Vincent Selhorst-Jones

Arithmetic Sequences & Series

Slide Duration:

Section 1: Introduction
Introduction to Precalculus

10m 3s

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

45m 11s

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

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
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
19:04
19:21
20:04
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

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

56m 31s

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

39m 54s

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

58m 26s

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

48m 49s

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

29m 20s

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

48m 35s

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

33m 24s

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

51m 42s

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

49m 37s

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

28m 49s

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

38m 41s

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

41m 7s

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

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
Solving Quadratics with Ease
9:56
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
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
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

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

34m 16s

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

49m 7s

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

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
8:25
Example of Quadratic Factors
9:26
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
Section 5: Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

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

47m 4s

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

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
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
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
2:08
Circle is 2 Pi Radians
2:31
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
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
9:43
Mnemonic: All Students Take Calculus
10:13
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
Section 7: Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2
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
Section 8: Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

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

49m 5s

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

27m 37s

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

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
Section 9: Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

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

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
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
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

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

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
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
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

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

58m 34s

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

53m 33s

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

48m 7s

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

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
Section 12: Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

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

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
Section 13: Counting & Probability
Counting

31m 36s

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

44m 3s

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

36m 58s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Section 15: Sequences, Series, & Induction
Introduction to Sequences

57m 45s

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

40m 27s

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

31m 36s

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

39m 27s

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

49m 53s

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

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
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
Section 16: Preview of Calculus
Idea of a Limit

40m 22s

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

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
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
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

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

45m 26s

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

10m 41s

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

10m 51s

Intro
0:00
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
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
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
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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### Arithmetic Sequences & Series

• A sequence is arithmetic if the difference between any two consecutive terms is constant:
 an−an−1 = d,
where d is a constant. We call d the common difference. Every "step" in the sequence has the same change. The difference can be positive or negative, so long as it's always the same.
• The formula for the nth term (general term) of an arithmetic sequence is
 an = a1 + (n−1)d.
• To find the formula for the general term of an arithmetic sequence, we only need to figure out its first term (a1) and the common difference (d).
• We can use the following formula to calculate the value of an arithmetic series. Given any arithmetic sequence a1, a2, a3, …, the sum of the first n terms (the nth partial sum) is
 n 2 ·(a1 + an).
• We can find the sum by only knowing the first term (a1), the last term (an), and the total number of terms (n). [Caution: Be careful to pay attention to how many terms there are in the series. It can be easy to get the value of n confused and accidentally think it is 1 higher or 1 lower than it really is.]

### Arithmetic Sequences & Series

For each sequence below, decide whether or not it is an arithmetic sequence.
 31,  33,   35,   37,   … ⎢⎢ 1, 7 9 , 5 9 , 1 3 ,   …
• A sequence is arithmetic if the difference between any two consecutive terms is constant. In other words, the difference between the first and second terms is the same as the difference between the second and third, the third and fourth, etc.
• To check if a sequence is arithmetic, we simply need to find the difference between any two consecutive terms, then see if that's the difference everywhere else in the sequence.
• Let's start with the first sequence:
 31,  33,   35,   37,   …
Begin by noticing that the exponents of the terms follow an arithmetic pattern: they add 2 each time. However, it doesn't matter if the exponents follow an arithmetic pattern: we need the terms themselves to follow an arithmetic pattern. To see if that is the case, we first need to write each term as a clear number:
 31,  33,   35,   37,   …       ⇒       3,   27,   243,   2187,   …
From here, it's pretty easy to see that the sequence is not arithmetic. The difference between any two terms does not remain constant. We can make this very clear by comparing the difference between the first and second with the difference between the second and third:
 33 − 31 = 9 − 3 = 6       ≠        35 − 33 = 243 − 27 = 216
Thus, since the difference does not stay the same for each consecutive pair, the sequence is not arithmetic.
• Now let's consider the second sequence:
 1, 7 9 , 5 9 , 1 3 ,   …
At first, it's difficult to tell if this sequence is arithmetic because the format of each term keeps changing. One way to deal with that would be to put them all in a common format (such as putting them all on a common denominator of 9). Alternatively, we can just check by looking at the differences:
 First to Second: 7 9 − 1     = 7 9 − 9 9 =     − 29

 Second to Third: 5 9 − 7 9 =     − 29

 Third to Fourth: 1 3 − 5 9 = 3 9 − 5 9 =     − 29
If we look at the difference between each pair of consecutive terms, we see that the difference is always −[(2)/(9)]. Thus the sequence is arithmetic.
First sequence is not arithmetic, second sequence is arithmetic.
Give the nth term formula an for the arithmetic sequence that has the first term (a1) and difference (d) given below.
 a1 = 47,        d=8
• The number given by a1 is the first term of the sequence. The value of d is how much each subsequent term changes by. That is a1 + d = a2, and so on for later terms.
• The nth term formula an is a formula where we can plug in n (the number of the term we want) and get out the value of the term for that nth term. For an arithmetic sequence, in the video lesson we saw that there is a simple formula if you know the first term and the difference. It is simply
 an = a1 + (n−1)d.
• Since we have a formula and already know a1 and d, we can just plug in:
 an = 47 + (n−1)·8
From there, we can easily simplify a bit:
 an = 47 + (n−1) ·8     =     47 + 8n − 8     =     8n+39
Therefore we have the general term formula an =8n + 39.

[If we want to check our answer, we know the first term is a1=47 and the difference is d=8, so we can easily write out the first few terms:
 47,   55,   63,   …
Now that we have the first few terms, check to make sure the formula gives the same values:
 n=1     ⇒     a1 = 8(1) + 39     =     47

 n=3     ⇒     a3 = 8(3) + 39     =     63
Great: our formula for an checks out, so we know our answer is correct.]
an = 8n + 39
Give the nth term formula an for the arithmetic sequence written below.
 103,  100,   97,   94,   91,   …
• For an arithmetic sequence, in the video lesson we saw that there is a simple formula if you know the first term and the difference. It is simply
 an = a1 + (n−1)d.
• This means we can easily find the formula if we know the first term and the difference. Looking at the sequence, the first term is clearly there, so we know a1 = 103. Now we just need to find the difference. To do that, just find the difference between two terms (remember, we find difference as "later minus earlier", for example, third minus second). To be sure we got it correctly, do it for two different pairs. Below we'll check first and second along with fourth and fifth:
 100 − 103     =     −3     =     91 − 94
Thus the difference is d=−3.
• Now that we know a1 = 103 and d=−3, we just plug in:
 an = a1 + (n−1)d     =     103 + (n−1)·(−3)
We can easily simplify a bit:
 an = 103 + (n−1)·(−3)     =     103 −3n + 3     =     −3n + 106
Therefore we have the general term formula an =3n + 106.

[If we want to check our answer, we already know the first few terms of the sequence, so we can check that our formula gives the same values:
 n=1     ⇒     a1 = −3 ·1 + 106     =     103

 n=4     ⇒     a4 = −3 ·4 + 106     =     94
Great: our formula for an checks out, so we know our answer is correct.]
an = −3n + 106
Find the value of the below sum.
 3+6+9+12+15+18+21+24+27+30
• We could find the value of the sum by just getting a calculator (or a piece of scratch paper) and adding everything up. It would take a little bit of time, but it could be done that way. However, instead of that, let's start by noticing that each term in the sum is effectively a term in an arithmetic sequence. Thus, we're working with an arithmetic series: a sum where every subsequent term has a common difference with its preceding term.
• From the video lesson, we learned that the sum of any arithmetic sequence is
 n 2 ·(a1 + an),
where n is the total number of terms in the series, a1 is the first term, and an is the last term.
• Looking at this problem, the first term and last term are obvious:
 a1 = 3,        an = 30
We also need to know the number of terms, n. We can approach this in two ways: one, we can simply count the number of terms by hand to find that there are 10 terms, thus n=10. However, for more difficult problems, it will be hard to simply count the terms by hand. Instead, we can find the number of terms by considering the common difference d and how many times that has been added to the starting term of a1 to get the final term of an. Notice, for this problem:
 an − a1     =     30 − 3     =     27        ⇒ 27 3 =     9
Thus, we added 3 to a1 a total of 9 times. Thus, that's 9 steps, but we have to count the first location we started at (a1), so we have a total of n=10. Either way we do it, we can now find the sum with the formula:
 n 2 ·(a1 + an)     = 10 2 ·(3 + 30)     =     5 ·33     =     165
165
Find the sum of the first 1000 positive, even integers.
• The question asks for us to sum up a portion of the positive, even integers. These begin as below:
 2,   4,   6,   8,   …
Notice that the even integers make up an arithmetic sequence, since we get each following term by adding 2.
• From the video lesson, we learned that the sum of any arithmetic sequence is
 n 2 ·(a1 + an),
where n is the total number of terms in the series, a1 is the first term, and an is the last term. Clearly, we have a1=2 and since we're adding up the first 1000 positive even integers, that means there must be n=1000 of the numbers total. The only potentially difficult part for this problem is figuring out what the last term (an) is.
• If you have difficulty finding the last term, try thinking about what the fifth term would be: 2,   4,   6,   8,   10. Thus, for the fifth term, we simply have 2·5 = 10. By this same logic, we can see that the 1000th term will be
 an = 2 ·1000     =     2000
Now that we know all the necessary values for the arithmetic series formula, we can just plug in:
 n 2 ·(a1 + an)     = 1000 2 ·(2 + 2000)     =     500 ·2002     =     1 001 000
1 001 000
Calculate the value of the below sum.
 53∑i=4 (8i + 2)
• (Note: If you are unfamiliar with using sigma notation (Σ) for compactly showing a sum/series, make sure to check out the lesson Introduction to Series. How to read and use the notation is carefully explained in that lesson, but it will be assumed you already understand it in the below steps.) Begin by noticing that the notation indicates an arithmetic series. It will have a difference of d=8 for every term because of the 8i in the sigma notation. Thus, we can apply what we know about finding arithmetic series.
• From the video lesson, we learned that the sum of any arithmetic sequence is
 n 2 ·(a1 + an),
where n is the total number of terms in the series, a1 is the first term, and an is the last term. To find the first term, just plug in the lowest value our sigma index can give: i=4.
 a1     ⇒     i = 4     ⇒     8 ·(4) + 2     =     34
To find the last term, just plug in the upper value our sigma index can give: i=53.
 an     ⇒     i = 53     ⇒     8 ·(53) + 2     =     426
• The trickiest part is probably figuring out what the the number of terms (n) is. To do this, notice that the first term has an index of i=4, while the last term has an index of i=53. Thus there are 53−4 = 49 steps between the two terms. However!, we must also remember to include the starting location, since it doesn't get counted as a step. Thus there are a total of n=50 numbers (49 steps plus 1 for "home"). Now that we know all the necessary values for the arithmetic series formula, we can just plug in:
 n 2 ·(a1 + an)     = 50 2 ·(34+426)     =     25·460     =     11  500
11 500
Calculate the value of the below.
 40∑i=1 (2+5i)  − 22∑k=4 (100−4k)
• (Note: If you are unfamiliar with using sigma notation (Σ) for compactly showing a sum/series, make sure to check out the lesson Introduction to Series. How to read and use the notation is carefully explained in that lesson, but it will be assumed you already understand it in the below steps.) Don't let yourself get freaked out by the appearance of two sigma (Σ) signs: it just means that we need to find each of the two sums, then subtract one from the other. Like in the previous problem, notice that each of the series are arithmetic, so we can use the formula for an arithmetic sequence:
 n 2 ·(a1 + an),
where n is the total number of terms in the series, a1 is the first term, and an is the last term.
• Begin by finding the value of the first series. The first term will occur at the lowest index value of i=1:
 a1     ⇒     i = 1     ⇒     2+5(1)     =     7
The last term occurs at the highest index value of i=40:
 an     ⇒     i = 40     ⇒     2+5(40)     =     202
Finally, the number of terms is the number of "steps" plus one for "home": 40−1=39 steps, then one more for the first term, giving n=40. Plug in to the formula to find the sum:
 40∑i=1 (2+5i)     = n 2 ·(a1 + an)     = 40 2 ·(7+202)     =     20 ·209     =     4180
• Next, we find the value of the second series. The first term will occur at the lowest index value of k=4:
 a1     ⇒     k = 4     ⇒     100−4(4)     =     84
The last term occurs at the highest index value of k=22:
 an     ⇒     k=22     ⇒     100−4(22)     =     12
Finally, the number of terms is the number of "steps" plus one for "home": 22−4=18 steps, then one more for the first term, giving n=19. Plug in to the formula to find the sum:
 22∑k=4 (100−4k)     = n 2 ·(a1 + an)     = 19 2 ·(84+12)     = 19 2 ·96     =    912
• From the above work, we now know
 40∑i=1 (2+5i) = 4180 ⎢⎢ 22∑k=4 (100−4k) = 912
Our goal was to find the value of the below
 40∑i=1 (2+5i)  − 22∑k=4 (100−4k),
so we just plug in the values that we found:
 40∑i=1 (2+5i)  − 22∑k=4 (100−4k)     =     4180   −   912     =     3268
3268
An arithmetic series has a first term of a1 = 3 and a common difference of d=4 between each pair of consecutive terms. If the total value of the series is 4095, what is the number of terms in the series, n? What is the value of the final term in the series, an?
• From the problem, we're adding up some number of terms from an arithmetic sequence that begins at a1=3 with a difference of d=4:
 3,   7,   11,   15,   19,   …
We know that if we add up some number of terms, we get the below:
 3  + 7  + 11  + … +  ?    =   4095
Our goal is to figure out how many terms we have to add up and what the value of that last term will be.
• From the video lesson, we learned that the sum of any arithmetic sequence is
 n 2 ·(a1 + an),
where n is the total number of terms in the series, a1 is the first term, and an is the last term. Thus we can plug in to that and we know that the value it should output is 4095:
 n 2 ·(3 + an) = 4095
The only issue is that the above equation has two unknowns: both n and an. To be able to solve the equation, we need to somehow put it in terms of only a single unknown. We need to somehow express n or an in terms of the other.
• Notice that each subsequent term is based off the previous terms. Consider the below:
 a2 = 7 = 4+3,            a3 = 11 = 2·4 + 3,            a4 = 15 = 3·4 + 3
We can see that getting to a given term is a case of adding up the difference the correct number of times. To get to the nth term from the first term, it will take (n−1) "steps". Thus, we must add the difference a total of (n−1) times to the starting value. Therefore, to get to our last term an, we have
 an = (n−1)·4 + 3,
which we can simplify to an =4n1.
• Now we can plug in to our arithmetic series formula to get an equation with a single unknown:
 n 2 ·(3 + an) = 4095     ⇒ n 2 ·(3 + [4n−1]) = 4095
From here, work towards solving for n:
 n 2 ·(3 + [4n−1]) = 4095     ⇒ n 2 ·(4n+2) = 4095     ⇒     2n2+n = 4095
We can solve using the quadratic formula, so set it up:
2n2 + n − 4095 = 0    ⇒     n = −1 ± √ 12 − 4·2·(−4095)

2·2
Working with a calculator, we get
 n = −1 ±181 4 ,
which provides us with two values for n:
 n = −1+181 4 = 180 4 = 45 ⎢⎢ n = −1−181 4 = −182 4 = − 91 2
Since n is the number of terms we must add together, it clearly makes no sense to have a negative (or fractional) value for n, so we throw out the second answer, leaving us with n=45.
• Finally, we can use this value to find the last term. Since
 an = (n−1)·4 + 3,
we can plug in n=45 to find the value of an:
 an = (45−1) ·4 + 3     =     44 ·4 + 3     =     179
[Since we put so much work into the problem, it would probably be a good idea to check our result since it is easy to check. Just use the formula for arithmetic series with the values of a1, n, and an:
 n 2 ·(a1 + an)     = 45 2 ·(3 + 179)     = 45 2 ·182     =     4095
Great! Since the formula gives the same sum as the problem told us it should have, we know our results for n and an are correct.]
n=45,    an = 179
A national spelling bee competition gives scholarships to those who finish in the top 20 spots. First place gets $5000, second place gets$4800, third gets $4600, and so on. How much scholarship money is given out in total? • Notice that the prizes are given as an arithmetic sequence, since each prize has a difference of d=−200 between it and the previous one. Because we're dealing with adding up the terms from an arithmetic sequence, we can use our formula:  n 2 ·(a1 + an) • From the problem, it's quite clear there are a total of 20 prize-winners, so n=20, and that first place gets$5000, so a1 = 5000. The only difficult part is figuring out the value of the final prize, an. To figure it out, remember it is the 20th prize so it is 19 steps away from first place (be sure to notice that it is 19 steps, not 20: first place does not count as a step). Thus we apply the difference 19 times and add that to the first term:
 an = a20 = 19·(−200) + 5000     =     1200
Therefore the last prize winner gets an=1200.
• Now that we know all the pertinent values, we can apply the formula:
 n 2 ·(a1 + an)     = 20 2 ·(5000+1200)     =     10 ·6200     =     62 000
\$62 000
An ancient coliseum has a central arena that is ringed by an audience seated in circular rows. The row closest to the arena has a total of 60 seats in it. The next one up has 65 seats, and the next up has 70 seats. If this pattern continues and the coliseum has a total of 23 rows, how many seats does the coliseum have?
• Notice that the number of seats that each row has increases with a common difference of d=5: this means we're working with an arithmetic sequence. We want to add up the number of seats in the first 23 rows, so we can do that by adding up the first 23 terms of the arithmetic sequence.
 n 2 ·(a1 + an)
• From the problem, we know there are a total of 23 rows, so n=23, and that the first row has 60 seats, so a1=60. The only difficult part is figuring out the number of seats in the final row, the 23rd row, a23. To figure it out, remember it is the 23rd row, so it is 22 steps away from first place (be sure to notice that it is 22 steps, not 23: first place does not count as a step). Thus we apply the difference 22 times and add that to the first term:
 an = a23 = 22·(5) + 60     =     170
Therefore the final row has an=170 seats.
• Now that we know all the pertinent values, we can apply the formula:
 n 2 ·(a1 + an)     = 23 2 ·(60 + 170)     = 23 2 ·230     =     2645
2645

*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.

### Arithmetic Sequences & 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 0:05
• Definition: Arithmetic Sequence 0:47
• Common Difference
• Two Examples
• Form for the nth Term 2:14
• Recursive Relation
• 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

### Transcription: Arithmetic Sequences & Series

Hi--welcome back to Educator.com.0000

Today, we are going to talk about arithmetic sequences and series.0002

Now that we have an understanding of sequences and series, we are ready to look at specific kinds of sequences.0006

The first that we will consider is an arithmetic sequence, a sequence where we add a constant number each step.0011

We will add some number, and we keep adding the same number every time we go forward a term.0018

Sequences of this form pop up all the time in real life, and we often need to add up their terms.0022

We will explore the creation of a formula for arithmetic series that will allow us to quickly and easily add up those terms.0027

Plus, by the end of this lesson, we will be able to add all of the numbers between 1 and 1000 in less time than it takes to put on a pair of shoes.0033

I think that is pretty cool; we will be able to do something really, really fast that seems like it would take a long time, just like that.0041

All right, a sequence is arithmetic if the difference between any two consecutive terms is constant.0048

We can show that with the recursive relationship an - an - 1 = d.0055

Notice: the nth term minus the n - 1 term (that is, the term one before the nth term) equals d.0061

Some term minus the term before it equals d.0069

And d is just some constant number; we call d the common difference.0073

Here are two examples of arithmetic sequences: they are arithmetic because every step in the sequence has the same change.0079

For example, here, 1 to 4, 4 to 7, 7 to 10, 10 to 13...it is + 3, + 3, + 3, + 3.0087

It is always the same amount that we change; it is always adding a constant number.0099

Over here, it is 5 to 3, 3 to 1, 1 to -1, -1 to -3...it is - 2, - 2, - 2, - 2; and that pattern would continue, as well.0103

In this case, our common difference is -2; we are adding -2 each time; or we can think of it as subtracting 2 each time.0116

The important thing is that it is always the same; the difference can be positive; the difference can be negative;0124

but it always has to be the same for each one--that is what makes it an arithmetic sequence.0128

How can we find the nth term? The definition for an arithmetic sequence is based on a recursive relation.0135

It is based on an = an - 1 + d, that some term is equal to its previous term, with d added to it.0141

So, how can we turn this formula into something for the general term--how can we get a general term formula out of this?0148

Remember: a recursive relation needs an initial term--we have to have some starting place.0154

There is nothing before our starting place to refer back to, so we actually have to be given the initial term directly.0158

Now, we don't know its value yet; so we will just call it a1; we will call it the first term--we will leave it as that.0164

Now, from an = an - 1 + d, we see that a1 relates to later terms.0171

At the most basic level, we have that a2 is equal to a1 + d.0177

The second term is equal to the first term, adding d onto it; that is what it means for it to be an arithmetic sequence.0183

We can take this out and continue looking at later terms.0188

a3 would be equal to a2 + d, based on this recurrence relation.0191

But we just figured out that a2 is equal to a1 + d, so we can swap out for a2:0196

a1 + d, which now gets us a1 + 2d when we add this d to that one there.0203

So, we have a1 + 2d for a3; when we work on a4,0210

well, a4 is going to be a3 + d, the previous term, adding d.0215

But once again, we have just figured out that a1 + 2d is what a3 is.0220

So, we can plug in for a3; we have a1 + 2d, and now we can add that d onto the 2d.0225

So, we end up getting that a1 + 3d is what a4 is equal to.0231

So, we notice that this pattern is going to keep going; we are just going to keep adding on more and more d's to our number.0238

So, a1 = a1; a2 = a1 + d; a3 = a1 + 2d;0245

a4 = a1 + 3d, and this pattern will continue down.0253

We see that the nth term is n - 1 steps from a1.0257

The first term is at 1, and the nth term is at n; so to get from the first term to the n term, we have to go forward n - 1 steps.0262

Since it is n - 1 steps away from a1, we will have added d for each of those steps;0270

so we will have added d that many times, or n - 1 times d.0276

That means that the an term is equal to a1 plus all of those steps times d.0280

So, the nth term, an, equals a1, our first term, plus n - 1 times d.0288

Thus, to find the formula for the general term of an arithmetic sequence, we only need to figure out the first term, a1, and the common difference, d.0294

With those two pieces of information, we automatically have the general term; we automatically have that nth term formula--pretty great.0306

How about if we want an arithmetic series formula?0314

Consider if we were told to add up all of the integers from 1 to 100; how could we find 1 + 2 + 3 + 4 + 5 + 6...+ 98 + 99 + 100?0316

How could we add that whole thing up? Well, we could do it by brute force, where we would just sit down0327

with a piece of paper or a calculator and just punch the whole thing out.0334

We could do it by hand; but that is going to take a long time.0337

And any time that we end up seeing something that is going to take us forever to do, we want to ask ourselves,0341

"Is there a way to be clever--is there an easier way that I can do this that will be able to take away some, or a lot, of the time and effort?"0345

How are we going to do that? We want to look for some sort of pattern that we can exploit.0353

We want to find a pattern that we can exploit--something that will keep happening--0357

something that we can rely on, that will keep us from having to add up all of these numbers,0361

because we can instead use this pattern to give us a deeper insight to what is going on.0365

So, if we look at this for a while, we might start to realize that there is a pattern in the numbers.0368

But that doesn't help us, because that is just adding the numbers.0374

But is there a way that addition itself has a pattern?0376

There is something that we could match up--something that we could create--and this is where we are getting really clever.0379

This is the hard part, where you really have to sit down and think about it for a long time.0383

And hopefully you just end up getting some "lightning bolt" of insight.0386

And hopefully, at some point, we will notice that here is 100; here is 1; if we add them together, we get 101.0390

But not only that--if we had 99 (let's use a new color)...if we use 99 and 2, we get 101.0396

If we add 98 and 3, we get 101; if we keep doing this, working our way in, we are going to keep adding things up to 101, 101, 101...0404

So, if we notice that we can add 1 and 100 to get 101; 2 and 99 to get 101; 3 and 98...we get 101; and so on and so on and so on...0415

what we can do is pair up each number from 1 to 50 with a number from 51 to 100.0425

And we will always be able to make 101 out of it.0430

We start out at the extremes, 1 and 100, and we work in: 2 and 99; 3 and 98; 4 and 97; until we finally make our way to 50 and 51.0433

So, we were able to figure out this pattern; there is something going on.0446

Now, we are finding something; now we have something that we can pull into a formula that will make this really easy.0449

With this realization in mind, let's look for an easy way to pair up the numbers.0455

The first thing: it is nice to give names to things in algebra--it lets us work with them more easily.0458

So, let us have s denote the sum of 1 to 100; so s is equal to 1 + 2 + 3...+ 99 + 100; it is all of those numbers added up together.0462

Now, notice: we can rewrite the order of those numbers, since order of addition doesn't matter.0472

1 + 2 + ... + 99 + 100, here, is the same thing as 100 + 99 + ... + 2 + 1.0476

We can swap the order, and we still have the same value in the end.0484

That is one of the nice things about the real numbers: order of addition doesn't matter.0489

Furthermore, we can add two equations together--that is elimination.0492

Remember: when we worked on systems of linear equations, if you have an equation, you can just add it0497

to another equation, because they are both working equations.0501

You can add the left sides and the right sides, and you know that everything works out; there is nothing wrong with doing that.0503

What we have is: we can add the top equation there and the bottom equation, the normal order and the reversed order; we can add them together.0508

What do we end up getting? Well, here we have a hundred and one, so we get 101; here we have 99 and 2, so we get 101;0516

here we have 2 and 99, so we get 101; here we have 1 and 100, so we get 101.0523

And we know that we are going to end up having 101 show up for every one of the values inside of here, as well.0527

How many terms are there total? Well, we had 1, 2, 3...99, 100; so we had 100 terms, left to right.0532

So, if we had that many terms total, well, even after we add them up, and each one of them becomes 101, then we have 100 terms total.0544

We have 100 terms on the right side; so if we have the same number appearing 100 times, we can just condense that with multiplication.0553

We can condense all of that addition with multiplication: 3 + 3 + 3 + 3 is the same thing as 3 times 4, 4 times 3.0561

So, if we have 101 appearing 100 times, then we can turn that into 100 times 101.0569

Our left side is still just 2s; so we have 2s = 100(101).0576

What we are looking for is the sum, s = ...up until 100; so we just divide both sides by 2 to get rid of this 2 here.0580

Divide both sides by 2; 100 divided by 2 gets us 50, so 50 times 101 means we have an answer of 5050.0588

So, that probably took about as much time as if we had added up 1 + 2, all the way up to 100.0596

If we had done that whole thing by hand, it would have taken a while.0601

And now, we have the beginning kernel to think, "We can just do this for anything at all, and it will end up working out!"0603

Indeed, that is what will work out.0609

We have this method in mind of being able to string all of the things in our arithmetic sequence together,0610

and then flip it and add them together and see what happens.0616

We can now figure out a general formula for any finite arithmetic series.0619

Let sn denote the nth partial sum--that is, the first n terms of the sequence, added together, of some arithmetic sequence.0623

So, sn = a1 + a2 + a3 + ... up until we get to + an - 1,0632

up until, finally, an is our end, because we have the nth partial sum; great.0639

Earlier, we figured out the general term for any arithmetic sequence is an = a1 + (n - 1)d.0645

So, we can swap out a1 for what it is in the general form, a2 for what is in the general term,0652

an - 1 for what it is in the general term, an for what it is in the general term.0658

This will get everything in terms of a1 and d and that n; great.0662

Thus, we can write out sn if we want to; we can write it out as sn = a1,0667

and then a2 would be a1 + d (2 minus 1, so 1 times d...a1 + d).0673

We work our way out: an - 1 would be a1 + (n - 2)d;0679

we plug in n - 1 for the general n term, so n - 1, minus 1...n minus 2 times d.0683

And finally, the an would be n plus n minus 1 times d.0689

Great; so we have this thing where the only thing showing up there is a1, n, and d.0693

We have far fewer things that we have to worry about getting in our way.0698

Furthermore, we can write sn in the opposite order; we are allowed to flip addition order.0702

So, we write it in the opposite order as sn =...the last thing now goes first...a1 + (n - 1)d.0706

a1 + (n - 2)d goes next; and then finally, we work our way down: a1 + d...a1...0713

so now, we have the equation in its normal order and the equation in its opposite order.0719

We can add these two equations for sn together; they are both equal; they are both fine equations.0723

There is nothing wrong with them, so we are allowed to use elimination to be able to add equations together.0728

We add them together, and we have our normal way of writing it, sn = a1 + ...0732

+ up until our an term, a1 + (n - 1)d.0738

And then, the opposite order is sn = a1 + (n - 1)d + ... up until a1.0741

We add these together; a1 + a1 + (n - 1)d ends up getting us 2a1 + (n - 1)d.0748

Over on the far end, we will end up having the exact same thing: a1 + (n - 1)d + a1 will get us 2a1 + (n - 1)d.0756

And we are going to end up getting the same thing for every term in the middle, as well.0766

All of those dots will end up matching up, as well, for the same reason that we added 1 and 100, then 2 and 99, then 3 and 98.0770

They all ended up matching up together; the same thing happens.0776

We will always end up having that be the value for each of the additions through our elimination.0779

So, notice, at this point, that we can do the following: we can write this 2a1 + (n - 1)d here: 2a1 + (n - 1)d.0785

Well, that is the same thing: we can split the 2a1 into two different parts.0793

So, we have a1 plus...and then we can just put parentheses: a1 + (n - 1)d.0798

Well, we already have a way of writing this out: a1 + (n - 1)d is the an term.0803

So, what we have is a1 + an; so we can write this as a1 + an.0809

We swap each one of them out; we now have that 2sn is equal to a1 + an + ... + a1 + an.0814

How many terms are there total? There are n terms here total, because we started at a1 here,0822

and we worked our way up until we finally got to an here: first term, second term, third term...up until the nth term.0828

The first term to the nth term--that means that we have a total of n terms.0835

So, a1 + an gets added to itself n times (n terms, so n times, since they are all identical).0839

At that point, we have 2sn = n(a1 + an).0845

And since what we wanted on its own was just sn, we divide 2sn by 2 on both sides of our equation.0850

And we get n/2(a1 + an); great.0857

Thus, we now have a formula for the value of any finite arithmetic series.0863

Given any arithmetic sequence, a1, a2, a3...the sum of the first n terms is n/2(a1 + an).0868

This works for any finite arithmetic sequence, starting at the first term and working up to the nth term.0884

So, we can find the sum by only knowing the first term, a1, the last term, an, and the total number of terms, n.0890

That is all we need, and we can just easily, just like that, find out what the value of a finite arithmetic series is--that is pretty great.0904

Before we go on, though, one little thing to be careful about: be careful to pay attention to how many terms are in the series.0911

It can be easy to get the value of n confused and accidentally think it is one higher or one lower than it really is.0918

We will see why that is the case in the examples; so just pay really close attention.0925

If you are working from a1 up until an, then that is easy, because it is 1, 2, 3, 4...up until the n.0928

So, it must be that there are n things there.0934

But it can start getting a little bit more confusing if you start at a number that isn't 1--0935

if you start at 5 and count your way up to 27, how many things did you just say out loud?0939

We will see what we are talking about there as we work through the examples.0943

All right, let's see some examples: Show that the sequence below is arithmetic; then give a formula for the general term, an.0947

First, to show that it is arithmetic, we need to show that it has a constant difference.0953

To get from 2.6 to 3.3, we add 0.7; to get from 3.3 to 4, we add 0.7; to get from 4 to 4.7, we add 0.7;0956

and we can see that this is going to keep going like this, so it checks out.0968

It is an arithmetic sequence, because there is a common difference; its common difference is 0.7.0972

To figure out the general term, an, we want to figure out what our a1 is.0979

a1 is just the first term, which is 2.6; so our general term, an, always ends up working like this.0983

It is the first term, plus (n - 1) times the common difference.0990

So now, we can just plug in our values: an =...we figured out that a1 is 2.6, plus (n - 1)...0995

that is just going in because it is the general term...times our difference of 0.7.1001

And there we are; there is our general term; there is the formula for the nth term.1007

Alternatively, if we wanted to, we could also simplify this a little bit more, so it isn't n - 1 (that part doesn't show up).1011

Sometimes it is useful to have it in this format; but other times we might want to simplify it.1017

So, if we decided to simplify it, we would have an = 2.6 + n(0.7), so 0.7n, minus 1(0.7), so minus 0.7;1021

so the 2.6 and the -0.7 interact, and we have 1.9 + 0.7n.1033

Alternatively, we could write it like this: either of these two ways is perfectly valid.1040

Either one of these two things is a formula for the general term.1045

Sometimes it will be more useful to write it one way, and sometimes it will be more useful to write it the other way.1048

So, don't be scared if you see one written in a different way than the other one; they are both totally acceptable.1051

The second example: Find the value of the arithmetic series below.1058

What is our difference? That will help us understand what is working on here.1062

The difference will not actually be necessary to use our formula for an arithmetic series,1066

but it will help us see what is going on just a little bit on our way to using it.1070

We have a difference of 5 each time; so it is + 5, + 5, + 5...difference = 5.1073

We need three things to know what the series' value is.1080

We need to know the first term; that is easy--we can see it right there: a1 = 7.1086

We need to know the last term; that is easy, as well: an = 107.1091

And we need to know what the number of terms is, n = ?.1096

So, how can we figure out how many terms there are?1101

We might be tempted to do the following: 107 - 7 comes out to be 100; and then we say,1104

"Oh, our difference is 5, so let's divide by 5," and so we get 20; so n must be 20...NO, that is not the case.1114

Now, to understand why this is not the case, we need to look at something.1123

Let's create a little sidebar here to understand what is going on a little better.1128

Look at...if we wanted to talk about the number from 1 to 25, if we wanted to count how many numbers there are between 1 and 25,1133

we count: 1, 2, 3, 4...25--pretty obvious: that means that the number of numbers is 25.1140

There are 25 things there; great--that makes sense.1148

What if we were talking about going from 25 to 50?1150

Well, we might say that we can count by hand...50 - 25...so then, there are a total of 25 terms, because 50 - 25 is 25...No.1155

Wait, what? Well, let's count it by hand: how does this work.1165

25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50; that is 26 things that we just counted out there.1170

So, what is going on? 25 counts as something we have to count.1189

When we count from 1 to 25, if we just subtracted 25 minus 1, if we subtracted 1 from 25, 25 - 1, that is 24.1195

But we don't say, "Oh, from 1 to 25, there must be 24 numbers, because 25 - 1 is 24"--no, we don't think like that.1202

We know that counting from 1 to 25, it is 25 things there.1207

So, counting from 25 to 50 means a difference of 25; that means that there are 25 steps to get to 50 from 25.1211

But we also have to count the first location that we started on.1220

We have to actually count to 25, as well; so that is a total of 26.1223

What we have is: we have 50 - 25 = 25; but then we have to have 25 + 1 = 26.1227

So, our number is 26 for how many things we ended up doing.1240

It is the same thing with 7 to 107; how many steps?1245

If we have a distance of 5 for each step, how many steps do we have to take from the 7?1250

Well, we have to take 20 steps, because 20 times 5 is 100.1256

So, if we take 20 5-distance steps from 7, we will make it to 107; so we have 20 steps that we take.1260

But we also have to count the 7, so we end up actually have n = 21.1268

And so, that is our value for n.1278

This is why you really have to think about this stuff carefully.1280

It is really easy to just say, "OK, I took that many steps, so that must be my value."1283

No, you have to really pay attention to make sure that you are counting also where you started.1287

But sometimes, you have to pay attention and think about it: "Did I already count where I started?"1291

So, you really have to be careful with this sort of thing.1295

It is easy, as long as you can get a1, an, and the number of steps, n.1296

But sometimes, it is hard to really realize just how many steps you have precisely.1300

So, be careful with that sort of thing.1304

All right, at this point, we are ready to use our formula: n/2(a1 + an) is the value1306

of the sum of all of those numbers, the sum of that finite arithmetic series.1313

Our n was 22, over 2; our a1 was 7; our an was 107, so + 107; so we get 21/2(114).1317

We punch that into a calculator, and we end up getting 1197; 1197 is our answer for adding that all up.1331

All right, the third example: here is my thing that I said at the beginning, when we talked through the introduction.1341

At this point, we are now able to add the numbers from 1 to 1000 in less time than it takes to put on a pair of shoes and tie them up.1346

If you are going to take off your shoes and test if that is really the case, now is the time to do it, before we start looking at this problem.1354

Are you ready? OK, let me read the problem, and then we will have things be a fair challenge between shoe-putting-on and massive addition.1360

Add all of the integers from 1 to 1000 (so 1 + 2 + 3...+ 999 + 1000).1369

All right, are you ready? Ready, set, shoes on now!1376

The first term is equal to 1; our last term is equal to 1000; the total number of terms we have from 1 to 1000 is simply 1000.1380

So, it is 1000, the number of terms, divided by 2 times the first term, plus the last term...1000, so 500 times 1001...is equal to 500,500; I am done!1388

It's pretty amazing how fast we can end up adding everything from 1 to 1000 in that little time.1401

This is the power of the series formula; this is the power of studying series--1409

the fact that we can add things that would take so long to work out by hand, like that.1413

We can do this stuff really, really quickly, once we work through this.1418

Our first term was 1; our last term was 1000; so a1 = 1; an = 1000.1422

How many things are there from 1 up to 1000? Well, that one is pretty easy; that one is 1000.1428

So, we have n/2, 1000/2, times 1 + 1000, so 500 times (I accidentally made a little bit of a typo as we were writing that out) 1001.1432

Multiply that out, and you get 500 thousand, 500; it is as simple as that.1445

The fourth example: Find the value of the sum below.1449

To do this, let's write out what this sigma notation ends up giving us.1453

i = 4 is our first place, so that is going to be 53 minus...oops, if it is going to be an i here and a k here, they have to agree on that.1458

So, that should actually read as a k, or the thing on the inside should read as an i, for this problem--I'm sorry about that.1468

53 - 4 times 4 is our first one; plus 53 - 4 times 5 (our next step up--our index goes up by one) plus 53 - 4 times 61474

(our index goes up one again); and it keeps doing this, until we get to our last upper limit for our sum, 53 minus 4 times 25; cool.1491

Now, at this point, we think, "OK, how can we add this up?"1503

Well...oh, this is an arithmetic sequence; it is 4 times some steadily-increasing, one-by-one thing.1505

So, it is an arithmetic sequence, an arithmetic series, that is appearing here.1513

If that is the case, what do we need?1517

We need to know the first term, the last term, and the number of terms that there are total.1518

If that is the case, all we really care about is this first term and this last term.1525

All of the stuff in the middle--we don't really need to work with it.1529

53 - 4 times 4, so 53 - 16, plus...up until our last term of 53 - 4 times 25 is 100; 53 - 16 comes out to be 37, plus...plus...53 - 100 comes out to be -47.1532

Our first term is 37; our last term is -47; the only real question that we have now is what is the value for n.1554

How many terms are there total? We are counting from 4 up to 25.1566

So, from 4 up to 25, how many steps do we have to take there?1570

25 - 4 means 21 steps; but notice, it is steps; there are 21 steps, but we also have to count the k = 4.1573

It is 21 steps above 4, so we also have to count the step at 4; so 21 + 1 counts where we start.1584

It is not just how many steps you take forward, but how many stones there are total, so to speak.1592

21 + 1 = 22 for our value of n; so we get n = 22; great.1598

n = 22; we know what the first one is; we know what the last one is; we are ready to work this out.1606

22 is our n, divided by 2; n/2 times the first term, 37, plus the last term, -47...we work this out.1611

22/2 is 11, times 37 + -47 (is -10)...we get -110; that is what that whole series ends up working out to be; cool.1622

All right, and we are ready for our fifth and final example.1635

An amphitheater has 24 seats in the third row, 26 in the fourth.1637

If this pattern of seat increase between rows is the same for any two consecutive rows, and there are 27 rows total, how many seats are there in total?1641

The first thing we want to do is understand how this is working.1650

Well, it is an amphitheater; we can see this picture here to help illustrate what is going on.1652

As we get farther and farther from the stage, it curves out more and more.1657

It is pretty small near the stage; as it gets farther and farther, it expands out and out.1660

So, that means there are more seats in every row, the farther back in the row we go.1665

Later rows will end up having more seats than earlier rows.1670

That is why we have 24 in the third, but 26 in the fourth.1673

We can see this: the early rows have fewer seats than the later rows, from how far they are from the stage.1676

OK, what we are looking for is how many seats there are total.1683

What we can do is talk about the third row having 24 seats; the fourth row has 26 seats; so we could think of this as a sequence,1689

where you know that it has the constant increase; the pattern of seat increase between rows is always the same.1698

What we have here is an arithmetic sequence; it makes sense, since that is what the lesson is about.1703

We can write 24 seats in the third row as a3 = 24.1708

We also know that it is 26 in the fourth row; so a4 = 26.1716

OK, so if that is the case, and the pattern of seat increase is always the same for two consecutive rows,1726

that means...to get from 24 to 26, we added +2; we have a common difference of positive 2.1734

So, if that is the case, what would the second row have to be?1741

Well, it would have to be -2 from the third row, so it would be at 22; 22 + 2 gets us to 24.1744

The same logic works for the first row, so the first row must be at 12 20, 22, 24, 26...that is the number of seats.1750

We see that we have a nice arithmetic sequence here.1758

What we are really looking to do is take a finite arithmetic series.1760

We are looking to figure out what is the 27th partial sum, because what we want to do1763

is add the number of seats in the first, second, third, fourth...up until the twenty-seventh row.1768

And we will be able to figure out all of those.1774

So, what we need to use the formula that we figured out: we need to know how many seats there are in the first row,1776

how many seats there are in the last row, and the total number of rows.1781

How many seats are there in the last row?1784

a27 is going to be our last row, because there are 27 rows total.1787

a27 is going to be the number in the first row, plus...27 - 1 (n - 1 is 27 - 1) times our difference (our difference is 2, so times 2).1791

This makes sense, because what we have here is that the 27th row1803

is going to be equal to our first row, 20, plus...how many steps is it to get from the first to the twenty-seventh row?1806

That is going to be 26 steps, times an increase of 2 for every row we go forward.1812

We work this out; that means that our 27th row is equal to 20 plus 26 times 2 is 52;1818

a27, our 27th row...the number of seats in our 27th row, 20 + 52, is 72 seats total.1825

So, at this point, we have 72 seats for our final row, 20 seats for our first row...1833

how many total rows are there? Well, that is going to be...if we are going from the first row,1839

up until the 27th row, then we can just count: 1, 2, 3, 4...counting up to 27.1843

That is easy; that is 27, so n = 27.1847

So, our formula is the number of terms total, divided by 2, times the first term plus the last term.1851

So, our number of terms total (number of rows total) is 27, divided by 2, times...1859

what is the first term, the first number of seats? 20, plus...what is the last number of seats, our last term? 72.1864

27/2 times 92...we work that out with a calculator, and we end up getting 1242 seats total in the amphitheater.1872

Great; there we are with the answer.1885

All right, in the next lesson, we will end up looking at geometric sequences and series,1886

which give us a way to look at this through multiplying instead of just adding.1890

All right, we will see you at Educator.com later--goodbye!1894