Vincent Selhorst-Jones

Variables, Equations, & Algebra

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

### Variables, Equations, & Algebra

• A variable is a placeholder for a number. It is a symbol that stands in for a number. There are generally two ways to use a variable:
• The variable is allowed to vary. As its value changes, it will affect something else (the output of a function, a different variable, or some other thing).
• The variable is a fixed value (or represents multiple possible fixed values) that we do not know (yet) or do not want to write out. Normally we can figure out the value by using information in the problem.
• We normally use lowercase letters to denote variables, but occasionally we will use Greek letters or other symbols.
• A constant is a fixed, unchanging number. Occasionally, we might use a symbol to refer to a constant (In such a case, we might refer to it as a variable, but we know that since it's a constant, the variable is fixed.).
• A coefficient is a multiplicative factor applied to a variable.
• An expression is a string of mathematical symbols that make sense used together. Often we will simplify an expression by converting it into something with the same value, but easier to understand (and usually shorter). For example, we might simplify the expression 7+1+2 into the equivalent 10.
• An equation is a statement that two expressions have the same value. We show this with the equals sign:  =. For example, the equation
 2x+7 = 47
says that the expression 2x+7 is equivalent (equal) to the expression 47. In other words, each side of the equation has the same value.
• If we have an equation (or other kinds of relationships as well), we can do algebra. The idea of algebra is that since each side is equivalent to the other side, if we do the exact same operation to both sides, the results must also be equivalent. This idea makes sense, but it's critically important to remember you must do the exact same thing to both sides when doing algebra. If you do different things on each side, you no longer have an equation. This is a common mistake, so don't let it happen to you!
• When you solve an equation, you are looking for what value(s) make(s) the equation true. Most often you will do this by isolating the variable on one side: whatever is then on the other side must be its value. You isolate the variable by doing algebra. Ask yourself, "What operation would help get this variable alone?", then apply that operation to both sides.
• It's critical to remember the order of operations when simplifying expressions and doing algebra. Certain operations take precedence over others. In order, it goes
1. Parentheses (things in parentheses go first),
2. Exponents and Roots,
3. Multiplication and Division,
• Distribution allows multiplication to act over parentheses. The number multiplying the parentheses multiplies each term inside the parentheses:
 3(5 + k + 7) = 3·5 + 3k + 3·7.
We can also use the distributive property in reverse to "pull out" a factor that appears in multiple terms:
 3x2 + 7x2 − 5x2 = (3 + 7 − 5) x2.
• We can use information from one equation in another equation through substitution. If we know that two things are equal to each other, we can substitute one for the other.
 x = 2z + 3,     5y = x−2        ⇒     5y = (2z+3) − 2.
When we substitute, we need to treat the replacement the exact same way we treated what was initially there. The best way to do this is to always put your substitution in parentheses.

### Variables, Equations, & Algebra

Consider this expression:
−47t + 8.
What term is the variable?
What term is the coefficient?
What terms are the constants (there are two of them)?
• The variable is a placeholder for a number. It is a symbol that stands in for a number. Which symbol in the expression is standing in for a number?
• A coefficient is a multiplicative factor on a variable. Since t is our variable, what number is multiplying it?
• A constant is a fixed, unchanging number. What numbers show up in the expression? [Notice that t is not a constant, because we don't know whether or not it can change.]
The variable is t.
The coefficient is −47.
The constants are −47 and 8.
Explain, in your own words, why we can solve the equation
 x+2 = 8
for x by subtracting 2 from each side.
• To solve an equation for something means to get that something alone on one side of the equals sign. In this case, if we're solving for x, we want to get x alone on one side.
• The idea of algebra is that if we do the exact same thing to both sides of an equation, we obtain another equation.
Answers may vary; We know that x+2 and 8 are equivalent because of the equals sign. If we do the exact same operation to each side (in this case, subtract 2), our "new" left and right sides will still be equal. Thus, if we subtract 2 from both sides, we will have x alone on the left and it will equal whatever is on the right. This gives us x=6.
It is this idea of doing the exact same thing to both sides that algebra is based upon.
Solve for x:
 2x − 14 = −8
• We want to isolate x on one side. We choose all of our steps based on that goal.
• Begin by adding 14 to both sides (this cancels out the −14 on the left).
• Divide the resulting equation by 2 on both sides (this cancels out the 2 in 2x on the left).
x = 3
Simplify (pay attention to the order of operations!):
 3 ·(42 − 10)
• The order of operations goes in the following order: parentheses, exponents and roots, multiplication and division, addition and subtraction.
• (42 − 10) is in parentheses, so we need to calculate it first.
• Inside of the parentheses, we have an exponent: 42, so that happens before the subtraction.
 (42 − 10) = (16 − 10) = (6)
• Now that we know what is inside the parentheses, we can multiply 3 by that number.
18
Simplify (make sure to use distribution!):
 5(2a − b + c) + 3a + 2b
• Distribution allows multiplication to act over parentheses.
 3( x+ 7) = 3x + 3·7
The thing multiplying outside the parentheses distributes to each term inside of the parentheses.
• Begin the problem by distributing the 5 to each term inside of the parentheses.
 5(2a − b +c) = 5·2a + 5·(−b) + 5·c
• Multiply out, then add the result to the rest of the expression
 (10a − 5b + 5c) + 3a + 2b
13a − 3b + 5c
Simplify (use distribution and the order of operations):
 12p + 3 ⎛⎝ 22 (p+q) − 3 ·2 p ⎞⎠ + 10q
• The order of operations goes in the following order: parentheses, exponents and roots, multiplication and division, addition and subtraction.
• Because we have parentheses nested inside of parentheses, those go first. However, we can't simplify it anymore since p and q are different variables.
• Next, we need to distribute on to (p+q), but we first need to know what 22 is. 22 = 4, so we multiply that on to (p+q).
• We continue with the order of operations and now multiply −3 and 2p to get −6p.
• At this point, we have shown
 12p + 3 ⎛⎝ 22 (p+q) − 3 ·2 p ⎞⎠ + 10q = 12p + 3 ⎛⎝ 4p+4q − 6p ⎞⎠ + 10q
• We continue to simplify inside the parentheses, finally getting (−2p + 4q), which we then multiply by the 3 in front.
• We are now at the last step in the order of operations: adding everything together.
6p + 22q
Solve for k:
 3k + 18 = 3(5−2)
• Start by simplifying the equation (pay attention to the order of operations):
 3k + 18 = 9
• We want to isolate k on one side. We choose the rest of our steps based on that goal.
• Get rid of the 18 on the left side by subtracting it from both sides.
• Get rid of the 3 on the 3k by dividing both sides by 3.
k = −3
Solve for x+2y:
 5x + 10 y −35 = 25
• When solving for something, we are isolating it on one side. In this case, we aren't isolating just one variable, we're isolating the whole expression x+2y. We choose the rest of our steps based on that goal.
• Get rid of the −35 on the left side by adding 35 to both sides. This gives
 5x + 10 y = 60.
• Notice that 5x + 10y is a multiple of x+2y. Both the terms have just been multiplied by 5. Thus, we can divide both sides of the equation by 5 to reach our goal.
x+2y = 12
Use substitution to solve for n:
 n = g2               g = −3 ·5 + 17
• To find n, we need to substitute the value of g in to n=g2.
• We have two options. First, we could just substitute in g immediately to get
 n = (−3 ·5 + 17)2,
then work out the answer. Alternatively, we could first work out the value of g, then plug it in (substitute it in). This seems slightly easier, so we will do that [although the other method is perfectly fine].
• Simplifying g, we get g = 2.
• Plug g=2 into n = g2 to obtain
 n = (2)2.
n=4
Solve for t in terms of s:
 r = 3s               q = −s+3               3t − 11s = 5q + 2r2
• The phrase "solve for t in terms of s" means that we need to isolate t on one side (solve for t') and the only other variable that is allowed to appear in the final equation is s (in terms of s').
• This means we need to eliminate the variables q and r from the equation. We can do this by using substitution. Since we can see that q and r are in terms of s in the other equations, we can replace them in the equation containing t.
• When we substitute in, it is crucial to wrap our substitution with parentheses or things will go very wrong. Correctly doing so, we get
 3t − 11s = 5(−s+3) + 2(3s)2.
• Notice that (3s)2 = 9s2. Simplifying the right side of the equation, we have
 3t − 11s = −5s +15 + 18s2.
• Now we isolate for t: add 11s to both sides, then divide both sides by 3.
t = 2s + 5 + 6s2.
Later on when we learn about polynomials, we'll learn that we often order terms by the exponent on the variable. While the above is correct, some teachers might like to see that now: t = 6s2 + 2s + 5. Notice how that is equivalent to the above, just in a different order.

*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.

### Variables, Equations, & Algebra

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

### Transcription: Variables, Equations, & Algebra

Hi--welcome back to Educator.com.0000

Today we are going to talk about variables, equations, and algebra.0002

What is a variable? We talk about them all the time: we want to think of a variable as just being a placeholder.0006

It is a placeholder for a number; it is a symbol that stands in for something that can come in later; it is standing in for a number.0011

Sometimes the variable will be able to vary; it is going to be able to change, depending on what we want to do.0019

And as the value of the variable changes, it will affect something else.0025

It might affect the output of the function; it might affect some other dependent variable,0028

if we see something like y = 3x, where you change x--we make it the independent variable.0033

So, we put in different things for x, and it causes our dependent variable, y, to change, varying on what we put in for x, or something else.0039

That is one way of looking at a variable: it is something that is allowed to vary, and it causes other things to shift around as it changes.0046

Other times, we are just using a variable as a fixed value we don't know yet.0053

Sometimes, it might even be multiple possible fixed values; it could be fixed values or a fixed value.0057

But the point is that it is something that we just don't know yet; it is a placeholder for something that we want to find out more about.0062

So, normally, we are going to be able to figure out what it is, based on the information given to us in the problem.0070

Otherwise, it is probably not going to be a very good problem, if we can't actually solve for what the variable is.0075

So, we will almost always have enough information to figure out what this variable is.0080

That is the other possibility: a variable is something that we just don't know yet.0085

It is a number that has been given a name, because we are trying to figure out more information about it.0089

It is like if a detective is trying to find out who committed a crime; they might talk about the perpetrator, and they might find facts out0093

about the perpetrator, until they have enough information to be able to figure out who the perpetrator actually is.0100

"Perpetrator" is just a placeholder for some other person, until they figure out who that person is.0106

Great; we can name variables any symbol that we want.0111

Normally, we are going to use lowercase letters to denote variables; but occasionally, we will use Greek letters or other symbols.0115

When we are working on word problems, we are going to choose our symbols...we want to choose the letter,0121

or maybe other symbol that we use, based on something that helps us remember what it is representing in this word problem.0125

What am I using this variable to get across?0132

There are a lot of them that we regularly use; and so, we will get an idea of what they are; here we go!0135

Any symbol could potentially be used for any meaning at all.0142

We could make a smiley face; and I sometimes do use a smiley face to represent a number.0145

But smiley face is a little bit harder to draw than x, so we tend to use letters that we are used to drawing.0150

Anything could potentially be used for anything else; but here is a list of common symbols used,0156

and what the meaning we normally associate with them is.0161

Occasionally, we will have different meanings associated with them, depending on the problem.0163

We might use y to talk about the number of yaks that there are at a farm;0167

but generally, we are going to use them as we see right here--all of this right here.0171

So, x is our most common one, probably our favorite variable of all.0176

We use it for general use, when we are talking about horizontal location or distance.0180

y is vertical location or distance; t normally stands in for time; n stands in for a quantity of some stuff.0185

θ--this is a Greek letter, theta; when we encounter Greek letters, I will talk about them a little bit more, but mainly, it is just going to be θ.0192

We draw θ by hand; you just make something sort of like you are drawing a zero or an O,0199

and then you just draw a line straight across the middle of it; that is θ.0205

r is radius; A is area; V is volume; and we often use a, b, c, and k to represent fixed, unchanging values--0209

values that are not going to vary and change into other things--things where we know0221

that they are going to just stay the same, but we don't know what they are yet.0224

Or we might decide on what they are later on.0228

Anyway, this gives us a general idea of what the normal stable variables we constantly encounter are.0231

Now, you might use x for something totally different than what we have here.0238

You might use r for something totally different; you are not stuck to just using this.0241

But we are going to see them in a lot of problems, and we want what we do to make sense to other people.0244

So, it is good to go along with these conventions, usually.0249

All right, what is a constant? A constant is a fixed, unchanging number.0253

It is a value that does not turn into another value.0258

So, we can have variables become different values; we might plug in x = 3, and then plug in x = 5, and then plug in x = 7.0261

But a constant only has one thing; it just stays the same.0269

So, any time we see a number, like 3 or 5.7 or -82 or anything that is just a number, it is a constant, because numbers do not change.0272

After all, we don't have to worry about 3 suddenly turning into 4.0281

It is just 3; it is going to be 3 today; it is going to be 3 tomorrow; it is going to be 3 forever.0284

3 doesn't suddenly jump around and become a new number.0288

Occasionally, we might refer to a symbol that is representing a number as a constant.0292

We might say a is a constant in this problem.0295

We might not know what value that symbol represents; but we know it cannot change--a constant is something that cannot change.0298

And other times, we might even refer to a symbol as a variable, and just know that that variable is fixed, that it is a constant variable.0305

It seems kind of like a contradiction in terms, but remember: we are using "variable" more for the idea of placeholder.0311

And while sometimes it varies, sometimes it can also just be a placeholder in general.0316

A constant is something that isn't going to move around; it is one number, and one number only.0321

It doesn't matter if it is a symbol, or if it is actually a number; but the idea is that it is something that is not going to change.0326

A coefficient is a multiplicative factor on a variable.0333

So, anything that has some number multiplying in front of it, and it is a variable, like 3 times x...its coefficient is 3.0336

Normally, it is just going to end up being a number; but occasionally, it is also going to involve other variables.0346

So, not all coefficients are constants, and not all constants are coefficients.0352

For example, if we have n times x, plus 7, we have n as the coefficient of x, because it is multiplied against x.0356

But 7 is not a coefficient, because it is not multiplying against any variable.0364

7 is a constant, though, because it is just a fixed number.0369

So, n is a coefficient, but it might not be a constant--it might be allowed to vary.0372

But it isn't going to be...n is probably not a constant, but it is definitely a coefficient.0377

And 7 is not a coefficient, but it is definitely a constant.0384

And we could even look at x as being a coefficient on n: we can look at it from n's point of view, or look at it from x's point of view.0387

So, a coefficient is a multiplicative factor; a constant is just something that doesn't change.0397

An expression is a string of mathematical symbols that makes sense.0403

What do I mean by "makes sense"? Well, you can put together a string of words in English that makes no sense,0407

like tree sound running carpet; that didn't make any sense, right?--tree sound running carpet--that was meaningless.0414

But it was a bunch of words; to be an expression in math means that you have to make sense.0423

So, to be an expression in English (passing this idea along as a metaphor) would mean that it has to make sense as a sentence.0428

A string of mathematical symbols that make sense together: 2 times 3 minus 5 could be an expression.0435

But (((((( times divide minus 4 times plus (...that doesn't make any sense; that is just a bunch of things that have been put down on paper.0444

They have just been written down, but they don't actually mean anything.0464

So, an expression has to make sense; that is one of the basic ideas behind it.0468

Often, we will need to simplify an expression by converting it into something0472

that has the exact same value, but is easier to understand, and often is just shorter.0475

For example, we might simplify 7 + 1 + 2 into the equivalent 10: 7 + 1 is 8, and 8 + 2 is 10; so 7 + 1 + 2 has the same value as 10.0480

They are both different expressions; they are different expressions, but they have the same value, so we can convert one to the other.0492

We can simplify it if we want to.0497

An equation is a statement that two expressions have the same value.0500

We show it with an equals sign: what is on the left side of the equals sign and what is on the right side of the equals sign--0504

we know that those two things are the same--they have the same value.0511

Each side of the equation might look very different: 3x + 82 looks very different that 110/2.0515

But that equals sign is telling us that what is on the left is the same as what is on the right.0522

It guarantees equality between the two sides.0526

Algebra...for being able to do algebra, we need to have some sort of relationship between two or more expressions.0532

In this course, our relationship is almost always going to end up being equality.0539

It is going to be based on having an equals sign between two expressions.0543

The two expressions will be equal to each other, and that gives us an equation to work with and allows us to do algebra and do some things.0547

We could potentially have a relationship that is not based on equality.0552

We could have an inequality, where one side is less than another side, or one side is greater that another side.0555

Or we could have another relationship that is different than either of those.0561

But for this course, we are going to almost entirely see equality; and that is going to make up pretty much all of the relationships we ever see in math.0565

They will be based around knowing equality between two things.0572

So, the two expressions will be equal to each other; and this gives us a starting point to work from.0576

The key idea behind math...not behind all math, but behind algebra...is simple; it is intuitive; and it is incredibly important.0582

If two things are the same, equal, then we can do the exact same operation to both things; and the results will have to be the same.0590

Let's look at it like this: if we have a carrot here, and then we have another carrot that is exactly like that first carrot,0600

so they are a perfect copy of one another...we have two carrots, and then we come along and pick up a knife,0610

and we cut up this carrot with the knife, and we cut up this carrot with the same knife--0617

the exact same knife being used on both of them--and we cut up both of the carrots in the exact same way;0621

we cut 1-inch sections, exactly the same, on both of them; we are going to end up having chopped carrots0627

from the first carrot, and from the second carrot; but we know, because we did the exact same way of cutting them up,0634

and we started with the exact same carrot--our chopped carrot piles will be exactly equal to one another.0645

Since we start with the same carrot, and then we do the exact same kind of thing to both of the carrots,0652

we will end up having the exact same pile of chopped carrot at the end.0659

Now, compare that to if we had a third carrot that was exactly the same as its other two carrot "brothers,"0663

but instead of using a knife on it, we decide to shove it into a blender.0669

We put it in a blender, and we run it for a minute.0677

Out of that blender, we are going to get a pile of carrot mush; we are going to have a carrot mush pile.0680

And that carrot mush pile is going to be nothing like those chopped carrots.0687

It doesn't matter that we just started with equality; we also have to do the same thing.0692

Starting with equality is important; but if we don't do the same thing to both objects--0697

we don't do the same thing to both sides of our equation--we end up with totally different things.0701

We no longer have that relationship of equality that we really want to be operating on.0706

If you shove the carrot into a blender, you are going to have something totally different than if you had chopped it up.0711

If we do the exact same chopping to the two carrots, we end up getting the same pile of carrots.0715

But if we do a totally different thing, like shove it into a blender, we have something totally different at the end.0722

We have this pile of carrot mush; that is nothing like what we have from the other two.0727

The idea here is that we have to have the same operation be applied to both.0732

Doing algebra is based around this idea of doing the same thing to both sides.0737

Now, of course, you have seen this idea before: but it is absolutely critical to remember.0742

You have to remember this fact: always do the exact same thing to both sides.0747

If you don't do the exact same thing to both sides, you are not doing algebra anymore; you are just making fantasies up.0755

You have to do the same thing: if you add 7 to one side, you have to add 7 to the right side.0761

If you square the left side, you have to square the right side.0766

If you say higgledy-piggledy to the left side, you have to say higgledy-piggledy to the right side.0768

The huge quantity of mistakes that students make are because they forgot to do the operation on both sides.0772

They used it only on one side, or they used slightly different operations on the two sides.0779

If you end up doing this, you are going to end up making mistakes; don't let this happen to you.0786

Pay close attention when you are doing algebra--make sure you are doing the exact same thing to both sides.0791

You have to follow all of the rules on both sides; otherwise, we are just making things up--we are no longer following algebra.0796

When you are asked to solve an equation, you are being asked to solve for something.0804

This usually means solving the equation for whatever variable is in it.0809

If more than one variable is present, you will be told which variable to solve for.0813

What does solving an equation mean? It means you are looking for the things that make the equation true.0817

You are told that this side equals this side; the stuff on the left equals the stuff on the right.0824

But they both have variables in them, or one side has variables in it, or one side has just one variable in it.0831

But the point is that, depending on what that variable is, or depending on what those variables are, that equation might no longer be true.0836

So, what you need to do is make sure that this is true.0843

You were told that it equals one another; so you have to figure out what variable, what value for my variable,0848

or what values for my variables, will make this equation continue to be true.0854

I was told it was true from the beginning; so I have to make sure that it stays true.0859

Most often, you will be able to figure out what the values are that make something true by isolating a variable (or variables) on one side.0864

You will isolate the variable on one side, and then whatever is on the other side must be the value of that variable.0871

How are you going to do that in general?0877

Normally, you are going to isolate the variable by doing algebra.0879

You will ask yourself what operation would help get this variable alone.0881

What would I have to do to this side to be able to get this variable on its own?0886

Then, you do that operation to both sides; you continue to apply these operations, asking yourself,0890

one time after another, "What could I do to get this variable alone?"0895

You keep asking yourself, "What could I do?"; you keep doing operations to both sides.0899

And then, you keep doing this until, eventually, the variable is alone on one side, and you have solved it.0904

You will get something in the form like x = ... of numbers--so you will know that x is equal to this stuff right here; you will have solved it.0908

Now, keep in mind: sometimes you will not solve something by directly doing algebra.0917

Algebra will probably be involved, but you might actually be doing something a little bit more creative.0921

For example, we will see stuff like this when we work on polynomials.0926

We will see cases where we are not just doing algebra; we are also trying to figure out some other stuff and think on a slightly higher level.0929

But the key idea is that we are figuring out what makes this equation true.0936

What are all the possible ways to make this equation be true?0941

That is the real heart behind solving an equation.0945

It just so happens that it is very often a good way to solve it by doing algebra and getting the variable alone,0949

because once you get the variable alone and on one side, that tells you what value would make that original equation true.0954

Order of operations: it is critical to remember the order of operations.0963

We have known about this for a long time, but it still matters today; and it is going to matter for as long as you are doing math.0966

Certain operations take precedence over others.0971

In order, it goes: parentheses (things in parentheses go first), then exponents and roots, multiplication and division, addition and subtraction.0974

Always pay attention to the order of operations.0982

If you forget to do the order of operations, and you do it in a different order, disaster will befall your arithmetic.0985

So, always make sure you are working based on this idea of the order of operations.0989

Also, I just want to point out something: exponents and roots are two sides of the same thing.0993

x2 reverses square root: x2, √x...if you take something,0998

and you square it, and then you take its square root, they reverse one another.1003

Multiplication and division reverse one another: if we multiply by 3, and then divide by 3, it reverses.1006

Addition and subtraction reverse one another: if we add 5, and then we subtract 5, they reverse one another.1013

So, exponents and roots--the reason why they go at the same time is because they are really two sides of the same thing.1019

They have some similar idea going on behind them.1024

We will talk about that more when we get into exponents more, later in the course.1027

Multiplication and division: they go together at the same time, because they are two sides of the same thing; they can reverse one another.1030

Addition and subtraction go together at the same time, because they are working together; they are, once again, things that can reverse one another.1035

So, that is why we have these things paired together.1041

Parentheses, exponents/roots, multiplication/division, addition/subtraction: always make sure1043

that you are working in that order, or at least that whatever you are doing goes along with that order.1047

Sometimes, you might be able to do things where you don't have to follow this order absolutely precisely,1052

because you might see something like 3 times 2, plus (7 - 5).1056

Well, because there is this plus sign in the middle, we know that we can actually do what is on the left side1063

and what is on the right side simultaneously, because they will never talk to each other1068

until both orders of operations have completely gone through on their two sides.1072

So, we can just skip right to 6 + 2 = 8; we don't have to do everything there.1076

But if you are not quite sure--if you are not really capable with the order of operations,1082

so that you can see this sort of thing right away, always go with the order of operations very carefully, very explicitly.1085

In the worst case, it will just take a little bit longer, but at least you will not make a mistake.1091

Distributive property: we do not want to forget about the distributive property.1096

It allows multiplication to act over addition when it is inside of parentheses.1099

So, if we have 3 times (5 + k + 7), then that is equal to 3 times the first one, plus 3 times the second one, plus 3 times the third one (7)...1103

so 3 times 5, plus 3 times k, plus 3 times 7; that is the distributive property.1115

Always make sure you distribute to all of the terms that are inside of the parentheses; we have to distribute to everything inside of the parentheses.1120

I see lots of students see something like this, and they say, "Oh, 3 times 5, plus k, plus 7!"1127

No, no, no, no, no! You have to do everything inside of the parentheses; otherwise, you are not distributing.1132

So, make sure that you are always distributing to everything in there--everything, when you are multiplying in there.1138

All right, we can also use the distributive property in reverse, so to speak; we can go backwards, in a way.1144

This idea is what allows us to combine like terms.1150

For example, if we have 3x2 + 7x2 - 5x2, well, we have x2 here,1153

x2 here, and x2 here; so we can just pick them all up,1157

and we can shove them in, because they are all multiplying.1162

We pick them all up; and it is times x2; so we have (3 + 7 - 5) times x2,1165

because if we did the distributive property again, we would get what we started with; so it must be the same thing.1170

Now, 3 + 7 - 5--well, that just comes out to be 5: 3 + 7 is 10, minus 5 is 5; we get 5x2.1174

And that is what we are using to allow us to combine like terms.1182

We are sort of pulling out the like term, doing the things, and then putting it back in.1185

At this point, we have gotten so used to doing it that we don't have to explicitly do this.1190

But for some problems, it will end up being a really useful thing to notice.1193

So, it is important to see that we can occasionally use the distributive property in reverse; sometimes it will help us see what is going on.1197

Substitution: this is a really important idea in math.1204

We can use information from one equation in another equation through substitution.1207

If we know that two things are equal to each other, we can substitute one for the other.1212

For example, if we know that x is equal to 2z + 3, and we also have this equation that 5y = x - 2,1215

well, we can say, "Oh, look, right here I have an x, and I also know that x is the same thing as saying 2z + 3."1223

So, we take this information, and we plug it in for x.1231

That is what gets us (2z + 3); we will replace that x; so we have 5y is also equal to (2z + 3) - 2.1235

When we substitute, we need to treat the replacement the exact same way we treated what was initially there.1244

The best way to do this is by putting your substitution in parentheses.1254

Notice how I took 2z + 3, and I put it in parentheses up here, even though right here, it didn't start in parentheses.1257

That is because I was substituting in for x; so I want to make sure 2z + 3 is treated the exact same way that x was treated.1263

So, I have to put it in parentheses to make sure that it gets treated the exact same way that x got done.1269

The best way to do this is always to just put your substitution in parentheses.1275

It won't always be necessary: for example, on that 5y = 2z + 3, we didn't actually have to put it in parentheses there.1279

But it will never cause us to make a mistake; it is never going to hurt us.1285

(2z + 3) is just the same thing as 2z + 3, in this case right up here.1290

And in other cases (like this one that we are about to talk about), it is absolutely necessary; otherwise we will make bad mistakes.1294

Consider this really common mistake: if we know that a is equal b + 2, and we know that c is equal to a2,1301

then we can say, "Oh, a is right here; a is right here; I will take b + 2, and I will substitute it in for a."1308

Lots of students will say, "Oh, well, it is a2, so it must be b2 + 2."1315

No, that is not the same thing: we need a to be all of what it is.1319

a is all of (b + 2), not just the b part; and c is similarly not going to be equal to b + 22.1323

This right here is not working, because it has to be over this and this; everything needs to be put together.1331

b2 is not going to work here, as well.1338

The thing that we have to do is: we have to have it in parentheses.1340

The parentheses cause us to treat that a the same way that we are going to treat (b + 2).1343

a2...since a is equal to b + 2, all of a has to be squared; all of that (b + 2) has to be squared.1349

And the way that we get all of it is by putting it in parentheses.1356

So, whenever you are substituting something in, make sure that whatever is getting substituted in gets plugged inside of parentheses.1360

Otherwise, lots of bad mistakes can happen.1369

Sometimes, when you see the problem, you will be able to say, "Oh, I don't actually have to plug it in in parentheses,"1372

at which point, yes, you might be right; sometimes it will make it a little bit faster.1376

But really, it is a possible risk that you are taking for just putting down ().1380

It is not that much effort to put down parentheses, and it is going to save you so many times.1386

So, I really recommend that you put all of your substitutions, any time you are substituting something in, in parentheses.1390

Let's do some examples: we want to simplify the following: 2 times 32 + 4((5 + 7)2 - 27).1397

Well, we have parentheses inside of parentheses; so first, let's work on the thing inside of the parentheses.1407

And then, inside of that, we have even more parentheses.1413

So, first we do 5 + 7; we bring everything down--each new horizontal line is a copy of what was above it, but just put in new ways of talking about it.1415

4 times the quantity...well, what does 5 + 7 become? 5 + 7 becomes 12, so 12 times 2 minus 27.1425

Now, we keep doing this inside of these parentheses: first 2 times 32, plus 4...12 times 2 becomes 24, minus 27.1434

2 times 32...still working inside of these parentheses...24 - 27 becomes -3.1445

Now, we have 4 times -3, so now there is no longer anything happening inside of the parentheses.1454

So, what is next on the order of operations? Parentheses, then exponents and roots.1458

So, 2 times 32...32 is 9, plus 4(-3).1462

Next, we have multiplication: 18 + 4(-3)...-12; finally, we are down to addition and subtraction: 18 + -12 becomes 6; our answer is 6.1469

One thing I would like to point out is: if we are really good at math, we might have been able to say,1482

"Oh, look, there is a plus sign between these two sides, so these two sides aren't going to be able to talk to each other1485

until they have done everything they have to do on their own two sides."1491

So, we could have gone right down to saying 2 times 32...that is the same thing as 2 times 9, which is the same thing as 18.1494

And then, we would have kept doing our stuff on the right side, but we could have been simultaneously doing everything on the left side,1502

because they are not able to talk to each other, because they have plus signs between them and everything else.1508

That is a more advanced trick, and you are probably at the point where you can start seeing this sort of thing.1514

But if you have difficulty with the order of operations, you end up making mistakes like this sometimes.1518

Be careful and go through it really carefully, and make sure you have that stuff completely understood.1522

You need that foundation before math is going to be able to work.1526

It is the grammar of math; it is like knowing the grammar of English.1528

If you don't put words in the right order, it is just nonsense.1532

If you don't follow the operations in the right order, it is just nonsense; we are not able to speak the same language1535

as everyone else is speaking in math, and what everyone else is expecting us to be able to do1539

when we are working on problems or solving things...or engineering bridges...whatever we are going to do with math.1544

All right, Example 2: Use the distributive property to simplify 5(x + x2) + 3(x + y) - 7(x2 + x + y).1549

So, 5 times (x + x2) becomes 5x + 5x2.1560

Plus 3(x + y) becomes + 3x + 3y; minus 7...oh, here is something we have to be careful about.1567

It is not just going to be minus 7x2, but minus 7 is the entire thing.1576

So, it is that -7 that gets distributed; it is easier to see this as +, and then a -7.1581

+ -7: -7x2 + -7x + -7y--we have to make sure we distribute that negative, as well.1587

We see a minus, but it means that the "negative-ness" has to be distributed to everything inside of there.1600

Now, at this point, so we can see things a little bit more easily, let's move things together.1605

5x2...and here is a little trick: if you are not sure...we have 1, 2, 3, 4, 5, 6, 71610

different terms here--lots of different terms here to have to work with.1616

We can say, "Let's mark off each one; we will make a little tick mark after we write it on the next line, so we don't get confused,1619

accidentally use the same thing twice, or not even use it once."1626

So, 5x2 +...what is another thing involving x2? -7x2.1629

Plus...what comes next? It looks like we can work on the x's next: 5x (tick there) + 3x (tick there) - 7x (plus -7x);1637

and then finally, we have the y's: + 3y + -7y.1652

Those tick marks just help us keep track of what we are doing.1658

They are not necessary, but it makes it easier to follow, so we don't accidentally make any mistakes.1660

5x2 + -7x2...those will combine to become -2x2.1665

5x + 3x + -7x...we have 8x - 7x; we have 1x, which we just write as x.1671

And 3y + -7y becomes -4y; -4y we can also just write as minus 4y; and there is our answer.1681

Third example: we want to solve for x, so the first thing we do is ask ourselves, "How can I get x by itself?"1692

How can I get it isolated on one side, where it is just the variable, and only one of the variable, and nothing else there?1699

So, we say, "Well, it is inside of a fraction; we want it to be on top, and we want it to be the only thing there."1705

So, we are going to have to somehow change this fraction; how do we change a fraction?1713

Well, multiply by x + 3, and that will destroy the denominator.1716

Great--but if we multiply by x + 3, then this 2 is going to get hit, and this 3 is going to get hit, by the x + 3.1721

We have to hit everything on both sides, so the 3 will get hit by x + 3; the fraction will get hit by x + 3; and the 2 will get hit by x + 3.1728

So, the first thing we want to do is have some way of being able to have it operate on fewer things.1735

Let's try to get it to operate just on the fraction, at least on one side.1739

What we will do is start by subtracting 2 from both sides; that will make it easier to have a simple time with that x + 3.1743

We won't have anything else getting in the way.1750

That gets us 1 = 5/(x + 3).1753

Now, we can multiply by (x + 3); and while we will still have to multiply the 1 (we have to multiply both sides),1759

we will have at least a little less stuff in the way.1765

We multiply by (x + 3) over here, and we multiply by (x + 3) over here; so (x + 3) times (x + 3) on the bottom...they cancel each other out.1768

(x + 3) times 1...that is just going to become (x + 3).1780

Since we canceled out the (x + 3) on the bottom, we have 5 here.1785

Now, we ask ourselves, "How can I get that x alone?"1788

Oh, it is not too hard from here: we just subtract 3 from both sides: minus 3, minus 3; we get x = 2, and there is our answer; great.1791

Example 4--the final example: this one is a little bit tough, but we can totally understand what is going on.1802

x = 2z; y = z + 4; we want to solve for a in terms of z.1808

So, we have a in this equation down here; and we have x2, and we have y, and we have x.1814

So, z doesn't currently show up in this equation; we want to solve for a in terms of z.1821

What that means is that we want to get a = .... with z's...z's are going to be inside of that stuff on the right side.1825

And we are going to have a by itself: that is what "solve for a in terms of z" means.1834

a equals stuff involving z; it may be multiple z's; it may be just one z; but it is going to be a = [stuff involving z].1839

But notice: it is not going to involve x; it is not going to involve y.1848

We are told to solve in terms of z, so it is going to be only in terms of z and other actual numbers--that is, constant numbers.1851

So, if we know that x = 2z and y = z + 4, we need to get that z stuff to show up here, and we need to get rid of the y and get rid of the x.1859

So, we will use substitution: x = 2z and y = z + 4.1866

So, right here, we have a y; here we have an x; here we have an x.1877

So, let's do substitution: we have the left side--it will still be the same: 2a - 26 =.1883

What comes in for x? 2z comes in; so (2z)2 + 4 - 2(3...what goes inside for y? (z + 4)), minus...1889

what goes inside for x? (2z); close up that.1908

What we have is our original equation, but we have now gotten rid of x and gotten rid of y, and we only have z's and a's in here.1913

Now, we are able to solve for a in terms of z.1920

So, let's simplify what we have on the right side.1923

2a - 26 =...we have this plus sign in the middle, so we can work out what is on the left and what is on the right simultaneously.1926

We don't have to worry about them interfering with each other, even though they don't show up at the same time in the order of operations.1935

The only time they will be able to talk to each other is when we get all the way down to +.1939

So, we can have stuff on the left and stuff on the right work simultaneously to make it a little bit faster.1943

(2z)2...we square the 2; we square the z; so 22 and z2,1948

plus 4 minus 2...we go inside...3 times z, plus 3 times 4, minus 2z.1957

2a - 26 =...square 2; we get 4; square z; we don't know what z is, so it just stays as z2;1971

plus 4 minus 2 times (3z + 12 now - 2z)...keep simplifying...2a - 26 = 4z2 + 4 - 2(3z - 2z becomes just 1z + 12).1980

We can now distribute this -2: 2a - 26 = 4z2 + 4...we distribute the -2, so remember, it is plus a -2; so we get + -2z + -24.2004

Now, we are in a position to be able to keep simplifying the right side to its most fundamental level.2023

It equals 4z2...we don't have any other z2s, so it is just 4z2.2029

Plus 4...well, let's put our constants in there again; so we will go to + -2z, so - 2z, and 4 + -24 becomes - 20.2033

At this point, we can now do our algebra.2043

We will add 26 to both sides, and we will get 2a = 4z2 - 2z + 26, so plus...oops, I accidentally wrote what I was saying...2045

minus 2z still; add 26; -20 plus 26 becomes just 6; divide both sides by 2: a = (4z2 - 2z + 6), all over 2.2060

And we can simplify that: 4z2 becomes 2z2, minus 2z; that becomes -z; plus 6--that becomes + 3.2077

So, it is a = 2z2 - z + 3; a equals stuff-just-involving-z.2086

Solve for a in terms of z; great.2095

I really want to point out that the reason we were able to get that right is because we put parentheses when we substituted in.2097

If we hadn't done that, we would not have had our square go onto both the 2 and the z.2105

We would not have had our 3 distribute to both the z and the + 4; we wouldn't have had our subtraction...2110

well, our subtraction actually still would have subtracted 2z.2115

But if we didn't put in those parentheses, we would have definitely made some mistakes.2118

It is absolutely critical to put in parentheses when we are substituting.2121

Otherwise, mistakes will just start popping up everywhere.2125

All right, I hope all that made sense; we will see you at Educator.com later--goodbye!2128