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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Introduction to Series

Slide Duration:

Table of Contents

I. 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
Put Your Substitution in Parentheses
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

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
Tip To Help You Remember the Signs
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
The Plane and Quadrants
18:50
Quadrant I
19:04
Quadrant II
19:21
Quadrant III
20:04
Quadrant IV
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
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
Grades
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
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
II. 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
The Basketball Factory
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
Some Comments
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

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
III. 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
Leading Coefficient Test
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

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
Factoring Quadratics
13:08
Quadratic Trinomials
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
Factoring Quadratics, Check Your Work
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
Factoring a Quadratic
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

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
Solving Quadratics with Ease
9:56
The Quadratic Formula
11:38
Derivation
11:43
Final Form
12:23
Follow Format to Use Formula
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
Example 2: Solve the Quadratic
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52
Properties of Quadratic Functions

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
An Alternate Form to Quadratics
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

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
Advantages of Synthetic Method
26:49
Advantages of Long Division
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

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
Addition and Subtraction
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
Factoring So-Called 'Irreducible' Quadratics
19:24
Revisiting the Quadratic Formula
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

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
Comments: Multiplicity
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
Comments: Existence Theorem
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
IV. Rational Functions
Rational Functions and Vertical Asymptotes

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
Irreducible Quadratic Factors
8:25
Example of Quadratic Factors
9:26
Multiple Quadratic Factors
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
V. Exponential & Logarithmic Functions
Understanding Exponents

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
Thinking about Logs like Inverses
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

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
VI. Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
Radians
2:08
Circle is 2 Pi Radians
2:31
One Radian
2:52
Half-Circle and Right Angle
4:00
Converting Between Degrees and Radians
6:24
Formulas for Degrees and Radians
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-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
Adjacent, Opposite, Hypotenuse
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-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
Other Quadrants
9:43
Mnemonic: All Students Take Calculus
10:13
Example 1: Convert, Quadrant, Sine/Cosine
13:11
Example 2: Convert, Quadrant, Sine/Cosine
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-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
Sign of Tangent in Quadrants
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-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
VII. 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
Addition and Subtraction Formulas

52m 52s

Intro
0:00
Addition and Subtraction Formulas
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(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
VIII. 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
IX. 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
Test Points for Shading
11:42
Example of Using a Point
12:41
Drawing Shading from the Point
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear 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
Shade Appropriately
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
X. Vectors and Matrices
Vectors

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
Talking About Specific Entries
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
Matrix Addition
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

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
XI. 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
XII. 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
XIII. 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
Addition Principle
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

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
XIV. 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
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

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
XV. 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
Diagonal Addition of Terms
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
XVI. Preview of Calculus
Idea of a Limit

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
This Game is About Limits
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

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
A Limit is About Figuring Out Where a Function is 'Headed'
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & 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
Two Good Ways to Think About This
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the 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
XVII. Appendix: Graphing Calculators
Buying a Graphing Calculator

10m 41s

Intro
0:00
Should You Buy?
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
Deciding on Buying
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
Which to Buy
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
Do Your Research
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
Tips for Purchasing
9:17
Buy Online
9:19
Buy Used
9:35
Ask Around
10:09
Graphing Calculator Basics

10m 51s

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

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
Adjust Viewing Window
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
Advanced Technique: Arbitrary Solving
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
Advanced Technique: Calculus
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

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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Lecture Comments (3)

1 answer

Last reply by: Professor Selhorst-Jones
Fri May 9, 2014 6:11 PM

Post by abendra naidoo on May 8, 2014

hello,
I came across this problem in my reading and I am stumped as to how the answer can have 2 terms with a plus in between:Find the exact sum of the first 14terms in the geom. sequence sqrt.2,2,2sqrt2,4�
The answer given is 254+127sqrt2.
Using formula Ssubn=Asub1(1-r^n)/1-r --- I can't see how you end up with a plus sign in the answer?
Thanks.

Introduction to Series

  • Given some sequence a1, a2, a3, a4, …, a series is the sum of the terms in the sequence (or a portion of them).
  • If the sequence is infinite, we call it an infinite series. It adds all of the terms together:
    a1 + a2 + a3 + a4 + …
    If the sequence is not infinite or we only wish to add up a finite number of its terms, we call it a finite series. Adding the first n terms of the sequence together is called the nth partial sum:
    a1 + a2 + a3 + …+ an
  • To compactly describe sums, we use sigma notation (sometimes also called summation notation).


     
    ai
    • ai is the thing being summed. Terms from the sequence given by ai are added together. This might be a sequence, but is more often an algebraic expression.
    • i is the index of summation. It increases by 1 for each "step" of summation. The index can be any symbol, but i is common.
    • Above, i=1 is the first value used for the index. This is the Lower Limit of Summation.
    • Above, n is the last value used for the index this is the Upper Limit of Summation.
    Sigma notation can be really confusing the first few times you use it. Check out the video to see an example of how it works and some picture diagrams.
  • To show an infinite series (one where the terms keep adding forever), we put `∞' on top of the sigma. This shows that the series has no upper limit and instead continues on forever.


     
    ai    =  a1 + a2 + a3 + a4 + a5 + …
  • Sometimes it is useful to reindex a series (or sequence). We might have a series that has the index begin at one value, but we want it to start at another. However, we can't just change the number for the lower limit, because that would affect the whole series. This means we have to think about how to alter every part of the sigma notation so we can get the index we want without changing the value of the series. There are two main ways of doing this:
    • Expand: Write the sigma notation out in its expanded form, then look for how we could rewrite the pattern with our chosen starting index in mind.
    • Substitution: How does the old index relate to the new index that we want? Set up that relation, then use substitution to find the new general term and upper limit.
  • Sums have various properties that we can occasionally use to our advantage. Below, let ai and bi be sequences, let c be a constant.
    • i=1nc = c·n
    • i(c ·ai) = c ·∑i ai
    • i(ai + bi) = ∑i ai + ∑i bi

Introduction to Series

Given the sequence below, find the 4th partial sum. Assuming the pattern continues, find the 6th partial sum as well.
25,   18,   11,   4,   …
  • The nth partial sum of a sequence is the summation of the first n terms of the sequence. For example, the 3rd partial sum of a sequence would add up the first, second, and third terms.
  • Finding the 4th partial sum is quite easy: we already have the first four terms, so just add them all together:
    25 + 18 + 11 + 4    =     58
  • Finding the 6th partial sum takes a little more work: we don't know the rest of the sequence yet, so we need to write out more terms. Looking at the sequence, we see the pattern is to subtract by 7 for every step. Thus our next two terms can be figured out as below:
    4 − 7 = −3     ⇒     −3 − 7 = −10
    Thus, continuing the sequence to the sixth term, we have
    25,   18,   11,   4,   −3,   −10,   …
  • Now that we know the first six terms, it's easy to find the 6th partial sum:
    25 + 18 + 11 + 4 + (−3) + (−10)     =     45
4th partial sum: 58,        6th partial sum: 45
Compute the value of the below sum:
(21+1)  + (22+2)  + … + (26+6)
  • To understand how to sum this up, we need to realize what the ellipsis (`...') means. In math, an ellipsis between terms or after terms means that the pattern continues as expected.
  • For this problem, the pattern is clear: the nth term is (2n + n). Since we don't have very many terms to add, it's easiest to just write out each term in the sum to compute the sum's value.
  • Write out each term following the pattern:
    (21+1)  + (22+2)  + (23+3)  + (24+4)  + (25+5)  + (26+6)
    From there, just work through the arithmetic:
    (2+1)  + (4+2)  + (8+3)  + (16+4)  + (32+5)  + (64+6)

    (3)  + (6)  + (11)  + (20)  + (37)  + (70)

    147
147
Expand the below sigma notation into a series of terms added together, but do not add the terms together. Just leave it in a format where the pattern of the terms is easy to see.
5

i=2 
( i2 + 3i)
  • Sigma notation (also called summation notation) allows us to compactly write a series. If you are unfamiliar with it, make sure to watch the video lesson: it is explained carefully and with helpful diagrams to show how it works.
  • Remember, the variable below the sigma (Σ) shows what changes, and the number the variable is equal to shows where to start. The number at the top is where we end, and the expression to the right gives how to create the terms.
  • We start by plugging in i=2 for the first term, then i=3, then i=4, then we finish at i=5 because that's the value at the top of the sigma.
    5

    i=2 
    ( i2 + 3i)     =     (22 + 3·2)  +  (32 + 3·3)  +  (42 + 3·4)  +  (52 + 3·5)
    Since the problem told us to leave it in a form where the pattern of the terms is obvious, we're done.
(22 + 3·2)  +  (32 + 3·3)  +  (42 + 3·4)  +  (52 + 3·5)
Compute the value of the below series:
8

k=3 
(2k−7)
  • Sigma notation (also called summation notation) allows us to compactly write a series. If you are unfamiliar with it, make sure to watch the video lesson: it is explained carefully and with helpful diagrams to show how it works.
  • Remember, the variable below the sigma (Σ) shows what changes, and the number the variable is equal to shows where to start. The number at the top is where we end, and the expression to the right gives how to create the terms.
  • For this problem, there aren't too many terms, so it's easiest to just expand out all the terms from the sigma notation then add them up by hand. [If the series had many terms, it would be quite time-consuming. Luckily, there are formulas for various types of series, as we will see in later lessons. If you have to add more than 10 or 20 terms together, you're probably better off trying to find a formula.] We start by plugging in k=3 and stepping up until we hit k=8:
    8

    k=3 
    (2k−7) =     (2·3−7)  + (2·4−7)  + (2·5−7)  + (2·6−7)  + (2·7−7)  + (2·8−7)
    From there, just work through the arithmetic:
    (6−7)  + (8−7)  + (10−7)  + (12−7)  + (14−7)  + (16−7)

    (−1)  + (1)  + (3)  + (5)  + (7)  + (9)

    24
24
Expand the below sigma notation into a series of terms added together, using an ellipsis (`...') to show the pattern continuing. Do not compute the sum, just leave it in a format where the pattern of the terms is easy to see.
47

v=−3 
(v3+8)
  • Sigma notation (also called summation notation) allows us to compactly write a series. If you are unfamiliar with it, make sure to watch the video lesson: it is explained carefully and with helpful diagrams to show how it works.
  • Remember, the variable below the sigma (Σ) shows what changes, and the number the variable is equal to shows where to start. The number at the top is where we end, and the expression to the right gives how to create the terms.
  • It would take a very long time and a lot of space to write out the entirety of the series. However, using an ellipsis, we can just write enough terms to convey the pattern, along with showing where it starts and stops. Write out the first three terms added together (or however many is needed to clearly see the pattern), then put an ellipsis, then the final term to show where it stops. For this problem, we start at v=−3, then stop at v=47.
    47

    v=−3 
    (v3+8)     =     ( (−3)3 + 8 )  +  ( (−2)3 + 8 )  +  ( (−1)3 + 8 )  +  … +  ( (47)3 + 8 )
    Since the problem told us to leave it in a form where the pattern of the terms is obvious, we don't want to simplify anything (since that would make the pattern less clear).
( (−3)3 + 8 )  +  ( (−2)3 + 8 )  +  ( (−1)3 + 8 )  +  … +  ( (47)3 + 8 )
Use sigma notation to represent the below sum. Write it in a such way so that the lower limit of summation (the number it "starts" at) is 1.
7  +  9  +  11  +  13  +  … +  33
  • Sigma notation (also called summation notation) allows us to compactly write a series. If you are unfamiliar with it, make sure to watch the video lesson: it is explained carefully and with helpful diagrams to show how it works. Remember, the variable below the sigma (Σ) shows what changes, and the number the variable is equal to is the lower limit of summation (where it "starts"). The number at the top is where we end, and the expression to the right gives how to create the terms.
  • Notice that writing something in sigma notation means we need a formula for any given term in the series. This is the same as figuring out the nth term for a sequence, so consider the terms of the series as a sequence:
    7,   9,   11,   13,   …
    With this in mind, it's fairly easy to see that the pattern adds 2 each time and starts at 7. We can express this as
    an = 2n+5
    [If you are unfamiliar with sequences and/or finding formulas for the nth term of a sequence, check out the previous lesson, Introduction to Sequences. It will be very helpful for understanding the more difficult problems about series.]
  • We set it up so that a1 = 7, so our starting index of i=1 in the sigma notation is already set. We just need to figure out what the upper limit of summation needs to be (where it "ends"):
    ?

    i=1 
    (2i+5)
    Looking at the series in the problem, we see that it must end at 33, so we can find what n value will create 33 from the an formula:
    33 = 2n+5     ⇒     28 = 2n    ⇒     14=n
    Thus our series must end on 14:
    14

    i=1 
    (2i+5)
    Finally, always make sure to check your series. It's easy to make a mistake, so expand your series to see the first couple terms and the last term to confirm it against the original series.
    14

    i=1 
    (2i+5)     =     (2·1 + 5)  + (2·2 +5)  + … + (2·14 + 5)     =     7  +  9  +  … +  33    
i=114 (2i+5)
Use sigma notation to represent the below sum. Write it in a such way so that the lower limit of summation (the number it "starts" at) is 1.
(−7)  +  (−4)  +  1  +  8  +  17  +  28  +  41  +  … +  392
  • Sigma notation (also called summation notation) allows us to compactly write a series. If you are unfamiliar with it, make sure to watch the video lesson: it is explained carefully and with helpful diagrams to show how it works. Remember, the variable below the sigma (Σ) shows what changes, and the number the variable is equal to is the lower limit of summation (where it "starts"). The number at the top is where we end, and the expression to the right gives how to create the terms.
  • Notice that writing something in sigma notation means we need a formula for any given term in the series. This is the same as figuring out the nth term for a sequence, so consider the terms of the series as a sequence:
    −7,   −4,   1,   8,   17,   28,   41,   …
    This is a fairly tough pattern to crack, but after realizing it's not based on a basic operation, we will eventually compare it to the pattern of n2:
    n2     ⇒     1,   4,   9,   16,   25,   36,   49,   …
    This is a challenging pattern to spot, but we can see that the sequence from the series matches up to the n2 sequence, it's just 8 below every term. Thus we can give the sequence for the series as
    an = n2 − 8
    [If you are unfamiliar with sequences and/or finding formulas for the nth term of a sequence, check out the previous lesson, Introduction to Sequences. It will be very helpful for understanding the more difficult problems about series.]
  • We set it up so that a1 = −7, so our starting index of i=1 in the sigma notation is already set. We just need to figure out what the upper limit of summation needs to be (where it "ends"):
    ?

    i=1 
    (i2−8)
    Looking at the series in the problem, we see that it must end at 392, so we can find what n value will create 392 from the an formula:
    392 = n2−8     ⇒     400 = n2    ⇒     ±20 = n
    Since the upper limit must be higher than the lower limit, we throw away the −20 and see that it ends on 20:
    20

    i=1 
    (i2−8)
    Finally, always make sure to check your series. It's easy to make a mistake, so expand your series to see the first couple terms and the last term to confirm it against the original series.
    20

    i=1 
    (i2−8)     =     (12 − 8)  + (22−8)  + … + (202 − 8)     =     (−7)  +  (−4)  +  … + 392    
i=120 (i2−8)
Use sigma notation to represent the below sum. Write it in a such way so that the lower limit of summation (the number it "starts" at) is 1. [Notice it is an infinite series.]
1

1
 +  1

4
 +  1

9
 +  1

16
 +  1

25
 + …
  • Sigma notation (also called summation notation) allows us to compactly write a series. If you are unfamiliar with it, make sure to watch the video lesson: it is explained carefully and with helpful diagrams to show how it works. Remember, the variable below the sigma (Σ) shows what changes, and the number the variable is equal to is the lower limit of summation (where it "starts"). Normally, the number at the top is where we end, but in this case the series never ends, so we will have an ∞ sign. Finally, the expression to the right gives how to create the terms.
  • Notice that writing something in sigma notation means we need a formula for any given term in the series. This is the same as figuring out the nth term for a sequence, so consider the terms of the series as a sequence:
    1

    1
    ,    1

    4
    ,    1

    9
    ,    1

    16
    ,    1

    25
    ,   …
    With this in mind, it's fairly easy to see that the pattern is just a fraction with 1 on top and a denominator given by n2. We can express this as
    an = 1

    n2
    [If you are unfamiliar with sequences and/or finding formulas for the nth term of a sequence, check out the previous lesson, Introduction to Sequences. It will be very helpful for understanding the more difficult problems about series.]
  • We set it up so that a1 = [1/1], so our starting index of i=1 in the sigma notation is already set. Normally we would need to figure out what the upper limit of summation needs to be (where it "ends"):
    ?

    i=1 
    1

    i2
    However, notice that this series is infinite: it continues the pattern forever. We show this by putting an ∞ on top of the sigma:


    i=1 
    1

    i2
    Finally, always make sure to check your series. It's easy to make a mistake, so expand your series to confirm it against the original series.


    i=1 
    1

    i2
        =     1

    12
     +  1

    22
     +  1

    32
     + …    =     1

    1
     +  1

    4
     +  1

    9
     + …   
i=1 [1/(i2)]
Reindex the below series so it starts at the index i=1.
100

j=15 
(4j+7)
  • Reindexing a series means changing the way the lower limit, upper limit, and expression are written, but still getting the same terms. For a very simple example, the series on the left has been re-indexed into the series on the right, but they still give the exact same terms and result:
    11

    k=10 
    k            =           2

    l = 1 
    (l+9)
    Make sure to check out the video lesson for more information and a much more comprehensive explanation.
  • There are two different ways to approach reindexing. First, we'll look at it from the point of view of expanding. Expand the sigma notation so that you can see the terms it is built on:
    100

    j=15 
    (4j+7)     =     (4·15 +7)  + (4·16 + 7)  + … + (4 ·100 + 7)

                  = 67  + 71  + … + 407
    Now that we see it in its expanded form, we can approach putting it in sigma notation like we normally would. We see that it has a pattern of adding 4 each term and starts at 67, so the general term is
    an = 4n + 63
    Next, we need to know when it stops, so we solve for the appropriate n to create an=407:
    407 = 4n+63     ⇒     344 = 4n     ⇒     86 = n
    Thus the sigma series will end at the 86th term. Putting this all together, we get
    86

    i=1 
    (4i+63)
  • Alternatively, we could approach reindexing by substitution. Notice that we want to have a starting index of i=1, but we currently have j=15. However, they must indicate the same first term, so we see that
    if i=1 and j=15,    then    j =14+i.
    With this in mind, we can now substitute for the bottom index and the index variable in the expression. For now, we'll leave the top index as a `?' since we haven't figured out what it must be yet. Plug in:
    100

    j=15 
    (4j+7)     ⇒    ?

    14+i=15 
    (4(14+i)+7)     =    ?

    i=1 
    (56+4i+7)     =    ?

    i=1 
    (4i+63)
    Finally, we need to figure out the top index. Notice that in terms of j, the final value was 100. Thus we can plug in j=100 to the conversion formula to find the matching i-value:
    j=100     ⇒     100 = 14+i     ⇒     86 = i
    Therefore the top index is 86 in the new i-format, which completes the sum:
    86

    i=1 
    (4i+63)
    [As an alternative to find the top index, we could also see that the original top index was 85 above the starting index (100−15 = 85), so the new top index must still be 85 above the new starting index (1+85=86).]
  • Whatever method you use to find the reindexed series, make sure to check that it still contains the same terms as the original series. It's easy to make a mistake, so double-check your answer:
    100

    j=15 
    (4j+7)    ⇒     67  + 71  + … + 407     ⇐    86

    i=1 
    (4i+63)    
i=186 (4i+63)
Using the functions below, calculate f(−5) and g(3).
f(x) =4

i=0 
xi           
           g(t) =t

k=1 
4k
  • Plugging into a function is just like we're used to, even if it's mixed up with a series. Simply replace the variable the function uses as input with the given number, then work out the arithmetic.
  • For f(−5), since the function is f(x), we plug in x=−5:
    f(−5) =4

    i=0 
    (−5)i
    From there, just calculate the series like usual:
    4

    i=0 
    (−5)i     =     (−5)0  + (−5)1  + (−5)2  + (−5)3  + (−5)4

                         =     1  + (−5)  + 25  + (−125)  + 625     =     521
    Thus f(−5) = 521.
  • For g(3), since the function is g(t), we plug in t=3:
    g(3) =3

    k=1 
    4k
    From there, just calculate the series like usual:
    3

    k=1 
    4k     =     4·1  + 4·2  + 4·3     =     4+8 + 12     =     24
    Thus g(3) = 24.
f(−5)=521,        g(3) = 24

*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.

Answer

Introduction to Series

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

  • Intro 0:00
  • Introduction 0:06
  • Definition: Series 1:20
  • Why We Need Notation 2:48
  • Simga Notation (AKA Summation Notation) 4:44
    • Thing Being Summed
    • Index of Summation
    • Lower Limit of Summation
    • Upper Limit of Summation
  • Sigma Notation, Example 7:36
  • Sigma Notation for Infinite Series 9:08
  • How to Reindex 10:58
    • How to Reindex, Expanding
    • How to Reindex, Substitution
  • 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

Transcription: Introduction to Series

Hi--welcome back to Educator.com.0000

Today, we are going to talk about an introduction to series.0002

In the previous lesson, we introduced the idea of a sequence--that is, an ordered list of numbers.0005

With this idea in mind, we can now discuss the concept of a series, summing up the terms of a sequence.0011

Now, at first, this might seem a little bit series.0018

It is just addition, and we have been doing addition since kindergarten; so why do we need to talk about addition in a special way?0020

But consider how long it would take to add up 100 terms by hand.0025

Could you imagine how much time it would take up if you had to add up 100 different things from a sequence?0029

What if it were a thousand terms, or ten thousand terms?0033

The study of series can make these seemingly colossal summation tasks, having to add up huge numbers of numbers, really, really easy.0037

We can just turn this stuff into being some trivially easy task once we figure out how to talk about series in a deep way.0044

Being able to easily add up lots of numbers has many applications.0051

You will see it in science, engineering, economics, computer programming, advanced math, and many other fields.0054

All of these things benefit greatly from the study of series,0061

from being able to talk about adding up a whole bunch of numbers in an easy way,0064

where we can talk about it in compact notation--all of these things are really important to a variety of fields.0068

So, this is a great thing to study; let's start by defining a series.0073

Given some sequence a1, a2, a3, a4...0078

a series is the sum of the terms in the sequence; it is just adding up the terms,0083

or maybe a portion of them--maybe not the entire sequence.0088

But it is just adding up terms from the sequence.0091

If the sequence is infinite, and we are adding up all of the terms from the sequence, we call it an infinite series.0093

It adds all of the terms together; so I am saying that it keeps going forever.0099

It is a1 + a2 + a3 + a4...going on forever...0104

so plus a5, a6, a7, a8...forever and ever and ever.0110

Since our sequence was infinite, we are just saying to keep adding them forever and ever and ever.0114

That is an infinite series.0119

On the other hand, we can talk about a sequence that is not infinite--or if we only wish to add up a finite number of its terms.0121

We aren't going to add them forever; we are just adding up a finite portion of them--just a number where we can count how many are there.0129

We call this a finite series; adding the first n terms of the sequence together is called the nth partial sum.0136

If we add a1 + a2 + a3, up until a1,0144

but notice: it just stop there--we don't go any farther past an--then that is the nth partial sum,0149

because we are adding in all of the terms, 1, 2, 3, up until we get to n; so it is the nth partial sum,0154

because it is only a part of our infinite sequence.0160

Consider if we had a sequence to find by the general term an = n + 2n over 3n.0164

That is 1 + 2 over 3, 2 + 22 over 32, 3 + 23 over 33,0170

4 + 24 over 34...what if we wanted to talk about its thirtieth partial sum?0176

In talking about its thirtieth partial sum, we will see very quickly why we need a special form of notation for series.0182

Why is writing this stuff out by hand going to be a real pain?0189

If we wanted to write this out, we would have to first show the pattern that occurs here.0192

We have this pattern that created each one of these various terms.0197

We would have to have that show up in our summation; otherwise, we wouldn't be able to realize what pattern is going on.0202

So, we are going to at least need the first three terms--at least.0208

Then, we could use an ellipsis, that ... , to say that the pattern continues on in this manner.0213

We can not have to write a massive amount, because we can use the ellipsis to say that the pattern keeps going.0218

But we are still going to have to show its stop at the thirtieth term.0223

At a very minimum, we are going to have to write out the first three terms, the last term, and an ellipsis in the middle.0226

We would have to write out (1 + 2)/3 + (2 + 22)/32 + (3 + 23)/33 + ...0232

+ (30 + 230)/330; that is the thirtieth partial sum, like we were talking about.0240

But that is a lot of stuff to write out; that is a lot of writing for a fairly simple series.0246

This isn't even that complicated; but if we had to write this on multiple lines, as we worked through a problem--0250

if we had to keep talking about this over and over as we did steps in a problem--your wrist is going to hurt after doing one of these problems.0256

And you are going to have to do a bunch of problems about summation.0261

This is why we need some sort of notation: we need notation to make it easier to compactly describe a series.0263

We don't have to write out this really, really long thing every time we want to talk about some summation from a sequence.0271

We need some easy way to be able to do it in a short way.0278

Enter sigma notation: to compactly describe sums, we use something called sigma notation.0281

It is also sometimes called summation notation.0287

Why is it called sigma notation?--because it uses the uppercase Greek letter sigma.0290

Sigma is this right here; if you are drawing it by hand (this is a computer-drawn picture of sigma,0295

made by a computer typesetting program), I would recommend writing it like this,0302

where you have these little vertical hooks on either end; and then it is just kind of a capital M on its side.0312

Just write it out like that; and that is great.0319

If you are feeling lazy, you can end up just sort of writing it like that, as well.0323

But it helps to write it like this, so that we can clearly see what it is.0326

But if you are doing a lot of problems that way, don't worry if it ends up getting not quite absolutely perfect each time,0329

because you will be able to recognize it from other things that are about to be seen.0335

All right, let's see how we use sigma notation.0338

Let's look at the anatomy of a series in sigma notation.0342

The first thing to look at is the thing being summed: the terms from the sequence given by ai, this thing to the right of the sigma, are added together.0346

This is what will be added together over and over with each step as we add up:0356

the first thing, and then the second thing, and then the third thing, and so on.0360

It might be a sequence, but it is much more often going to be an algebraic expression--something like 3 times i plus 10 or 2 to the i or i!,0362

something that we could plug in a number and be able to churn out a value.0373

You will often see algebraic expressions; but once in a while, you will see a sequence like ai.0376

Next, the index of summation; i is the index--it increases by 1 for each step of summation.0382

We do the first thing, and then we add the second thing in the next step;0391

and then we add the next thing in the next step; and we add the next thing in the next step, and so on, and so forth.0394

So, every time we step forward, the index will click up by 1.0399

We start at some value, and then we go to the value one above that, then the value one above that, then the value one above that, and so on.0403

So, every step, it will increase by 1.0409

The index can be any symbol; but i is very common, and that is what we will end up using for the most part in this course.0412

So, that is the thing that ends up changing; and notice how i will generally occur over here in the series, as well.0419

That is saying that this is the thing that changes; this is what will change with each step--this little thing right here.0425

The lower limit of summation: this is the first value used for the index.0432

This is where the series starts; in this case, since i = 1, our first value would be plugging in a 1 for the i.0436

The upper limit of summation: this is the last value used for the index, where the series ends.0444

We would step up until we eventually got to some value n.0450

All right, let's see it in action: we have this pictorial summary of how it works.0454

This is a great picture; so check against this, if you get used confused in a later point; this is a great slide right here.0458

Let's see it in action: we have sigma...it's i = 3 for the starting index, and then 7 is the upper limit, and we have 2i - 1 as the actual expression.0464

The very first thing: we have the i right here, and we start at i = 3.0475

So, the first value that we plug in for our i is 2 times 3, because our lower limit of summation was that 3.0479

So, 2 times 3 minus 1 is the first thing.0486

Then, we go on to our next step: plus...we increase our index by 1.0489

Our index started at 3, so it goes from 3...plus 1...to 4; so 2 times 4, minus 1.0493

Then, we go on to our next step; it will be 2 times...we were at 4 in the last step, so it increases to 5; so we are at 4 + 1 is 5.0500

2 times 5 minus 1, plus...our index increases by 1 again, so 2 times 6 minus 1, plus...0508

and now, finally, we have just hit our final, upper limit.0516

So, at this point, we stop; this will be the last one, once we click up to this final upper limit for how high our index goes to.0520

That is the last one in the series; so it is 2 times 7 minus 1.0528

And at this point, we could simplify it out if we wanted to.0533

We could get a value from this; but that is also a good way to see how it expands.0535

And that is all we are looking for right now--seeing how it expands.0539

But if you wanted to, you could simplify each one of these.0541

And you could also combine them and be able to get a value for that series.0543

So far, we have only seen how to use sigma notation for finite sums--that is to say, partial sums, not sums that are going on forever.0548

Since they have all had some upper limit of summation, something on top of this sigma, they have all had to eventually stop.0555

They have had some top limit to how far they step out.0561

If we want to show an infinite series, one where the series' terms keep adding forever and ever,0564

we simply put an infinity symbol on top of the sigma.0570

This shows that the series has no upper limit, and instead continues on forever.0573

So, if we had a sigma with i = 1 on the bottom, our index is i, and our lower limit is 1,0577

and our upper limit is infinity, that says really "just keep going."0583

It is not that we actually get to infinity and stop; we can't ever get to infinity.0586

Infinity is just the idea of going on forever; so we start at 1, and then on to 2, and then on to 3, and then on to 4, and then on to 5, and then on to 6...0589

And it just keeps going forever and ever.0598

So we would have...since it is ai, we would have a1 first, because it was a 1 for our lower limit,0600

then a2, and then + a3, and then + a4, and then + a5...0605

and it will just keep going on forever, because it is an infinite series that we are working with.0609

If we wanted to see in specific, here is an example that uses specific numbers.0614

We have some series with i = 1 on the bottom; it is an infinite series, because of the infinity on top.0618

We have 1/2 to the i; so our first i is 1; that is 1/2 (is our first term, one-half), plus 1/2...0623

our next step with the i will be now at a 2, so 1/2 to the 2, 1/4...and then our next step would be i at 3;0633

so 1/2 to the 3; that is 8; at our next step, i will be at 4, so that is 1/2 to the 4, so 1/16.0641

And then, i will be at 5, so it is 1/2 to the fifth, so 1/32; and that pattern will just keep going.0648

We will keep adding them on forever and ever.0653

Sometimes it is useful to re-index a series (or a sequence, sometimes).0657

We might have a series that has the index begin at one value, but we want it to start at another value.0662

For example, we might want a transformation like this one here, where we have, in our original sequence,0667

i = 7 as our starting index, and it has some upper limit, and it has some expression that we are actually working with.0672

And we want to transform it to k = 1; we want our starting index to be k = 1.0678

And we are going to, of course, also need some upper limit and something here.0684

Now, of course, we can't just change the number of the lower limit.0688

If we just went and changed the lower limit, that would affect the whole series.0691

We have to pay attention to how the thing being summed, and our upper limit, end up making the transformation, as well.0696

They are going to change over the process; they are not going to be the same thing on the left side and the right side.0703

They are going to end up becoming different things; otherwise we would have altered the whole thing by starting...0710

we start at one starting place, and then we just change the starting place, and we don't do anything else;0715

well, if you start in a different starting place, you are going to have different values coming out.0718

So, since we want to end up having the same value come out of our two ways of talking about it,0721

we have to alter every part of the sigma notation, so that we can get the index we want without changing the value that comes out of it.0725

You are probably asking why we care; if we have it written in one way, why not leave it that way?0734

Well, there are a bunch of reasons to do it.0738

For example, if you are working on a proof, it can sometimes help to re-index it.0740

But specifically to this course, a lot of formulas that we end up working with will be given in the form of something with an i = 1 on the bottom.0743

So, since a lot of our formulas will have i = 1 on the bottom, we will have to start at this index of something = 1,0752

some symbol = 1, to be able to work with some of the formulas we have.0758

A lot of the formulas that you end up seeing are in this format.0762

So, it is really helpful to be able to re-index, so we get to the format that we already have a formula for,0764

as opposed to having to figure out an entirely new formula for some new, different index.0768

So, how do we actually re-index?0773

The most essential way is to expand the sigma notation into a written-out series,0775

then see how we could rewrite the pattern with our chosen starting index in mind.0779

So, for example, if we want to go from i = 7 to k = 1, we can just start by...0783

a series is just a shorthand way of writing out something + something + something...+ something.0791

And even that is a shorthand way of writing out every single term.0797

So, we can expand it into that format; if it is i = 7 first, then we would have 3(7) - 16;0800

the next thing would be + 3 times 1 step up to 8, minus 16; plus...it would continue in this form.0807

And 3 times...our final upper limit is 22; so 3(22) - 16; great.0815

If we want, we can figure out how to write this sigma over here, with just that in mind.0822

But we can also simplify things to see if we can see another easier-to-see pattern.0827

So, 3(7) would give us 21, minus 16, plus...3 times 8...16, 24...minus 16 + ... 3(22) is 6, minus 16...0831

so, 21 - 16 is 5, plus 24 - 16 (that is 8), plus...plus 66 - 16 (that is 50).0845

We look at this, and we might realize, "Oh, what is doing is going up by 3 each time."0854

And that makes sense, since we started at 3i.0858

It is going up at 3 each time; if that is the case, and we want to get this k = 1, then that means0860

that we know that here, we are at k = 1; here we are at k = 2; here we are at k =...0865

we will actually leave that as a question mark right now, because we don't quite know yet.0870

We will talk about that in just a moment--what number we are at; we will see why.0873

k = 1; if it is + 3 each time, we are going to want some 3k + some number.0877

So, if we are at 5 here, then that would be 3k + 2, because 1 for k...3 times 1 plus 2 does get us 5, so that checks out.0882

Here, with 3k + 2, it works, as well; so 3k + 2 is here; we plug in 2; 3(2) is 6, plus 2 is 8; that checks out, as well.0889

So, it looks like it is going to be some 3k + 2 that will be here; we will have 3k + 2 for that part right there.0897

Now, what is the upper limit of summation going to end up being?0905

Well, if we are at 3k + 2, and it equals 50, we have to figure out what value would end up coming out here.0908

So, k = ?; we could solve for this: 3k = 48; k...divided by 3 now on both sides...we get 16.0915

So, we know that our upper limit is going to have to be 16.0922

Another way of doing it is to notice what the difference is between our upper limit and our lower limit.0925

In our original, we had 22 as our upper and 7 as our lower; so, 22 - 7 means 15.0930

Oops, I'm sorry; not 15...oh, it is 15; 22 - 7 comes out to be 15.0936

Over here, we know that we are going to end up having that difference of 15, as well.0944

Since we start at k = 1, 1 plus that same 15 (because we are going to have to have the same number of steps,0949

however we phrase it) comes out to be 16, which is the same thing we figured out that it has to be to work there.0958

With that in mind, we now know that our lower limit with our index is going to be k = 1,0965

because that is what we wanted to figure out in the first place.0972

Our top, our upper limit of summation, will be 16; we have figured out two different ways that that has to be the case,0974

because 16 - 1 is 15, which is the same as 22 - 17; or alternately, we can solve for it0980

using the formation that we came up with for that general term.0985

And then, 3k + 2 is what is actually going in that is being added on each term.0988

And so, that is one way of doing it.0994

We worked this out by expanding first; expanding is a very specific way to see it.0995

It is a great thing to do if you get confused by the problem that you are working on.1000

There is another way to do this, though; we can also re-index by thinking in terms of substitution.1004

How do our old index and the new index we are creating relate to each other?1008

We can think in terms of substitution: consider, once again, if we wanted to convert from i = 7, with all of the same thing, to k = 1.1012

Well, since we have i = 7 and k = 1 both at their starting places, how is i related to k?1020

Well, i is the same thing as k + 6; 1 + 6 gives us 7.1026

So, we now have this relationship where i is equal to k + 6.1031

With this in mind, we can substitute to figure out what the upper limit is, and then to figure out what the general term is.1036

To figure out the upper limit, we know that i = 22 is what we are going to plug in, since that is its upper limit.1043

And then, we are going to...over here, we know i = k + 6, so we plug in 22 for i: we have 22 = k + 6.1051

Subtract 6 on both sides, and we get 16 = k for its upper limit.1062

So, at this point, we can now write that our sigma is going to end up being...upper limit of 16;1072

the lower starting index is at k = 1; now, what is it going to end up being?1080

Well, here it was 3i - 16; so we have 3; what is our i in terms of k?1085

Well, i is k + 6; and then, we continue on with the rest of it...minus 16.1093

We have substituted out the i here for the k + 6 here, into what i was in our initial version for the sigma notation.1100

So, at this point, we can expand this and work this out, and simplify it if we want.1111

We could also just leave it as it is, and it would be just fine.1114

The upper limit and lower limit and our index--they will all stay the same throughout.1117

We work this out: 3 times k plus 6 gets 3k +...3 times 6 is 18; minus 16; and so, we get 16 for our upper limit; k = 1 for our index and lower limit.1121

3k + 18 simplifies to 3k + 2, which is exactly what we had the first time when we did this through expanding it.1135

So, substitution and expanding both work the same way.1146

Substitution is probably a little bit faster and easier, but it is a little bit more complicated to see what is going on, to really understand.1148

The important thing is to figure out how your i...whatever your initial symbol and the new symbol that you are changing to are...1153

how they are related to each other; and you probably start with the lower limit,1159

and then ask what your upper limit will have to be, to have that same relationship between old index and new index.1162

And then, what is your expression going to end up being, if you just plug in your connection between the two indexes?1168

All right, working with series, there are various properties that we can occasionally use to our advantage.1177

Let's look at some properties of sums: below, let ai and bi be sequences.1181

They are things that are liable to change; but c will simply always be a constant.1187

Our first one is that the sum from i = 1 to n of c is equal to c times n.1191

Remember: c is just a constant.1197

Why is this the case? Well, remember: if we had the sum of i = 1 to n of c,1198

well, c doesn't change any time, because it is a constant; it isn't affected by the index.1206

So, it is just going to be c, plus c, plus c...plus c, that many times.1210

If we have that, how many times did it show up?1216

Well, we went from i = 1 up until n, so 1, 2, 3, 4...we count up to n; we have a total of n terms in there.1219

If c adds to itself n times, then that is just c times n.1227

So, that is equal to c times n; and that is where we get that property for how sums work.1233

If it is just a constant being added through a series, we can just multiply it by the number of terms in that series.1239

Next, we have the summation of some summation...and this will end up working for any summation limit;1246

the only thing that we have to care about is the index.1253

You can start with any lower limit and any upper limit, including an infinite sum; it is still going to end up working out the same.1255

So, c times ai...we can do this where we pull out the constant, and we bring it out to the front.1261

And it is c times the summation of ai.1267

So, why is this the case? Well, notice: our first term would be c times a1.1270

Plus...our next term would be c times a2; plus...our next term would be c times a3.1276

And it is going to continue on in this pattern.1283

It might end; it might not end; we will just leave it as dots there.1285

But notice: we have a c on each one of our terms.1288

So, if we want, since we have a c here, a c here, a c here, and that thing is going to end up continuing1291

(there is going to be a c on each one of these, because we see it is c times ai),1295

we can pull out all of the c's; we pull them out, so it is c times a1 + a2 + a3 + ....1299

So, it is c multiplied against that entire series; it is not going to have any difference in how we do it.1310

So, this entire series, here to here--we can think of it as c times a new series of just the ai's changing.1315

And that is how we end up getting the same thing here.1323

The final idea: if we have addition, ai + bi, well, we can end up breaking that into two separate series, added together.1327

And once again, this would end up working with any limit; so that is why we don't have a lower limit and don't have an upper limit--just an index.1335

It is because this will work with any limits whatsoever, even an infinite series.1340

We could write this...if it is series of i, ai + bi, well, that is going to be a1 + b1,1346

plus a2 + b2, plus a3 + b3...1353

And it is just going to continue on in that format.1360

Well, order of addition doesn't matter; a1 + b1 + a2 + b21363

is the same thing as a1 + a2, then plus b1 + b2.1368

So, what we do is order all of our a1's first; we will have all of the a1's in our series show up first.1371

And then, we will add on our b1's: b1 + b2 + b3...1381

And that will also go on the same as it would have on its own.1388

So, at that point, what we have done is just re-ordered how we are looking at the series, since we have expanded it,1393

and now we have re-ordered the way we are looking at it.1397

But we haven't changed anything; so at this point, we can just pull it into two separate series, i...1399

so, we will have ai here, plus...and then the bi.1404

There is the a portion of the series and the b portion of the series.1410

So, we can either have them intermixed together, or we can spread them apart and work on each of them on its own.1413

And that is how we end up having this final property.1419

All right, we are ready for some examples.1422

The first example here: Given the sequence below, find the third partial sum.1424

If we are going to find the third partial sum, that is as easy as just adding up the first three terms.1429

That is 0, 3, and 8; 0 + 3 + 8; we end up getting 11; it's done; it was easy.1433

The next one: If we are looking for the seventh partial sum...well, so far, we only have 0, 3, 8, 15, 24...1443

so we are going to have to figure out what the pattern will continue on to.1448

Our first step is to figure out how this pattern ends up working.1452

We look at this through addition: 0 to 3, 3 to 8, 8 to 15...well, the addition is changing each time.1455

So, we could think in terms of some recursive relationship, but that is not really going to make it easier.1460

Multiplication? Multiplication doesn't really work, either.1464

But we look at this for a while, and we might realize that this looks kind of like 1, 4, 9, 16, 25...1466

0, 3, 8, 15, 24...that is just - 1 on each one of the terms.1476

So, we can think of this as being, in general, n2 - 1.1480

If that is the case, then we can figure out what the next terms are.1486

If it is n2 - 1, the next term to follow is going to end up being 36 - 1, because we are at 5 here; 52 - 1 is 24.1489

So, the next will be 62 - 1; that is going to get us 35.1497

The next one will be 72 - 1, 49 - 1, or 48.1502

At this point, the seventh partial sum will be adding up these first seven terms.1507

So, 0 + 3 + 8 + 15 + 24 + 35 + 48...we toss that all into a calculator, and we end up getting 133; there we go.1511

The next one: Expand the below sigma notation into a series of terms; add them together using an ellipsis.1530

We break it apart into the thing where we are not using sigma notation.1536

We are just writing it as one term after another, with some pattern occurring.1539

So, what would our very first term be?1542

Our very first term would start at i = 3; so we plug in a 3 for our i first; that would be 3! over 73.1544

Plus...our next term would end up being...we click up one, from 3 to 4; that will be 4! over 74.1554

Plus...our next one: we click up another one to 5! over 75, plus..., plus...we are going to end up finishing here at 15.1564

We could also start with something before 15; for example, we could write out 14!,1574

because that would be in the expansion: 714, plus 15! over 715.1579

There we go; we have managed to write the whole thing out, using ellipsis.1589

We have expanded the sigma notation into a series of terms.1591

And we don't have to necessarily write out all of these.1596

We could probably get away with not writing it.1599

We could certainly get away with not writing this one right here, the 14!/714.1601

And we might even be able to get away with not writing the 5!/75.1605

It helps us see the pattern, but it is not absolutely necessary.1609

But by including more terms, sometimes the reason we want to expand things is so we can manipulate how the terms work.1612

So, sometimes it is useful to include more things at the start and the end, so that we can end up seeing how things are interacting.1618

It will sometimes allow things to work out better.1625

And we will see that sometimes, when we are working in proofs.1627

Example 3: Condense the sum below into a series expressed using sigma notation.1631

Notice: it is an infinite series.1635

The first thing: let's notice how these connect to each other; how do we get 1/2 - 1/4 + 1/8 - 1/16?1637

Well, we can break this into a sequence, first, that just has a series going on.1643

So, let's look at this underlying sequence; how do we get from 1/2 to -1/4? Well, we multiply by -1/2.1647

How do we get from -1/4 to positive 1/8? We multiply by -1/2.1653

How do we get the next one? We multiply by -1/2.1658

So, with this in mind, how can we create a general term an?1661

That is going to be an =...since we are multiplying by -1/2 each time, it will be -1/2 to the...1665

is it going to be to the n? Well, here we are going to end up starting with something else, actually.1672

It is n - 1, because here is n = 1; we want to not have anything at the beginning, which we could write as times 1/2... so -1/2 to the n - 1, times 1/2.1677

Alternatively, another equivalent way to write this out: we could write this as -1 to the n - 1 over 2 to the n - 1, times 1/2.1687

The 2's in the denominator will end up compacting together.1699

And we could also write this as -1 to the n - 1, over 2 to the n.1701

Both of these are just fine ways to write it.1710

We will end up getting the same thing, whether we write it using this general term, or we end up writing it with this general term.1711

We will end up getting the same series.1718

What is our first term? We decided to set it at some n = 1; let's use an index of i,1720

just because it is what we are used to, although we could use n.1724

Sometimes, you might have a little bit of confusion, since we are used to talking about nth terms with n.1727

So, it might be weird to use an index of n; but you will often see it used, as well.1731

I like i, so I am going to use i.1734

i = 1, because that is our first location; what do we go up to?1737

Well, it is an infinite series, because it goes on forever, and we are never told that it stops.1740

So, we use an infinity symbol on top; and then, let's do this one first.1745

We could write this as -1/2 to the n - 1, times 1/2; and that whole thing has the series applied to it.1749

Or equivalently, this would also be equal to the series--the same upper limit of going on forever, and the same starting index.1759

We could also write this as -1 to the n - 1 over 2 to the n.1770

Either way we end up working it out, both of these are just fine.1775

They are going to give us the exact same answer; they are just two different ways to write it out.1780

Depending on the specific problem, it might be useful to write it one way or the other.1783

But any teacher, if this was just the question, should accept both of these.1787

All right, calculate the value of sigma with a lower limit of 0 and an upper limit of 4.1791

These are indexed where the expression is v! - 5v.1796

The first thing we do is plug in v = 0 for our first 0.1800

Our very first term is going to be 0! - 5(0).1805

Our index is v, so that is the thing being swapped out.1810

Plus...we step up our index to the next level, 0 up to 1 now; so 1! - 5(1).1813

Plus...the next thing, stepping up from 1, is 2; so 2! - 5(2).1822

Plus...the next thing, from 2 to 3: 3! - 5(3); and then, plus...3 to 4...4! - 5(4).1829

Finally, at this point, we notice that that is our upper limit, so we stop.1841

We step from our lower limit, 0, 1, 2, 3, 4...our upper limit, so that is where we stop.1845

And we do each of the steps in between and add them all together.1851

So now, it is just a matter of simplifying to actually get the value.1853

What is 0!? Remember: 0! is simply defined to be equal to 1.1856

All of the other numbers, factorial, are that number, times each of the positive integer numbers underneath it.1861

But 0! is just simply defined to be 1.1868

So, 0! gets us 1; minus...5 times 0 is 0, plus 1! is just 1 times itself; so 1 minus...5 times 1 is 5, plus 2!...1872

2 times 1 is 2, minus 5(2) is 10, plus 3! is 3 times 2 times 1, so it is 6; minus 5(3) is 15;1883

plus 4! is 4 times 3 times 2 times 1, 24; minus 5(4) is 20.1893

We work this out; we have 1 here, plus -4, plus -8, plus -9, plus +4.1901

We see that we have a +4 here and a -4 here; they cancel out.1912

1 + -9 gets us -8, so we have -8 and -8, or -16, once we simplify the whole thing out.1917

Great; the fifth example: Re-index the series below so that it starts at the index k = 1.1924

We are looking to swap the index i = 5 out for k = 1, but have the exact same value come out of the series.1930

Our sigma notation will change; the upper limit will end up changing, and the expression here will end up changing to some extent,1937

so that we can achieve this without affecting the value of the series.1943

We talked about two different ways to do this when we went through the lesson.1947

The first way we will approach this is through expanding it.1951

We will start by doing it through expanding it; and then later, we will look at doing it through substitution.1957

There are two alternative ways; whichever one makes more sense to you, that is the one that you probably want to end up using for your own work.1964

All right, we want to start by expanding this.1969

If we are going to expand i = 5, 37 - i, that is going to be...our first thing would be 37 - 5, so 32.1972

Plus...then the next would be 6 for our i; so 37 - 6 would be 31,1981

plus...then at 7, we will have 30, plus...plus...and then finally,1990

when we get up to an upper limit of 20, 7 - 20 is -13, so it will be 3-13.1997

OK, that is what we see that we end up getting out of this: 32 + 31 + 30...+ 3-13.2004

So, we see that the top steps down each time.2011

We want it to figure out how we can end up showing this.2015

We start with k = 1 for this first slot; use k = 2, k = 3...and here is k = we-don't-know-what-yet.2017

We can actually figure out that it is going to have to be...20 - 5 is 15, so 1, our starting lower limit here, plus 15 would be 16.2026

We know it will end up having it come out to 16; but let's work through it the other way.2034

So, what we have here...it looks like we could write this as 33 - k.2038

33 - 1 would get us 32; the next one, 33 - 2, would get us 31; that works out.2045

At k = 3 on our third term, we would have 33 - 3; it would be 30; that ends up working out.2054

So, if we are going to have 3-13 = 33 - k, we are now going to know that -13 has to be equal to 3 - k.2059

So, we have k = 16 as our upper limit, as well; great.2069

That ends up making sense; we can also figure it out, once again, from:2076

our starting upper limit was 20; our starting lower limit was 5.2079

So, that means that there is a difference of 15; so if we have k = 1, then 1 + 15 has to come out to be 16, as well.2083

So, that is going to be our upper limit; there are two ways of looking at it.2093

Either way, just make sure that you are careful with this sort of thing.2095

It is easy to get confused the first few times you work with it.2097

So now, we know what our general term is; it is 33 - k.2100

We know what our upper limit is, and we know what our index and starting term is, and lower limit.2103

So, we can write the whole thing out: we have 16 as our upper limit; k = lower limit of 1; 3 to the 3 - k; great.2109

So, that is our answer; and that is doing it through expanding as our method.2120

Alternatively, we could do this through substitution.2125

We could have also set this up by noticing that we have i = 5 here, and we want to start at k = 1.2128

So, at the first place, we have i = 5; and then, at our second place, we have k = 1.2133

So, if we have i = 5 here and k = 1 here, then we can see that i is equal to k + 4.2139

So, if i is equal to k + 4, in general (this is just going to hold true in general), then what about our upper limit?2146

Our upper limit: it is going to have to end up being the case that when i is equal to 20...what will our k end up being?2154

We will have 20 = k + 4, which means 16 is equal to k for the upper limit.2160

16 = k for our upper limit; now we just plug in through substitution.2167

So, 16 is our upper limit; we just figured that out; k = 1 is our index and lower limit.2172

So, it is going to be the same thing we started with, 37 - i, so 3 to the 7 minus...2178

but we are not going to use i; we are going to use i = k + 4.2186

So, we are going to swap out this i here for k + 4.2190

We substitute that in, and we have k + 4.2194

We can now work this out and simplify it, since it is the upper limit and lower limit and index...2199

none of those will change; we are just simplifying what is on the inside.2205

3 to the 7 - (k + 4), so minus k and minus 4...7 - 4 gets us 3, and minus k gets us 3 - k in the whole.2208

So, we end up getting the exact same thing, either way we end up approaching this: through substitution or through expanding.2218

We can end up re-indexing the series, so we can have it change at a different starting index.2224

They both work fine; whichever one makes more sense to you, that is the one I would recommend using.2229

All right, the final example: Calculate the value of the series below; use summation properties to make the math less tedious.2233

We could just do this by hand; we could say, "All right, we have 3 times 1 minus 5, plus...plus...all the way up to 3 times 15 minus 5."2240

OK, and then we could calculate what 3 times 1 minus 5 is, and what 3 times 2 minus 5 is, and what 3 times 3 minus 5 is.2252

And we could do this by hand or through the calculator; but that is kind of a pain.2258

That is going to be a lot of writing; that is going to be a lot of calculation.2262

Luckily, there are some clever ways to use the summation properties that we learned early to make this at least...2265

not easy...not fast (it will be easy), but less tedious, so we have to do less writing.2270

How can we do this? Well, first, remember: we can split based on addition.2276

We can see this as 3i + -5; and we can split this into i = 1; the limits to our summation will never end up changing,2280

but we can split this into a 3i +...the same sum over here...15i = 1 on -5; cool.2290

We also were allowed to pull out constants; we have this constant of a 3 here; we have this constant of a -1 here, effectively.2299

We can pull those out front; we can now write this as 3 times the summation of 15, i = 1 of i, plus...2306

Now, I am sorry...plus -1, so let's just write that as minus...summation...15...i = positive 1 (I got confused by the negative) on 5; great.2319

So, 3 times...well, there are no cool properties that we have learned yet, although we will learn in the very next lesson,2330

how to easily sum this one up; but we would have 1 plus 2 plus 3 plus...plus 15.2338

And over here, we will have minus...well, we could do 5 + 5 + 5...but it is just going to show up 15 times.2345

We have i = 1 to 15; we talked about this before.2351

Since a constant is just showing up 15 times, that is going to be 15 times the constant.2354

So, it will be 15 (the number of terms that are there) times the constant that is showing up over and over, so 15 times 5.2359

We can now work out with a calculator 1 + 2 + 3 +...up until 15.2366

That ends up working out to 120; so we have 3(120) - 15(5) is 75; 3 times 120 is 360, minus 75; that comes out to equal 285, and there is our answer.2372

We still had to do a little bit of writing this out.2389

The slowest part here is going to be adding by hand: 1 + 2 + 3 +...to 15, or by using a calculator.2391

But either way, that is much better than having to multiply numbers, and then subtract 5, and then add that, each time, over and over.2399

So, we can use these summation properties to split things up into ways that make them easier to work with.2406

All right, cool; in the next lesson, we will end up looking at our first specific kind of sequence and series, arithmetic.2411

And then later on, we will work with geometric, which will let us apply these ideas about series into one specific thing,2416

where we can actually start creating some formulas to make things really easy and fast.2421

All right, we will see you at Educator.com later--goodbye!2424