ≤ and ≥ get solid lines, while the strict inequalities of < and > get dashed lines.
• Once the lines are plotted on the graph (and solid or dashed, according to the inequality type), shade each one according to its inequality. This can be done in one of two ways. First, if it's in the slope-intercept structure (like ours are), you can look at the "direction" of the inequality sign. If the y is greater (y ≥ or y > ), then we shade above the line. If the y is lesser (y ≤ or y < ), then we shade below the line. Alternatively, if you're not sure of the above or the inequality is not in slope-intercept, choose a "test point" on the graph. Plug it in to the inequality. If it works, shade the side that the test point is on. If it does not satisfy the inequality (it "breaks"), shade the opposite side.
• Once you have shaded each of the inequality lines appropriately, look for where the shadings overlap. The overlap is the set of solutions.
Determine if the point (4,  −7) is a solution to the system.
 −3x−y < 6               4x+2y ≥ 2
• One way to approach this problem would be to graph all the solutions to the system of inequalities, then plot the point, and look to see if it is within the solution set on the graph. If it is, then it's a solution. If not, then it is not a solution. However, a much easier way to do it is to just check each of the two inequalities with the point. If it satisfies each inequality, then it's a solution. If it fails to satisfy one or both, then it is not a solution.
• Check to see if it satisfies each of the inequalities. [Remember, (4,  −7) means the combination of x=4 and y=−7.]
 −3x−y < 6     ⇒     −3(4) − (−7) < 6     ⇒     −12 + 7 < 6     ⇒         ⇒     − 5 < 6

 4x+2y ≥ 2     ⇒     4(4) + 2(−7) ≥ 2     ⇒     16 − 14 ≥ 2     ⇒     2 ≥ 2
• Thus, since the point satisfies ("is true in") each of the inequalities, it is a solution to the system. [Notice that just because it is a solution to the system does not mean it is the only solution to the system. There are infinitely many solutions, this point is just one of them.]
(4,  −7) is a solution to the system
Give the system of linear inequalities that produces the below graph.
• Begin by identifying the equation for each line first. Just treat it as a line equation to begin with, even though they will later become inequalities.
• Let's start with the red, upper line. We see it has a y-intercept of b=7. From there, it crosses the x-axis by dropping down −7 and moving right 7. Since slope is the change in the vertical divided by the change in the horizontal, we get
 m = −7 7 =     −1
Putting these two things together, we have y=−x + 7 for the red line equation.
• Now we will work on the blue, lower line. We see it has a y-intercept of b=−6. From there, it crosses the x-axis by rising up 6 and moving right 2. Since slope is the change in the vertical divided by the change in the horizontal, we get
 m = 6 2 =     3
Putting these two things together, we have y=3x −6 for the blue line equation.
• Now that we know equations to describe the lines, we need to figure out the appropriate inequality. For the red, upper line, we see that it is solid, so we will use either ≤ or ≥ . Notice that the solution set is always below the red line, so the y value must be less, giving us
 y ≤ −x + 7
For the blue, lower line, we see that it is dashed, so we will use either < or > . Notice that the solution set is always above the blue line, so the y value must be greater, giving us
 y > 3x −6
y ≤ −x + 7               y > 3x −6
Graph the solutions to the system.
 y ≥ 0               y <− 1 2 x + 4               y < 3x +9
• It helps to begin by putting the inequalities into slope-intercept form (y=mx+b, where m is the slope and b is the y-intercept). Of course, since we're dealing with inequalities, we won't be able to precisely fit the form (since it has an =' sign), but we can use the same structure. For this problem, everything is already in the slope-intercept structure, so we can move on to the next step. [If you're confused by how to graph y ≥ 0, remember, y=0 would just graph as a horizontal line at height 0.]
• Plot out a line based on each inequality. While plotting the line on the graph, treat it as if it were a normal equation that you were graphing. The only thing to keep in mind is that ≤ and ≥ get solid lines, while the strict inequalities of < and > get dashed lines.
• Once the lines are plotted on the graph (and solid or dashed, according to the inequality type), shade each one according to its inequality. This can be done in one of two ways. First, if it's in the slope-intercept structure (like ours are), you can look at the "direction" of the inequality sign. If the y is greater (y ≥ or y > ), then we shade above the line. If the y is lesser (y ≤ or y < ), then we shade below the line. Alternatively, if you're not sure of the above or the inequality is not in slope-intercept, choose a "test point" on the graph. Plug it in to the inequality. If it works, shade the side that the test point is on. If it does not satisfy the inequality (it "breaks"), shade the opposite side.
• Once you have shaded each of the inequality lines appropriately, look for where the shadings overlap. The overlap is the set of solutions.
Graph the solutions to the system.
 2x+3y >−12               3x−y ≥ − 8               −3x+6y < 30
• It helps to begin by putting the inequalities into slope-intercept form (y=mx+b, where m is the slope and b is the y-intercept). Of course, since we're dealing with inequalities, we won't be able to precisely fit the form (since it has an =' sign), but we can use the same structure.
• Converting them to slope-intercept structure, we get the following:
 2x+3y >−12     ⇒     3y >−2x − 12     ⇒     y >− 2 3 x −4

 3x−y ≥ − 8     ⇒     −y ≥ −3x − 8     ⇒     y ≤ 3x + 8

 −3x+6y < 30     ⇒     6y < 3x + 30     ⇒     y < 1 2 x + 5
• Plot out a line based on each inequality. While plotting the line on the graph, treat it as if it were a normal equation that you were graphing. The only thing to keep in mind is that ≤ and ≥ get solid lines, while the strict inequalities of < and > get dashed lines.
• Once the lines are plotted on the graph (and solid or dashed, according to the inequality type), shade each one according to its inequality. This can be done in one of two ways. First, if it's in the slope-intercept structure (like ours are), you can look at the "direction" of the inequality sign. If the y is greater (y ≥ or y > ), then we shade above the line. If the y is lesser (y ≤ or y < ), then we shade below the line. Alternatively, if you're not sure of the above or the inequality is not in slope-intercept, choose a "test point" on the graph. Plug it in to the inequality. If it works, shade the side that the test point is on. If it does not satisfy the inequality (it "breaks"), shade the opposite side.
• Once you have shaded each of the inequality lines appropriately, look for where the shadings overlap. The overlap is the set of solutions.
Give the system of linear inequalities that produces the below graph.
• Begin by identifying the equation for each line first. Just treat it as a line equation to begin with, even though they will later become inequalities.
• Let's start with the red, vertical line. This is possibly the most difficult because we can't give an equation in slope-intercept form. We have to realize that it has a fixed value of x=−5, while the y value is allowed to "roam freely". Thus, it can be described with the equation x=−5
• Now we will work on the blue, horizontal line. It has a y-intercept of b=−3 and, since it is horizontal, a slope of m=0. Thus it has the equation y=−3. [We can also see this by the same logic as we used above: y is fixed, while x is allowed to "roam freely".]
• Finally, we finish with the green, diagonal line. It has a y-intercept of b=−2. It crosses the x-axis from there by going up 2 while going backwards by −1. Since slope is rise over run, we have
 m = 2 −1 =     −2,
giving the line equation y = −2x −2.
• Now that we have equations to describe all of the lines, we need to figure out the appropriate inequality for each. For the red, vertical line, we see that it is dashed, so we will use either < or > . Notice that the solution set is always right of the red line, so the x value must be greater, giving us
 x > −5
For the blue, horizontal line, we see that it is solid, so we will use either ≤ or ≥ . Notice that the solution set is always above the blue line, so the y value must be greater, giving us
 y ≥ −3
For the green, diagonal line, we see that it is dashed, so we will use either < or > . Notice that the solution set is always above the green line, so the y value must be greater, giving us
 y >−2x −2
x > −5               y ≥ −3               y >−2x −2
Find the maximum and minimum values of the objective function z=3x+2y given the below constraints.
 y ≥ 0               y ≤ x               y ≤ −2x + 12
• We will approach this problem by the method of linear programming. We have various constraints on our variables, and the maximum/minimum to the objective function must come from a point within the space determined by those constraints. Specifically, from the theory of linear programming, we know the max/min must occur at one of the vertices (corners) of the space determined by the constraints.
• While it is not strictly necessary to graph the constraints, it can sometimes be helpful to assist us in understanding the problem. Graphing them carefully isn't the important part: just a rough sketch so that we can better understand what we're doing. Here's a graph of the constraints and the space they determine:
• We know that the minimum and maximum of the objective function (z) occur at the vertices of the constraint space (at the corners). Thus, we want to solve for what those points are. Find these intersections by solving each pair of constraints as system of equalities. First, we see where the lines y=0 and y=x intersect:
 0 = x     ⇒     y = 0
Next, where the lines y=0 and y = −2x + 12 intersect:
 0 = −2x + 12     ⇒     x = 6     ⇒     y = 0
Finally, the last pair of y=x and y=−2x+12:
 x = −2x + 12     ⇒     x=4     ⇒     y = 4
• Thus, our three vertices are located at the points
 (0, 0)               (6, 0)               (4,  4)
Now plug each of these points into the objective function z to find the value at those points:
 (0, 0)     ⇒     z = 3(0) + 2(0)     =     0

 (6, 0)     ⇒     z = 3(6) + 2(0)     =     18

 (4,  4)     ⇒     z = 3(4) + 2(4)     =     20
Therefore we see that plugging in (0, 0) gives the smallest number, while (4, 4) gives the largest number.
Maximum at (4, 4),    Minimum at (0, 0)
A laboratory is working on a chemical process and needs at least 50 kg of element type A and 4 kg of element type B. However, the lab cannot buy either element in its pure form. Instead, the lab must buy ores that contain the elements in various combinations. There are two types of ore the lab can buy, composed as follows:
 Commonium:  90% A, 10% B              Rarite:  20% A, 80% B
It costs $5 to purchase one kilogram of commonium and$20 to purchase one kilogram of rarite. How much of each ore should the laboratory purchase so that they can work on the chemical process while minimizing their expenditures?
• First, let's begin by naming variables:
 c = kilograms of commonium               r = kilograms of rarite
Once we have our variables in place, we can come up with an equation for the thing we're trying to minimize. We're trying to minimize the price (P) of the ore purchase. The total price is
 P = 5c + 20r
However, from the problem, we have constraints on the amount of commonium and rarium we must purchase: we need enough of the ores to hit our required totals of elements A and B.
• Working on the constraints, we know that we need at least 50 kg of type A. Thus, it must be that the total amount of A obtained from the ores is greater than or equal to 50 kg:
 50 ≤ 0.9c + 0.2r
Similarly, we need at least 4 kg of type B, so the total amount obtained from the ore must be greater than or equal:
 4 ≤ 0.1c + 0.8r
Those are all the constraints that the problem directly gave us, but there are two very important, implied constraints: we cannot have a negative amount of ore. While it was never directly stated, the idea is obvious-it is impossible to have negative kilograms of ore. However, it is important to state this mathematically so we can work with it as a constraint:
 c ≥ 0               r ≥ 0
• Since we have a set of linear constraints and an objective function (the price, P) that we are trying to minimize, we do this with linear programming. We need to find the vertices to the space of possible choices. To help us visualize the problem, let's graph the space of possible choices. To make it easier to graph, put the constraints in a form similar to slope-intercept (let's arbitrarily choose c to be the independent variable [what x usually is]):
 50 ≤ 0.9c + 0.2r     ⇒     0.2 r ≥ 50 − 0.9 c     ⇒     r ≥ −4.5c + 250

 4 ≤ 0.1c + 0.8r     ⇒     0.8 r ≥ 4 −0.1c     ⇒     r ≥ −0.125c+ 5
• Looking at our visual reference, we see that there are only two vertices: where the red line (from r = −4.5c + 250) intersects the vertical axis and where it intersects the horizontal axis. The intersection for the vertical axis will occur at c=0:
 r = −4.5(0) + 250     ⇒     r = 250
The intersection for the horizontal axis will occur at r=0:
 (0) = −4.5 c + 250     ⇒     4.5 c = 250     ⇒     c ≈ 55.556
• Thus we have two vertices to check in our price function: c=0,  r=250:
 P = 5(0) + 20 (250)     =     5000
c=55.556,  r=0:
 P = 5(55.556) + 20 (0)     =     277.78
Thus, it clearly is much less expensive to purchase nothing but commonium ore.
To minimize expenditures, the lab should purchase 55.556 kg of commonium ore and none of the rarite ore.
A factory manufactures two types of widgets: type A and type B. They sell each type A for $135 and sell each type B for$55. Manufacturing a type A widget takes 5 units of raw material and 2 hours of production labor. Manufacturing a type B widget takes 2 units of raw material and 1 hour of production labor. The factory has a total of 1000 units of raw material and a total of 440 hours of production labor. How many of each widget type should the factory manufacture to maximize its income? (Assume they sell everything they build.) What will that income be?
• Begin by naming variables:
 a = number of Type A               b = number of Type B
Once we have the variables in place, we can create an equation to describe the thing we're trying to maximize. The amount of income (I) the factory brings in is based on the number of each type of widget produced:
 I = 135a + 55b
From the problem, we have constraints on the number of widgets that can be produced. We must mathematically name these to go on.
• We know that each widget produced uses up some amount of raw material. We only have 1000 units to work with, so the total amount of raw material used must be less than or equal to that:
 1000 ≥ 5a + 2b
Similarly, there are only 440 hours of production labor to go around, so the total amount used must be less than or equal:
 440 ≥ 2a + b
Those are all the constraints that the problem directly gave us, but there are two very important, implied constraints: we cannot produce a negative number of widgets. It is important to state this mathematically so we recognize it as one of our constraints:
 a ≥ 0              b ≥ 0
• Since we have a set of linear constraints and an objective function (the price, I) that we are trying to maximize, we do this with linear programming. We need to find the vertices to the space of possible choices. To help us visualize the problem, let's graph the space of possible choices. To make it easier to graph, put the constraints in a form similar to slope-intercept (let's arbitrarily choose a to be the independent variable [what x usually is]):
 1000 ≥ 5a + 2b     ⇒     2b ≤ 1000 −5a     ⇒     b ≤ −2.5a + 500

 440 ≥ 2a + b     ⇒     b ≤ 440 −2a     ⇒     b ≤ −2a + 440
• Looking at our visual reference, we see that there are a total of four vertices. b = −2a + 440 and a=0:
 b = −2 (0) + 440     ⇒     b = 440     ⇒     a = 0
b = −2.5a + 500 and b = −2a + 440:
 −2.5a + 500 = −2a + 440     ⇒     a=120     ⇒     b = 200
b = −2.5a + 500 and b=0:
 0 = −2.5a + 500     ⇒     a = 200     ⇒     b = 0
a=0 and b=0
• Finally, we check all of our vertices using the income function: a=0,  b=440
 I = 135(0) + 55 (440)     =     24  200
a = 120,  b=200
 I = 135 (120) + 55(200)     =     27  200
a = 200,  b=0
 I = 135(200) + 55(0)     =     27  000
a=0,  b=0
 I = 135 (0) + 55(0)     =     0
Thus, of all the vertices, the one that gives us the maximum income is a=120, b=200.
Income is maximized by producing 120 type A widgets and 200 type B widgets. The income will be \$27  200.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Systems of Linear Inequalities

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 0:04
• Inequality Refresher-Solutions 0:46
• Equation Solutions vs. Inequality Solutions
• Essentially a Wide Variety of Answers
• Refresher--Negative Multiplication Flips 1:43
• Refresher--Negative Flips: Why? 3:19
• Multiplication by a Negative
• The Relationship Flips
• 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
• How to Show The Difference Between Strict and Not Strict Inequalities
• Dashed Line--Not Solutions
• Solid Line--Are Solutions
• Test Points for Shading 11:42
• Example of Using a Point
• Drawing Shading from the Point
• Graphing a System 14:53
• Set of Solutions is the Overlap
• Example
• Solutions are Best Found Through Graphing 18:05
• Linear Programming-Idea 19:52
• Use a Linear Objective Function
• Variables in Objective Function have Constraints
• Linear Programming-Method 22:09
• Rearrange Equations
• Graph
• Critical Solution is at the Vertex of the Overlap
• Try Each Vertice
• Example 1 24:58
• Example 2 28:57
• Example 3 33:48
• Example 4 43:10

### Transcription: Systems of Linear Inequalities

Hi--welcome back to Educator.com.0000

Today, we are going to talk about systems of linear inequalities.0002

In the previous lesson, we worked with systems of linear equations.0005

In this lesson, we will work with systems of linear inequalities.0009

Remember: equations are based on equals signs--left side = right side.0012

But inequalities are relationships based on less than (I should perhaps use something different),0016

less than or equal, greater than or equal, or greater than.0023

Since we haven't really worked with inequalities before in this course, we will first have a brief refresher.0028

After that, we will consider systems of linear inequalities and how their solutions are based on all the inequalities at the same time.0033

Finally, we will see how systems of linear inequalities can be applied through linear programming, which will allow us to solve optimization problems.0039

Let's go: when solving a linear equation in one variable, we always find exactly one solution.0046

For example, if we have x + 1 = 0, we subtract by 1 on both sides, and we get x = -1.0051

So, on the number line, we would see that the answer to this is when x is -1 on the number line.0056

On the other hand, when we solve a linear inequality in one variable, we find an infinite variety of solutions.0063

So, if we had x + 1 ≥ 0, we subtract 1 on both sides, and we get x ≥ -1, which means we have x equal to -1 as one of the solutions;0070

but we also are allowed to go anywhere above that.0080

+3 would be an answer to this; +52 would be an answer to this; -1/2 would be an answer to this.0083

Everything from -1 on up is an answer to this--everything there will satisfy our inequality.0089

This idea of a wide variety of answers will be important to us as we work with systems of linear inequalities, so we want to keep this in mind.0096

Mostly, when you are doing algebra with an inequality, it is just the same as doing it with an equation.0104

We can do any arithmetic operation--addition, subtraction, multiplication, and division--on both sides, just like normal algebra.0109

However, there is one special thing to be aware of: if you use algebra to multiply or divide by a negative number, the relationship flips.0117

For example, if we have -x + 1, we can subtract 1 on both sides, and we get -x < -1.0126

At this point, we want to make our x positive, so we multiply both sides by -1.0131

That causes our inequality symbol to flip to the other way; so we go from less than to greater than now.0136

So, we now have x > 1; this is really important to remember.0146

Any time you multiply or divide by a negative number, it will cause0153

your less than/greater than/less than or equal to/greater than or equal to/whatever you are dealing with...0157

it will flip; your inequality symbol will flip to the other direction.0163

It is really easy to miss this flip; it is a really easy mistake to make, so just be careful with this.0167

Every time you are doing either multiplication or division, and it is a negative number that you are dealing with--0172

negative anything that you are multiplying on both sides--you want to make sure you remember about this flipping thing.0178

Notice that we can also get the same thing as we have here if we just added x to both sides.0184

If we had had +x and +x, we would have had 1 < x, which is equivalent to saying x > 1.0188

They are just the same statement, but two different ways of saying it.0197

What causes this to happen--why do we see this negative flipping?0200

You probably learned about it in algebra for a very long time, but you might not have a great understanding of it.0204

So, let's actually get this: consider if we had some a and b, where a is less than b.0208

On our number line, a would come before b; a would be closer to the 0 than b would be.0213

So, what would it look like if we multiplied a and b by -1?0219

Well, that would cause this mirroring around the 0; they would pop to the side opposite, depending on the distance that they originally were from the 0.0223

So, -b and -a appear here; but notice that -b is now farther to the left than -a.0232

Just as b was farther to the right, it is now going to be farther to the left.0238

So, there is still a relationship, but now the relationship has flipped.0242

Because b was more positive than a--b was originally more right than a--we know that -b has to be more negative,0245

has to be more left, than -a; thus, we have this -a > -b,0252

because originally a is less than b, but now -b is less than -a, so that causes our sign to effectively flip,0258

because -b < -a is the same thing as saying -a > -b.0266

Great; that is what is going on.0274

You want to stick to basic operations; when you are working with inequalities, you want to try to keep using addition, subtraction, multiplication, and division.0276

You can use more complicated algebra, like raising both sides to a power or taking a square root or taking a logarithm or taking some trigonometric function.0284

But they can cause little problems to come up.0293

Consider if you have x2 > 0; now, that does not imply that x is greater than 0.0296

We might think, "x2 > 0, so I will take the square root of both sides."0302

√x2 > ±0...well, that is just 0, so we have x > 0, right?0306

No, that is not true at all: over here, x = -3 is true--it checks out: (-3)2 is greater than 0, because 9 is greater than 0.0312

So, that is great; but -3 > 0--that is completely false; that is not a true statement.0322

So, we do not have this ability to just toss out other, more complicated algebra things0330

without really thinking about all of the implications that are going to happen here.0335

So, this idea here doesn't work, because we are dealing with an inequality.0339

We can't just take square roots, like we can when we are dealing with equations, which is how we built up all that previous theory.0344

So, when you have an inequality, and you want to do something more than just a basic operation on both sides,0350

you really have to be careful and think about what you are doing.0355

That means that it is best to stick to basic arithmetic;0359

so we really just want to stick with addition, subtraction, multiplication, and division,0363

not because it is impossible to do this other stuff--not because you wouldn't get it--0367

but just because it is easy to make a mistake doing it.0370

You can end up making mistakes, because you haven't thought about absolutely all of the ramifications.0373

So, you want to stick to the basic operations, because it is easy to see what happens.0378

And remember: even with the basic operations, when we have multiplication and division,0381

there is still a little bit of danger there, because when you do it with a negative number, it causes flipping of your inequality.0385

So, with all of this in mind, it is really just best to stick to basic arithmetic.0391

Happily, since we are only going to be working with linear inequalities, that means we can get through everything0395

just using these basic operations--addition, subtraction, multiplication, and division.0400

They are going to be enough to manipulate our linear inequalities into forms that make it easier for us to understand.0404

All right, that finishes our refresher; now we are ready to actually talk about linear inequalities in two variables.0409

A linear inequality in two (or more) variables tells us about a relationship between those variables.0415

Consider -x + y ≥ 1: we can make it easier to interpret by just having y on one side and other stuff involving x on the other.0420

So, we add x to both sides: + x, + x; and we get y ≥ x + 1.0429

This gives us a relationship between x and y--how they are connected.0435

If x is equal to 1, we know that y has to be greater than or equal to 2.0439

If x is 1, then y has to be greater than or equal to 1 + 1, so y must be greater than or equal to 2.0444

Similarly, if x were equal to -9, then that would mean that y is greater than or equal to -8.0451

So, depending on what the x is, we will change the requirements on our y.0456

It is a relationship: it is not this definite single static defined thing.0459

It is this relationship of "as x shifts, y has to shift in certain ways to accommodate the shifting x."0464

Now, notice: unlike a linear equality, where a single x would give us just a single y, a single x in an inequality gives infinitely many possible y's.0470

x = 1 means y is greater than or equal to 2; so x = 1 means that y = 2 is good; y = 5 is good; y = 47 is good; y = 1 billion is good.0482

We have this stack as possibilities, as long as they follow that inequality relationship.0493

If x equals 1, y still can't be equal to -3; so it is not that anything goes--it just has to follow this inequality relationship.0497

But if it does, there is an infinite variety of things that it could end up being.0505

The best way to understand inequalities is by graphing them.0509

This gives us a visual reference to see all of the possible solutions.0512

So, for example, if we had y ≥ x + 1, we could graph it like this.0515

Because it is a greater than or equal to, everything on our actual line is allowed as a solution,0521

because y = x + 1 is going to be true by "greater than or equal."0530

That "or equal" means that we can have equality; the line itself will be true.0534

But then, in addition to that, everything above the line is also going to be true.0538

Everything in here is going to end up being true; and you should notice that it doesn't just stop at the edge there.0544

Once again, it is like how we don't draw arrows to show that it keeps going.0550

We have gotten used to assuming that the graph keeps going beyond the edges of our graphing window.0553

We are going to also be assuming that our shading goes past the edges; it doesn't stop suddenly--it goes forever, infinitely, up.0557

So, everything above the line is a solution, because if y is greater than or equal to x + 1, then consider at x = 0:0566

then 1 is true for y, but so would 2 be true for y...0574

Let's use a different color, just so we can see this a little bit easier.0577

2 would be true for y; 3 would be true for y; 4 would be true for y; 5 would be true for y; 6 would be true for y; and anything above that.0580

So, as long as we are above this line with our y-value for a given x-value, we will end up being true; the y > x + 1 is true.0587

So, a graph is a really great way to get across all of these possibilities, because it is not just this single line.0597

It is certainly not just a single point; it is this huge variety--it is this large area of possibilities.0602

This brings up a question: what does the line become if the inequality is strict--less than or greater than--0609

as opposed to not strict, which is less than or equal to, or greater than or equal to?0615

So, if we have a strict inequality, less than or greater than (it is purely less than, or it is purely greater than),0619

we show this by changing how we draw the line on the graph.0628

So, if it is a strict graph, like y > x + 1, we show that it is strict by using a dashed line.0631

It is a little bit hard to see here; but notice that the line is actually dashed, like this.0639

So, that says that we are saying that we are not including the line; we are only including the stuff above the line; the line itself is not allowed.0644

On the flip side, if it is a not-strict inequality, like greater than or equal to, we show it by using a solid line,0653

saying that the line itself is actually going to be an answer, as well.0661

If you are on the line, that is an allowed answer, in addition to all of the stuff that is above it.0665

So, we show that the points on the line are not solutions to our inequality with a dashed line, when it is a strict inequality.0671

We show that they are solutions with a solid line when it is a not-strict inequality.0684

And then, we just shade the appropriate side.0692

In both of these cases, we have greater than and greater than or equal to; so they both had the part that was above it.0694

Now, we don't necessarily have a very great way to see how to shade which way.0702

You will get a sense of if it is up, or if it is down, as you work with these more.0707

You can pay attention to what the y is, and if it is less than or greater than.0710

But a really great way to be sure of which side to shade is by using a test point.0713

You can shade by seeing how this test point gets affected.0719

Would this test point satisfy our inequality, or would it fail to satisfy it?0723

If the point satisfies the inequality (you just choose some point--it is completely your choice--0729

you choose some point, and if it satisfies that inequality), then the whole side does,0735

because we have to shade in an entire side.0739

So, if one point on one side satisfies, that entire side has to satisfy.0741

On the flip, if the point does not satisfy the inequality, then the other side will satisfy the inequality.0745

So, if your point fails to satisfy, the opposite side will satisfy it.0755

This makes it easy for us to shade: for example, if we have y < -5/2x + 3, we could try the test point (0,0).0759

(0,0) is almost always a great test point, because you can plug in 0's really, really easily.0771

Let's see: would that end up working out, if we have y < -5/2x + 3?0777

We plug in 0 for y; we plug in 0 for our x; plus 3...so we have...is 0 less than 3?0783

Sure enough, that checks out; so our test point is good.0792

If our test point is good, that means that the entire side must be good; so we just shade this side to say anything below that line is going to be true.0795

And notice: how did we get this line in the first place?0808

We just take our inequality; we turn it into a line, as if it had been an equation, y = -5/2x + 3;0810

and then, we graph that, because we know that that is going to be our line of demarcation.0817

It is going to be the place where the shading changes over.0821

And then, it gets dashed, because it is a strict inequality; so that is why we have that dashed line.0824

And then, we shade off of that line that we drew.0830

Alternatively, we could have tested a different point, like, let's say, (4,4).0834

Maybe we had wanted to test (4,4); well, if we had tested that, it would be 4 for our y, less than -5/2 times 4, plus 3.0838

So, 4 < -5/2(4)...so that gets us -10...+ 3; so is 4 less than -7?0848

No, that fails to be true; so this side cannot be the side that get shaded in; that side is not going to end up being true.0865

But really, I recommend using the point (0,0).0873

You could use any point; you could use (1,0), (-5,3)...whatever point is useful for your specific case.0875

But (0,0) is almost always going to be something...you want to choose something that is definitely on one side or the other.0881

You don't want to choose something on the line itself.0886

But (0,0), as long as it is not on one of your lines, is going to be a really good test point for using.0888

All right, if you are graphing a system of inequalities, you do it in the same way.0893

You graph each of the lines as if they were equalities; and you use either a dashed line0898

(if it is a strict inequality, less than or greater than); and you use a solid line if it is a not-strict inequality0904

(less than or equal, or greater than or equal); and then you shade in each inequality appropriately.0912

And wherever the shadings overlap is the set of solutions to the system.0917

So, let's see how that would work out here.0921

For this one, y < 2/3x - 2, that is our red dashed line--dashed because it is a less than.0922

So, let's try the test point (0,0) for that: (0,0)...we plug that in; we have 0 < 2/3(0) - 2.0930

So, 0 < -2 is not true; 0 is greater than -2; so this fails.0945

So, that means that the side opposite our test point is going to have to be what we shade in for the red one.0955

Our test point was in the very center, in the origin; so we are going to be opposite that side.0961

So, this is going to be what we shade in for the red line: y < 2/3x - 2 is true for everything in the shading, but not on the dashed line itself.0966

Then, if we have y ≥ -2x + 1, let's use that same test point; it is not on that line, either.0979

So, (0,0): 0 ≥ -2(0) + 1; so 0 ≥ 1; is that true?0985

No, 0 is not greater than or equal to 1; so that fails here, as well; 0 is not greater than 1.0998

That fails; so by the same logic, we are going to have to be opposite that side, greater than or equal;1006

so we are going to be above it; so we shade in the side that is opposite.1013

Also, you can think of this as being that y is greater, so it must be above the line;1019

and the red one was "y is less than," so it must be below it.1023

But the test points are a nice way to be absolutely sure of it.1026

And notice how we have overlapping; we have red and blue overlapping in this bottom-right portion.1029

So, that means the part that really, truly makes up the answers is everything in here.1034

And that is going to keep going out forever, as long as it is in there.1043

Everything on the blue line actually would be an answer, but things on the red dashed line...1047

The blue solid line here is going to be answers; but the red dashed line here, since it is dashed, fails to be answers,1052

because it doesn't satisfy our inequality y < 2/3x - 2, because that is strict.1064

So, if it is on the blue line in that shaded portion, it is an answer; if it is on the dashed red line, it is not an answer.1070

And there we are--we can see how the system comes together; we can see all of the solutions to it at once.1077

All right, now: solutions are best found through graphing.1086

Graphing is usually the best way to understand the solutions to a system of linear inequalities, as long as it is in two dimensions.1089

For dealing with really high numbers of dimensions, it gets kind of hard to graph.1095

But if we are in two dimensions, it is a good way to understand what is going on.1098

It gives us a way to visually comprehend what we are seeing.1101

It lets us see all of the requirements that the system places on the variables.1105

So, this system of inequalities makes certain requirements of our variables.1110

And by graphing it, we are able to see all of the requirements at the same time.1116

Now, with a linear equality--if we had a linear equality (an equals sign between the left and right side),1119

we could use substitution or elimination, because we were solving for a single answer.1126

A linear equality is solving for one answer.1131

But with a system of inequalities, we aren't going to get just one answer; we are going to have infinitely many answers, because we have shadings.1135

As long as we have overlapping shadings for our system, that entire area of overlap,1145

where all of our inequalities overlap together--that is going to be all of our answers.1150

And because it is an area, there are infinitely many things inside of that area.1155

Now, it is possible for us to have no answers whatsoever, if the shadings don't overlap--1158

if one side shades up, and the other side shades down, and they never touch each other.1163

That is never going to have any answers, because the two can never agree; they are inconsistent.1168

But as long as they end up agreeing in some portion--some amount of area, inside of that area we will have infinitely many answers.1172

So, when working with a system of inequalities, you will almost always want to graph,1178

because you can graph it and then shade in the solutions, and you are able to see what is going on,1183

and get an intuitive, visual sense of how this thing is coming together.1188

All right, now we are able to talk about linear programming: an excellent application of linear inequalities is linear programming.1193

It is not quite like computer programming; but it helps us optimize...1201

It is actually quite different than computer programming; but in any case,1205

it helps us optimize systems and make the best choice, given certain requirements.1208

What does it mean to optimize? Well, we start with a linear objective function.1214

We have something that is our objective, something that is sort of looking at the variables that we are dealing with.1218

And we are trying to either maximize the objective function, or we are trying to minimize the objective function.1224

So, we might want to try to maximize something like profit; or we might want to try to minimize something like cost.1229

These are useful things; business is something where linear programming will definitely pop up.1237

For example, we could have an objective function like z = 2x + 7y.1241

So, if we have z = 2x + 7y, it uses x; it uses y; it is an objective function.1247

We get some number out of 2x + 7y; we want to try to maximize or minimize this quantity, z.1252

Now, notice: if there are no restrictions on x and y, z will be at its biggest when x is flying out to infinity and y is flying out to infinity.1263

It will be minimized when x is flying to negative infinity and y is flying to negative infinity.1273

So, we need to have some limitations on what x and y are going to be for us to actually be able to find any maximum or minimum, for that to be meaningful.1277

And indeed, when you are working with linear programming, the variables in the objective function,1285

these variables x and y, are going to have various constraints on them--things that keep them from being able to go anywhere.1289

And those constraints will be given as a system of linear inequalities.1295

So, we might have constraints like x + 3y must be less than or equal to 8,1300

2x + y must be less than or equal to -4, and -x + 2y must be greater than or equal to -3.1304

So, our objective will be to find the maximum or minimum of our objective function1309

(remember: our objective function, in this case, was our z = 2x + 7y) while we are obeying these constraints.1315

We can't break these constraints; so we have to stick within these constraints,1321

but we are trying to get z to be the biggest or the smallest thing we possibly can get out of it.1325

So, once we know what our objective function is (z in the previous slide), the first step is to graph the system of linear inequalities.1331

To do that, we probably want to put them in a form that we can easily graph.1337

So, x + 3y ≤ 8--we can convert that into y ≤ -1/3x + 8/3: we subtract x and divide by 3.1340

We can convert 2x + y ≤ -4 into -2x - 4 by subtracting 2x.1351

And -x + 2y ≥ -3--we add x to both sides and divide by 2; so we have things that are easy for us to graph.1358

We have slope; we have the y-intercept at this point; so we can graph these.1366

We graph these, and we get this; and we also shade in, and according to our shading,1370

we find out that what is inside of there is going to be what is allowed by these constraints.1375

So, the shaded area is the location of all feasible solutions--and by feasible, we mean those that are allowed by the constraints,1381

those that are possible things that we can even begin to consider using.1388

So, the location of our optimum value, whether it is a maximum or a minimum, must be inside of this portion.1392

It has to be inside of this shaded thing or on the lines, because these were all less than or equal or greater than or equal.1399

They are all not strict, so we can be on the lines, as well.1405

So, since we can be on the lines, it has to be somewhere either on the lines or inside of that shaded portion.1408

We know that for sure; otherwise, it won't have followed the constraint, so we can't even look at it for trying to use our objective function.1413

Here is the critical idea: the theory of linear programming tells us that, if the solution exists--1421

if there is some optimum maximum or minimum--if we can maximize or minimize our objective function--1427

if that exists, then it occurs at a vertex of the set of feasible solutions.1434

Now, a vertex of the set is one of the corners of the set, where two or more of them intersect.1441

That is going to be one of the vertices; a vertex occurs at a corner.1451

Now, we can find the locations of the vertices by solving for intersection points.1455

We can see where the red line intersects the blue line, where the green line intersects the blue line, or where the green line intersects the red line.1459

We are just using what we learned from systems of linear equations to be able to figure out where two lines intersect each other.1466

We can figure out where these vertices are located.1472

Once we know all of the vertices, we just try each of them in our objective function.1475

We can figure out each of these vertices from what we learned in the previous lesson.1480

We can understand what is going on from what we have seen in this lesson.1483

And then, from there, we just plug them into our objective function;1486

and whichever ends up coming out to be highest or lowest ends up being our maximum or our minimum.1489

We will see this process done in Example 3 and in Example 4.1494

All right, we are ready for some examples.1498

The first one: Give the system of linear inequalities that produces the below graph.1501

So, the first thing: let's figure out what are the lines that we are seeing drawn here.1505

Before we even worry about dashed...is it greater than? Is it less than? Is it less than or equal to? Is it greater than or equal to?...1509

before we even worry about that, let's just figure out what equation could draw each of these lines.1516

For our red one right here, we see that it is a horizontal line; it is at y height of -2.1521

So, that would be made up by y = -2; that equation would draw that line.1527

Of course, it is not going to be a less than or equal or a greater than or equal, because it was dashed.1532

But we know that that line would be drawn by y = -2; we will come back to figuring out what it is later.1538

Our green one is going to be set at a horizontal location of 4; it has no slope whatsoever (it is vertical).1543

So, that is going to be x = 4; we have set our horizontal location at 4, and our y is allowed to freely roam up and down.1551

Finally, the only one that might even be slightly difficult is this blue dashed line, which goes from (0,4) to (2,0).1559

If that is the case, we can figure out what the slope of our blue one is.1568

It manages to go down four; it has a vertical change of -4 over a horizontal change of +2.1572

It goes from a height of 4 to a height of 0 by going 2 steps to the right.1580

That means it has a slope of -2; we also see that its y-intercept is right here at 4.1583

So, its y-intercept is 4, so we have y = -2x + 4.1589

At this point, we have equations that would produce each of these lines.1597

So now, it is a question of what symbol goes in there; we can't use equals, because that would be an actual line.1601

We need inequalities; so now it is a question of less than, greater than, less than or equal to, or greater than or equal to, or each of these.1607

All right, we are going to swap out that equals sign for something that can actually be used as an inequality.1616

So, the first step: the red has to shade above, because we have stuff above it,1622

because if we go below it, that fails to be true; so it must be above it.1628

That means that y has to be strictly greater than (because it is a dashed line) -2.1633

If we plug in 0 > -2, that works out; so we know that that side is correct; that is the side that we want.1648

For the green one, we see that it is one the left side of that.1655

So, if it is on the left side, we know that the horizontal values that we are allowed to use must be less than 4.1659

Is it less than, or less than or equal to? It is less than or equal to, because it is a solid line: less than or equal to 4.1665

We can also check out a test point: if we plug in (0,0), then we have 0 ≤ 4; yes, that is true.1673

So, we know that that is correct for that shading.1679

And then finally, our blue one: for this one, we know that it is shading up.1681

So, if it is shading up, then it must be that our y-values are greater than1688

(and it is going to be strictly greater, because it was a dashed blue line) -2x + 4.1693

If we want to check that with a test point, we could try a test point, just like we did with the others.1702

This one is a little bit harder to do in our head, since it involves more things.1707

0 is greater than -2(0) + 4; 0 > 4; that test point fails, so indeed, it has to be shaded on this side--we chose the correct one.1711

I'm sorry; I didn't mean to make that look like a greater than or equal to.1723

We chose the correct symbol: y > -2x + 4, so in total, these are the inequalities that make that system be graphed like that.1725

Great; the second example: Graph the solutions to the system below.1737

The first thing we want to do--we probably want to convert this into a form where we can easily graph these.1741

Divide by -3; since we are dividing by a negative number, the sign flips.1747

We have y ≤ -2; next, 3x + y < 3, so y <...subtracting 3x...-3x...+3...we don't have to worry about flipping, because it is addition or subtraction.1751

And then finally, 6x - 4y...-4y < -6x - 4; divide by -4: y...it flips because we divided by a negative number, so -6/-4 becomes +3/2x; -4/-4 becomes positive 1.1766

At this point, we have enough for us to graph this thing; so let's draw some axes in.1784

We will make a tick-mark spacing; it is just a length of 1, so 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5.1795

Great; at this point, we can draw each one of these in.1818

We will draw them in as lines first; and then, from there, we will figure out how to shade.1821

y ≤ -2; we use a solid line, because it is less than or equal, so it is not strict.1826

At height -2, we are going to have a line, because we said y ≤ 2; so we graph in the line as if it was y = -2.1832

We have just set a vertical height of -2, and our x is allowed to freely roam.1842

Next, the red one: y < -3x + 3, so our first point is at the y-intercept of 3.1847

If we go over one step, we will have our slope of -3 kick in; we will go down 3, so we can draw this one in.1853

All right; oops, I should have had that be dashed, because it was a less than, so it is strict; I will dash that with the eraser.1865

And then finally, our green one: I won't forget to dash this one.1879

3/2x + 1; so we have a y-intercept at 1; if we go over 2, we go up 3, because our slope is 3/2: over 2, up 3; 1, 2, 3.1882

We draw this one in; all right, and now let's figure out our shading.1894

y ≤ -2; well, if it is less than or equal to -2, we have to have shading below that; so we shade below that here,1909

because we can use a test point, like (0,0), but we also see that y has to be below -2; otherwise that is going to not be true.1925

So, we shade in below that.1933

For our red one, y has to be less than -3x + 3; once again, the y's will have to be below this value.1934

We can use a test point, like (0,0); if we plug that in, we will have 0 < -3(0) + 3, which means 0 < +3, which is, indeed, true.1940

So, we shade towards that test point of (0,0).1950

And then finally, we have a green one: y > 3/2 + x; 3/2 + x is going to get us this one here.1956

So, that is going to cause us to shade up; if we plug in a test point of (0,0), we have 0 > 3/2 times 0, which means 0 > 1.1975

Oops, if 0 is greater than 1...that would fail to be true, so we are shading the opposite way; we are shading away from our original test point.1987

Our original test point was (0,0), so we are shading in the opposite direction of that.1997

Let's extend our red so we can see a little bit better where we are going.2001

At this point, we see that the only place that ends up having green, red, and blue together is this section over here.2005

Here (where I am shading in with a zigzag purple)--this section over here, and continuing out in that arc, will end up being the answers to our solution.2013

We see the solution graphed in that way.2025

All right, the third example: Find the maximum and minimum values of the objective function z = 2x + 7, given the below constraints.2028

The first thing we do is change them into a form so we can easily graph them and have a better understanding of what is going on.2037

Just like we had before, this is the same example that we were working with2045

when we were talking about the idea of linear programming, maximizing and minimizing objective functions.2048

So, this is the same one that we had before: y ≤ -1/3x + 8/3; 2x + y becomes y ≤ -2x - 4;2054

and then, we have y ≥ -1/2x...sorry, not -1/2; it gets canceled out as it adds over, so +1/2x - 3/2; great.2067

So, let's just give a rough sketch, so we can see what is going on.2082

We don't have to worry about having this perfectly precise.2085

The idea is just to be able to see this, because unlike graphing the solutions to an inequality,2087

where we never really figure out the thing, so we want to have a solid graph; in this case,2093

the graph is just a reference, so we can have a better understanding of how the solving is working.2096

So, -1/3x + 8/3; I will start at around some height of about 3.2101

And so, let's actually put in marks, so we have some rough sense of where we are going here.2109

1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5; great.2116

OK, the red one: positive 8/3, so we are almost at the 3; and then 1/3...it will slowly slope down to...2126

by the time it gets here, it will be about here; you could drop down...and it is going to be a solid line,2134

because it was less than or equal to; so it is like that.2140

And then, y ≤ -2x - 4; at -4...and then, for every one it goes to the left, it will go up 2, 2, 2...2143

Also, it is solid once again, because it was a less than or equal.2158

And then, the final one is also going to be solid; it is less than or equal to 1/2x - 3/2, so halfway between the -1 and the -2.2162

At 1/2x...so there...OK; it is not a perfect graph, but it gives us a rough idea of what we are looking for.2170

We are looking for something that is going to be in this part, including the lines.2181

Now, the theory of linear programming--what we learned about that--the critical idea--was that we know2187

that the answer is going to be (if there is an answer at all) on one of these vertices.2192

For the maximum and the minimum, each will show up as a vertex on this triangle (or whatever kind of picture we have).2198

It is going to show up as a vertex on this image--one of the corners to our set of allowable points/feasible solutions.2205

So, let's figure out what it is: let's do the red and the blue together first.2213

If that is the case, we can treat this as y = -1/3x + 8/3, and y = -2x - 4,2217

because for this, we are trying to figure out where the lines intersect, because we are looking for vertices.2224

-1/3x + 8/3 = -2x - 4; where do our red line and our blue line intersect?2228

Multiply everything by 3; at this point, I am just going to use blue.2240

Multiply everything by 3 for ease: -x + 8 cancels out those fractions; -6x - 12...2242

We add 12 to both sides; we get 20 over here; add x to both sides; we get -5x; divide by -5 on both sides; we get -4 = x.2250

We can plug this into one of the two functions; let's go with the -2x - 4, because that will be easier.2262

Once again, we can use it as an equality, because we are looking for a line intersection.2267

We aren't worried about the inequality effect; we are worried about "as if it were a line," because we only care about the vertex right now.2271

So, we can plug that in, and we will have that y = -2...plug in our x, -4; minus 4; y =...-2(-4) gives us + 8, minus 4; so y = 4.2277

Great; so one of our vertices is (-4,4).2290

Next, what if we had our red intersect our green?2295

-1/3x + 8/3--if that was the line--would be equal to (because we are looking for the intersection) 1/2x - 3/2.2299

So, where do those intersect? Let's multiply everything by 6, just so we can get rid of these fractions; I think that makes it easier to look at.2309

So, multiply everything by 6; I will just choose red for the color I will use here.2316

That gets us -2x plus...8/3 cancels out the 3 portion of the 6, which means it is going to be 2(8), 16.2322

Now, we are multiplying by 6 still, so it is going to give us 3x, multiplying by 6, minus 9, because 3/2 times 6 becomes 9.2330

At this point, we can solve this out: add 2x to both sides; add 9 to both sides.2338

Adding 9 to both sides, we get 25 =...add 2x to both sides...5x/5; 5 = x.2342

We can now plug this into either one of them; to me, the y = 1/2x - 3/2 looks ever so slightly friendlier, so I will plug into that one.2348

So, y = 1/2x, this one right here, so y = 1/2(5) - 3/2; so 5/2 - 3/2 = 2/2, which equals 1.2358

At this point, we have another vertex; 5 = x and y = 1; so that is going to occur where the red and green intersect over here, (5,1).2375

And then finally, let's look at where the blue and the green would intersect, the one that we haven't looked at yet.2390

-2x - 4 = 1/2x - 3/2; multiply everything...2396

I'm sorry; let's change that into colors, just so we can keep our color scheme going.2404

1/2x - 3/2...we will multiply everything by 2 to get rid of that fraction; so -4x - 8 = x - 3.2409

Put our x's together; put our constants together; add 3 to both sides; we get -5.2420

Add positive 4x to both sides; we get 5x; divide by 5 on both sides; we get -1 = x.2424

We can now plug that into either one; this one, to me, looks friendlier, so y = -2(-1) - 4; y = +2 (because the negatives cancel out) - 4; y = -2.2431

At this point, we have y = -2 and -1 = x; so a point here will be (-1,-2).2447

We have found the three vertices to this; now, we need to go use that information.2458

They are at (-4,4), (5,1), (-1,-2); so now, we want to evaluate them in our objective function, z = 2x + 7y,2464

and figure out which one makes z biggest, which one makes z smallest, and which one is just somewhere in the middle.2473

All right, so where are our maximum and minimum?2479

Our vertices are (-4,4), (5,1), (-1,2); we have (-4,4) here, (5,1), and (-1,-2) here.2481

All right, so let's start with (-4,4); we plug that in for z = 2x + 7y.2494

So, what will our z end up coming out to be?2503

2 times our x component is -4; plus 7 times our y component is positive 4, so that gets us -8, plus 7 times 4 (28), which equals 20.2506

So, we get 20 out of (-4,4).2517

Let's try our next one, (5,1): plug that in--we get z = 2 times our x component of 5,2519

plus 7 times our y component of 1 which equals 10 + 7, so we get 17.2527

And then, finally, we plug in our (-1,-2), and that gets us z = 2 times x component, -1, plus y component, 7(-2).2535

So, we get -2 - 14, which is equal to -16.2552

All right; at this point, we can see who is our winner for maximum; that is going to be z = 20, which occurred at (-4,4).2558

So, (-4,4) is the maximum for our objective function.2566

And (-1,-2) put out -16, so that was the minimum value that we ended up seeing.2571

And (5,1) gave us +17, but that is neither a maximum nor a minimum, so we just forget about it.2581

All right, the final example: A car lot needs to choose what cars to order for selling on their lot.2589

They have two options: the Nice, which they purchase at 20000 and sell at 30000, and the Superfine, which they purchase at 50000 and sell at 65000.2596

If they have a total budget of 2.9 million dollars for purchasing cars, and 100 spots on the lot for new cars,2606

how many of each car should they buy to maximize profit?2613

Assume that they manage to sell whatever they purchase.2616

The idea here is that they purchase a car, and then they mark it up and sell it.2619

And so, the difference between those two is how much money they make.2623

But they have limits on how much money they start with for purchasing cars, and how many spots they have to actually put the cars on the lot.2626

So, they have to figure out what is the best combination of cars to purchase, to be able to maximize how much money they make out of it.2633

So, with all of this in mind, let's start setting some things up.2640

First, we are going to have to figure out how many numbers of Nice they buy.2643

Let's say N is going to be the symbol that we use for saying the number of Nice cars that they buy.2647

The Nice cars that they buy will be N, however many that is.2655

And then, the Superfines that they buy--S will symbolize the number of Superfines that they buy.2659

OK, now another thing that we have here is a lot of really big numbers: 30000, 20000, 50000, 65000, 2.9 million dollars.2666

Now, we could work with the numbers as they actually are, and everything would end up working out.2678

But we would have this extra factor of 100 showing up; all of these things are divisible by 1000.2681

So, I am going to say, "Why don't we make things a little bit easier on ourselves?"2686

And since we are dealing with lots of money--we are dealing with all of these big things,2690

measured in the thousands and tens of thousands, let's convert everything.2694

For ease, let's work in terms of thousands of dollars; let's talk about everything in terms of how many K it is--2699

how many thousand (K, kilo, thousand)--how many thousand dollars everything is.2708

OK, that will make things a little bit easier, just in how many 0's we have to write down.2713

So, next, let's figure out how much profit they make off of a single Nice,2718

because what we want to do is maximize the profit; so we need to create some function p2724

that is going to be p = the profit, so we need to come up with some equation that describes p2729

in terms of how many Nice and how many Superfines they buy--how much N and S we have to go around--2735

because we know for sure that they will sell all of them.2740

So, how much profit do they make if they buy such a number of Nice and such a number of Superfines?2742

First, we have to figure out how much profit they make off of selling a Nice and how much profit they make off of selling a Superfine.2747

A Nice: if they buy a Nice, it costs them 20K, 20000; it costs them 20000, which is 20K in our new form.2753

They can sell that for 30K, which would net them a profit of the amount that they sold it for, minus the amount that they had to buy it for.2766

That would net them a profit of 10K.2778

For the Superfines, they cost them 50K to buy; they sell them at 65K; so each one they sell is going to net them a profit of 15K.2782

OK, with this in mind, let's write p in green for American money, since we are dealing with U.S. dollars here.2800

p = 10 times N, the number of Nices that we sell, plus 15 times S, the number of Superfines that we sell,2810

because remember: for each Nice we sell, we manage to make 10K; for each Superfine we sell, we manage to make 15K.2822

So, the total profit is going to be 10N + 15S...K, but we don't have to worry about the K's now.2829

We won't worry about it when we are actually dealing with the numbers--the fact that they have the unit of thousands of dollars.2836

All right; but there are some restrictions here.2842

We were told that there is a restriction of 2.9 million dollars for purchasing cars.2845

They don't just have unlimited amounts of money to buy cars.2852

If they had unlimited amounts of money, they would want to stock their entire car lot with nothing but Superfines.2855

They would just want to fill the thing all up with Superfines; but they have a cost to these things.2860

So, we have to deal with that; so if it is 2.9 million for purchasing cars, then...2865

well, each Nice that we buy is going to be 20N; and each Superfine that we buy is going to cause us to spend 50S.2873

So, 20N + 50S is how much we have spent on cars.2884

Now, they can spend up to 2.9 million; so 20N + 50S has to be less than or equal to their total budget.2887

And their total budget is 2.9 million, which we can write out as 2900, because 2.9 million is the same thing as 2900 thousand.2895

So, since we divided everything by 1000, we are now dealing with 2900.2904

All right, another piece of information that we had was that there are only 100 spots on the lot.2908

There is a maximum number of spots that the can fit cars into.2914

So, if that is the case--if they only have 100 spots to fit cars into--well then, the number of Nices that we are buying,2918

plus the number of Superfines that we are buying, has to be less than or equal to 100,2924

because they can only fit up to 100 of these cars on the lot.2929

So, that is two requirements so far.2932

Now, there are two hidden requirements that we might not see, but they will make it easier to actually graph the thing.2934

No, the lowest number of Nices that they can buy, or the lowest number of any car that they can buy, is 0.2947

So, we know that however many Nices that they buy, it has to be greater than or equal to 0.2952

And similarly, S must be greater than or equal to 0.2958

We have four requirements on this: the amount of money that have to spend (20N + 50S has to be less than or equal to 2900);2961

N + S has to be less than or equal to 100 (the number of spots on the lot); and the fact that they can't buy negative cars--2968

N has to be greater than or equal to 0, and S has to be greater than or equal to 0.2974

And our objective function that we are trying to maximize, because we are trying to maximize our profit, is this p = 10N + 15S.2977

We want to maximize p; at this point, we bring to bear the power of linear programming.2987

First, let's convert some of this stuff into things that we can easily graph.2993

So, if we are going to graph this, so we can see what is going on, we have to choose something to be horizontal and something to be vertical.2998

So, just arbitrarily, let's make N the horizontal and S the vertical; N gets to be horizontal; S is vertical.3009

With this idea in mind, we can now talk about ordered pairs that come in the form (N,S), just as (x,y) is (horizontal,vertical).3022

Since we made N horizontal and we made S vertical, it is going to come in (N,S).3031

Now, there is no reason it couldn't be flipped; but we just chose one, and we stick to it.3036

And as long as we stick to it, it will work out.3039

OK, at this point, if that is the case, then since S is the vertical, we normally solve for y (the vertical) in terms of other stuff;3041

so we want to solve for S (the vertical) in terms of other stuff, because that is what we are used to.3047

N + S ≤ 100 means that S has to be less than or equal to -N + 100.3051

20N + 50S ≤ 2900; 50S ≤ -20N + 2900.3058

We divide both sides by 50; we have S ≤ -2/5N + 2900/50 (becomes...give me just a second...58; I had to use a calculator for that one).3068

So, 2900/50 becomes 58; we have S ≤ -2/5N + 58.3084

So, with this in mind, we have enough to be able to see what we are allowed to go for here.3091

If N has to be greater than or equal to 0, then we have this graph...3099

If N has to be greater than or equal to 0, this is actually going to be the green one; so for a second, forget about that mistake.3104

N has to be greater than or equal to 0, so we set N equal to 0 and draw a line.3114

S is allowed to go wherever it wants; everything on this side is fine by our N > 0.3119

S has to be greater than or equal to 0: we set a line S = 0; everything greater than that is fine by S ≥ 0.3126

Now, the interesting parts are actually going to end up being the S ≤ -N + 100...3136

If that is the case, we can get a good sense of this; let's say 100 is up here; at -1, it marches down like this at a 45-degree angle.3141

And we know that S has to be less than or equal to -N + 100, so it will be below this.3152

And S ≤ -2/5N + 58; it is going to start at a y-intercept height of 58, but it goes much less steeply down at -2/5.3159

So, it goes like that; and it also will allow for the part below it.3172

So, the total set of things that is allowed is this highlighted part in yellow.3177

However, the only things that are really going to be interesting are going to be our vertices--the vertex here, here, here, and here.3184

We will circle those in yellow, just to make them a little bit more obvious.3197

But it is probably becoming pretty hard to see through the loudness of those colors.3200

We have an idea of those corners.3203

Now, notice: (0,0) is one of the points--we have an origin corner.3205

Is it possible to make any money if we plug that into p =...0 for N and 0 for S, if we buy no cars to sell?3209

No, we can't make any money; so we only have three things to care about:3217

where red intersects green, where red intersects blue, and where blue intersects purple.3220

So, if that is the case, let's look at where red intersects green first.3227

That is -2/5N + 58 gives us N ≥ 0.3231

If that is the case, if it is N ≥ 0, that is a little bit difficult, because we are set up for solving for S.3241

So, where is that going to intersect? Well, at N ≥ 0, we want to figure out what that comes out to be.3248

If S is equal to -2/5 times 0 plus 58, if we know that this is going to be equal to S, and N is greater than or equal to 0,3259

we can treat it as N = 0, because we are only dealing with lines here.3269

So, we plug that in here; we have -2/5 times 0, plus 58, equals S; so 58 is S, and we have an N of 0.3272

So, the point that we get out of this is going to be N = 0, 58 = S, so (0,58) is one of our points so far.3285

Next, let's look at where the blue intersects the purple.3294

-N + 100 =...the purple had a height of S = 0, no height at all, so that is just going to be 0, so 100 = N.3299

That means we have (100,0), because if we were to plug that into N + S = 100,3311

since we are dealing with the lines now--if we plugged in that, we would have 100 + S = 100; therefore, S = 0.3321

So, we have (100,0); notice that these are the two extremes possible for the car lot to sell.3328

It could sell all Superfines--it could sell nothing but Superfines--and it has a maximum amount of money that it can spend, 2.9 million.3334

So, it can't buy its lot entirely full of Superfines; it could only have 58 Superfines on the lot before it would run out of money to purchase cars with.3342

On the other hand, it could buy 100 of the Nices; and it would still have money left over, but it has run out of spots on the lot.3351

So, it could buy 100 Nices and no Superfines, and have extra money left over;3358

or the lot could buy all Superfines and run out of money before it runs out of spots.3363

It would have 58 Superfines, but still, that gives us another 42 spots left for Nices.3370

So, those are the two extremes possible, so far.3375

Finally, we could also have the possibility of if we have the blue and the red intersect.3378

If we had -N + 100 equal to -2/5N + 58, we can multiply everything by 5, just to make it a little bit easier on us.3385

-5N + 500 = -2N + 58...we turn to a calculator to figure out what 58 times 5 is.3402

We get 290 out of that; we add 5n to both sides; we subtract 290 on both sides.3411

500 - 290 gets us 210; we add 5n to both sides; that gets us 3n.3423

That gets us...dividing by 3...I'm sorry; not 80, but 70, equals N.3429

We use that to figure out how many Superfines that would bring us.3436

By our lot equation of S ≤ -N + 100, we want to be on the line (equals), so S = -N + 100; so S = -70 + 100, or S = 30.3440

The three possibilities are (0,58), (100,0), and (70,30).3457

At this point, we just want to actually plug each one of them in and figure out which of these works best.3466

If we sell nothing but Superfines--we spend all of our money on Superfines--then that is 10 times 0, plus 15 times 58, just so we keep up the same pattern.3477

15 times 58 becomes 870; so we would make a total of 870 thousand dollars at the lot if we bought nothing but Superfines.3490

Next, we will do the one that we arbitrarily made the blue point: the profit out of that one would be 10(100) + 15(0), the number of Superfines.3501

So, the profit that we would make off the lot would be 1000 + 0.3512

So, if we sold nothing but Nices, we would be able to get 1000 thousand, or 1 million, off of our sales.3516

But what if we sold a combination of Nice and Superfine?3526

If we sold a combination, the one we arbitrarily made the green point, then our profit would be 10(70) + 15(30).3529

10 is the profit from the Nices; 15 is the profit from the Superfines; we are selling 70 Nices and 30 Superfines.3541

We know that it is possible, because it is on our vertex.3547

That gets us 700 + 15(30) is 300 + 150; 450; so this is a total profit of 1150 thousand from 70 and 30.3550

So, the best, the optimum solution, for maximum profit, is to sell 70 Nices and 30 Superfines.3567

And if you are curious, that came out to be a profit of 1150 thousand, so the total profit,3582

the maximum possible profit for the lot, will end up being 1150 million dollars.3588

All right, cool; that finishes for linear inequalities.3596

There is this idea of linear programming, and there is also just being able to figure out what the things that our linear inequality allows for are.3600

Linear programming is a great thing that we can use all of our information about linear inequalities for.3605

But you can also just work with linear inequalities.3609

All right, we will see you at Educator.com later--goodbye!3611