  Vincent Selhorst-Jones

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

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
18:50
19:04
19:21
20:04
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

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

56m 31s

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

39m 54s

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

58m 26s

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

48m 49s

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

29m 20s

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

48m 35s

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

33m 24s

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

51m 42s

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

49m 37s

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

28m 49s

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

38m 41s

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

41m 7s

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

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
9:56
11:38
Derivation
11:43
Final Form
12:23
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
26:49
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
19:24
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
7:41
8:55
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

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

34m 16s

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

49m 7s

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

44m 56s

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

35m 17s

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

47m 4s

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

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
2:08
2:31
2:52
Half-Circle and Right Angle
4:00
6:24
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

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

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Section 7: Trigonometric Identities
Pythagorean Identity

19m 11s

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

23m 16s

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

52m 52s

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

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

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

25m 43s

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

56m 40s

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

49m 5s

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

27m 37s

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

34m 25s

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

55m 40s

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

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
11:42
Example of Using a Point
12:41
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: Vectors and Matrices
Vectors

1h 9m 31s

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

35m 20s

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

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

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

58m 34s

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

53m 33s

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

48m 7s

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

38m 16s

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

40m 43s

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

57m 37s

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

31m 36s

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

44m 3s

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

36m 58s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

57m 45s

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

40m 27s

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

31m 36s

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

39m 27s

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

49m 53s

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

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
Section 16: Preview of Calculus
Idea of a Limit

40m 22s

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

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

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

45m 26s

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

10m 41s

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

10m 51s

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

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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### Geometric Sequences & Series

• A sequence is geometric if every term in the sequence can be given by multiplying the previous term by some constant number r:
 an = r·an−1.
We call r the common ratio (since r = [(an)/(an−1)]). Every "step" in the sequence multiplies by the same number. The number can be anything, so long as it is always the same for each step.
• The formula for the nth term (general term) of a geometric sequence is
 an = rn−1 ·a1.
• To find the formula for the general term of a geometric sequence, we only need to figure out its first term (a1) and the common ratio (r).
• We can use the following formula to calculate the value of a geometric series. Given any geometric sequence a1, a2, a3, …, the sum of the first n terms (the nth partial sum) is
 Sn = a1 · 1−rn1−r .
• We can find the partial sum (Sn) by only knowing the first term (a1), the common ratio (r), and how many terms are being added together (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.]
• Unlike an arithmetic series, we can also consider an infinite series when working with a geometric sequence. If |r| < 1, then
 S∞   =  a1 · 1 1−r .

### Geometric Sequences & Series

For each sequence below, decide whether or not it is a geometric sequence.
 31,   33,   35,   37,   … ⎢⎢ −4,   6,   −9, 27 2 ,   …
• A sequence is geometric if the ratio between any two consecutive terms is a constant. In other words, there exists some constant value r that we can multiply to get from one term to the following term.
• To check if a sequence is geometric, we simply need to find the ratio between any two consecutive terms, then see if every pair of consecutive terms in the sequence has the same ratio.
 31,  33,   35,   37,   …
Don't be fooled by the fact that the exponents come as an arithmetic sequence (constant difference), we care about the ratio between the terms. Check to see what the ratio between each consecutive pair of terms is:
 3331 = 9 3533 = 9 3735 = 9
Since the ratio between any two terms is always the same (r=9), we have that the sequence is geometric.
• Now let's consider the second sequence:
 −4,   6,   −9, 27 2 ,   …
At first glance, it's difficult to see that there is some common ratio (or, equivalently, that there exists some number we can multiply by to get to the next term at each step). While we don't immediately see a value for r, check the ratios between each consecutive pair of terms:
6

−4
= − 3

2
−9

6
= − 3

2
,               27 2

−9
= − 3

2
Thus, now that we've actually calculated the ratios, we see that the sequence does have a common ratio of r=−[3/2]. Thus it is a geometric sequence.
Both sequences are geometric.
Give the nth term formula an for the geometric sequence that has the first term (a1) and common ratio (r) given below.
 a1 = 5,        r = 4
• The number given by a1 is the first term of the sequence. The value of r is what we multiply by to get each subsequent term. That is r·a1 = 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 a geometric sequence, we saw in the video lesson that there is a simple formula if you know the first term and the common ratio. It is simply
 an = rn−1 ·a1.
• Since we have a formula and already know a1 and r, we can just plug in:
 an = rn−1 ·a1     =     4n−1 ·5

[If we want to check our answer, we know the first term is a1=5 and the common ratio is r=4, so we can easily write out the first few terms:
 5,   20,   80,   …
Now that we have the first few terms, check to make sure the formula gives the same values:
 n=1     ⇒     a1 = 41−1 ·5     =     5

 n=3     ⇒     a3 = 43−1 ·5     =     80
Great: our formula for an checks out, so we know our answer is correct.]
an = 4n−1 ·5
Give the nth term formula an for the geometric sequence written below.
 343,   98,   28,   8,   …
• For a geometric sequence, we saw in the video lesson that there is a simple formula if you know the first term and the common ratio. It is simply
 an = rn−1 ·a1.
• This means we can easily find the formula if we know the first term (a1) and the common ratio (r). Looking at the sequence, the first term is clearly there, so we know a1 = 343. Now we just need to find the common ratio. To do that, just find the common ratio between two terms (remember, we find the ratio as "later divided by earlier", for example, third over second). To be sure we got it correctly, do it for two different pairs. Below we'll check first and second along with third and fourth:
 98 343 = 27 = 8 28
Thus the common ratio is r=[2/7].
• Now that we know a1 = 343 and r=[2/7], we just plug in:
 an = rn−1 ·a1     = ⎛⎝ 2 7 ⎞⎠ n−1 ·343

[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 = ⎛⎝ 2 7 ⎞⎠ 1−1 ·343    =     343

 n=4     ⇒     a4 = ⎛⎝ 2 7 ⎞⎠ 4−1 ·343    =     8
Great: our formula for an checks out, so we know our answer is correct.]
an = ([2/7])n−1 ·343
Find the value of the below sum.
 2  + 2·3  + 2·32  + 2·33  + … + 2·310
• 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 whole lot 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 a geometric sequence. Thus, we're working with a geometric series: a sum where every subsequent term has a common ratio with its preceding term. We see this because every subsequent term is the same as the previous, just multiplied by r=3.
• From the video lesson, we learned that the sum of any geometric sequence is
 a1 · 1−rn1−r ,
where a1 is the first term, r is the common ratio between terms, and n is the total number of terms in the series.
• Looking at the problem, it's easy to see that the first term is a1 = 2. We know it's a geometric series because we noticed that each term is the same as the previous, just multiplied by the common ratio or r=3. The hardest part is figuring out what n is-how many terms there are total. We might be tempted to think there are 10 terms because we have 310 in the last term. That is not the case. Notice that we don't start at 31, we actually start with no 3's at all: that is a1 = 2 ·30  [since 30 = 1, this is equivalent to a1 = 2]. Thus, we actually have n=11 terms, because the first term starts at 30, and we have to count that first term.
• Now that we know all the pertinent values, we can use the formula:
 a1 · 1−rn1−r =     2 · 1−3111−3 =     2 · −177  146 −2 =     177  146
177  146
Calculate the value of the below sum. [Round your answer to the nearest whole number.]
 50∑i=1 ⎛⎝ 100 ·(1.03)i ⎞⎠
• (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 a geometric series. It will have a common ratio of r=1.03 for every term because of the (1.03)i in the sigma notation. That means every subsequent term will be multiplied by another (1.03), so it follows the rules of a geometric series/sequence.
• From the video lesson, we learned that the sum of any geometric sequence is
 a1 · 1−rn1−r ,
where a1 is the first term, r is the common ratio between terms, and n is the total number of terms in the series. To find the first term of the series, just plug in the lowest value the index can give: i=1.
 a1     ⇒     i = 1     ⇒     100 ·(1.03)1     =     103
We already noticed that the common ratio is r=1.03 since every subsequent term in the series multiplies by an additional (1.03) [this happens because the exponent goes up 1 for every subsequent term]. For this problem, it's not too difficult to figure out how many terms there are (n), because we just need to know how many numbers 1→ 50 is. That's pretty clearly n=50, so we know the number of terms. [Still, be very careful when figuring out the number of terms n in general: it's very easy to make a mistake and over- or undershoot by 1.]
• Now that we know all the necessary values for the geometric series formula, we plug in:
 a1 · 1−rn1−r =     103 · 1−1.03501−1.03 =     103 · −3.3839 −0.03 ≈     11 618
11 618
Calculate the value of the below sum.
 21∑k=7 −6400 · ⎛⎝ − 1 2 ⎞⎠ k
• (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 a geometric series. It will have a common ratio of r=−[1/2] for every term because of the (−[1/2])k in the sigma notation. That means every subsequent term will be multiplied by another (−[1/2]), so it follows the rules of a geometric series/sequence.
• From the video lesson, we learned that the sum of any geometric sequence is
 a1 · 1−rn1−r ,
where a1 is the first term, r is the common ratio between terms, and n is the total number of terms in the series. To find the first term of the series, just plug in the lowest value the index can give: k=7.
 a1     ⇒     k=7     ⇒     −6400 · ⎛⎝ − 1 2 ⎞⎠ 7 =     50
We already noticed that the common ratio is r=−[1/2] since every subsequent term in the series multiplies by an additional (−[1/2]) [this happens because the exponent goes up 1 for every subsequent term]. Don't worry about the fact that r is negative: the formula will work as long as r ≠ 1.
• 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 k=7, while the last term has an index of k=22. Thus there are 22−7 = 15 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=16 terms (15 steps plus 1 for "home"). Now that we know all the necessary values for the arithmetic series formula, we can just plug in:
a1 · 1−rn

1−r
=     50 · 1− ⎛⎝ − 1 2 ⎞⎠ 16

 1− ⎛⎝ − 1 2 ⎞⎠
=    50 · 65 535 65 536

 3 2
=     546 125

16 384
[(546 125)/(16 384)]    ≈     33.3328
Find the sum of the infinite geometric series below.
 ∞∑i=0 ⎛⎝ 4 5 ⎞⎠ i
• An infinite geometric series is one where we consider what happens as we work towards adding an infinite number of terms together. For this problem, we would sum the below:
 1  + 4 5 + 16 25 + 64 125 + …
Notice that we never stop adding terms. The pattern of adding more and more terms continues forever. We want to know what value the series will converge to as we add ever more terms (what value the series "settles toward" in the long run).
• From the video lesson, we learned that an infinite geometric series converges if |r| < 1. If that is the case, then the sum converges to
 a11−r .
Thus, to find the sum of the infinite geometric series, we only need to know two things: r and a1.
• Looking at the series, we see that with each "step" the next term will be multiplied by an additional [4/5]. This comes from the the ([4/5] )i in the series: since i increases by 1 for every subsequent term, the exponent will increase by 1, causing an additional [4/5] multiplication to occur. Thus
 r = 4 5
To figure out a1, we need to see what the first term the series produces is. Don't be fooled into thinking that the first term is [4/5]-we need to pay attention to what value the lowest index will create. The lowest index for the sigma notation (Σ) is i=0, so use that to find a1.
 ⎛⎝ 4 5 ⎞⎠ i ⇒     i=0     ⇒     a1 = ⎛⎝ 4 5 ⎞⎠ 0 =     1
• Now that we know the values of r and a1, we plug in:
a1

1−r
=     1

 1− 4 5
=     1

 1 5
=     5
[Notice that we were only able to plug in to the above formula because |r| < 1. If we had |r| ≥ 1, then the infinite geometric series would have no sum, because it would never converge to a single value.]
5
Find the sum of the infinite geometric series below.
 9  − 9 2 + 9 4 − 9 8 + 9 16 − …
• An infinite geometric series is one where we consider what happens as we work towards adding an infinite number of terms together. Notice that we never stop adding terms. The pattern of adding more and more terms continues forever. We want to know what value the series will converge to as we add ever more terms (what value the series "settles toward" in the long run).
• From the video lesson, we learned that an infinite geometric series converges if |r| < 1. If that is the case, then the sum converges to
 a11−r .
Thus, to find the sum of the infinite geometric series, we only need to know two things: r and a1.
• Looking at the terms of the series, notice that the denominator increases by a factor of 2 with each step. This helps us find r, but we also need to pay attention to the signs. We can rewrite the series to make it clearer:
 9  + ⎛⎝ − 9 2 ⎞⎠ + 9 4 + ⎛⎝ − 9 8 ⎞⎠ + 9 16 + …
Thus on every "step" the term is also multiplied by a negative. Putting this together with the denominator growing by a factor of 2 each time, we get
 r = − 1 2 .

Finding a1 is quite simple for this problem, since a1 is just the first term. Looking at the series, we see
 a1 = 9.
• Now that we know the values of r and a1, we can plug in. Don't worry about the fact that r is a negative number: the formula works for positive and negative values of r, they just need to have |r| < 1.
a1

1−r
=     9

 1− ⎛⎝ − 1 2 ⎞⎠
=     9

 3 2
=     9·2

3
=     6
[Notice that we were only able to plug in to the above formula because |r| < 1. If we had |r| ≥ 1, then the infinite geometric series would have no sum, because it would never converge to a single value.]
6
Looking for employment, you manage to find two different businesses that are interested in hiring you. Each of the businesses offers a job with a starting salary and a guaranteed raise every year, as below.
 Job One:
 $70 000 starting salary, with an annual raise of 2%  Job Two: $50 000 starting salary, with an annual raise of 5%
If you work the job you choose for the next 30 years, which of the two jobs will earn you more money in total over the course of those 30 years?
• Begin by understanding what the question is asking. You have a choice of two different jobs, and you want to know which one will make you more money in total over the course of 30 years. In other words, we want to know how much you'll make for each of those 30 years, then add them all up:
 ⎛⎝ Year one salary ⎞⎠ + ⎛⎝ Year two salary ⎞⎠ +   …  + ⎛⎝ Year thirty salary ⎞⎠
Since we're just adding up a bunch of terms, we're working with a sequence. Let's get a better sense of how the terms are related.
• Consider Job One' first. During year one, you will make \$70 000. The following year, you get a 2% raise. Mathematically, we can express that as
 Job One, year two:    70 000 + (0.02)·70 000     =     (1.02) ·70 000
Thus, we see that a 2% raise is equivalent to multiplying the value we previously had by (1.02). [Why? Because the raise increases the salary to 102% of what it had been.] Furthermore, we can now realize that the salary increase will have this affect at every "step": just multiply the previous term by (1.02). This allows us to re-write each term for Job One' in this format
 70 000,     (1.02)·70 000,     (1.02)2·70 000,    (1.02)3·70 000,    (1.02)4·70 000,    …

By similar logic, we see that for Job Two' the 5% raise is equivalent to multiplying the previous year's salary by (1.05). This allows us to re-write each term for Job Two' as
 50 000,     (1.05)·50 000,     (1.05)2·50 000,    (1.05)3·50 000,    (1.05)4·50 000,    …
• Notice that we can now see both sums are geometric series (this should come as no surprise, since we're working on a section devoted to that). From the video lesson, we learned that the sum of any geometric sequence is
 a1 · 1−rn1−r ,
where a1 is the first term, r is the common ratio between terms, and n is the total number of terms in the series. For Job One', the first salary is a1,One = 70 000 and the common ratio is rOne=1.02. For Job Two', the first salary is a1,Two = 50 000 and the common ratio is rTwo = 1.05. For both of the jobs, the number of years we're adding up is n=30.
• Now that we know all the necessary values, we can plug in to the formula to find each sum, then compare them:
 Job One:        a1 · 1−rn1−r =     70 000 ⎛⎝ 1−(1.02)301−1.02 ⎞⎠ ≈     2 839 766

 Job Two:        a1 · 1−rn1−r =     50 000 ⎛⎝ 1−(1.05)301−1.05 ⎞⎠ ≈     3 321 942
Thus we see that `Job Two' manages to make more money total over the 30 year period.
Job Two
A geometric figure is made by doing the following. [The number of each step corresponds to the diagram below.] (1.) Begin by creating an equilateral triangle. Shade in the inside of this triangle. (2.) Draw another equilateral triangle inside of the previous one. Draw it such that the corners of the new triangle are on the midpoints of each segment from the previous triangle. Erase the shading inside of this triangle. (3.) Draw another equilateral triangle inside of the previous one. Draw it such that the corners of the new triangle are on the midpoints of each segment from the previous triangle. Shade in the inside of this triangle. (4.) Draw another equilateral triangle inside of the previous one. Draw it such that the corners of the new triangle are on the midpoints of each segment from the previous triangle. Erase the shading inside of this triangle.

Notice that this is a process of creating interior, equilateral triangles and alternating between shading and erasing. If we continue this process forever, what portion of the original, starting triangle will wind up being shaded in?  • We need to know what portion of the previous triangle each subsequent triangle makes up. To figure this out, consider part (2.) of the diagram. Notice that the new (unshaded) triangle must be [1/4] of the original (shaded) triangle. We can show this by symmetry: the new (unshaded) triangle must be the same size as each of the remaining three triangles (all still shaded). This means a total of four, equal-sized triangles. Thus, the new (unshaded) triangle must be a quarter of the original triangle. [If you don't see how to prove the symmetry, you can also work out what the area of the new (unshaded) triangle must be since you know each of its vertices are on the midpoints of the sides of the original triangle. This is much harder though, and it's much easier to argue symmetry since the four triangles inside must all be equilateral triangles.]
• By the same logic, we can see that this will happen at every step in the process. The area covered by each subsequent triangle is [1/4] of the previous triangle's coverage. If we say the first, original triangle had an area of 1, then we can consider the areas of the sequence of subsequent triangles:
 Area of triangles, in order:       1, 1 4 , ⎛⎝ 1 4 · 1 4 ⎞⎠ , ⎛⎝ 1 4 · 1 4 · 1 4 ⎞⎠ ,    …
Simplifying a bit, we see we can also express this as
 Area of triangles, in order:       1, ⎛⎝ 1 4 ⎞⎠ , ⎛⎝ 1 4 ⎞⎠ 2 , ⎛⎝ 1 4 ⎞⎠ 3 , ⎛⎝ 1 4 ⎞⎠ 4 ,     …
• Remember, our goal is to find the area of the triangle that is shaded in over the long-run of the process. Notice that the coverage of each triangle alternates in its effect: some shade in, while others erase what they cover. To express this idea mathematically, let's consider shading area as positive, while erasing shaded area as negative. With this in mind, we can express the effect each triangle has in terms of it being a positive addition of shaded area or it subtracting from the shaded area:
 Effect on shaded area, in order:       1,    − ⎛⎝ 1 4 ⎞⎠ , ⎛⎝ 1 4 ⎞⎠ 2 ,     − ⎛⎝ 1 4 ⎞⎠ 3 , ⎛⎝ 1 4 ⎞⎠ 4 ,     …
• Now that we know what each triangle does to the shaded area, we can find how much will wind up being shaded by adding them all together:
 Shaded area:       1  − ⎛⎝ 1 4 ⎞⎠ + ⎛⎝ 1 4 ⎞⎠ 2 − ⎛⎝ 1 4 ⎞⎠ 3 + ⎛⎝ 1 4 ⎞⎠ 4 −  …
Since the process continues on forever, the sum will go on infinitely. At this point it's probably clear that we're dealing with an infinite geometric series. From the video, we learned that an infinite geometric series converges if |r| < 1. If that is the case, then the sum converges to
 a11−r .
Thus, to find the sum of the infinite geometric series, we only need to know two things: r and a1.
• Looking at how the series works, we see that the sign alternates every time and so it is multiplied by a factor of r=−[1/4]. Furthermore, the first term in the series is a1 = 1, the area of the entire triangle. Since |−[1/4]|< 1, the series converges and we can apply the formula:
a1

1−r
=     1

 1−(− 1 4 )
=     1

 5 4
=     4

5
[4/5]

*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.

### Geometric 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 0:06
• Definition 0:48
• Form for the nth Term 2:42
• Formula for Geometric Series 5:16
• Infinite Geometric Series 11:48
• Diverges
• Converges
• 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

### Transcription: Geometric Sequences & Series

Hi--welcome back to Educator.com.0000

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

The other specific kind of sequence we will look at in this course is the geometric sequence,0006

a sequence where we multiply by a constant number for each step.0011

In the previous lesson, we looked at the arithmetic sequence, which is where you add by a constant number.0013

Now, we are looking at geometric, where you multiply by a constant number every step.0018

Just like arithmetic sequences, geometric sequences commonly appear in real life.0022

Since geometric sequences are based on ratios, since we are always multiplying by the same thing,0027

and ratios occur a lot in the world, they give us a way to describe a wide variety of things.0030

In this lesson, we will begin by going over what a geometric sequence is, and how we can talk about them in general.0036

Then, we will look into formulas for geometric series to make adding up a bunch of terms really easy and fast; let's go!0041

We will start with a definition: a sequence is geometric if every term in the sequence can be given0047

by multiplying the previous term by some constant number, r.0053

an is equal to r times an - 1; that is, some term is equal to the previous term, multiplied by r.0058

We call r the common ratio, because we can express it as an divided by an - 1; that is, some term divided by the previous term.0073

And so, we have a ratio in the way that we are building it.0084

Here are two examples of geometric sequences: 3, 6, 12, 24...continuing on; 4/5, -4/25, 4/125, -4/625...continuing on.0087

They are geometric, because each step multiplies by the same number.0099

For example, in this one, every step we go forward, we are multiplying by 2: 3 to 6--times 2; 6 to 12--times 2; 12 to 24--times 2.0103

And this is going to continue on forever, as long as we keep going with that sequence.0112

Over here, 4/5 and -4/25...it is not quite as simple, but it is basically the same thing, multiplying by -1/5.0116

That is how we get from 4/5 to -4/25, if we are going to multiply.0124

To get to 4/125, once again, we multiply by -1/5; to get from 4/125 to -4/625, we multiply by -1/5.0128

And this is going to keep going, every time we keep stepping.0139

Every step is multiplying by the same number: we are multiplying by the same number each step.0143

The number can be anything: it can positive; it can be larger than one; it can be less than one; it can be negative.0148

It doesn't matter, so long as it is always the same value for every step.0153

The definition of a geometric sequence is based on the recursive relation an = r(an - 1).0159

That is, every term is equal to the previous term, multiplied by r.0165

How can we turn this into a formula for the general term, where we don't have to know what the previous term is--0170

we can just say, "I want to know the nth term," plug it into a formula, and out will come the nth term?0173

Remember: a recursive relation needs an initial term.0179

So, while this relation that defines a geometric sequence is useful, we still need a little bit more.0181

We need this initial term to know where we start--what our very first term is--because previous to that...there is nothing previous.0187

So, we just have to state that as one specific term.0193

Since we don't know its value yet, we will just leave it as a1, our first term.0197

From an = r(an - 1), we see that a1 relates to later terms as: a2 will be equal to r(a1).0202

The second term will be equal to the first term, times r; this will continue on.0210

The third term, a3, will be equal to r times a2.0215

But we just showed that a2 is equal to r(a1), so we can plug that in there;0218

and we will have r times r(a1), so we end up getting r2(a1) = a3.0224

We can continue on here; a4 is going to be equal to r times a3.0233

But we just figured out that a3 is equal to r2 times a1.0237

So, we replace a3, and we end up having r times r2 times a1;0241

so now we have that r3(a1) = a4.0246

And we will see that this pattern will just keep going like this.0251

So, we have that a1 is equal to a1 (there is no big surprise there).0254

a2 is equal to r times a1; a3 is equal to r2 times a1.0258

a4 is equal to r3 times a1; and we see that the pattern is just going to keep going like this.0263

The nth term is n - 1 steps away from a1; it is n - 1 steps to get from a1 to n.0268

If you start on the first stepping-stone, and you go to the nth stepping-stone, you have to take n - 1 steps forward.0277

We start at a1; to get to the an, we have to go n - 1 steps forward.0283

Since every step means multiplying by r, that means we have multiplied by r n - 1 times, which is r raised to the n - 1 power.0287

Thus, we have that an = rn - 1(a1).0296

So, to find the formula for the general term of a geometric sequence, we only need to figure out what the first term is, and the common ratio.0301

As soon as we figure out a1 and our value for r, we have figured out what the general term is, what the an term is--pretty great.0308

What if we want to find the nth partial sum of a geometric sequence (that is, adding up0316

the first n terms of the sequence, a1 + a2 + a3 up until + an)?0320

Well, we could just add it all up by hand for small values of n.0327

If it was n = 2, so it was just a1 and then a2, it is probably not that hard to just figure it out by hand.0331

If it was n = 3, we could probably do it by hand; if it was n = 10, n = 100, n = 1000,0336

this gets really, really tiresome, really quickly, as the value of n gets larger and larger.0341

So, how can we create a formula--how can we just have some formula where we can plug some stuff in,0345

and we will immediately know what that nth partial sum is?0350

Well, let's do two things: first, let's give the sum a name.0355

So, we will call our nth partial sum sn, the sum for the nth partial sum.0358

Second, let's use the form for the nth term of a geometric sequence.0364

Remember: we just figured out that the general form for an is rn - 1(a1).0367

We can use this general term to put of the terms in this series into a format that will involve a1.0376

So, we have sn = a1 + a2 (which is r times a1) + a30382

(which is now r2 times a1), up until we get to + an (which is now rn - 1(a1)).0390

Great; but at the moment, we can't do anything with just this.0399

This isn't quite enough information; we can't combine the various r's, because they all have different exponents.0402

r to the 0, what is effectively here; r1; r2; r3; up until rn - 1--0410

they don't talk the same language, because they don't have the same exponent.0417

So, since they can't really communicate with each other, we can't pull them out all at once.0420

We could pull out all of the a1's, but then we would still be left with all of these different kinds of r's.0423

So, we don't really have a good thing that we can do right now.0428

What we want, what we are really looking for, is a way to somehow get rid of having so many things to add up.0431

We want fewer things to add up; if we only had a few things to add up and compute, it would be easy for us to calculate these values.0439

So, that is what we want to figure out how to do.0445

This is the really clever part; this is basically the part of the magic trick where suddenly we pull the rabbit out of the hat.0449

And so, you might see this and think, "How would I figure this out?"0455

And it is a little bit confusing at first; but just like, as you study magic more and more (if you were to study magic),0458

you would eventually realize, "Oh, that is how they got the rabbit into the hat," or "this is how the trick works"--0463

as you work with math more and more, you will be able to see, "Oh, that is how we can make these sorts of things."0468

So, don't be worried by the fact that you would not be able to think of this immediately.0472

The people who made this proof at first didn't think of it immediately.0477

They thought about it for a while; they figured out different things, and maybe they tried something that did not work;0481

and eventually they stumbled on something and said, "Oh, if I do this, it works,"0486

and they were able to come up with this really easy, cool, clever way to do it.0489

But it is not something that you just have immediately.0492

It is something that you have to think about, until eventually you can "pull your own rabbit out of a hat."0494

But first, you have to get the rabbit into the hat.0498

Anyway, here is the clever part: what is it?0500

We have sn = a1 + r(a1) + r2(a1) + ... + rn - 1 (a1).0503

The really cool trick that we do is say, "What if we multiplied r times sn?"0515

Well, that would end up distributing to everything in here, since we are multiplying it on both sides of the equation.0522

We would have r(a1) + r2(a1) + r3(a1)...0528

all the way until we get up to + rn(a1).0534

Now, notice: we now have matching stuff going on.0537

Here is r times a1; here is r times a1; there is a connection here.0540

Here is r2 times a1; here is r2 times a1; there is a connection there.0544

Here is r3 times a1; and somewhere in the next spot in the decimals is r3 times a1, as well.0548

Here is rn - 1 times a1; and in the next back spot in the decimals, here is rn - 1 times a1.0555

We have all of this matching going on; well, with this idea, since we have all of this matching,0560

we can subtract this equation, the rsn equation, this one right here,0565

from the sn equation by using elimination.0569

From when we talked about systems of linear equations: if we have two equations, we can subtract, and we can add them together.0572

We are just using elimination to do this.0578

So, we have our sn equation here, and then we subtract by minus r times sn.0580

So now, we have this matching pattern going on: r times a1 matches to -r times a1; they cancel each other out.0586

r2 times a1 matches to minus r2 times a1; they cancel each other out.0592

Everything in the dots here cancels out with all of the negatives in the dots here.0596

We finally get to rn - 1 times a1 in our top equation,0600

which cancels out with minus rn - 1 times a1 in our bottom equation.0603

The only things that end up being left over are minus rn times a1 and +a1 here.0608

We get sn -rsn is equal to a1 - rn(a1).0617

So, through this immense cleverness (and once again, this isn't something that you would be expected to just know immediately,0624

and be able to figure out really easily--this is the part that takes the really long thinking.0629

This is the really clever part; this is what takes hours of thought, by just thinking, "I wonder if there is a clever way to do this."0632

And eventually, you end up stumbling on it.0637

So, through immense cleverness, we have shown that sn - rsn = a1 - rn(a1).0640

That is pretty great: we have gotten this from what we just figured out.0648

Now, our original goal was to find the value of the nth partial sum, which was sn.0653

Using the above, we can now solve for sn: we just pull out the sn,0658

so we have sn times 1 minus r; we can also pull out the a1 over here on the right side.0663

We divide both sides by 1 - r, and we get that the nth partial sum is equal to0667

the first term, a1, times the fraction 1 - rn over 1 - r.0672

So, we now have a formula to find the value of any finite geometric series at all, really easily.0678

All that we need to know is the first term, a1; the common ratio, r,0683

which shows up on the top and the bottom; and how many terms are being added together total (the n exponent on our top ratio).0688

It is pretty great--a really, really powerful formula that lets us do a lot of what would be very tedious, very slow, difficult addition, just like that.0696

But what if the series was not finite?0705

So far, we have only talked about if we are doing a finite sum of the sequence.0707

But what if we had an infinite sequence, and we wanted to add up infinitely many of the terms, so we kept adding terms forever and ever and ever?0711

To understand this better, let's consider some geometric sequences and what happens as we take partial sums using more and more and more terms.0718

First, we will look at 3, 9, 27, 81, 243...so it is times 3 each time; it is a geometric sequence.0725

So, our first partial sum, s1, would be just the 3.0732

Our next partial sum would be s2, so we add on the 9, as well; we get 12.0737

Our next partial sum would be s3; we add on the 27, as well; we get 39.0741

The next partial sum is s4; we add on the 81; we get 120.0745

The next partial sum: we add on 243; we get 363; and this is just going to keep going on in this pattern.0749

It is going to keep adding more and more and more.0754

We look at this, and we notice that, as the partial sums use more and more terms, it continues to grow at this really fast rate.0756

In fact, the rate is going to get faster and faster and faster as we add more and more terms.0763

We see its rate of growth increasing as it goes to larger sums.0766

So, if we add terms to the series forever, it is not going to really get to anything.0770

It is going to blow out to infinity; it is going to just "blast off" to infinity.0774

There is no stopping this thing; it is not going to give us a single value.0778

It never stops growing; we say that such an infinite series, one that never stops growing, that doesn't go to a single value, "diverges."0781

As we add more and more terms, it continues to change forever.0789

It diverges from giving us a single, nice, clean value, because it instead just blows off to infinity.0792

It keeps moving around on us; it doesn't stay still; it doesn't go to something; it just goes off, so this would be a divergent series.0798

On the other hand, we could consider another partial sum from this below geometric sequence: 1, 1/2, 1/4, 1/8, 1/16...0807

What we are doing each time here is dividing by 2, or multiplying by 1/2; so we see that this is a geometric sequence.0815

Our first partial sum would be s1 = 1; we just add in that first term.0821

The next one, s2, would be 1.5, because we added 1/2, so we are at 1.5.0826

The next one, s3: we add in 1/4; that is 0.25 added in; that becomes 1.75.0831

The next one, s4: we add in an 8; that becomes 1.875.0836

The next one, s5: we add in 1/16; that becomes 1.9375.0840

And it would continue in this way; but we notice that it is not really growing the same way.0844

This time, the sums are continuing to grow, but the rate of growth is slowing down with each step.0849

It is not increasing out like the previous one (it was blowing out somewhere; it was becoming really, really big).0856

But with this one, we see it settling down; as we add more and more terms, it is going to a specific value.0861

It is going to 2; the infinite sum, this infinite series, is going to a very specific value; it is working its way towards 2.0868

If you keep adding more, you will see even more, as it gets to 1.99, 1.999, 1.9999999...0876

As you keep adding more and more terms, you will see that it is really just working its way to a single value.0883

It is slowing down as it gets to 2.0887

In this case, we say that such an infinite series converges; it is converging on a specific value.0889

As we add more and more terms, it works its way towards a single value.0895

There is this single value that it is working towards.0899

From the two examples we have seen, we see that whether a series converges or diverges is based on the common ratio of its underlying sequence.0902

If the common ratio is large, it causes the sequence to always grow.0911

It keeps growing, because that ratio keeps multiplying it to get larger and larger and larger, moving around.0917

Specifically, if the absolute value of r is greater than or equal to 1, the partial sums will always be changing,0922

because the size of our terms never shrinks down; so the series will diverge.0927

On the other hand, if the common ratio is small, it causes the sequence to shrink down.0933

If we have a small common ratio, it is going to make it smaller and smaller and smaller with every term we work on.0937

Since it gets smaller and smaller and smaller, we have that if the absolute value of r is less than 1,0944

the rate of change for the series will slowly disappear to nothing, because every time we go to the next term,0948

since r is less than 1, it makes it smaller; and then it makes it smaller; and then it makes it smaller, and makes it smaller,0954

and makes it smaller, and makes it smaller; so every time it is getting smaller.0959

So, every time, the rate of growth is going down to less and less and less.0962

And so, over the long term, over that infinite number of terms, it ends up converging to a single value.0965

So, as a general rule, an infinite geometric series will converge if and only if the absolute value of r is less than 1.0971

So, the absolute value of r being less than 1 means that the infinite geometric series converges.0979

If the infinite geometric series converges, then the absolute value of r must be less than 1; they are equivalent things for a geometric sequence.0983

All right: assuming that the absolute value of r is less than 1 for a geometric sequence, how can we figure out a formula for its corresponding infinite geometric series?0990

Well, we already figured out a formula that is true for any finite geometric series.0999

Remember: sn, the nth partial sum of any finite geometric series,1003

is a1 times (1 - rn)/(1 - r).1007

We just figured out that formula; that is pretty cool.1011

Not only that, but we also know that, as we look at partial sums containing more and more terms,1013

as a partial sum has more and more terms, they have to be growing closer and closer to the value that the infinite series will converge to.1018

As we put in more and more terms into our partial sum, it has to be getting closer to this value that it is going to converge to.1027

Think about why that is: if the partial sums were not getting closer to a specific value,1032

if they were moving around away from the specific value,1036

then it couldn't be converging to that, because it would always be changing around.1039

If it is going to converge to a single value, it has to be working its way towards it.1042

If it is working its way towards it, it must always be getting closer to the thing.1046

If it wasn't always getting closer, if it was sometimes jumping away, it wouldn't be working its way towards it; it would be going somewhere else.1049

Since we know that it is converging, we know that it must be working its way, as we have more and more terms in our partial sum.1054

As we put in more terms in our partial sum, we will be closer to the value that we are converging on.1065

As we add many more and more and more terms in our partial sum, we are going to be closer to the thing that we are converging to.1069

What we are asking ourselves is, "As we have really large values for n, what value are we getting close to?"1076

What happens to the formula we figured out, our nth partial sum formula,1082

sn = a1 times (1 - rn)/(1 - r), as the number of terms we have, our n, goes off to infinity?1086

As the value for the number of terms we have, our n, becomes infinitely large, what will happen to this formula?1093

Whatever happens to this formula is what we have to be converging to, because of the argument1098

that we just talked about, about how it has to be getting closer as we put in more and more terms.1102

Notice: the only term on the right side affected directly by the n is rn; there is no other term that directly has an n connected to it.1107

We can ask ourselves what happens to rn as our n grows very large.1115

Also, remember: r is less than 1; the absolute value of r has to be less than 1.1119

These two things combine as we ask ourselves...as n goes to infinity, and we are looking at our rn here,1125

since the absolute value of r is less than 1, we have that as n goes to infinity, rn has to go to 0.1133

So, it is going to shrink down to 0 as n grows infinitely large.1140

Why is that the case? Well, since the absolute value of r is less than 1, every step has to make it smaller.1144

For example, if we look at .9 raised to the 100, we get that that is less than .0001.1150

Why is this occurring? Well, since the absolute value of r is less than 1, we know that it is this fractional thing--1157

that effectively, every time we iterate it, every time we hit a term with this common ratio, it takes a little bite out of it.1164

Whether it is .1 or 1/2 or 3/5 or 922/1000, it is going to take a bite out of whatever the term that it is being multiplied against is.1170

As it takes infinitely many bites, since it is always shrinking it down, it means that it is always working its way towards this value of 0.1181

Infinitely many bites away gets us to having nothing.1189

Therefore, because rn is going to 0 as n goes to infinity, we have this part right here shrinking down to a 0 in our formula.1194

So, we get the following formula for an infinite geometric series: the infinite sum is equal to a1 times 1/(1 - r),1202

assuming that the absolute value of r is less than 1.1210

If the absolute value of r is greater than or equal to 1, we couldn't even talk about this in the first place,1212

because our series would be diverging, because it would always be growing and changing around on us.1217

But if we have that the absolute value of r is less than 1, all we need to know is our first term, a1, and the rate that it is growing at.1220

And we work it out through this formula, and we know what it will converge to over the long run.1228

All right, cool--we are ready for some examples.1232

The first one: Show that the sequence below is geometric; then give a formula for the general term (that is, the an, the nth term).1234

We have 7, 35, 175, 875...so what we want to ask ourselves is, "What number are we multiplying by each time?"1241

How do we get from 7 to 35? Well, we multiply by 5.1247

Let's check and make sure that that works: 35 to 175--yes, if we use a calculator (or do it in our heads,1250

or write it out by hand), we realize that, yes, we can multiply by 5 to get from there to there.1256

The same thing: 175 to 875: we multiply by 5; so we see that this is continuing.1259

Yes, it is geometric; that checks out.1263

Now, we want to give a formula for the general term.1266

We talked, in the lesson, about how an is equal to (let me write it the way we had it last time):1268

r, the rate that we are increasing at, to the n - 1, times a1.1276

What is our a1? Well, a1 is equal to 7, because it is the first term.1281

What is our r? r is equal to 5, since it is the number we are multiplying each time.1286

So, r = 5; r = 5; a1 = 7; so an is equal to 5n - 1 times 7.1290

And if we wanted to check this out, we could do a really quick check.1299

We could plug in...let's look at a2; that would be equal to 52 - 1 times 7,1303

so 5 to the 1 times 7; 5 times 7 is 35; we check that against what our second term was.1310

And indeed, that checks out; so it looks like we have our answer--there is our answer.1316

All right, the next example: Find the value of the finite geometric series below.1320

Notice: this does have an end--we stop at 3072, so it is not an infinite one.1326

If it were infinite, it would go out to infinity, so we wouldn't actually be able to find a value.1330

All right, how do we figure this out?1333

The first thing: what is the rate that we are increasing at?1336

To get from 3 to 6, we multiply by 2; to get from 6 to 12, we multiply by 2; so at this point, we realize that r equals 2.1338

What is the value a1? That is 3.1345

What we are looking for, remember: the formula we are going to do is: the nth partial sum, sn,1348

is equal to a1 times 1 minus the rate, raised to the nth power, divided by 1 minus the rate.1353

So, the only thing we have to figure out, that is left, is what our n is; n = ?.1359

How many value are we going to be at?1365

Here we are at a1, but this is a?; this is an, right over here.1367

What would that have to be? Well, we could set this up, using the formula that we talked about before, our general formula for the general term.1374

an is equal to rn - 1 times a1.1382

We plug in our an; that is 3072; 3072 =...our rate is 2, raised to the n - 1, times...a1 is 3.1388

So, we divide both sides by 3, because we are looking to figure out our n.1398

Divide both sides by 3, and we get 1024 = 2n - 1.1401

Now, you might just know immediately that 1024 is the tenth power of 2: 210 = 1024.1407

So, we would see that it is 10 steps to get forward; we would multiply 10 steps forward, so that would mean that our n is equal to 11.1415

We have figured out that to get 210...that is 10 steps; we multiplied all of them on the 3; we started at 31424

as our first stepping-stone; we stepped forward 10 times...so the first stepping-stone, plus 10 steps forward,1430

means a total of 11 stepping-stones; so we have n = 11.1435

However, alternatively, we could just figure this equation right here out by using the work that we did with logarithms long ago in this class.1439

1024 = 2n - 1...well, what we can do is just take the log of both sides: log(1024) = log(2n - 1).1448

One of the properties of logs is that we can pull down exponents; that is why this is so useful.1459

We have n - 1 times log(2); log(1024) over log(2) equals n - 1.1463

log(1024)/log(2) is equal to 10; it equals n - 1, which tells us that n equals 11.1475

So, you could work this out just through raw algebra and using logarithms;1486

or you could work it out, if you recognized 210 as equal to 1024, if you just kept dividing 1024 with a calculator1489

until you saw how many steps it was; either way will end up getting us this value, that n equals 11.1495

All right, great; now we have everything that we need for our sum: sn, the nth partial sum1503

(in this case, the 11th partial sum of what this sequence would be) is going to be equal to...1509

a1 is 3 (our first term), times 1 minus our rate (it multiplied by 2 on each one of them),1517

raised to the 11th power (because the number of terms we have total is 11), divided by 1 minus the rate (2, once again).1525

We work this out: 3 times 1 - 211 is -2047; 1 - 2 is -1, so the -1 cancels out with the negative on top.1534

We have positive 2047 times 3; and we end up getting 6141; that is what we get once we add up all of those terms.1545

Great; the third example: Find the value of the below sum.1554

The sum is in sigma notation: from i = 0 up until 10 of 500 times 1/2 raised to the i minus 3 to the i divided by 64.1558

Previously, when we talked about sigma notation, series notation, we talked about how summations have properties1569

where we can separate some of the things in sigma notation, and we can pull out constants; great.1575

So, let's start by separating this: we can write this as Σ...it still has the same upper limit; the limits will not change...1580

the same index and lower limit of 500, times 1/2 to the i...minus...so what we are doing is separating around this subtraction...1587

minus...the series...same limits...of 3 to the i over 64.1599

So, we can separate, based on addition and subtraction, into two separate series.1607

Furthermore, we can also pull out constants and just multiply the whole thing.1610

We have 500 times the series; 10i equals 0, 1/2 to the i, minus...we pull out the 1/64, since that is just a constant, as well...1613

on the series, 10i equals 0; 3 to the i...cool.1627

So, at this point, we can now use our formulas.1632

Remember: our formula was that the nth sum is equal to the first term, times 1 minus rn, over 1 - r.1634

So, each one of these is going to end up having different rates.1643

But they are going to end up having the same n.1646

Our n is going to be based off of going from 0, our starting index, all the way up to 10, our ending index.1648

What is our n if we go from 0 to 10?1656

Remember: we have to count that first step, so 0 up until 10 isn't 10; it is 11.1659

1 to 10 is 10, so 0 to 10 must be one more, 11; so we have n = 11.1664

All right, back into this: we have 500 times...let's use that formula...the series from i = 0 to 10 of 1/2 to the i...1670

well, what would it be for our a1--what would be the first term?1678

Well, if we plugged in i = 0 on (1/2)i, (1/2)0, or anything raised to the 0, is just 1; so our a1 is 1 there.1681

Times 1 minus...what is the rate? Well, we are multiplying by 1/2 each time, because it is 1/2 with an exponent on it.1690

So, 1/2 is our rate, raised to the n; our n that we figured out was 11, divided by 1 minus the same rate, 1/2.1695

Great; minus...over here, 1/64: we apply that formula again; so, the series from i = 0 to 10 of 3i:1705

what is our a1--what is the first term?1714

Well, 3 to the 0, because our starting index is 0 once again--the first term of this series would be 3 to the 0, because it is the first thing that would show up.1716

3 to the 0 is just 1 again; 1 minus...what is our rate? Our rate is 3, because we multiply by 3,1726

successively, for each iteration, because it is an exponent.1732

3 to the...what is our number of terms? 11; n = 11...one minus three.1735

Great; now we can work through a calculator to figure out what these things are.1741

We will save the incredibly boring actual doing of the arithmetic, but it shouldn't be too difficult.1744

This will become 500 times 2047, over 1024, minus 1 over 64, times 88573.1749

We can combine these things; actually, let's distribute our constants first.1765

2047...well, we will distribute just the multiplication; they are all constant numbers now.1769

255 thousand on our left, 875, divided by 256, minus 88573 over 64;1773

we have to put the right side on a common denominator, but once again, I will spare working out every single aspect.1793

This would simplify to -98417 divided by 256; and there is our answer, exactly precise.1799

If we wanted to, we could have approximated to decimals at some point, and we would get -384.44, approximately.1810

Notice: probably back here, when we were working at these steps, where we have these giant fractions,1819

there is a good chance, if you are working with a scientific calculator or a graphing calculator, that it doesn't show it perfectly in fractions.1824

You would end up getting fractional numbers; and for the most part, most teachers and textbooks would be fine with giving an answer in decimals instead.1829

So, either one of these two answers--the specific exact fraction, -98417/256, or approximately -384.44--both of them are perfectly fine.1835

The fourth example: Many lessons ago, when we first learned about exponential functions,1847

we had a story where a clever mathematician asked for one grain of rice on the first square of a chessboard,1851

then two on the second square, four on the third square, eight on the fourth square, 16, 32, 64, 128...1857

he keeps asking for doubling amounts of grains of rice.1872

So, if he had actually been paid on all of the squares, just as he was initially promised, how many grains of rice would he have total?1876

So, what we have here is a finite geometric series, because it is doubling each time.1885

What is our first term? Well, a1 was on the very first square; he had one grain, so our first term is going to be 1.1891

What is the rate that it increases at each time?1900

It goes 1, 2, 4, 8, 16...so it is doubling each time; so that means that the rate is multiplied by 2.1902

And what is the number of terms total? If we are going from the first square of a chessboard to...1908

a chessboard is 8 x 8, so the total squares on an 8 x 8 chessboard...we have 8 by 8, so that means we have 64 squares total.1913

So, that is going to be 64 times that this is going to happen; we have 64 terms here.1926

If we are going to figure this out, the nth partial sum, the 64th partial sum,1933

is going to be the first term, a1, times 1 minus the rate, 2, raised to the number of terms, 64, over 1 minus the rate, 2.1937

We work this out: we end up getting 1 - 264 over -1, which is the same thing as negative 1 - 264,1948

which we will end up seeing comes out to be very positive.1962

This is our precise answer for the number of grains; but let's see what that is approximately.1964

That comes out to be approximately 1.845x1019 grains of rice.1968

That is a big, incredibly huge number of grains of rice.1978

It might be a little bit hard to see just how many grains of rice that is: 1019--we are not used to working with scientific notation that large.1982

So, let's turn it into some words that we might understand.1989

That would be about the same as 18 quintillion (that is a quintillion: it goes million, billion, trillion, quadrillion, quintillion) grains of rice.1991

And we are, once again, not really used to working with numbers as incredibly large as the quintillion scale.2003

So, let's write that as 18 billion billion grains of rice; that is an incredible number of grains of rice--that is so many grains of rice!2008

How much rice is that? That is about more than the rice that has ever been produced by humanity, over all of humanity's time existing!2022

It might be approximately on the same scale as the amount of rice humanity has ever created.2030

I am talking about all of humanity ever creating this over the whole history of the world.2036

All of the rice that has ever been made...18 billion billion grains...that is probably somewhere on the same scale.2041

It might be a little bit more; it might be a little bit less; but that is the sort of scale.2046

All of the rice that humanity has ever created: we get to very, very large numbers very quickly2049

with these exponential functions when we are working in geometric stuff; that is something to keep in mind.2054

18 billion billion grains is as much rice as humanity has ever made--pretty amazing.2058

The final example: a super ball is dropped from a height of 3 meters.2065

On every bounce, it bounces 4/5 of the previous height.2068

If allowed to bounce forever, what is the total back-and-forth distance that the ball travels over?2071

Notice that it is total, so it is up and down.2076

Let's draw a picture to help ourselves see what is going on here.2079

We start...we have this ball; we drop the ball, and it falls three meters.2081

Then, it bounces; it is a super ball, so it bounces up, and it will go up by how much?2087

It will go up by 3 times 4/5, because it goes up to 4/5 of its previous height on every bounce.2092

Now, it is going to fall down; well, what height did it just fall down from?2099

It is going to fall down from the same thing, 3 times 4/5.2102

Then, it is going to bounce up again; it is going to bounce up by how much?2106

Well, it is going to bounce up to 3 times 4/5 times 4/5, because it is 4/5 of the previous height it came from.2108

3 times 4/5 times 4/5...well, we could just write that as 3 times (4/5)2.2113

Then, it is going to fall down, once again, from that same height; so it is 3 times (4/5)2.2118

And then, it is going to just continue on in this manner of going a little bit less each time:2124

3 times (4/5)3, and then down again: 3 times (4/5)3, and just continuing on in this manner forever.2129

We want to figure out the total back-and-forth distance that the ball travels over.2140

Notice: what we have here is ups and downs.2144

So, I am going to go with calculating the ups first: how much is from this up plus this up plus this up, going on forever and ever and ever and ever?2147

Notice: what that means we are dealing with is an infinite sum, because we are saying,2158

"If it were to continue bouncing forever, what would be the amount of distance it would travel over?"2162

That is equal to the first value, times 1 over 1 minus the rate; that was what we figured out that the infinite sum comes out to be.2167

It is the first value, times 1 over 1 minus the rate.2176

All right, in this case, for our up bounces, all of the ups are going to end up being...our first value2179

is a1, so our first value is this one right here, 3 times 4/5.2194

We might be tempted to think that it is 3 meters, but remember: this is all of the ups--we are looking at the ups first.2198

And we will see why at the end--why I chose to look at the ups first.2203

So, all ups is going to be...3 times 4/5 is our first value that we end up having occur.2207

And then, that is going to be times 1 over 1 minus...what is the rate? 4/5, because it is 4/5 on every bounce.2213

It goes up to 4/5 of its previous height, so the rate for the next one would be 4/5 of that, and 4/5 of that, and 4/5 of that.2221

Notice: there was one other requirement on being able to use this.2227

The absolute value of our rate must be less than 1.2230

But since it is 4/5, that ends up checking out; OK.2233

Keep going with this: that is 3 times 4/5; let's write it as 12/5; 12/5 times 1 over...1 - 4/5 is 1/5;2237

well, 1 divided by 1/5 is 12/5 times...1 over 1/5 is 5; so we have 5 times 1/5 on the bottom;2251

that cancels out; so we have 12 meters total of bouncing for our up values.2259

OK, but that is only part of it; now we also have to figure out what all of the down values are--2267

what is the down value here? the down value here? the down value here?...going out this way infinitely.2273

Well, we also have this part right here; but notice: all of these green things end up having a matching up value.2279

They each match up: the purple and the green, the purple ups and the green downs; they all match up.2292

The only one who is out of the normal case of matching up is that red first drop down.2297

So, what that means is that we can match all of our downs; they have a matching 12 meters,2303

because it matches all of the ups; but then we just have to add on the initial 3-meter drop at the beginning.2311

So, how many downs do we have in total?2319

Well, that 3-meter drop, plus the matched value (because it matches to the 12-meter value for all of the ups),2321

12 meters, plus 3: we get 15 meters, so the total number is going to be equal to the 12 from our ups,2327

plus the 15 from our downs; 12 + 15 means 27 meters in total.2345

Great; there we are--we finished that one.2354

All right, now we have a pretty good understanding of how sequences work.2355

We have worked through geometric sequences and arithmetic sequences.2358

We understand how series work; we have a pretty good idea of how sequences work; great.2361

All right, we will see you at Educator.com later--goodbye!2365