$Vincent Selhorst-Jones$

Vincent Selhorst-Jones

Introduction to Sequences

Slide Duration:

Table of Contents

Section 1: Introduction

Introduction to Math Analysis

10m 3s

Intro

0:00

Title of the Course

0:06

Different Names for the Course

0:07

Precalculus

0:12

Math Analysis

0:14

Trigonometry

0:16

Algebra III

0:20

Geometry II

0:24

College Algebra

0:30

Same Concepts

0:36

How do the Lessons Work?

0:54

Introducing Concepts

0:56

Apply Concepts

1:04

Go through Examples

1:25

Who is this Course For?

1:38

Those Who Need eExtra Help with Class Work

1:52

Those Working on Material but not in Formal Class at School

1:54

Those Who Want a Refresher

2:00

Try to Watch the Whole Lesson

2:20

Understanding is So Important

3:56

What to Watch First

5:26

Lesson #2: Sets, Elements, and Numbers

5:30

Lesson #7: Idea of a Function

5:33

Lesson #6: Word Problems

6:04

What to Watch First, cont.

6:46

Lesson #2: Sets, Elements and Numbers

6:56

Lesson #3: Variables, Equations, and Algebra

6:58

Lesson #4: Coordinate Systems

7:00

Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope

7:02

Lesson #6: Word Problems

7:10

Lesson #7: Idea of a Function

7:12

Lesson #8: Graphs

7:14

Graphing Calculator Appendix

7:40

What to Watch Last

8:46

Let's get Started!

9:48

Sets, Elements, & Numbers

45m 11s

Intro

0:00

Introduction

0:05

Sets and Elements

1:19

Set

1:20

Element

1:23

Name a Set

2:20

Order The Elements Appear In Has No Effect on the Set

2:55

Describing/ Defining Sets

3:28

Directly Say All the Elements

3:36

Clearly Describing All the Members of the Set

3:55

Describing the Quality (or Qualities) Each member Of the Set Has In Common

4:32

Symbols: 'Element of' and 'Subset of'

6:01

Symbol is ∈

6:03

Subset Symbol is ⊂

6:35

Empty Set

8:07

Symbol is ∅

8:20

Since It's Empty, It is a Subset of All Sets

8:44

Union and Intersection

9:54

Union Symbol is ∪

10:08

Intersection Symbol is ∩

10:18

Sets Can Be Weird Stuff

12:26

Can Have Elements in a Set

12:50

We Can Have Infinite Sets

13:09

Example

13:22

Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times

14:08

This Set Has Infinitely Many Distinct Elements

14:40

Numbers as Sets

16:03

Natural Numbers ℕ

16:16

Including 0 and the Negatives ℤ

18:13

Rational Numbers ℚ

19:27

Can Express Rational Numbers with Decimal Expansions

22:05

Irrational Numbers

23:37

Real Numbers ℝ: Put the Rational and Irrational Numbers Together

25:15

Interval Notation and the Real Numbers

26:45

Include the End Numbers

27:06

Exclude the End Numbers

27:33

Example

28:28

Interval Notation: Infinity

29:09

Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other

29:14

Always Use Parentheses

29:50

Examples

30:27

Example 1

31:23

Example 2

35:26

Example 3

38:02

Example 4

42:21

Variables, Equations, & Algebra

35m 31s

Intro

0:00

What is a Variable?

0:05

A Variable is a Placeholder for a Number

0:11

Affects the Output of a Function or a Dependent Variable

0:24

Naming Variables

1:51

Useful to Use Symbols

2:21

What is a Constant?

4:14

A Constant is a Fixed, Unchanging Number

4:28

We Might Refer to a Symbol Representing a Number as a Constant

4:51

What is a Coefficient?

5:33

A Coefficient is a Multiplicative Factor on a Variable

5:37

Not All Coefficients are Constants

5:51

Expressions and Equations

6:42

An Expression is a String of Mathematical Symbols That Make Sense Used Together

7:05

An Equation is a Statement That Two Expression Have the Same Value

8:20

The Idea of Algebra

8:51

Equality

8:59

If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same

9:41

Always Do The Exact Same Thing to Both Sides

12:22

Solving Equations

13:23

When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something

13:33

Look For What Values Makes the Equation True

13:38

Isolate the Variable by Doing Algebra

14:37

Order of Operations

16:02

Why Certain Operations are Grouped

17:01

When You Don't Have to Worry About Order

17:39

Distributive Property

18:15

It Allows Multiplication to Act Over Addition in Parentheses

18:23

We Can Use the Distributive Property in Reverse to Combine Like Terms

19:05

Substitution

20:03

Use Information From One Equation in Another Equation

20:07

Put Your Substitution in Parentheses

20:44

Example 1

23:17

Example 2

25:49

Example 3

28:11

Example 4

30:02

Coordinate Systems

35m 2s

Intro

0:00

Inherent Order in ℝ

0:05

Real Numbers Come with an Inherent Order

0:11

Positive Numbers

0:21

Negative Numbers

0:58

'Less Than' and 'Greater Than'

2:04

Tip To Help You Remember the Signs

2:56

Inequality

4:06

Less Than or Equal and Greater Than or Equal

4:51

One Dimension: The Number Line

5:36

Graphically Represent ℝ on a Number Line

5:43

Note on Infinities

5:57

With the Number Line, We Can Directly See the Order We Put on ℝ

6:35

Ordered Pairs

7:22

Example

7:34

Allows Us to Talk About Two Numbers at the Same Time

9:41

Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ

10:41

Two Dimensions: The Plane

13:13

We Can Represent Ordered Pairs with the Plane

13:24

Intersection is known as the Origin

14:31

Plotting the Point

14:32

Plane = Coordinate Plane = Cartesian Plane = ℝ²

17:46

The Plane and Quadrants

18:50

Quadrant I

19:04

Quadrant II

19:21

Quadrant III

20:04

Quadrant IV

20:20

Three Dimensions: Space

21:02

Create Ordered Triplets

21:09

Visually Represent This

21:19

Three-Dimension = Space = ℝ³

21:47

Higher Dimensions

22:24

If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power

22:31

We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers

22:41

Hard to Visualize Higher Dimensional Spaces

23:18

Example 1

25:07

Example 2

26:10

Example 3

28:58

Example 4

31:05

Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro

0:00

Introduction

0:07

Midpoint: One Dimension

2:09

Example of Something More Complex

2:31

Use the Idea of a Middle

3:28

Find the Midpoint of Arbitrary Values a and b

4:17

How They're Equivalent

5:05

Official Midpoint Formula

5:46

Midpoint: Two Dimensions

6:19

The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle

6:38

Arbitrary Pair of Points Example

7:25

Distance: One Dimension

9:26

Absolute Value

10:54

Idea of Forcing Positive

11:06

Distance: One Dimension, Formula

11:47

Distance Between Arbitrary a and b

11:48

Absolute Value Helps When the Distance is Negative

12:41

Distance Formula

12:58

The Pythagorean Theorem

13:24

a²+b²=c²

13:50

Distance: Two Dimensions

14:59

Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem

15:16

Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)

16:21

Slope

19:30

Slope is the Rate of Change

19:41

m = rise over run

21:27

Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)

22:31

Interpreting Slope

24:12

Positive Slope and Negative Slope

25:40

m=1, m=0, m=-1

26:48

Example 1

28:25

Example 2

31:42

Example 3

36:40

Example 4

42:48

Word Problems

56m 31s

Intro

0:00

Introduction

0:05

What is a Word Problem?

0:45

Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols

0:48

Requires Us to Think

1:32

Why Are They So Hard?

2:11

Reason 1: No Simple Formula to Solve Them

2:16

Reason 2: Harder to Teach Word Problems

2:47

You Can Learn How to Do Them!

3:51

Grades

7:57

'But I'm Never Going to Use This In Real Life'

9:46

Solving Word Problems

12:58

First: Understand the Problem

13:37

Second: What Are You Looking For?

14:33

Third: Set Up Relationships

16:21

Fourth: Solve It!

17:48

Summary of Method

19:04

Examples on Things Other Than Math

20:21

Math-Specific Method: What You Need Now

25:30

Understand What the Problem is Talking About

25:37

Set Up and Name Any Variables You Need to Know

25:56

Set Up Equations Connecting Those Variables to the Information in the Problem Statement

26:02

Use the Equations to Solve for an Answer

26:14

Tip

26:58

Draw Pictures

27:22

Breaking Into Pieces

28:28

Try Out Hypothetical Numbers

29:52

Student Logic

31:27

Jump In!

32:40

Example 1

34:03

Example 2

39:15

Example 3

44:22

Example 4

50:24

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

The Basketball Factory

2:12

The Importance of Information

2:45

One-to-One

4:04

Requirement for Reversibility

4:21

When a Function has an Inverse

4:43

One-to-One

5:13

Not One-to-One

5:50

Not a Function

6:19

Horizontal Line Test

7:01

How to the test Works

7:12

One-to-One

8:12

Not One-to-One

8:45

Definition: Inverse Function

9:12

Formal Definition

9:21

Caution to Students

10:02

Domain and Range

11:12

Finding the Range of the Function Inverse

11:56

Finding the Domain of the Function Inverse

12:11

Inverse of an Inverse

13:09

Its just x!

13:26

Proof

14:03

Graphical Interpretation

17:07

Horizontal Line Test

17:20

Graph of the Inverse

18:04

Swapping Inputs and Outputs to Draw Inverses

19:02

How to Find the Inverse

21:03

What We Are Looking For

21:21

Reversing the Function

21:38

A Method to Find Inverses

22:33

Check Function is One-to-One

23:04

Swap f(x) for y

23:25

Interchange x and y

23:41

Solve for y

24:12

Replace y with the inverse

24:40

Some Comments

25:01

Keeping Step 2 and 3 Straight

25:44

Switching to Inverse

26:12

Checking Inverses

28:52

How to Check an Inverse

29:06

Quick Example of How to Check

29:56

Example 1

31:48

Example 2

34:56

Example 3

39:29

Example 4

46:19

Variation Direct and Inverse

28m 49s

Intro

0:00

Introduction

0:06

Direct Variation

1:14

Same Direction

1:21

Common Example: Groceries

1:56

Different Ways to Say that Two Things Vary Directly

2:28

Basic Equation for Direct Variation

2:55

Inverse Variation

3:40

Opposite Direction

3:50

Common Example: Gravity

4:53

Different Ways to Say that Two Things Vary Indirectly

5:48

Basic Equation for Indirect Variation

6:33

Joint Variation

7:27

Equation for Joint Variation

7:53

Explanation of the Constant

8:48

Combined Variation

9:35

Gas Law as a Combination

9:44

Single Constant

10:33

Example 1

10:49

Example 2

13:34

Example 3

15:39

Example 4

19:48

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

Leading Coefficient Test

22:07

Even Degree, Positive Coefficient

22:13

Even Degree, Negative Coefficient

22:39

Odd Degree, Positive Coefficient

23:09

Odd Degree, Negative Coefficient

23:27

Example 1

25:11

Example 2

27:16

Example 3

31:16

Example 4

34:41

Roots (Zeros) of Polynomials

41m 7s

Intro

0:00

Introduction

0:05

Roots in Graphs

1:17

The x-intercepts

1:33

How to Remember What 'Roots' Are

1:50

Naïve Attempts

2:31

Isolating Variables

2:45

Failures of Isolating Variables

3:30

Missing Solutions

4:59

Factoring: How to Find Roots

6:28

How Factoring Works

6:36

Why Factoring Works

7:20

Steps to Finding Polynomial Roots

9:21

Factoring: How to Find Roots CAUTION

10:08

Factoring is Not Easy

11:32

Factoring Quadratics

13:08

Quadratic Trinomials

13:21

Form of Factored Binomials

13:38

Factoring Examples

14:40

Factoring Quadratics, Check Your Work

16:58

Factoring Higher Degree Polynomials

18:19

Factoring a Cubic

18:32

Factoring a Quadratic

19:04

Factoring: Roots Imply Factors

19:54

Where a Root is, A Factor Is

20:01

How to Use Known Roots to Make Factoring Easier

20:35

Not all Polynomials Can be Factored

22:30

Irreducible Polynomials

23:27

Complex Numbers Help

23:55

Max Number of Roots/Factors

24:57

Limit to Number of Roots Equal to the Degree

25:18

Why there is a Limit

25:25

Max Number of Peaks/Valleys

26:39

Shape Information from Degree

26:46

Example Graph

26:54

Max, But Not Required

28:00

Example 1

28:37

Example 2

31:21

Example 3

36:12

Example 4

38:40

Completing the Square and the Quadratic Formula

39m 43s

Intro

0:00

Introduction

0:05

Square Roots and Equations

0:51

Taking the Square Root to Find the Value of x

0:55

Getting the Positive and Negative Answers

1:05

Completing the Square: Motivation

2:04

Polynomials that are Easy to Solve

2:20

Making Complex Polynomials Easy to Solve

3:03

Steps to Completing the Square

4:30

Completing the Square: Method

7:22

Move C over

7:35

Divide by A

7:44

Find r

7:59

Add to Both Sides to Complete the Square

8:49

Solving Quadratics with Ease

9:56

The Quadratic Formula

11:38

Derivation

11:43

Final Form

12:23

Follow Format to Use Formula

13:38

How Many Roots?

14:53

The Discriminant

15:47

What the Discriminant Tells Us: How Many Roots

15:58

How the Discriminant Works

16:30

Example 1: Complete the Square

18:24

Example 2: Solve the Quadratic

22:00

Example 3: Solve for Zeroes

25:28

Example 4: Using the Quadratic Formula

30:52

Properties of Quadratic Functions

45m 34s

Intro

0:00

Introduction

0:05

Parabolas

0:35

Examples of Different Parabolas

1:06

Axis of Symmetry and Vertex

1:28

Drawing an Axis of Symmetry

1:51

Placing the Vertex

2:28

Looking at the Axis of Symmetry and Vertex for other Parabolas

3:09

Transformations

4:18

Reviewing Transformation Rules

6:28

Note the Different Horizontal Shift Form

7:45

An Alternate Form to Quadratics

8:54

The Constants: k, h, a

9:05

Transformations Formed

10:01

Analyzing Different Parabolas

10:10

Switching Forms by Completing the Square

11:43

Vertex of a Parabola

16:30

Vertex at (h, k)

16:47

Vertex in Terms of a, b, and c Coefficients

17:28

Minimum/Maximum at Vertex

18:19

When a is Positive

18:25

When a is Negative

18:52

Axis of Symmetry

19:54

Incredibly Minor Note on Grammar

20:52

Example 1

21:48

Example 2

26:35

Example 3

28:55

Example 4

31:40

Intermediate Value Theorem and Polynomial Division

46m 8s

Intro

0:00

Introduction

0:05

Reminder: Roots Imply Factors

1:32

The Intermediate Value Theorem

3:41

The Basis: U between a and b

4:11

U is on the Function

4:52

Intermediate Value Theorem, Proof Sketch

5:51

If Not True, the Graph Would Have to Jump

5:58

But Graph is Defined as Continuous

6:43

Finding Roots with the Intermediate Value Theorem

7:01

Picking a and b to be of Different Signs

7:10

Must Be at Least One Root

7:46

Dividing a Polynomial

8:16

Using Roots and Division to Factor

8:38

Long Division Refresher

9:08

The Division Algorithm

12:18

How It Works to Divide Polynomials

12:37

The Parts of the Equation

13:24

Rewriting the Equation

14:47

Polynomial Long Division

16:20

Polynomial Long Division In Action

16:29

One Step at a Time

20:51

Synthetic Division

22:46

Setup

23:11

Synthetic Division, Example

24:44

Which Method Should We Use

26:39

Advantages of Synthetic Method

26:49

Advantages of Long Division

27:13

Example 1

29:24

Example 2

31:27

Example 3

36:22

Example 4

40:55

Complex Numbers

45m 36s

Intro

0:00

Introduction

0:04

A Wacky Idea

1:02

The Definition of the Imaginary Number

1:22

How it Helps Solve Equations

2:20

Square Roots and Imaginary Numbers

3:15

Complex Numbers

5:00

Real Part and Imaginary Part

5:20

When Two Complex Numbers are Equal

6:10

Addition and Subtraction

6:40

Deal with Real and Imaginary Parts Separately

7:36

Two Quick Examples

7:54

Multiplication

9:07

FOIL Expansion

9:14

Note What Happens to the Square of the Imaginary Number

9:41

Two Quick Examples

10:22

Division

11:27

Complex Conjugates

13:37

Getting Rid of i

14:08

How to Denote the Conjugate

14:48

Division through Complex Conjugates

16:11

Multiply by the Conjugate of the Denominator

16:28

Example

17:46

Factoring So-Called 'Irreducible' Quadratics

19:24

Revisiting the Quadratic Formula

20:12

Conjugate Pairs

20:37

But Are the Complex Numbers 'Real'?

21:27

What Makes a Number Legitimate

25:38

Where Complex Numbers are Used

27:20

Still, We Won't See Much of C

29:05

Example 1

30:30

Example 2

33:15

Example 3

38:12

Example 4

42:07

Fundamental Theorem of Algebra

19m 9s

Intro

0:00

Introduction

0:05

Idea: Hidden Roots

1:16

Roots in Complex Form

1:42

All Polynomials Have Roots

2:08

Fundamental Theorem of Algebra

2:21

Where Are All the Imaginary Roots, Then?

3:17

All Roots are Complex

3:45

Real Numbers are a Subset of Complex Numbers

3:59

The n Roots Theorem

5:01

For Any Polynomial, Its Degree is Equal to the Number of Roots

5:11

Equivalent Statement

5:24

Comments: Multiplicity

6:29

Non-Distinct Roots

6:59

Denoting Multiplicity

7:20

Comments: Complex Numbers Necessary

7:41

Comments: Complex Coefficients Allowed

8:55

Comments: Existence Theorem

9:59

Proof Sketch of n Roots Theorem

10:45

First Root

11:36

Second Root

13:23

Continuation to Find all Roots

16:00

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

Irreducible Quadratic Factors

8:25

Example of Quadratic Factors

9:26

Multiple Quadratic Factors

9:49

Mixing Factor Types

10:28

Figuring Out the Numerator

11:10

How to Solve for the Constants

11:30

Quick Example

11:40

Example 1

14:29

Example 2

18:35

Example 3

20:33

Example 4

28:51

Section 5: Exponential & Logarithmic Functions

Understanding Exponents

35m 17s

Intro

0:00

Introduction

0:05

Fundamental Idea

1:46

Expanding the Idea

2:28

Multiplication of the Same Base

2:40

Exponents acting on Exponents

3:45

Different Bases with the Same Exponent

4:31

To the Zero

5:35

To the First

5:45

Fundamental Rule with the Zero Power

6:35

To the Negative

7:45

Any Number to a Negative Power

8:14

A Fraction to a Negative Power

9:58

Division with Exponential Terms

10:41

To the Fraction

11:33

Square Root

11:58

Any Root

12:59

Summary of Rules

14:38

To the Irrational

17:21

Example 1

20:34

Example 2

23:42

Example 3

27:44

Example 4

31:44

Example 5

33:15

Exponential Functions

47m 4s

Intro

0:00

Introduction

0:05

Definition of an Exponential Function

0:48

Definition of the Base

1:02

Restrictions on the Base

1:16

Computing Exponential Functions

2:29

Harder Computations

3:10

When to Use a Calculator

3:21

Graphing Exponential Functions: a>1

6:02

Three Examples

6:13

What to Notice on the Graph

7:44

A Story

8:27

Story Diagram

9:15

Increasing Exponentials

11:29

Story Morals

14:40

Application: Compound Interest

15:15

Compounding Year after Year

16:01

Function for Compounding Interest

16:51

A Special Number: e

20:55

Expression for e

21:28

Where e stabilizes

21:55

Application: Continuously Compounded Interest

24:07

Equation for Continuous Compounding

24:22

Exponential Decay 0<a<1

25:50

Three Examples

26:11

Why they 'lose' value

26:54

Example 1

27:47

Example 2

33:11

Example 3

36:34

Example 4

41:28

Introduction to Logarithms

40m 31s

Intro

0:00

Introduction

0:04

Definition of a Logarithm, Base 2

0:51

Log 2 Defined

0:55

Examples

2:28

Definition of a Logarithm, General

3:23

Examples of Logarithms

5:15

Problems with Unusual Bases

7:38

Shorthand Notation: ln and log

9:44

base e as ln

10:01

base 10 as log

10:34

Calculating Logarithms

11:01

using a calculator

11:34

issues with other bases

11:58

Graphs of Logarithms

13:21

Three Examples

13:29

Slow Growth

15:19

Logarithms as Inverse of Exponentiation

16:02

Using Base 2

16:05

General Case

17:10

Looking More Closely at Logarithm Graphs

19:16

The Domain of Logarithms

20:41

Thinking about Logs like Inverses

21:08

The Alternate

24:00

Example 1

25:59

Example 2

30:03

Example 3

32:49

Example 4

37:34

Properties of Logarithms

42m 33s

Intro

0:00

Introduction

0:04

Basic Properties

1:12

Inverse--log(exp)

1:43

A Key Idea

2:44

What We Get through Exponentiation

3:18

B Always Exists

4:50

Inverse--exp(log)

5:53

Logarithm of a Power

7:44

Logarithm of a Product

10:07

Logarithm of a Quotient

13:48

Caution! There Is No Rule for loga(M+N)

16:12

Summary of Properties

17:42

Change of Base--Motivation

20:17

No Calculator Button

20:59

A Specific Example

21:45

Simplifying

23:45

Change of Base--Formula

24:14

Example 1

25:47

Example 2

29:08

Example 3

31:14

Example 4

34:13

Solving Exponential and Logarithmic Equations

34m 10s

Intro

0:00

Introduction

0:05

One to One Property

1:09

Exponential

1:26

Logarithmic

1:44

Specific Considerations

2:02

One-to-One Property

3:30

Solving by One-to-One

4:11

Inverse Property

6:09

Solving by Inverses

7:25

Dealing with Equations

7:50

Example of Taking an Exponent or Logarithm of an Equation

9:07

A Useful Property

11:57

Bring Down Exponents

12:01

Try to Simplify

13:20

Extraneous Solutions

13:45

Example 1

16:37

Example 2

19:39

Example 3

21:37

Example 4

26:45

Example 5

29:37

Application of Exponential and Logarithmic Functions

48m 46s

Intro

0:00

Introduction

0:06

Applications of Exponential Functions

1:07

A Secret!

2:17

Natural Exponential Growth Model

3:07

Figure out r

3:34

A Secret!--Why Does It Work?

4:44

e to the r Morphs

4:57

Example

5:06

Applications of Logarithmic Functions

8:32

Examples

8:43

What Logarithms are Useful For

9:53

Example 1

11:29

Example 2

15:30

Example 3

26:22

Example 4

32:05

Example 5

39:19

Section 6: Trigonometric Functions

Angles

39m 5s

Intro

0:00

Degrees

0:22

Circle is 360 Degrees

0:48

Splitting a Circle

1:13

Radians

2:08

Circle is 2 Pi Radians

2:31

One Radian

2:52

Half-Circle and Right Angle

4:00

Converting Between Degrees and Radians

6:24

Formulas for Degrees and Radians

6:52

Coterminal, Complementary, Supplementary Angles

7:23

Coterminal Angles

7:30

Complementary Angles

9:40

Supplementary Angles

10:08

Example 1: Dividing a Circle

10:38

Example 2: Converting Between Degrees and Radians

11:56

Example 3: Quadrants and Coterminal Angles

14:18

Extra Example 1: Common Angle Conversions

-1

Extra Example 2: Quadrants and Coterminal Angles

-2

Sine and Cosine Functions

43m 16s

Intro

0:00

Sine and Cosine

0:15

Unit Circle

0:22

Coordinates on Unit Circle

1:03

Right Triangles

1:52

Adjacent, Opposite, Hypotenuse

2:25

Master Right Triangle Formula: SOHCAHTOA

2:48

Odd Functions, Even Functions

4:40

Example: Odd Function

4:56

Example: Even Function

7:30

Example 1: Sine and Cosine

10:27

Example 2: Graphing Sine and Cosine Functions

14:39

Example 3: Right Triangle

21:40

Example 4: Odd, Even, or Neither

26:01

Extra Example 1: Right Triangle

-1

Extra Example 2: Graphing Sine and Cosine Functions

-2

Sine and Cosine Values of Special Angles

33m 5s

Intro

0:00

45-45-90 Triangle and 30-60-90 Triangle

0:08

45-45-90 Triangle

0:21

30-60-90 Triangle

2:06

Mnemonic: All Students Take Calculus (ASTC)

5:21

Using the Unit Circle

5:59

New Angles

6:21

Other Quadrants

9:43

Mnemonic: All Students Take Calculus

10:13

Example 1: Convert, Quadrant, Sine/Cosine

13:11

Example 2: Convert, Quadrant, Sine/Cosine

16:48

Example 3: All Angles and Quadrants

20:21

Extra Example 1: Convert, Quadrant, Sine/Cosine

-1

Extra Example 2: All Angles and Quadrants

-2

Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro

0:00

Amplitude and Period of a Sine Wave

0:38

Sine Wave Graph

0:58

Amplitude: Distance from Middle to Peak

1:18

Peak: Distance from Peak to Peak

2:41

Phase Shift and Vertical Shift

4:13

Phase Shift: Distance Shifted Horizontally

4:16

Vertical Shift: Distance Shifted Vertically

6:48

Example 1: Amplitude/Period/Phase and Vertical Shift

8:04

Example 2: Amplitude/Period/Phase and Vertical Shift

17:39

Example 3: Find Sine Wave Given Attributes

25:23

Extra Example 1: Amplitude/Period/Phase and Vertical Shift

-1

Extra Example 2: Find Cosine Wave Given Attributes

-2

Tangent and Cotangent Functions

36m 4s

Intro

0:00

Tangent and Cotangent Definitions

0:21

Tangent Definition

0:25

Cotangent Definition

0:47

Master Formula: SOHCAHTOA

1:01

Mnemonic

1:16

Tangent and Cotangent Values

2:29

Remember Common Values of Sine and Cosine

2:46

90 Degrees Undefined

4:36

Slope and Menmonic: ASTC

5:47

Uses of Tangent

5:54

Example: Tangent of Angle is Slope

6:09

Sign of Tangent in Quadrants

7:49

Example 1: Graph Tangent and Cotangent Functions

10:42

Example 2: Tangent and Cotangent of Angles

16:09

Example 3: Odd, Even, or Neither

18:56

Extra Example 1: Tangent and Cotangent of Angles

-1

Extra Example 2: Tangent and Cotangent of Angles

-2

Secant and Cosecant Functions

27m 18s

Intro

0:00

Secant and Cosecant Definitions

0:17

Secant Definition

0:18

Cosecant Definition

0:33

Example 1: Graph Secant Function

0:48

Example 2: Values of Secant and Cosecant

6:49

Example 3: Odd, Even, or Neither

12:49

Extra Example 1: Graph of Cosecant Function

-1

Extra Example 2: Values of Secant and Cosecant

-2

Inverse Trigonometric Functions

32m 58s

Intro

0:00

Arcsine Function

0:24

Restrictions between -1 and 1

0:43

Arcsine Notation

1:26

Arccosine Function

3:07

Restrictions between -1 and 1

3:36

Cosine Notation

3:53

Arctangent Function

4:30

Between -Pi/2 and Pi/2

4:44

Tangent Notation

5:02

Example 1: Domain/Range/Graph of Arcsine

5:45

Example 2: Arcsin/Arccos/Arctan Values

10:46

Example 3: Domain/Range/Graph of Arctangent

17:14

Extra Example 1: Domain/Range/Graph of Arccosine

-1

Extra Example 2: Arcsin/Arccos/Arctan Values

-2

Computations of Inverse Trigonometric Functions

31m 8s

Intro

0:00

Inverse Trigonometric Function Domains and Ranges

0:31

Arcsine

0:41

Arccosine

1:14

Arctangent

1:41

Example 1: Arcsines of Common Values

2:44

Example 2: Odd, Even, or Neither

5:57

Example 3: Arccosines of Common Values

12:24

Extra Example 1: Arctangents of Common Values

-1

Extra Example 2: Arcsin/Arccos/Arctan Values

-2

Section 7: Trigonometric Identities

Pythagorean Identity

19m 11s

Intro

0:00

Pythagorean Identity

0:17

Pythagorean Triangle

0:27

Pythagorean Identity

0:45

Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity

1:14

Example 2: Find Angle Given Cosine and Quadrant

4:18

Example 3: Verify Trigonometric Identity

8:00

Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem

-1

Extra Example 2: Find Angle Given Cosine and Quadrant

-2

Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro

0:00

Main Formulas

0:19

Companion to Pythagorean Identity

0:27

For Cotangents and Cosecants

0:52

How to Remember

0:58

Example 1: Prove the Identity

1:40

Example 2: Given Tan Find Sec

3:42

Example 3: Prove the Identity

7:45

Extra Example 1: Prove the Identity

-1

Extra Example 2: Given Sec Find Tan

-2

Addition and Subtraction Formulas

52m 52s

Intro

0:00

Addition and Subtraction Formulas

0:09

How to Remember

0:48

Cofunction Identities

1:31

How to Remember Graphically

1:44

Where to Use Cofunction Identities

2:52

Example 1: Derive the Formula for cos(A-B)

3:08

Example 2: Use Addition and Subtraction Formulas

16:03

Example 3: Use Addition and Subtraction Formulas to Prove Identity

25:11

Extra Example 1: Use cos(A-B) and Cofunction Identities

-1

Extra Example 2: Convert to Radians and use Formulas

-2

Double Angle Formulas

29m 5s

Intro

0:00

Main Formula

0:07

How to Remember from Addition Formula

0:18

Two Other Forms

1:35

Example 1: Find Sine and Cosine of Angle using Double Angle

3:16

Example 2: Prove Trigonometric Identity using Double Angle

9:37

Example 3: Use Addition and Subtraction Formulas

12:38

Extra Example 1: Find Sine and Cosine of Angle using Double Angle

-1

Extra Example 2: Prove Trigonometric Identity using Double Angle

-2

Half-Angle Formulas

43m 55s

Intro

0:00

Main Formulas

0:09

Confusing Part

0:34

Example 1: Find Sine and Cosine of Angle using Half-Angle

0:54

Example 2: Prove Trigonometric Identity using Half-Angle

11:51

Example 3: Prove the Half-Angle Formula for Tangents

18:39

Extra Example 1: Find Sine and Cosine of Angle using Half-Angle

-1

Extra Example 2: Prove Trigonometric Identity using Half-Angle

-2

Section 8: Applications of Trigonometry

Trigonometry in Right Angles

25m 43s

Intro

0:00

Master Formula for Right Angles

0:11

SOHCAHTOA

0:15

Only for Right Triangles

1:26

Example 1: Find All Angles in a Triangle

2:19

Example 2: Find Lengths of All Sides of Triangle

7:39

Example 3: Find All Angles in a Triangle

11:00

Extra Example 1: Find All Angles in a Triangle

-1

Extra Example 2: Find Lengths of All Sides of Triangle

-2

Law of Sines

56m 40s

Intro

0:00

Law of Sines Formula

0:18

SOHCAHTOA

0:27

Any Triangle

0:59

Graphical Representation

1:25

Solving Triangle Completely

2:37

When to Use Law of Sines

2:55

ASA, SAA, SSA, AAA

2:59

SAS, SSS for Law of Cosines

7:11

Example 1: How Many Triangles Satisfy Conditions, Solve Completely

8:44

Example 2: How Many Triangles Satisfy Conditions, Solve Completely

15:30

Example 3: How Many Triangles Satisfy Conditions, Solve Completely

28:32

Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely

-1

Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely

-2

Law of Cosines

49m 5s

Intro

0:00

Law of Cosines Formula

0:23

Graphical Representation

0:34

Relates Sides to Angles

1:00

Any Triangle

1:20

Generalization of Pythagorean Theorem

1:32

When to Use Law of Cosines

2:26

SAS, SSS

2:30

Heron's Formula

4:49

Semiperimeter S

5:11

Example 1: How Many Triangles Satisfy Conditions, Solve Completely

5:53

Example 2: How Many Triangles Satisfy Conditions, Solve Completely

15:19

Example 3: Find Area of a Triangle Given All Side Lengths

26:33

Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely

-1

Extra Example 2: Length of Third Side and Area of Triangle

-2

Finding the Area of a Triangle

27m 37s

Intro

0:00

Master Right Triangle Formula and Law of Cosines

0:19

SOHCAHTOA

0:27

Law of Cosines

1:23

Heron's Formula

2:22

Semiperimeter S

2:37

Example 1: Area of Triangle with Two Sides and One Angle

3:12

Example 2: Area of Triangle with Three Sides

6:11

Example 3: Area of Triangle with Three Sides, No Heron's Formula

8:50

Extra Example 1: Area of Triangle with Two Sides and One Angle

-1

Extra Example 2: Area of Triangle with Two Sides and One Angle

-2

Word Problems and Applications of Trigonometry

34m 25s

Intro

0:00

Formulas to Remember

0:11

SOHCAHTOA

0:15

Law of Sines

0:55

Law of Cosines

1:48

Heron's Formula

2:46

Example 1: Telephone Pole Height

4:01

Example 2: Bridge Length

7:48

Example 3: Area of Triangular Field

14:20

Extra Example 1: Kite Height

-1

Extra Example 2: Roads to a Town

-2

Section 9: Systems of Equations and Inequalities

Systems of Linear Equations

55m 40s

Intro

0:00

Introduction

0:04

Graphs as Location of 'True'

1:49

All Locations that Make the Function True

2:25

Understand the Relationship Between Solutions and the Graph

3:43

Systems as Graphs

4:07

Equations as Lines

4:20

Intersection Point

5:19

Three Possibilities for Solutions

6:17

Independent

6:24

Inconsistent

6:36

Dependent

7:06

Solving by Substitution

8:37

Solve for One Variable

9:07

Substitute into the Second Equation

9:34

Solve for Both Variables

10:12

What If a System is Inconsistent or Dependent?

11:08

No Solutions

11:25

Infinite Solutions

12:30

Solving by Elimination

13:56

Example

14:22

Determining the Number of Solutions

16:30

Why Elimination Makes Sense

17:25

Solving by Graphing Calculator

19:59

Systems with More than Two Variables

23:22

Example 1

25:49

Example 2

30:22

Example 3

34:11

Example 4

38:55

Example 5

46:01

(Non-) Example 6

53:37

Systems of Linear Inequalities

1h 13s

Intro

0:00

Introduction

0:04

Inequality Refresher-Solutions

0:46

Equation Solutions vs. Inequality Solutions

1:02

Essentially a Wide Variety of Answers

1:35

Refresher--Negative Multiplication Flips

1:43

Refresher--Negative Flips: Why?

3:19

Multiplication by a Negative

3:43

The Relationship Flips

3:55

Refresher--Stick to Basic Operations

4:34

Linear Equations in Two Variables

6:50

Graphing Linear Inequalities

8:28

Why It Includes a Whole Section

8:43

How to Show The Difference Between Strict and Not Strict Inequalities

10:08

Dashed Line--Not Solutions

11:10

Solid Line--Are Solutions

11:24

Test Points for Shading

11:42

Example of Using a Point

12:41

Drawing Shading from the Point

13:14

Graphing a System

14:53

Set of Solutions is the Overlap

15:17

Example

15:22

Solutions are Best Found Through Graphing

18:05

Linear Programming-Idea

19:52

Use a Linear Objective Function

20:15

Variables in Objective Function have Constraints

21:24

Linear Programming-Method

22:09

Rearrange Equations

22:21

Graph

22:49

Critical Solution is at the Vertex of the Overlap

23:40

Try Each Vertice

24:35

Example 1

24:58

Example 2

28:57

Example 3

33:48

Example 4

43:10

Nonlinear Systems

41m 1s

Intro

0:00

Introduction

0:06

Substitution

1:12

Example

1:22

Elimination

3:46

Example

3:56

Elimination is Less Useful for Nonlinear Systems

4:56

Graphing

5:56

Using a Graphing Calculator

6:44

Number of Solutions

8:44

Systems of Nonlinear Inequalities

10:02

Graph Each Inequality

10:06

Dashed and/or Solid

10:18

Shade Appropriately

11:14

Example 1

13:24

Example 2

15:50

Example 3

22:02

Example 4

29:06

Example 4, cont.

33:40

Section 10: Vectors and Matrices

Vectors

1h 9m 31s

Intro

0:00

Introduction

0:10

Magnitude of the Force

0:22

Direction of the Force

0:48

Vector

0:52

Idea of a Vector

1:30

How Vectors are Denoted

2:00

Component Form

3:20

Angle Brackets and Parentheses

3:50

Magnitude/Length

4:26

Denoting the Magnitude of a Vector

5:16

Direction/Angle

7:52

Always Draw a Picture

8:50

Component Form from Magnitude & Angle

10:10

Scaling by Scalars

14:06

Unit Vectors

16:26

Combining Vectors - Algebraically

18:10

Combining Vectors - Geometrically

19:54

Resultant Vector

20:46

Alternate Component Form: i, j

21:16

The Zero Vector

23:18

Properties of Vectors

24:20

No Multiplication (Between Vectors)

28:30

Dot Product

29:40

Motion in a Medium

30:10

Fish in an Aquarium Example

31:38

More Than Two Dimensions

33:12

More Than Two Dimensions - Magnitude

34:18

Example 1

35:26

Example 2

38:10

Example 3

45:48

Example 4

50:40

Example 4, cont.

56:07

Example 5

1:01:32

Dot Product & Cross Product

35m 20s

Intro

0:00

Introduction

0:08

Dot Product - Definition

0:42

Dot Product Results in a Scalar, Not a Vector

2:10

Example in Two Dimensions

2:34

Angle and the Dot Product

2:58

The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors

2:59

Proof of Dot Product Formula

4:14

Won't Directly Help Us Better Understand Vectors

4:18

Dot Product - Geometric Interpretation

4:58

We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are

7:26

Dot Product - Perpendicular Vectors

8:24

If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other

8:54

Cross Product - Definition

11:08

Cross Product Only Works in Three Dimensions

11:09

Cross Product - A Mnemonic

12:16

The Determinant of a 3 x 3 Matrix and Standard Unit Vectors

12:17

Cross Product - Geometric Interpretations

14:30

The Right-Hand Rule

15:17

Cross Product - Geometric Interpretations Cont.

17:00

Example 1

18:40

Example 2

22:50

Example 3

24:04

Example 4

26:20

Bonus Round

29:18

Proof: Dot Product Formula

29:24

Proof: Dot Product Formula, cont.

30:38

Matrices

54m 7s

Intro

0:00

Introduction

0:08

Definition of a Matrix

3:02

Size or Dimension

3:58

Square Matrix

4:42

Denoted by Capital Letters

4:56

When are Two Matrices Equal?

5:04

Examples of Matrices

6:44

Rows x Columns

6:46

Talking About Specific Entries

7:48

We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries

8:32

Using Entries to Talk About Matrices

10:08

Scalar Multiplication

11:26

Scalar = Real Number

11:34

Example

12:36

Matrix Addition

13:08

Example

14:22

Matrix Multiplication

15:00

Example

18:52

Matrix Multiplication, cont.

19:58

Matrix Multiplication and Order (Size)

25:26

Make Sure Their Orders are Compatible

25:27

Matrix Multiplication is NOT Commutative

28:20

Example

30:08

Special Matrices - Zero Matrix (0)

32:48

Zero Matrix Has 0 for All of its Entries

32:49

Special Matrices - Identity Matrix (I)

34:14

Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries

34:15

Example 1

36:16

Example 2

40:00

Example 3

44:54

Example 4

50:08

Determinants & Inverses of Matrices

47m 12s

Intro

0:00

Introduction

0:06

Not All Matrices Are Invertible

1:30

What Must a Matrix Have to Be Invertible?

2:08

Determinant

2:32

The Determinant is a Real Number Associated With a Square Matrix

2:38

If the Determinant of a Matrix is Nonzero, the Matrix is Invertible

3:40

Determinant of a 2 x 2 Matrix

4:34

Think in Terms of Diagonals

5:12

Minors and Cofactors - Minors

6:24

Example

6:46

Minors and Cofactors - Cofactors

8:00

Cofactor is Closely Based on the Minor

8:01

Alternating Sign Pattern

9:04

Determinant of Larger Matrices

10:56

Example

13:00

Alternative Method for 3x3 Matrices

16:46

Not Recommended

16:48

Inverse of a 2 x 2 Matrix

19:02

Inverse of Larger Matrices

20:00

Using Inverse Matrices

21:06

When Multiplied Together, They Create the Identity Matrix

21:24

Example 1

23:45

Example 2

27:21

Example 3

32:49

Example 4

36:27

Finding the Inverse of Larger Matrices

41:59

General Inverse Method - Step 1

43:25

General Inverse Method - Step 2

43:27

General Inverse Method - Step 2, cont.

43:27

General Inverse Method - Step 3

45:15

Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro

0:00

Introduction

0:12

Augmented Matrix

1:44

We Can Represent the Entire Linear System With an Augmented Matrix

1:50

Row Operations

3:22

Interchange the Locations of Two Rows

3:50

Multiply (or Divide) a Row by a Nonzero Number

3:58

Add (or Subtract) a Multiple of One Row to Another

4:12

Row Operations - Keep Notes!

5:50

Suggested Symbols

7:08

Gauss-Jordan Elimination - Idea

8:04

Gauss-Jordan Elimination - Idea, cont.

9:16

Reduced Row-Echelon Form

9:18

Gauss-Jordan Elimination - Method

11:36

Begin by Writing the System As An Augmented Matrix

11:38

Gauss-Jordan Elimination - Method, cont.

13:48

Cramer's Rule - 2 x 2 Matrices

17:08

Cramer's Rule - n x n Matrices

19:24

Solving with Inverse Matrices

21:10

Solving Inverse Matrices, cont.

25:28

The Mighty (Graphing) Calculator

26:38

Example 1

29:56

Example 2

33:56

Example 3

37:00

Example 3, cont.

45:04

Example 4

51:28

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

Addition Principle

3:40

Example

4:18

Multiplication Principle

5:42

Example

6:24

Pigeonhole Principle

8:06

Example

10:26

Draw Pictures

11:06

Example 1

12:02

Example 2

14:16

Example 3

17:34

Example 4

21:26

Example 5

25:14

Permutations & Combinations

44m 3s

Intro

0:00

Introduction

0:08

Permutation

0:42

Combination

1:10

Towards a Permutation Formula

2:38

How Many Ways Can We Arrange the Letters A, B, C, D, and E?

3:02

Towards a Permutation Formula, cont.

3:34

Factorial Notation

6:56

Symbol Is '!'

6:58

Examples

7:32

Permutation of n Objects

8:44

Permutation of r Objects out of n

9:04

What If We Have More Objects Than We Have Slots to Fit Them Into?

9:46

Permutation of r Objects Out of n, cont.

10:28

Distinguishable Permutations

14:46

What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?

14:48

Distinguishable Permutations, cont.

17:04

Combinations

19:04

Combinations, cont.

20:56

Example 1

23:10

Example 2

26:16

Example 3

28:28

Example 4

31:52

Example 5

33:58

Example 6

36:34

Probability

36m 58s

Intro

0:00

Introduction

0:06

Definition: Sample Space

1:18

Event = Something Happening

1:20

Sample Space

1:36

Probability of an Event

2:12

Let E Be An Event and S Be The Corresponding Sample Space

2:14

'Equally Likely' Is Important

3:52

Fair and Random

5:26

Interpreting Probability

6:34

How Can We Interpret This Value?

7:24

We Can Represent Probability As a Fraction, a Decimal, Or a Percentage

8:04

One of Multiple Events Occurring

9:52

Mutually Exclusive Events

10:38

What If The Events Are Not Mutually Exclusive?

12:20

Taking the Possibility of Overlap Into Account

13:24

An Event Not Occurring

17:14

Complement of E

17:22

Independent Events

19:36

Independent

19:48

Conditional Events

21:28

What Is The Events Are Not Independent Though?

21:30

Conditional Probability

22:16

Conditional Events, cont.

23:51

Example 1

25:27

Example 2

27:09

Example 3

28:57

Example 4

30:51

Example 5

34:15

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

Radius

0:32

Equation of a Circle

0:44

Example: Standard Form

1:11

Graphing Circles

1:47

Example: Circle

1:56

Center Not at Origin

3:07

Example: Completing the Square

3:51

Example 1: Equation of Circle

6:44

Example 2: Center and Radius

11:51

Example 3: Radius

15:08

Example 4: Equation of Circle

16:57

Ellipses

46m 51s

Intro

0:00

What Are Ellipses?

0:11

Foci

0:23

Properties of Ellipses

1:43

Major Axis, Minor Axis

1:47

Center

1:54

Length of Major Axis and Minor Axis

3:21

Standard Form

5:33

Example: Standard Form of Ellipse

6:09

Vertical Major Axis

9:14

Example: Vertical Major Axis

9:46

Graphing Ellipses

12:51

Complete the Square and Symmetry

13:00

Example: Graphing Ellipse

13:16

Equation with Center at (h, k)

19:57

Horizontal and Vertical

20:14

Difference

20:27

Example: Center at (h, k)

20:55

Example 1: Equation of Ellipse

24:05

Example 2: Equation of Ellipse

27:57

Example 3: Equation of Ellipse

32:32

Example 4: Graph Ellipse

38:27

Hyperbolas

38m 15s

Intro

0:00

What are Hyperbolas?

0:12

Two Branches

0:18

Foci

0:38

Properties

2:00

Transverse Axis and Conjugate Axis

2:06

Vertices

2:46

Length of Transverse Axis

3:14

Distance Between Foci

3:31

Length of Conjugate Axis

3:38

Standard Form

5:45

Vertex Location

6:36

Known Points

6:52

Vertical Transverse Axis

7:26

Vertex Location

7:50

Asymptotes

8:36

Vertex Location

8:56

Rectangle

9:28

Diagonals

10:29

Graphing Hyperbolas

12:58

Example: Hyperbola

13:16

Equation with Center at (h, k)

16:32

Example: Center at (h, k)

17:21

Example 1: Equation of Hyperbola

19:20

Example 2: Equation of Hyperbola

22:48

Example 3: Graph Hyperbola

26:05

Example 4: Equation of Hyperbola

36:29

Conic Sections

18m 43s

Intro

0:00

Conic Sections

0:16

Double Cone Sections

0:24

Standard Form

1:27

General Form

1:37

Identify Conic Sections

2:16

B = 0

2:50

X and Y

3:22

Identify Conic Sections, Cont.

4:46

Parabola

5:17

Circle

5:51

Ellipse

6:31

Hyperbola

7:10

Example 1: Identify Conic Section

8:01

Example 2: Identify Conic Section

11:03

Example 3: Identify Conic Section

11:38

Example 4: Identify Conic Section

14:50

Section 15: Sequences, Series, & Induction

Introduction to Sequences

57m 45s

Intro

0:00

Introduction

0:06

Definition: Sequence

0:28

Infinite Sequence

2:08

Finite Sequence

2:22

Length

2:58

Formula for the nth Term

3:22

Defining a Sequence Recursively

5:54

Initial Term

7:58

Sequences and Patterns

10:40

First, Identify a Pattern

12:52

How to Get From One Term to the Next

17:38

Tips for Finding Patterns

19:52

More Tips for Finding Patterns

24:14

Even More Tips

26:50

Example 1

30:32

Example 2

34:54

Fibonacci Sequence

34:55

Example 3

38:40

Example 4

45:02

Example 5

49:26

Example 6

51:54

Introduction to Series

40m 27s

Intro

0:00

Introduction

0:06

Definition: Series

1:20

Why We Need Notation

2:48

Simga Notation (AKA Summation Notation)

4:44

Thing Being Summed

5:42

Index of Summation

6:21

Lower Limit of Summation

7:09

Upper Limit of Summation

7:23

Sigma Notation, Example

7:36

Sigma Notation for Infinite Series

9:08

How to Reindex

10:58

How to Reindex, Expanding

12:56

How to Reindex, Substitution

16:46

Properties of Sums

19:42

Example 1

23:46

Example 2

25:34

Example 3

27:12

Example 4

29:54

Example 5

32:06

Example 6

37:16

Arithmetic Sequences & Series

31m 36s

Intro

0:00

Introduction

0:05

Definition: Arithmetic Sequence

0:47

Common Difference

1:13

Two Examples

1:19

Form for the nth Term

2:14

Recursive Relation

2:33

Towards an Arithmetic Series Formula

5:12

Creating a General Formula

10:09

General Formula for Arithmetic Series

14:23

Example 1

15:46

Example 2

17:37

Example 3

22:21

Example 4

24:09

Example 5

27:14

Geometric Sequences & Series

39m 27s

Intro

0:00

Introduction

0:06

Definition

0:48

Form for the nth Term

2:42

Formula for Geometric Series

5:16

Infinite Geometric Series

11:48

Diverges

13:04

Converges

14:48

Formula for Infinite Geometric Series

16:32

Example 1

20:32

Example 2

22:02

Example 3

26:00

Example 4

30:48

Example 5

34:28

Mathematical Induction

49m 53s

Intro

0:00

Introduction

0:06

Belief Vs. Proof

1:22

A Metaphor for Induction

6:14

The Principle of Mathematical Induction

11:38

Base Case

13:24

Inductive Step

13:30

Inductive Hypothesis

13:52

A Remark on Statements

14:18

Using Mathematical Induction

16:58

Working Example

19:58

Finding Patterns

28:46

Example 1

30:17

Example 2

37:50

Example 3

42:38

The Binomial Theorem

1h 13m 13s

Intro

0:00

Introduction

0:06

We've Learned That a Binomial Is An Expression That Has Two Terms

0:07

Understanding Binomial Coefficients

1:20

Things We Notice

2:24

What Goes In the Blanks?

5:52

Each Blank is Called a Binomial Coefficient

6:18

The Binomial Theorem

6:38

Example

8:10

The Binomial Theorem, cont.

10:46

We Can Also Write This Expression Compactly Using Sigma Notation

12:06

Proof of the Binomial Theorem

13:22

Proving the Binomial Theorem Is Within Our Reach

13:24

Pascal's Triangle

15:12

Pascal's Triangle, cont.

16:12

Diagonal Addition of Terms

16:24

Zeroth Row

18:04

First Row

18:12

Why Do We Care About Pascal's Triangle?

18:50

Pascal's Triangle, Example

19:26

Example 1

21:26

Example 2

24:34

Example 3

28:34

Example 4

32:28

Example 5

37:12

Time for the Fireworks!

43:38

Proof of the Binomial Theorem

43:44

We'll Prove This By Induction

44:04

Proof (By Induction)

46:36

Proof, Base Case

47:00

Proof, Inductive Step - Notation Discussion

49:22

Induction Step

49:24

Proof, Inductive Step - Setting Up

52:26

Induction Hypothesis

52:34

What We What To Show

52:44

Proof, Inductive Step - Start

54:18

Proof, Inductive Step - Middle

55:38

Expand Sigma Notations

55:48

Proof, Inductive Step - Middle, cont.

58:40

Proof, Inductive Step - Checking In

1:01:08

Let's Check In With Our Original Goal

1:01:12

Want to Show

1:01:18

Lemma - A Mini Theorem

1:02:18

Proof, Inductive Step - Lemma

1:02:52

Proof of Lemma: Let's Investigate the Left Side

1:03:08

Proof, Inductive Step - Nearly There

1:07:54

Proof, Inductive Step - End!

1:09:18

Proof, Inductive Step - End!, cont.

1:11:01

Section 16: Preview of Calculus

Idea of a Limit

40m 22s

Intro

0:00

Introduction

0:05

Motivating Example

1:26

Fuzzy Notion of a Limit

3:38

Limit is the Vertical Location a Function is Headed Towards

3:44

Limit is What the Function Output is Going to Be

4:15

Limit Notation

4:33

Exploring Limits - 'Ordinary' Function

5:26

Test Out

5:27

Graphing, We See The Answer Is What We Would Expect

5:44

Exploring Limits - Piecewise Function

6:45

If We Modify the Function a Bit

6:49

Exploring Limits - A Visual Conception

10:08

Definition of a Limit

12:07

If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.

12:09

We Are Not Concerned with f(x) at x=c

12:49

We Are Considering x Approaching From All Directions, Not Just One Side

13:10

Limits Do Not Always Exist

15:47

Finding Limits

19:49

Graphs

19:52

Tables

21:48

Precise Methods

24:53

Example 1

26:06

Example 2

27:39

Example 3

30:51

Example 4

33:11

Example 5

37:07

Formal Definition of a Limit

57m 11s

Intro

0:00

Introduction

0:06

New Greek Letters

2:42

Delta

3:14

Epsilon

3:46

Sometimes Called the Epsilon-Delta Definition of a Limit

3:56

Formal Definition of a Limit

4:22

What does it MEAN!?!?

5:00

The Groundwork

5:38

Set Up the Limit

5:39

The Function is Defined Over Some Portion of the Reals

5:58

The Horizontal Location is the Value the Limit Will Approach

6:28

The Vertical Location L is Where the Limit Goes To

7:00

The Epsilon-Delta Part

7:26

The Hard Part is the Second Part of the Definition

7:30

Second Half of Definition

10:04

Restrictions on the Allowed x Values

10:28

The Epsilon-Delta Part, cont.

13:34

Sherlock Holmes and Dr. Watson

15:08

The Adventure of the Delta-Epsilon Limit

15:16

Setting

15:18

We Begin By Setting Up the Game As Follows

15:52

The Adventure of the Delta-Epsilon, cont.

17:24

This Game is About Limits

17:46

What If I Try Larger?

19:39

Technically, You Haven't Proven the Limit

20:53

Here is the Method

21:18

What We Should Concern Ourselves With

22:20

Investigate the Left Sides of the Expressions

25:24

We Can Create the Following Inequalities

28:08

Finally…

28:50

Nothing Like a Good Proof to Develop the Appetite

30:42

Example 1

31:02

Example 1, cont.

36:26

Example 2

41:46

Example 2, cont.

47:50

Finding Limits

32m 40s

Intro

0:00

Introduction

0:08

Method - 'Normal' Functions

2:04

The Easiest Limits to Find

2:06

It Does Not 'Break'

2:18

It Is Not Piecewise

2:26

Method - 'Normal' Functions, Example

3:38

Method - 'Normal' Functions, cont.

4:54

The Functions We're Used to Working With Go Where We Expect Them To Go

5:22

A Limit is About Figuring Out Where a Function is 'Headed'

5:42

Method - Canceling Factors

7:18

One Weird Thing That Often Happens is Dividing By 0

7:26

Method - Canceling Factors, cont.

8:16

Notice That The Two Functions Are Identical With the Exception of x=0

8:20

Method - Canceling Factors, cont.

10:00

Example

10:52

Method - Rationalization

12:04

Rationalizing a Portion of Some Fraction

12:05

Conjugate

12:26

Method - Rationalization, cont.

13:14

Example

13:50

Method - Piecewise

16:28

The Limits of Piecewise Functions

16:30

Example 1

17:42

Example 2

18:44

Example 3

20:20

Example 4

22:24

Example 5

24:24

Example 6

27:12

Continuity & One-Sided Limits

32m 43s

Intro

0:00

Introduction

0:06

Motivating Example

0:56

Continuity - Idea

2:14

Continuous Function

2:18

All Parts of Function Are Connected

2:28

Function's Graph Can Be Drawn Without Lifting Pencil

2:36

There Are No Breaks or Holes in Graph

2:56

Continuity - Idea, cont.

3:38

We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'

3:52

Continuity - Definition

5:16

A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does

5:18

Discontinuous

6:02

Discontinuity

6:10

Continuity and 'Normal' Functions

6:48

Return of the Motivating Example

8:14

One-Sided Limit

8:48

One-Sided Limit - Definition

9:16

Only Considers One Side

9:20

Be Careful to Keep Track of Which Symbol Goes With Which Side

10:06

One-Sided Limit - Example

10:50

There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits

11:16

Normal Limits and One-Sided Limits

12:08

Limits of Piecewise Functions

14:12

'Breakover' Points

14:22

We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other

15:34

Example 1

16:40

Example 2

18:54

Example 3

22:00

Example 4

26:36

Limits at Infinity & Limits of Sequences

32m 49s

Intro

0:00

Introduction

0:06

Definition: Limit of a Function at Infinity

1:44

A Limit at Infinity Works Very Similarly to How a Normal Limit Works

2:38

Evaluating Limits at Infinity

4:08

Rational Functions

4:17

Examples

4:30

For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator

5:22

There are Three Possibilities

6:36

Evaluating Limits at Infinity, cont.

8:08

Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?

10:06

Two Good Ways to Think About This

10:26

Limit of a Sequence

12:20

What Value Does the Sequence Tend to Do in the Long-Run?

12:41

The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity

12:52

Numerical Evaluation

14:16

Numerically: Plug in Numbers and See What Comes Out

14:24

Example 1

16:42

Example 2

21:00

Example 3

22:08

Example 4

26:14

Example 5

28:10

Example 6

31:06

Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro

0:00

Introduction

0:08

The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing

0:16

Instantaneous Slop

0:22

Instantaneous Rate of Change

0:28

Slope

1:24

The Vertical Change Divided by the Horizontal

1:40

Idea of Instantaneous Slope

2:10

What If We Wanted to Apply the Idea of Slope to a Non-Line?

2:14

Tangent to a Circle

3:52

What is the Tangent Line for a Circle?

4:42

Tangent to a Curve

5:20

Towards a Derivative - Average Slope

6:36

Towards a Derivative - Average Slope, cont.

8:20

An Approximation

11:24

Towards a Derivative - General Form

13:18

Towards a Derivative - General Form, cont.

16:46

An h Grows Smaller, Our Slope Approximation Becomes Better

18:44

Towards a Derivative - Limits!

20:04

Towards a Derivative - Limits!, cont.

22:08

We Want to Show the Slope at x=1

22:34

Towards a Derivative - Checking Our Slope

23:12

Definition of the Derivative

23:54

Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point

23:58

Differentiation

24:54

Notation for the Derivative

25:58

The Derivative is a Very Important Idea In Calculus

26:04

The Important Idea

27:34

Why Did We Learn the Formal Definition to Find a Derivative?

28:18

Example 1

30:50

Example 2

36:06

Example 3

40:24

The Power Rule

44:16

Makes It Easier to Find the Derivative of a Function

44:24

Examples

45:04

n Is Any Constant Number

45:46

Example 4

46:26

Area Under a Curve (Integrals)

45m 26s

Intro

0:00

Introduction

0:06

Integral

0:12

Idea of Area Under a Curve

1:18

Approximation by Rectangles

2:12

The Easiest Way to Find Area is With a Rectangle

2:18

Various Methods for Choosing Rectangles

4:30

Rectangle Method - Left-Most Point

5:12

The Left-Most Point

5:16

Rectangle Method - Right-Most Point

5:58

The Right-Most Point

6:00

Rectangle Method - Mid-Point

6:42

Horizontal Mid-Point

6:48

Rectangle Method - Maximum (Upper Sum)

7:34

Maximum Height

7:40

Rectangle Method - Minimum

8:54

Minimum Height

9:02

Evaluating the Area Approximation

10:08

Split the Interval Into n Sub-Intervals

10:30

More Rectangles, Better Approximation

12:14

The More We Us , the Better Our Approximation Becomes

12:16

Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity

12:44

Finding Area with a Limit

13:08

If This Limit Exists, It Is Called the Integral From a to b

14:08

The Process of Finding Integrals is Called Integration

14:22

The Big Reveal

14:40

The Integral is Based on the Antiderivative

14:46

The Big Reveal - Wait, Why?

16:28

The Rate of Change for the Area is Based on the Height of the Function

16:50

Height is the Derivative of Area, So Area is Based on the Antiderivative of Height

17:50

Example 1

19:06

Example 2

22:48

Example 3

29:06

Example 3, cont.

35:14

Example 4

40:14

Section 17: Appendix: Graphing Calculators

Buying a Graphing Calculator

10m 41s

Intro

0:00

Should You Buy?

0:06

Should I Get a Graphing Utility?

0:20

Free Graphing Utilities - Web Based

0:38

Personal Favorite: Desmos

0:58

Free Graphing Utilities - Offline Programs

1:18

GeoGebra

1:31

Microsoft Mathematics

1:50

Grapher

2:18

Other Graphing Utilities - Tablet/Phone

2:48

Should You Buy a Graphing Calculator?

3:22

The Only Real Downside

4:10

Deciding on Buying

4:20

If You Plan on Continuing in Math and/or Science

4:26

If Money is Not Particularly Tight for You

4:32

If You Don't Plan to Continue in Math and Science

5:02

If You Do Plan to Continue and Money Is Tight

5:28

Which to Buy

5:44

Which Graphing Calculator is Best?

5:46

Too Many Factors

5:54

Do Your Research

6:12

The Old Standby

7:10

TI-83 (Plus)

7:16

TI-84 (Plus)

7:18

Tips for Purchasing

9:17

Buy Online

9:19

Buy Used

9:35

Ask Around

10:09

Graphing Calculator Basics

10m 51s

Intro

0:00

Read the Manual

0:06

Skim It

0:20

Play Around and Experiment

0:34

Syntax

0:40

Definition of Syntax in English and Math

0:46

Pay Careful Attention to Your Syntax When Working With a Calculator

2:08

Make Sure You Use Parentheses to Indicate the Proper Order of Operations

2:16

Think About the Results

3:54

Settings

4:58

You'll Almost Never Need to Change the Settings on Your Calculator

5:00

Tell Calculator In Settings Whether the Angles Are In Radians or Degrees

5:26

Graphing Mode

6:32

Error Messages

7:10

Don't Panic

7:11

Internet Search

7:32

So Many Things

8:14

More Powerful Than You Realize

8:18

Other Things Your Graphing Calculator Can Do

8:24

Playing Around

9:16

Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro

0:00

Graphing Functions

0:18

Graphing Calculator Expects the Variable to Be x

0:28

Syntax

0:58

The Syntax We Choose Will Affect How the Function Graphs

1:00

Use Parentheses

1:26

The Viewing Window

2:00

One of the Most Important Ideas When Graphing Is To Think About The Viewing Window

2:01

For Example

2:30

The Viewing Window, cont.

2:36

Window Settings

3:24

Manually Choose Window Settings

4:20

x Min

4:40

x Max

4:42

y Min

4:44

y Max

4:46

Changing the x Scale or y Scale

5:08

Window Settings, cont.

5:44

Table of Values

7:38

Allows You to Quickly Churn Out Values for Various Inputs

7:42

For example

7:44

Changing the Independent Variable From 'Automatic' to 'Ask'

8:50

Finding Points of Interest

9m 45s

Intro

0:00

Points of Interest

0:06

Interesting Points on the Graph

0:11

Roots/Zeros (Zero)

0:18

Relative Minimums (Min)

0:26

Relative Maximums (Max)

0:32

Intersections (Intersection)

0:38

Finding Points of Interest - Process

1:48

Graph the Function

1:49

Adjust Viewing Window

2:12

Choose Point of Interest Type

2:54

Identify Where Search Should Occur

3:04

Give a Guess

3:36

Get Result

4:06

Advanced Technique: Arbitrary Solving

5:10

Find Out What Input Value Causes a Certain Output

5:12

For Example

5:24

Advanced Technique: Calculus

7:18

Derivative

7:22

Integral

7:30

But How Do You Show Work?

8:20

Parametric & Polar Graphs

7m 8s

Intro

0:00

Change Graph Type

0:08

Located in General 'Settings'

0:16

Graphing in Parametric

1:06

Set Up Both Horizontal Function and Vertical Function

1:08

For Example

2:04

Graphing in Polar

4:00

For Example

4:28

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Math Analysis Introduction to Sequences

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Section 15: Sequences, Series, & Induction: Lecture 1 | 57:45 min

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Lecture Comments (4)

1 answer

Last reply by: Professor Selhorst-Jones
Fri Jun 5, 2020 11:46 AM

Post by Chessdongdong on June 3, 2020

For the Examples of the Definition of a Sequence, does the sequence need to add up arithmetically as root(x), root(2x), root(3x) doesn't add up

1 answer

Last reply by: Professor Selhorst-Jones
Mon Jun 3, 2013 11:17 AM

Post by Vanessa Munoz on June 2, 2013

on example 3, second part, it could have been the sum, the addition pattern was wrong

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Introduction to Sequences

A sequence is an ordered list of numbers. We can write out a sequence as
a₁, a₂, a₃, a₄, …, a_n, …
We call each of the entries in the sequence a term. Above, a₁ is the first term, a₂ is the second term, and so on. Any symbol can be used to denote the sequence, while the subscript (small number to the right) tells us which term it is in the sequence.
If a sequence goes on forever without stopping, it is called an infinite sequence. Most of the sequences we will work with will be infinite sequences. If a sequence does stop, it is called a finite sequence.
a₁, a₂, a₃, a₄, …, a_k
We call the number of terms in a finite sequence its length. The length of the above sequence is k.
If we know a formula for the n^th term (this is also called the general term), we can easily find any term. Plug the appropriate value for n into the formula, then work out what that term is. For example, if we want to find the seventh term, we would plug in n=7.
A sequence can also be defined recursively: each term is based on what came before. The sequence is built on a recursion formula that shows how each term is built from preceding terms. To use a recursion formula, we need a "starting" place before we can make a sequence. This is called the initial term (or terms, if multiple are needed).
Given a recursion formula and initial term(s), it can be possible to find a formula for the n^th term. Similarly, it can be possible to transform an n^th term formula into a recursion formula and initial term(s). Still, there is no guarantee we can do this. Sometimes it will be easy, sometimes hard, and sometimes impossible.
Very, very often you will be given the first few terms of a sequence and told to either give more terms, or figure out a formula for the n^th term. To do this, you will have to find some pattern in the sequence, then exploit it.
When trying to recognize a pattern in a sequence, try to think in terms of how to get from one term to the next. Establish a hypothesis by looking at a₁→ a₂, then test it against a₂→ a₃, a₃→ a₄, and any other terms given. Once you've figured out the pattern, it's easy to find further terms in the sequence. Finding a formula for the n^th term can be tricky, though. Think carefully about how you can put the pattern in an equation, then make sure to check some terms after you create the formula.
When trying to find the pattern in a sequence, there are a variety of common pattern types that appear. Here are some important ones to keep in mind:
- Addition/Subtraction: add k every term.
- Mutliplication/Division: multiply by k every term.
- Squares (n²): 1, 4, 9, 16, 25, 36, 49, …
- Cubes (n³): 1, 8, 27, 64, 125, 216, …
- Factorials (n!): 1, 2, 6, 24, 120, 720, …
- Alternating Signs: Alternating signs can created by (−1)ⁿ⁺¹ or (−1)ⁿ.
If most of the terms in a sequence are presented in a certain format, like fractions, try to figure out a way to put all the terms in that format. It can be easier to see patterns if everything is in the same format. Furthermore, if the format can be clearly broken into multiple parts (in a fraction ^[¯]/^[¯], we can break it into numerator and denominator), it can help to figure out patterns for each part separately.
It can sometimes help to write the number of the term above or below each term (n=1, n=2, etc). This helps you keep track of numerical location, which often makes it easier to identify patterns.
In the end, there is no one way to identify all patterns. Try to take a broad view of the sequence and look for repetitions or similarities to other patterns you've seen. If you still can't figure it out, see if there's an alternative way to write the terms out. Persevere, and be creative.

Introduction to Sequences

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: Sequence 0:28

Infinite Sequence
Finite Sequence
Length

Formula for the nth Term 3:22
Defining a Sequence Recursively 5:54

Initial Term

Sequences and Patterns 10:40

First, Identify a Pattern
How to Get From One Term to the Next

Tips for Finding Patterns 19:52

More Tips for Finding Patterns
Even More Tips

Example 1 30:32
Example 2 34:54

Fibonacci Sequence

Example 3 38:40
Example 4 45:02
Example 5 49:26
Example 6 51:54

Math Analysis Online

Section 1: Introduction
	Introduction to Math Analysis	10:03
	Sets, Elements, & Numbers	45:11
	Variables, Equations, & Algebra	35:31
	Coordinate Systems	35:02
	Midpoints, Distance, the Pythagorean Theorem, & Slope	48:43
	Word Problems	56:31
Section 2: Functions
	Idea of a Function	39:54
	Graphs	58:26
	Properties of Functions	48:49
	Function Petting Zoo	29:20
	Transformation of Functions	48:35
	Composite Functions	33:24
	Piecewise Functions	51:42
	Inverse Functions	49:37
	Variation Direct and Inverse	28:49
Section 3: Polynomials
	Intro to Polynomials	38:41
	Roots (Zeros) of Polynomials	41:07
	Completing the Square and the Quadratic Formula	39:43
	Properties of Quadratic Functions	45:34
	Intermediate Value Theorem and Polynomial Division	46:08
	Complex Numbers	45:36
	Fundamental Theorem of Algebra	19:09
Section 4: Rational Functions
	Rational Functions and Vertical Asymptotes	33:22
	Horizontal Asymptotes	34:16
	Graphing Asymptotes in a Nutshell	49:07
	Partial Fractions	44:56
Section 5: Exponential & Logarithmic Functions
	Understanding Exponents	35:17
	Exponential Functions	47:04
	Introduction to Logarithms	40:31
	Properties of Logarithms	42:33
	Solving Exponential and Logarithmic Equations	34:10
	Application of Exponential and Logarithmic Functions	48:46
Section 6: Trigonometric Functions
	Angles	39:05
	Sine and Cosine Functions	43:16
	Sine and Cosine Values of Special Angles	33:05
	Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D	52:03
	Tangent and Cotangent Functions	36:04
	Secant and Cosecant Functions	27:18
	Inverse Trigonometric Functions	32:58
	Computations of Inverse Trigonometric Functions	31:08
Section 7: Trigonometric Identities
	Pythagorean Identity	19:11
	Identity Tan(squared)x+1=Sec(squared)x	23:16
	Addition and Subtraction Formulas	52:52
	Double Angle Formulas	29:05
	Half-Angle Formulas	43:55
Section 8: Applications of Trigonometry
	Trigonometry in Right Angles	25:43
	Law of Sines	56:40
	Law of Cosines	49:05
	Finding the Area of a Triangle	27:37
	Word Problems and Applications of Trigonometry	34:25
Section 9: Systems of Equations and Inequalities
	Systems of Linear Equations	55:40
	Systems of Linear Inequalities	1:00:13
	Nonlinear Systems	41:01
Section 10: Vectors and Matrices
	Vectors	1:09:31
	Dot Product & Cross Product	35:20
	Matrices	54:07
	Determinants & Inverses of Matrices	47:12
	Using Matrices to Solve Systems of Linear Equations	58:34
Section 11: Alternate Ways to Graph
	Parametric Equations	53:33
	Polar Coordinates	48:07
	Polar Equations & Functions	38:16
Section 12: Complex Numbers and Polar Coordinates
	Polar Form of Complex Numbers	40:43
	DeMoivre's Theorem	57:37
Section 13: Counting & Probability
	Counting	31:36
	Permutations & Combinations	44:03
	Probability	36:58
Section 14: Conic Sections
	Parabolas	41:27
	Circles	21:03
	Ellipses	46:51
	Hyperbolas	38:15
	Conic Sections	18:43
Section 15: Sequences, Series, & Induction
	Introduction to Sequences	57:45
	Introduction to Series	40:27
	Arithmetic Sequences & Series	31:36
	Geometric Sequences & Series	39:27
	Mathematical Induction	49:53
	The Binomial Theorem	1:13:13
Section 16: Preview of Calculus
	Idea of a Limit	40:22
	Formal Definition of a Limit	57:11
	Finding Limits	32:40
	Continuity & One-Sided Limits	32:43
	Limits at Infinity & Limits of Sequences	32:49
	Instantaneous Slope & Tangents (Derivatives)	51:13
	Area Under a Curve (Integrals)	45:26
Section 17: Appendix: Graphing Calculators
	Buying a Graphing Calculator	10:41
	Graphing Calculator Basics	10:51
	Graphing Functions, Window Settings, & Table of Values	10:38
	Finding Points of Interest	9:45
	Parametric & Polar Graphs	7:08

Transcription: Introduction to Sequences

Hi--welcome back to Educator.com.0000

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

For the most part, a sequence is simply an ordered list of numbers.0005

While the idea is simple, there are a huge variety of uses for sequences.0009

They come up in a wide variety of fields, and they are an extremely important tool in advanced mathematics.0013

In this lesson, we will learn what a sequence is, various ways to describe them, and how to find patterns that they may be based on.0018

Let's go: the definition of a sequence: a sequence, in math, means pretty much the same thing as it does in English.0027

It is an order of things; specifically, in this case, it is an ordered list of numbers.0034

We could write a sequence as a₁, a₂, a₃, a₄...a_n...0041

We call each of the entries in the sequence a term.0048

This would be a₁, the first term, because it is the first in the sequence.0050

a₂ is the second term, because it is the second in the sequence.0056

a₃ would be the third term, because it is the third in the sequence.0061

So, any symbol can be used to denote the sequence.0065

In this case, we are just the using the lowercase letter a, but we could use any letter, or any symbol, that we wanted.0069

a is a common, convenient one, though.0074

The subscript (that is the small number to the right: here, this little number 4, a₄) tells us which term it is in the sequence.0076

This 4 here tells us that it is the fourth term in the sequence.0087

Here are some examples: we could have 1, 2, 3...(and the ... just says "keep going in this manner; it continues on").0094

2, 9, 16...; √x, √2x, √3x...a sequence is just some ordered list of things--an ordered list of numbers, in this case.0103

So, even here, it ends up being an ordered list of numbers.0116

It is using a variable, but once we set x as some number, it is going to end up just being an ordered list of numbers there, as well.0119

All right, if a sequence goes on forever without stopping, it is called an infinite sequence.0127

Most of the sequences that we are going to work with are infinite sequences.0132

That is something where it is a₁, a₂, a₃, a₄...0135

and then there is nothing after those dots; it just says that it keeps going, and there is no stop to this thing.0139

On the other hand, we could have a finite sequence, if the sequence does stop.0145

In that case, we have a₁, a₂, a₃, a₄, and there is that ...0149

that says to continue in this manner; but then, we actually stop at a_k.0154

Notice how there is nothing after the a_k; it is just blank after that.0158

That says that we have reached the endpoint; there is not ... to tell us to keep going in this manner.0162

We get to a_k, and we just stop at a_k, because there is nothing after a_k.0167

So, it says that this is the end of our sequence.0171

We call the number of terms in a finite sequence its length.0174

In this case, in the length of the above sequence, we would have k, because we have a_k here and we have a₁ here.0179

So, that means we are counting our first term, our second term, our third term...all the way up until our k^th term.0185

1 up until k...if we count from 1 to k, whatever k is, that means we are going to have0190

a total of k things, so we have a length of k for that finite sequence.0195

We can often talk about some formula that allows us to find the n^th term, also called the general term.0201

If we know such a formula, we can easily find any term.0208

By plugging in different values for n, since we know what the n^th term is going to be,0211

well, if we say some value for n, we can find the term that is that value.0216

If we plug in n = 3, we can find the third locations.0221

So, as long as we know that a_n equals some stuff, some algebraic formulation,0224

then if we set n equal to 1, we would get the first term, a₁.0232

If we set n equal to 17, we get the 17^th term, a₁₇.0238

Notice how the n = 17 replaces where the n would have been; it is a subscript n;0245

but since we swapped it out for 17, we now have a, subscript 17.0251

So, because we have some algebraic formulation for the way a_n works,0256

for the way this n^th term works, the way this general term works,0260

we can plug in our value for n, use this algebraic formulation, and churn out some number to know what a for any term is going to be.0264

For example, if we know that a_n = 7n - 5, then we have the sequence a₁, a₂, a₃, a₄...0273

Well, notice here: in a₁, that means n = 1; so we swap out the n in 7n - 5 for 7(1) - 5.0281

7 - 5 gets us 2, so we now know that a₁ is equal to 2.0290

The same thing for a₂: we know that, at a₂, we have n = 2.0296

So, we swap out 7n - 5 to 7(2) - 5; 7(2) is 14; 14 - 5 is 9.0299

So now, we have that the second entry, the second term, in our sequence is 9.0307

a₃: we have n = 3, so we swap out; we have 7(3) - 5, 21 - 5, 16; so our third entry, a₃, is equal to 16.0312

At a₄, we have n = 4; 7(4) - 5...7 times 4 is 28; 28 - 5...so we have 23 for our fourth term, as well.0322

So, by knowing the general term, a_n = 7n - 5, we are able to find any term.0331

We can find any term if we know this general form a_n = some algebraic format, like 7n - 5.0340

We can also define a sequence based on terms that came previously.0351

We just figured out a way to just say that the absolute thing is going to be this; this will be this, based on this formula.0354

We have this definite, general term.0360

But we can also define it based on terms that came previously.0363

This is called defining a sequence recursively.0367

In this, the sequence is built on a recursion formula that shows how each term is based on preceding terms.0371

Recursive: we are looking backwards to something that came previously.0377

For example, if we have the recursion formula a_n = a_{n - 1} + 7, what is that saying?0381

It is that the n^th term--that is a_n, right here--is created by looking at the previous term.0387

Well, what would be one before n? n - 1; so a_{n - 1} is going to be the term just before the n^th term.0392

So, a_{n - 1}...and then adding 7 to it is that + 7 business right there.0401

a_n = a_{n - 1} + 7: some term is equal to the previous term, plus 7.0407

Since this is true for any n at all, the recursion formula tells us that every term is equal to the term before it, plus 7.0414

We didn't say n has to be some specific value; we haven't nailed down what the value of it is going to be.0422

So, since this is true for any n, the recursion formula tells us that every term (because it is true for any n,0427

so a_n = a_{n - 1} + 7 for any value of n) will be equal to the term before, plus 7.0433

However, there is one special term that doesn't have a term before it.0442

Our recursion formula was based on looking at the one behind you and adding 7.0446

But in this case, there is one number that isn't going to have anything behind it.0450

The person who is the first in line (not really a person--a number)--whatever term is first, our first term: there is nothing behind it.0454

There is nothing to look at behind that term.0461

So, if that is the case, we need something to start from.0464

A recursion formula on its own is not enough to obtain a sequence; we need some sort of starting place before we can make a sequence.0466

We need to know what that first term is, what that seed is that our recursion formula will grow off of.0473

This is called the initial term (or terms, if we need multiple of them).0479

So, using the initial term a₁ = 2, with the previous recursion formula a_n = a_{n - 1} + 7,0484

then our first term is going to be a₁, right here.0491

Well, we were just told that a₁ = 2; so that means that we have 2 here.0494

Then, from there on, we have a_n = a_{n - 1} + 7.0499

So, that says that to get a term, you take the previous term, and you add 7.0505

So, to get from 2 to the next term, we get + 7, so 2 + 7 gets us 9.0509

To get to the next one, we have + 7; so that gets us 16.0515

To get to the next term, we have + 7; that gets us 23.0518

Writing out exactly what happens: if we want to know what a₂ is going to be equal to0521

(a₂ is this one right here), then a₂ = a_{2 - 1}, or 1, + 7.0527

So, a₁ is equal to 2; we figured that out here; then + 7...so we get a₂ = 9, which is what we got right here.0535

And the same thing is going on for figuring out a₃: a₃ is going to be equal to a₂ + 7.0547

a₄ is going to be equal to a₃ + 7, because our recursion formula is telling us to go that way.0552

Given a recursion formula and initial terms, it can be possible to find a formula for the n^th term.0559

That absolute plug-in-a-number-for-our-n...using the general term, it just puts out what the value is.0564

There are sometimes ways to be able to do this; if you have a recursion formula and initial terms,0571

you can sometimes transform it into a formula for the n^th term.0578

Similarly, it can be possible to transform an n^th term formula into a recursion formula and initial terms.0582

So, if we know the general term, we can go to the recursion formula.0589

If we go to the recursion formula, we can go to the general form.0592

However, we can't always end up doing this.0596

There is no guarantee that we can do this; we very often can, especially at this level in math.0598

But sometimes it is not going to be so easy to do; sometimes it is going to be really hard.0603

Sometimes, it will be totally easy; but sometimes it is going to be hard, and sometimes it is going to be impossible.0607

It will depend on the specific sequence that we are working with.0612

Some sequences are really easy to talk about in a recursion formula, but basically impossible to talk about as a general term, an n^th term format.0615

Other ones are really easy to talk about in that n^th term format, that general term, but really hard,0623

practically impossible, to talk about in a recursion formula.0628

So, we won't necessarily be able to switch between the two, but we will often be able to switch between the two.0630

And if a problem ever asks us to switch between the two, it will certainly be possible.0635

Very, very often, you will be given the first few terms of a sequence and told to either give more terms or figure out a formula for the n^th term.0640

To do this, you will have to figure out some pattern in the sequence, and then exploit it.0647

You will have to look at the pattern, look at the sequence, and ask how these things are connected.0651

What is a pattern in here that I can use to create a formula?0656

However, before we learn how to do this, before we learn how to find patterns and sequences,0660

I want to point out that there is technically no guarantee that a sequence must have a pattern.0665

There is no guarantee that we will have patterns in our sequences.0671

For example, the below is a perfectly legitimate sequence with no pattern that we are going to be able to find in it.0676

47, then -3, then .0012, then π raised to the negative fifth power, then 17, then 1, then 1 again, then 800, and then a whole bunch of other numbers.0681

There is no pattern going on here; there is no rhyme; there is no reason.0693

There is nothing that we are going to be able to figure out to create some formula here.0696

So, there is no guarantee that a sequence has to have some pattern that we are going to be able to create a formula from.0700

Things can be really confusing with sequences, and we won't be able to figure out a way to generate a formula.0706

But good news: all of the sequences at this level in math will have patterns.0712

Technically, a sequence is not required to have a pattern; but at this level in math, all of the sequences that we see are definitely going to have patterns.0717

We will always be able to find patterns in the sequences that we are working with.0725

So, we don't have to worry about problems being unsolvable, because we can always rely on the fact0728

that there is going to be some pattern in there somewhere; there will always be a pattern that we can find, if we look hard enough.0733

If the problem is about finding patterns, there is definitely going to be a pattern for us to find.0740

We just have to look carefully and be really creative.0743

It won't necessarily be easy to find the pattern; but it will be in there somewhere.0746

It is not going to be something like this, where it is really, really hard for us to be able to see a pattern,0749

because there is simply no pattern; there are going to be cases...0755

in all of the things that we are looking at, it is always going to be the case that we are going to be able to find a pattern somehow.0758

We don't have to worry about the stuff where there is just no pattern whatsoever.0763

All of the problems that we will be asked to do, we will be able to do.0767

To find more terms in a sequence, or figure out a formula for the n^th term,0771

the first thing that we have to do is identify a pattern in the sequence.0774

If we want to find a formula, if we want to be able to talk about more terms,0777

the first thing that we have to figure out is what goes on in the sequence--how does the sequence work?0780

Consider these two sequences: 17, 12, 7, 2... and 2, 6, 18, 54...continuing on with both of them.0785

Let's look at the first one: 17, 12, 7, 2: what we can do is say, "Well, what is the connection here between 17 and 12,0794

and how is that related to 12, 7; and how is that related to 7 to 2?"0801

Well, looking at this for a little while, we probably realize that what we are doing,0804

to get from 17 to 12, is subtracting by 5; what we are doing to get from 12 to 7 is subtracting by 5;0808

what we are doing to get from 7 to 2 is subtracting by 5.0813

So, we know that this pattern of subtract by 5, subtract by 5, subtract by 5...it is going to continue on,0816

because it showed up everywhere in the sequence so far.0820

Similarly, over here, how do we get from 2 to 6, from 6 to 18, from 18 to 54?0823

Looking at it for a while, we probably realize that what we are doing is multiplying by 3.0828

2 times 3 gets us 6; 6 times 3 gets us 18; 18 times 3 gets us 54; so this pattern will continue on,0832

throughout the rest of the sequence, because it showed up in all of the sequence that we saw.0840

We have figured out what the patterns are.0843

We notice that the sequence on the left subtracts by 5 every term; the one on the right multiplies by 3.0846

At this point, it would be easy to find more terms.0850

If we wanted to find the next term in the sequence 17, 12, 7, 2, we would just subtract 5 again; 2 - 5 would get us -3.0853

And we would be able to keep going, if we wanted to.0862

Similarly, over here, if we wanted to figure out the next term in 2, 6, 18, 54, we would just have to multiply by 3 again.0864

So, what would come after that? 54 times 2 is 150 + 4(3) is 12, so 162.0872

So, we would get 162 as the next one, and we would be able to continue on in that manner, if we wanted to find any more terms in the sequence.0880

It is easy to find more terms, and it would also be easy to find a recursion formula.0887

The red one that we were doing, the minus 5 one, is just going to be a_{n - 1} - 5.0892

And the green one, the second one, with multiplication, times 3...that would be a_{n - 1} times 3.0899

So, it wouldn't be very hard to figure out recursion formulas, because they are really deeply connected to the pattern we saw.0904

However, if we want a formula for the n^th term, it is going to take a little more thought,0909

because we have to be able to figure out how this works for all of them.0913

It is not just describing the pattern of how we get from one to the next, or how we get from this one to the next one.0916

It is describing how we get to any of them, without being able to have any sort of reference points.0923

We can't just say - 5, - 5, - 5; we have to figure out a way of collecting all of the - 5s that happened, or all of the times 3s that happened.0928

So, we think about it for a while; and we will be able to figure this out.0934

We would be able to get a_n = 22 - 5n; we would be able to realize that,0937

since what we are doing is subtracting by 5 a bunch of times, it is going to be - 5 times n;0942

and then we need to figure out what number we are subtracting 5 from.0947

And we want to make sure to check the first few terms.0949

Always check the first few terms, once you think you have figured out a formula for the general term.0953

Once you think you know what the n^th term is going to be, make sure you check that what you figured out is right,0957

because it is easy to make a mistake and be off by a little bit, to be off by one number in the sequence.0963

So, just make sure that you try and check.0968

For example, if we have figured out what a₁ is going to be, well, that would be 22 - 5(1).0970

22 - 5(1) is 22 - 5, or 17; that checks out with what we have here.0975

If we did a₂, then that would be equal to 22 - 5(2); 22 - 5(2) is 22 - 10; 22 - 10 is 12; that checks out with what we have here.0980

So, it looks like it ends up working out.0990

Similarly, for the green one, our multiplication one, a_n = 2(3)^{n - 1}:0992

well, let's think about this--does this end up working out?0999

We realize that it has to be something about 3 to the some exponent, because every step, we are multiplying by some 3.1001

So, if we stack all of those 3s together, it is going to be 3 to the some sort of exponent.1009

The question is what that exponent should be: it is 3^{n - 1}, because this very first one hasn't been affected by the 3 at all.1014

Let's check and make sure that that is the case.1021

If we plug in a₁ = 2(3)^{1 - 1}, then that would be 2(3)⁰; 3⁰ is just 1;1023

any number raised to the 0 is just 1; so we get 2; 2 checks out.1032

If we did a₂, then we would have a₂ is 2(3)^{2 - 1}, or to the 1; so it equals 6; that checks out.1036

We can see that this is going to continue to work; so our general formula works out.1045

But make sure you check it and think about it; it can be a little bit difficult,1049

but as long as you check it, you can be sure that what you have is going to work.1052

When trying to recognize a pattern in a sequence, try to think in terms of how to get from one term to the next.1056

How do you get from this first term to the next term?1062

How do you get from the first term to the second term?1066

Establish a hypothesis, some guess at what you think the pattern is, by looking a₁ to a₂.1068

You want to start with a hypothesis--you think that this is probably how the pattern works.1074

You look at a₁ to a₂, and then you want to test that hypothesis against the following ones:1078

a₂ to a₃, a₃ to a₄, and any other terms that are given.1083

You come up with thinking that the pattern here, the way we get from one stepping-stone1087

to the next stepping-stone, is that we do some operation.1091

Add some number, multiply by some number...it is doing some sort of thing; how is it working out?1094

Figure out what you think is going on for the way that the pattern works.1099

And then, test and make sure that that works on the way that we get to the next one and the way that we get to the next one,1101

or that it just works for the n^th number location.1106

Once you have some hypothesis, test it against all of the other ones.1109

If it works, great; you just figured out the pattern--now you are ready to figure out some way1112

to formulate that general form, that general term, the n^th term.1116

If it doesn't work, go back to the beginning: figure out a new hypothesis, and then try again.1120

It is all about figuring out something that you think might work, and then testing it against the information you have.1128

Keep testing until you get something that actually ends up working out,1134

at which point, you have found the pattern; now you are ready to start working your way towards a general term formula.1137

Once you figure out the pattern, it is easy to find further terms in the sequence.1143

That is the easiest part; you just continue with that pattern to generate any more terms that they tell you to generate.1146

Finding a formula for the n^th term: that could be tricky--a formula for the n^th term can be a little bit tricky.1153

What you want to do is think carefully about how you can put the pattern into an equation1159

and make sure to check some terms after you create the formula.1164

That checking is really, really important; think carefully about how you can put that pattern into an equation,1166

and then check after you have come up with some sort of equation that you think will probably work.1173

It is really important to check, because it is really easy to make mistakes, especially the first couple of times you are doing it.1177

We will also see this a whole bunch of times in the examples.1182

We are going to work with a whole bunch of examples here.1184

So, that will really help to cement our understanding of how to do this.1186

We have lots of examples to make this clear.1188

All right, how do we find patterns?--that can sometimes be a tricky thing.1191

When trying to find the pattern in a sequence, the two most common pattern types that appear are addition/subtraction,1195

where we just add some k every term (k could be a positive number; k could be a negative number;1202

that allows us to add or subtract, depending on what the k is; but we are just adding the same k, adding some constant number,1207

every time we do a step); and then the other one is multiplication/division, where we multiply by some k every term.1213

If it is multiplication, it is just some constant number k; and we can also effectively divide.1221

If it is a fraction as our k, we are effectively doing division there, as well.1225

So, we are just multiplying by some constant number every term.1229

Every step is either going to be addition or multiplication.1231

This is the largest portion of the patterns; many, many of the patterns that we are going to work with,1235

at this level, and really at any level, are going to be connected to addition/subtraction or multiplication and division.1240

So, these are the first two that you want to keep in your head.1245

A large number of patterns can be figured out just by keeping these two types in mind.1248

Always think first in terms of addition/subtraction, multiplication/division.1254

Check to see if you see those first.1257

However, those are not the only kinds of patterns that you end up seeing.1259

These two pattern types are not enough to figure out the pattern for all sequences.1262

In that case, it can help to keep various other patterns in mind.1267

A good one to keep in mind is the squares: n²: 1², 2², 3², 4², 5², 6², 7²...1270

But often, you won't see them as a number squared, because then it would be really easy to recognize the pattern.1278

Instead, you see it as 1, 4, 9, 16, 25, 36, 49...and continuing on, if they still have even more terms.1283

So, it just helps to keep that structure of numbers in the back of your head.1291

It is really useful to be able to remember what all of those perfect squares are, what all those squares are1295

that you are used to working with, that you have seen in previous algebra classes.1299

Just keep them in your mind, and look for things that look somewhat like that, or along those lines.1303

Another one that often shows up is cubes; this one is less often than squares, but it does show up.1309

n³ is 1³, 2³, 3³, 4³, 5³, 6³, etc.1314

Once again, you very often won't see it as a number cubed, because then it would be easy to see that the pattern is cubed.1319

And it is supposed to be a little more challenging than that.1324

So instead, it is normally 1, and then 8, and then 27, and 64, then 125, and 216, and so on, and so on.1326

So, you are probably less used to using cubes; it is really important to just pay attention to at least these first four.1334

Memorize 1, 8, 27, and maybe 64, maybe 125; keep at least those first few terms in mind,1340

because you want to be prepared to say, "Well, I am not used to seeing this pattern; I am not used to seeing something like this;1348

but oh, maybe it is cubes, because I see that the first three are like that," and then you can check the other ones1353

by hand, and make sure that that does work out.1358

You don't have to memorize the whole thing, but you do have to be ready,1360

when you see the pattern, to be able to think that maybe that is going to be something.1363

You have to be prepared to recognize it; you don't have to know the whole pattern, but you have to be prepared to recognize it.1366

Finally, factorials: n!: 1!, 2!, 3!, 4!, 5!, 6!, etc., etc.: once again, you aren't normally going to see that as factorials written out.1371

You will instead often see it as something like 1, then 2, then 6, 24, 120, 720...a really good one to memorize is 1, 2, 6, 24, 120.1383

That is 1!, then 2!, 3!, 4!, 5!; so if you can keep that pattern in your head--just keep that one in the back of your head--1393

then that will end up showing up a lot, and if you are not prepared to recognize that pattern,1402

that kind of problem would be really, really hard, because you won't be able to see that pattern when it shows up.1405

And in case you forgot how a factorial works, factorials multiply the number that is factorial by every integer below the number.1410

So, for example, 5! would be 5 times 4 times 3 times 2 times 1, which works out to 120.1419

And we define...we just specifically define 0! = 1 for ease.1426

It helps things out; it helps a lot of other things in math work out.1431

So, 0! = 1; we just set it that way and try not to get worried about how that doesn't make sense, compared to how the other one works out.1434

It actually sort of makes sense; but it is better to just remember and memorize 0! = 1.1443

That one will come up occasionally.1449

All right, sometimes the sign will change with each term; it will flip between positive and negative, positive/negative, positive/negative.1451

You will see a positive on one, and the next one will be negative, and the next one will be positive, and the next will be negative.1458

And the next one will be positive, and then negative, and so on, and so forth.1463

And you will see this flipping pattern; if that is the case, it might be one of these two following--one of these two methods, these two types of patterns.1466

-1 raised to the n + 1: notice that, if we have n at 1, then we will have -1 squared, which is going to come out to be a positive 1.1474

And then, the next one would be -1 to the 2 + 1; n = 2, so -1 to the 3 is going to end up being -1.1482

And then, we get +1 and -1 and +1 and -1, because -1 raised to some integer is just going to multiply by -1 that many times.1489

So, it will flip between positive and negative, positive/negative, as our n's step up, one at a time.1496

Similarly, -1 raised to the n gives us the exact same pattern, flipping and turning.1501

It just starts, instead of starting at +1, at -1; and then, it will be +1 and -1, then +1, then -1, then +1.1506

So, if you see a flipping sign pattern, and you don't see something else to be able to cause that to happen,1513

in the sequence that you are working with, whatever pattern you are working with,1520

these two right here are a really good thing to keep in the back of your mind for a way to just cause a flipping sign to appear.1523

If most of the terms in the sequence are presented in a certain format--for example,1532

they are all in fractions--try to figure out a way to put all of the terms in that format.1535

So, if you see a certain format in your sequence, put all of the terms into that format.1539

If most of your terms are in fractions, make all of your terms in fractions.1545

It can be a lot easier to see patterns if everything is in the same format.1549

So, if you get all of your sequence terms in the same format, it normally causes patterns to appear more readily.1555

Furthermore, if the format can clearly be broken into multiple parts--for example,1562

if we have a fraction, _/_, we can break it into the numerator (the part on top) and the denominator (the part on the bottom)--1565

in that case, we can clearly talk about how all of our numerators1575

behave by some pattern, and all of our denominators behave by some pattern.1577

If you can break the thing into multiple parts, if a format can be broken into multiple parts--1581

for example, a fraction is a numerator over a denominator, every single time--it can help to figure out patterns for each part separately.1587

Figure out the numerator pattern on its own; figure out the denominator pattern on its own.1596

Sometimes, that will really help clarify things; you don't have to worry about trying to figure out the whole fraction.1599

You can instead break it apart piecemeal and then just put them back together once you recognize each pattern on its own.1603

It is important to note that all of these different pattern types that we have talked about so far-- they don't necessarily occur in isolation.1609

While that will sometimes happen (you will sometimes just have addition; you will sometimes just have multiplication;1619

you will sometimes just have switching signs), we are often going to end up working with sequences1624

that use multiple pattern types at once, so it will be up to us to figure out that it is using this pattern and this pattern and this pattern,1629

and then figure out some way to merge all of those three patterns together, once we are trying to describe the whole thing.1637

They might even end up involving patterns that you haven't seen before.1642

They might do something that you don't immediately recognize; we have to come up with a new way to describe it.1645

So, keep a lookout for something that looks weird, that is totally new to the way that you are doing things.1650

And you might end up having to learn a new method of describing things, which will just take longer to think through.1654

It can sometimes help to write the number of the term above or below each term,1660

to write n = 1 and then n = 2, and so on and so forth.1664

By writing the number of the term above or below, you are able to keep track of numerical location.1669

By being able to see what number we are at--are we at the first term? Are we at the fifth term?--1676

by being able to have this clear reference point of "this is term #1; this is term #5,"1680

you will be able to see how the number of the term relates1685

to the values inside of the actual term inside of the sequence--the values of that term.1688

The location of the term will normally be related to the values inside of the term.1694

That will often make it easier to identify patterns, by being able to see that this has number location 5;1698

and because of that, we see that the general form is working in this certain way.1704

So, writing the numbers above or below can really help you see that sort of thing.1708

In the end, there is no one way to identify all patterns.1712

I want you to try to take a broad view of the sequence and look for repetitions or similarities to other patterns that you have seen.1716

Don't try to focus on it always being the same thing, because it won't.1723

Each sequence is probably going to have its own special pattern.1726

You will start getting used to certain ways that patterns interact, or certain types of patterns.1729

And it will be easier to recognize them on future ones.1733

And a lot of this stuff will show up in a whole bunch of different places, like standardized tests,1736

later in different math classes, in science classes...so what you are learning in this class will definitely be applicable for a long time to come.1741

But it is not necessarily always going to be the same thing.1748

So, just take a broad view of what is going on.1751

Don't think that it is definitely going to work in one way, because you don't really know until you have figured out what the pattern is.1754

So, look at the thing carefully; think, "How does one term interact with the next term?"1760

How are these two terms related to each other?1766

Is there some sort of general pattern that is occurring as I look through all of the numbers, all of my terms, at once?1768

Try to think in really large, broad strokes before you try to come up with a very specific pattern showing up.1774

And if you still can't figure it out, if you are looking at it for a long time and you can't figure it out,1780

see if there is an alternative way to write the terms out.1784

That was what I was talking about with...if most of them are in fractions, put all of them in fractions.1787

Figure out if there is some way to write the terms in something, so that they all have this new alternative way,1791

because maybe that will help you see what is going on better.1796

And just in general, persevere; be creative.1799

Figuring out patterns is not something that is just a step-by-step method, and you have gotten to the answer.1802

It is something where you have to look at it and sort of think for a while.1807

Just be clever; have a little bit of luck; and just work at it.1809

If you really can't figure it out for a long time, go to the next problem and come back to that problem later.1812

Sometimes just a little bit of time will cause it to "bounce around" in your head,1817

and you will be able to see the pattern easily, when earlier it was really, really difficult.1821

So, just stick with it, and as you work with it more and more, it will make more and more sense.1825

All right, we are ready for some examples.1829

Given the n^th term, write the first four terms of each sequence.1831

Assume that each sequence starts at n = 1.1836

So, that n^th term, that general term, is a_n = stuff.1839

In this case, for our first one, we have a_n = 3n - 2.1843

Our first term, the a₁, is going to be when n = 1.1849

We plug in a₁ = 3(1) - 2.1855

Similarly, a₂ is when n = 2, so we have a₂ = 3(2) - 2.1860

a₃ = 3(3) - 2; a₄ = (I'll write that a little below; there is not quite enough room) 3(4) - 2.1867

We work this out: a₁ = 3(1) - 2, 3 - 2, so we get a₁ = 1.1880

a₂: 3(2) is 6; 6 - 2 is 4; a₃ = 3(3), 9, minus 2; 9 - 2 is 7.1886

a₄ = 3(4) is 12, minus 2 is 10.1896

So, we have the first four terms here: a₁ = 1, a₂ = 4, a₃ = 7, a₄ = 10.1902

Also, I want to point out; notice how a₁, to get to a₂...we added 3.1909

To get to a₃, we added 3; to get to a₄, we added 3.1913

Each term here has this + 3 step, which we are going to end up seeing from this 3 times n,1916

because the 3 times n...the n is what term location we are at, so as we go up more term locations,1922

we are going to end up seeing more times that we have ended up adding on this number 3 to the thing.1928

All right, the next one: b_n = -1ⁿ/(n + 3); our first one, a₁, is going to be (-1)¹/(1 + 3).1935

We swap out all of the n's that occur for whatever we have here, a₁.1947

Next, a₂ = -1 squared, -1 to the 2, divided by 2, plus 3.1952

a₃ is equal to -1 to the 3, over 3 + 3.1959

a₄ is going to be equal to -1 to the 4, over 4 + 3.1966

We can work this out here: we have a₁ = (-1)¹, which is still just -1; 1 + 3 is 4, so we have -1/4.1973

a₂: (-1)², -1 times -1...that cancels to just positive, so we just have a positive 1, divided by 2 + 3, 5.1984

a₃ = (-1)³; -1 to an odd exponent is going to end up leaving a negative after.1992

So, we have -1 over 3 + 3, 6.1998

a₄ = (-1)⁴, to an even exponent; it is going to cancel out; we are going to have a positive.2002

So, we have 1/(4 + 3) is 7: so a₁ = -1/4; a₂ = +1/5; a₃ = -1/6; a₄ = 1/7.2008

We found the first four terms.2020

Finally, c_n = 47...oh, oops; that whole time shouldn't have been a_n, because it was b_n.2022

So, it actually should have been not a₂, not a₁...any of these...2030

It should have been b₁, b₂, b₃, b₄, because it has a different name than the sequence at the top.2036

That is why we are using a different letter--because it is a different sequence for this problem.2043

b₁, b₂, b₃, b₄...it is easy to end up forgetting that we are changing symbols sometimes.2048

So, pay attention to the symbol of the sequence that you are working with.2055

All right, the last one: c_n = 47; the thing to notice here is...does this side, the right side, end up involving n at all?2058

It doesn't; as the n changes, the right side doesn't notice the n change.2065

So, c₁ is going to be equal to 47; c₂ is going to be equal to 47;2069

c₃ is going to be equal to 47; c₄ is going to be equal to 47.2074

So, whatever value we end up using for n, it is always going to end up coming out to 47.2080

So, the first term is 47; the second term is 47; the third term is 47; the fourth term is 47.2084

We always end up getting 47, because it is just a constant sequence.2089

All right, the second example: the Fibonacci sequence is a well-known, recursively-defined sequence.2094

It is given by the recursion formula and the initial terms below; write out the first 12 terms.2100

Its recursion formula is a_n = a_{n - 1} + a_{n - 2}, and a₁ = a₂, which equals 1.2105

Right away, we know that our first two terms are 1 (a₁ = 1), and then a₂ also equals 1; so 1, 1.2114

If we want to figure out the next term, we have a_n = a_{n - 1} + a_{n - 2}.2125

So, if we want to figure out what a₃ is going to be, then that is going to be equal to a_{3 - 1},2130

so a₂, plus a_{3 - 2}, a₁.2135

a₃...we don't know what a₃ is, but we do know what a₁ and a₂ are.2140

They are both 1; so we have 1 + 1; a₃ = 2.2144

So, our next term is 2; what comes after that?2150

If we want to figure out a₄, that would be a_{4 - 1}, a₃, plus a_{4 - 2}, a₂.2155

What is a₃? We just figured out a₃ = 2, so we have 2 +...a₂, once again, is 1; that equals 3.2164

a₄ = 3, so the next thing is going to be a 3.2175

Let's do one more of these, and then we will see what the general pattern here, going on, is.2181

a₅ is equal to...not the general term, but how this pattern is working, on the whole...2186

a₅ is equal to a₄, the previous term, plus the term previous to that one, a₃.2191

So, a₅ = a₄ + a₃: we just figured out that a₄ = 3 and a₃ = 2.2199

So, we have a₅ = 5; there is our next term, 5.2206

What comes after that--how do we get this?2213

Well, notice: a₅ = a₄ + a₃; so a₅ is equal to the previous term, plus the term previous to that.2215

a₄ equaled a₃ + a₂; a₄ is equal to the previous term, plus the term previous to that.2221

a₃ = a₂ + a₁, so the previous term, plus the term previous to that.2227

What we are doing to make the next term: it is looking at the previous term and the previous previous term.2232

To make whatever comes after the 5, it is going to be: add 3 and 5 together, and that will make the next thing.2237

So, 3 + 5 gets us 8; then, the next one is going to, once again, be: take the 5 and the 8; add them together.2243

5 + 8 gets us 13; we see that this relationship here is that any term is equal to the previous term and the previous previous term, added together.2251

So, that is why we needed a₁ and a₂, because we needed 2 terms,2263

so that we could talk in terms of the previous previous term.2267

We needed that larger start; we needed two initial terms before we would be able to get started.2269

So, at this point, we just add things together: 8 + 13 is 21; 21 + 13 is 34; 21 + 34 is 55; 34 + 55 is 89; 55 + 89 is 144.2274

And it will continue on in this manner.2294

And there we are; there is the Fibonacci sequence; there are the first 12 terms of the Fibonacci sequence.2296

Cool; the Fibonacci sequence has some other interesting properties.2304

You might end up studying it more in the class that you are currently in, or in a future class.2306

It is a pretty cool thing; but it is enough for us now just to understand how the thing works out.2310

All right, the third example: Find the n^th term for each sequence below, and assume that the sequence starts at n = 1.2315

The first thing that we always have to do, if we are looking at a sequence, and we want to figure out what the n^th term,2322

the a_n, the general term, is (it equals some formulaic algebra expression), then it is going to be...2326

we have to figure out the pattern, and then we use that pattern to come up with an equation.2335

So, what is the pattern in this first sequence? -3, 1, 5, 9, 13.2338

Well, notice: how do we get from -3 to 1? We can just add 4.2344

How do we get from 1 to 5? We add 4 again--it looks like our pattern probably is going to work out.2348

5 to 9--we add 4; 9 to 13--we add 4; 13 to whatever is next--it is probably going to be add 4, add 4, add 4.2353

Since the pattern worked for everything that we have seen so far in the sequence,2361

we can assume that the pattern is definitely just "add 4."2365

So, if that is going to be the case, we know that it has to be something of the form a_n = 4n +...we don't know yet.2369

We don't know what it is going to be, so we will just leave it as a question mark.2377

We could also use a variable like normal, but in this case, the variable I have decided to use is ?.2379

a_n = 4n...that represents this step, step, step, step: +4, +4, +4, +4.2384

Every step brings +4; every term you move on to brings this adding by 4.2391

And so, 4 times n will allow us to represent how many steps we have taken, multiplied by 4; and that brings that many 4s to the table.2396

So now, we just need to figure out what the question mark is.2403

Well, notice: we know that a₁ = -3; so that means we can have a₁ = 4(1) + ?.2405

We know that a₁ equals -3; so -3 = 4 (4 times 1 is just 4) + ?; so we have -7 (solving for question mark) = ?.2414

So, at this point, plugging that in, we have that a_n, the general term, is equal to 4n - 7.2426

It is always a good idea to check this sort of thing out; that probably will end up working out, but let's make sure.2436

We already checked a₁; that is how we figured it out.2440

Let's check and make sure that a₂ ends up working out.2442

a₂ = 4(2) - 7, 8 - 7, which equals 1; and that checks out.2445

Next, a₃ = 4(3) - 7 = 12 - 7 = 5; and that checks out.2452

And we can see, going along, this method--that this is going to end up working--2460

because we have this 4n here, so every time we go forward another term, we are going to end up adding another 4,2464

which also represents our pattern of + 4 each time; so it makes sense--we have our answer.2470

The general term for that sequence is a_n = 4n - 7.2476

All right, the next one: 2, 5, 10, 17, 26.2480

The first thing we want to do is figure out what the pattern is.2485

If it is addition, then we would have something like + 3.2488

All right, to get from 2 to 5, it is + 3; to get from 5 to 10, it is + 5; to get from 10 to 17, it is + 7...that is not going to end up working out.2492

It can't be addition as our pattern; we see that pretty quickly.2502

So, let's try another really common one, which is multiplying.2506

Well, to get from 2 to 5, you have to multiply by 5/2; to get from 5 to 10, we multiply by...that is not going to work.2508

We see very quickly that multiplication is just not friendly here; it is not going to work out in a very good way.2517

So, multiplication is out; at that point, we see that addition fails; multiplication fails.2522

So now, this is really where we get creative, and we start thinking.2531

What is going to be able to get us the answer here?2536

What is going on here--what is the pattern 2, 5, 10, 17, 26...?2539

And this is the part where we lean back, and we just think for a while.2545

We think, "What does this look like? What have I seen that looks even vaguely similar to the way this grows...2549

the fifth term is 26...how does this work?2555

We might try coming up with some pattern the first time that doesn't end up working.2557

That is OK; the important thing is just to keep looking and keep going, pondering and thinking: how is this related to something else?2560

Remember: we talked about some of the other patterns that are likely to show up--squares, cubes, factorials...2566

Those are good ones to start trying out if you can't figure it out yet.2571

So, let's look at squares: what are the squares?2574

Well, the squares would end up going...if we had simply n², then that would end up giving is the sequence:2576

1 (1² is 1), then 2² is 4, then 3² is 9; 4² is 16; 5² is 25...2582

How does 1, 4, 9, 16, 25 relate to 2, 5, 10, 17, 26?2594

Oh, they are very similar; we are just adding 1.2601

If we add 1, +1 would get us 2; +1 would get us 5; +1 would get us 10; +1 would get us 17; +1 would get us 26.2605

We have figured out what the general term here is--what the n^th term is.2617

It is the formula a_n = n² (because it is the number squared), but then we also have to add 1: a_n = n² + 1.2621

That seems to give us the formula for our general term for this sequence.2632

Let's check and make sure that that is, indeed, the case.2637

We do a quick check; if we plug in a₁ = 1² + 1, then we get 1 + 1, which equals 2.2639

That checks out; if we plug in for our second term, a₂, then we would have 2² + 1.2647

2² is 4; 4 + 1 is 5; that checks out.2654

Our next one: a₃ = 3² + 1; 9 + 1 = 10; that checks out; great.2658

At this point, it seems that our n² + 1 ends up working out; it ends up following this sort of like the squares method,2666

but a little bit more, just adding one each time; we see that the pattern that we figure out ends up being the same.2673

Notice: this pattern isn't really so much a pattern about a recursive thing going on.2679

It is not really so much about adding the same number each time, multiplying by some number each time...2683

We could figure it out as a recursion formula, but it is easier to think of it just in terms of this absolute:2690

here is the general term; here is how any given location ends up working.2695

It is the number of the location, squared, plus 1.2698

All right, the fourth example: Given the recursion relationship below, write the first five terms; then give the n^th term.2702

We have a_n = -3 times a_{n - 1}; that is to say, some term2708

is equal to -3 times the previous term; a_n is some term; a_{n - 1} is the term 1 back, going backwards by 1.2713

So, it is -3 times the previous term; and we have to have some starting place to begin with;2722

otherwise we won't be able to always look at previous terms.2727

So, we start at 2; then, the next one...if we want to talk about a₂, the second term,2730

it would be -3 times a₁, so a₂ is equal to -3 times...a₁ is just 2, so times 2; that equals -6.2736

So, a₂ = -6; notice: all that we really did there was just multiply by -3.2745

So, we can probably just end up doing this in our head.2751

2, then -6; what would come after that?2754

2, then, -6; then we would multiply by -3 again, -3 times the previous term, to get our next term.2758

-6 times -3 gets us +18; to get the next term, 18 times -3 is going to end up being -54, because it is -3.2765

The next term is going to end up being...times another -3...positive 162.2777

And it will continue on in this matter; so we have now figured out the first 5 terms.2782

Great; but we also have to give the n^th term, so how can we figure out what the n^th term is?2787

Let's switch colors for this; the n^th term...notice: 2, -6, 18, -54, 162...it is kind of hard to see a pattern there really obviously.2792

We see that every time, it is multiplying by -3, because we were told very explicitly that the recursion formula says to multiply by -3.2801

So maybe we could make that -3 show up so that we could see that a little more easily.2808

If we do that, we could write its 2 first; we have 2 show up at the beginning.2812

It doesn't have any -3s multiplying by it yet.2818

But next is going to be (-3)¹ times...well, we wouldn't be able to figure out immediately that it is going to be to the 1.2820

So, -3 times 2; then the next one would be -3 times it again, so now we have (-3)² times 2.2827

And the next one would be (-3)³ times 2, and the next one would be (-3)⁴ times 2.2837

And it would just continue on in this manner.2843

So remember: we want to get the whole thing to look like a similar format.2846

Everything has these -3's on it, raised to some exponent, for the most part.2849

Most of them have the exponents, and all of them end up having the -3 business here,2854

with the exception of this one, which has neither exponents nor -3.2861

So, we need to get them to all have this similar format.2864

We have -3 to the what?--what number can we multiply by?--what can we always multiply by?2867

We can always multiply by -3 to the 0; it is (-3)⁰ times 2, because that is just 1.2872

Over here, we have -3 to the 1, times 2; -3 to the 2, times 2; -3 to the 3, times 2; -3 to the 4, times 2; and so on.2879

So, if that is the case, we can match this up to n =...1 is here; n = 2 is here; n = 3 is here; n = 4 is here; n = 5 is here...2893

What changes each time? The only thing that ends up changing is 0, 1, 2, 3, 4...everything else is the same.2903

n = 1 gets us 0; n = 2 gets us 1; so what are we doing to the n? We are subtracting 1 each time.2910

So, we see that the general term can be given as a_n = -3 raised to the n - 1 times 2.2916

And there we go; if we want to check that, let's just check for the third term.2928

Just randomly, to make sure that everything ends up working out: the third term, a₃, is equal to -3, raised to the 3 - 1 times 2.2934

That is -3...3 - 1 is squared, times 2; -3 squared equals 9, times 2...9 times 2 is 18.2944

And that checks out with what we already figured out is the third term.2956

Great; so we figured out our general term, our n^th term formula; it works out perfectly.2959

The fifth example: Find the n^th term for the sequence below; assume that the sequence starts at n = 1.2965

The first thing: most of it ends up being in this fraction, fraction, fraction, fraction...not a fraction.2971

We want to get everything in terms of these fractions, so let's get everything into fraction format.2979

1 + 2 over 1 is how we will replace that; and then, 2 + 3 over 2, and so on and so on.2984

The rest of them are now in fractions.2991

We can write this as...here is our n = 1; here is our n = 2; n = 3; n = 4; n = 5; n = 6.2994

Notice: everything ends up changing; all of the numbers in each of these terms end up being different from the previous and the next term.3003

But we end up seeing some connections here: 1 matches to the 1's here; 2 matches to the 2's here; 3 matches to the 3's here;3012

4 matches to the 4's here; 5 matches to the 5's here; 6 matches to the 6's here.3022

And we see that this other number is just + 1 each time.3028

1 + 1 gets us 2; 2 + 1 gets us 3; 3 + 1 is 4; 4 + 1 is 5; 5 + 1 is 6; 6 + 1 is 7.3032

So now, we have an easy way to figure out what the n^th term is.3040

The n^th term is going to be equal to a_n =...well, it seems to be...3043

this one is just our value of n; plus some fraction...the bottom is also the value of n, and the top is n + 1.3049

And there we go: do a quick check, because it is always a good idea to do a check.3060

Let's check it out: a₁ would be equal to 1 + 2, over 1; so we get 1 + 2...that checks out with what we already had as the first term.3065

If we wanted to try another one, like a₂ = 1 + 2 + 1, over 1, which would be 1 + 3, over 1...3078

oops, sorry: not over 1; not over 1; it is divided by n as well; sorry about that mistake; divided by 2; 1 + 3 over 2.3087

Oh, I did it on both of them; it is important to end up using your formula in the check.3096

It is also an n here; 2 + 2 + 1 over 2; 2 + 3 over 2; and that does check out with what we had here.3100

Great; the last example: Find the n^th term for the sequence below; assume the sequence starts at n = 1.3108

We see, right away, that this thing kind of changes its format a fair bit in these two.3115

It is totally different in these two.3122

Before we even really start looking for patterns, we want to get everything into the same format.3124

That will just make it easier to see patterns.3127

So, how can we get them into the same format?3130

We first certainly need to have them as fractions.3132

So, as fractions, we have 1; we can always divide by 1, so 1/1; then 3/1, and then 3²/2, 3³/6, 3⁴/24, 3⁵/120.3135

The first thing that you are probably noticing is that we have 3⁵, 3⁴, 3³, 3², 3...3151

Well, we could write this as 3¹; how can we write 1 out?3158

Well, remember: 3 to the 0 equals 1; any number raised to the 0 comes out to be 1.3161

So, we could rewrite the top as 3⁰/1; 3¹/1; 3²/2; 3³/6; 3⁴/24; 3⁵/120, and so on.3166

At the top part, we now see pretty clearly that there is a pattern.3192

It is that the number increases by 1 each time from the exponent on the 3; it starts at 0, so that would be 3^{n - 1}.3194

But what about this bottom part, 120, 24, 6, 2, and then we have 1 and 1 here...?3201

1, 1, 2, 6, 24, 120...well, if we worked backwards, we might recognize factorials.3208

Remember: we talked about factorials earlier in the lesson; that looks like factorials.3215

So, 5 times 4 times 3 times 2 times 1 is 120; 4 times 3 times 2 times 1 is 24; so we have 5! on the far right...3220

We can write this as 3⁵/5!; we will keep going, so let's work backwards...3⁴/4!;3228

3³/3!; 3²/2!; 3¹/1!...that would just get us 1.3240

And here is the most confusing part of all: 3⁰ over...well, what can we do that will end up having factorials involved, and still get us 1?3254

Now we have to go back and remember: there is a very specific thing about factorials.3261

The way that factorials work is that 0! is just defined to equal 1.3265

So, that means we could also write this as 0!; so we have maintained a pattern.3270

We are going to end up seeing patterns, because all of these problems are going to be based on patterns.3275

So, we know that it is a factorial pattern; it is no surprise that it is going to keep going.3279

We have on the bottom 0!, 1!, 2!, 3!, 4!, 5!...on the top: 3⁰, 3¹, 3², 3³, 3⁴, 3⁵.3282

Let's compare that to the numbers n = 1, n = 2, n = 3, n = 4, n = 5, n = 6, our first location, our second location, third location, etc.3291

With that in mind, we see that the top exponent is always equal to the value of the location.3302

At n = 5, we get a 4 exponent, so it is always minus 1.3311

Similarly, the 0!, 1!, 2!, 3!, 4!, 5!...it is always the number of the location, minus 1, to get to the number in the factorial.3315

In the third location, at n = 3, it is a 2! (3 minus 1); in the sixth location, it is a 5!, 6 minus 1.3327

So, we see that they are both based off of this n - 1 business; so that means we can finally set our a_n equal to...3335

it is 3^{n - 1}, over (n - 1)!; and there is our answer.3343

There is the general term; and it is always, always a good idea to check your work with this sort of thing;3354

So, let's just do a quick check to make sure that this ends up coming out.3359

At a₁, we would have 3^{1 - 1}/(1 - 1)!; so that is 3⁰/0!, so we get 1/1,3365

which equals 1, which checks out with what we initially had.3378

Let's try jumping forward to a slightly larger number, so we can check against something else.3382

At a₄, we would be at 3^{4 - 1}/(4 - 1)!; so we have 3³/3!.3386

Everything is in this 3³, so we don't have to simplify that to 27; we can just leave it as it is.3400

3³/3!...that comes out to be 3 times 2 times 1, or 6; so 3³/6 is what we have for our fourth location.3406

1, 2, 3, 4^th location: 3³/6; that ends up checking out.3414

We end up seeing that our general term makes sense.3420

All right, we have a really good understanding of how sequences work, and how we can get general term,3423

n^th term, formulas from looking at them for a while.3427

Recognizing patterns is a really useful skill; it will show up in a whole bunch of different things in math.3430

Even if you end up thinking that this is a little bit difficult now, trust me: it is going to end up paying dividends later on.3435

You are going to end up using this stuff a lot, finding patterns in a variety of stuff,3440

whether it is in science class, math class, economics...whatever you end up studying.3443

You are going to end up having to find patterns of some sort.3448

Even in English, patterns are really important things; you are going to talk about themes in a book, so patterns really matter.3451

This sort of thing is really important.3456

We have a good understanding for sequences; now we are ready for the rest of this section.3458

We will have a whole bunch of ideas, working from this base of sequences.3461

All right, we will see you at Educator.com later--goodbye!3464

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Math Analysis Introduction to Sequences

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Math Analysis Introduction to Sequences

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Introduction to Sequences

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