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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Polar Equations & Functions

Slide Duration:

Table of Contents

I. Introduction
Introduction to Precalculus

10m 3s

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

45m 11s

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

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
Put Your Substitution in Parentheses
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

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

48m 43s

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

56m 31s

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

39m 54s

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

58m 26s

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

48m 49s

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

29m 20s

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

48m 35s

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

33m 24s

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

51m 42s

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

49m 37s

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

28m 49s

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

38m 41s

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

41m 7s

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

39m 43s

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

45m 34s

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

46m 8s

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

45m 36s

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

19m 9s

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

33m 22s

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

34m 16s

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

49m 7s

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

44m 56s

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

35m 17s

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

47m 4s

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

40m 31s

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

42m 33s

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

34m 10s

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

48m 46s

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

39m 5s

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

43m 16s

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

33m 5s

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

52m 3s

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

36m 4s

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

27m 18s

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

32m 58s

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

31m 8s

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

19m 11s

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

23m 16s

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

52m 52s

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

29m 5s

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

43m 55s

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

25m 43s

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

56m 40s

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

49m 5s

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

27m 37s

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

34m 25s

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

55m 40s

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

1h 13s

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

41m 1s

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

1h 9m 31s

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

35m 20s

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

54m 7s

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

47m 12s

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

58m 34s

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

53m 33s

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

48m 7s

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

38m 16s

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

40m 43s

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

57m 37s

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

31m 36s

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

44m 3s

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

36m 58s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

57m 45s

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

40m 27s

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

31m 36s

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

39m 27s

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

49m 53s

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

1h 13m 13s

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

40m 22s

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

57m 11s

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

32m 40s

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

32m 43s

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

32m 49s

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

51m 13s

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

45m 26s

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

10m 41s

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

10m 51s

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

10m 38s

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

9m 45s

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

7m 8s

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

  • We can set up equations and functions using r and θ exactly the same way as we did for x and y. Generally, θ is the independent variable (like x was), while r is the dependent variable (like y was).
  • We always assume that θ is in radians ([(π)/2],  [(7π)/4], etc.).
  • We graph polar equations/functions in the same way we graph "normal" stuff: plug in values, plot points, and connect with curves to make a graph. Since θ is the independent variable, we plug in some value for it, then see what distance r we get out at that angle.
  • Just like graphing with rectangular equations, you don't need to plot a huge number of points-merely enough to sketch the graph. Many polar equations involve trigonometric functions. The "interesting" points are when the trig function produces a zero or an extreme value (sin, cos ⇒ ±1). Figure out where these interesting values will occur and use them to help plot the graph.
  • If at anytime you're unsure how the graph will behave, just plot more points. The easiest way to work out your uncertainty is by calculating more points.
  • Occasionally you will see an equation that only uses one variable: that's fine! It means that variable is fixed, while the other can change freely.
  • Sometimes we'll want to convert an entire equation or function from polar to rectangular, or vice-versa. We can do so with the same conversion formulas we figured out and used in the previous lesson. Since these formulas were based off any x, y, r, θ, they work fine for substituting in equations.
    • Polar ⇒ Rectangular:    x = r cosθ              y = rsinθ
    • Rectangular ⇒ Polar:     r2 = x2 + y2           tanθ = y/x
  • If you have access to a graphing calculator, it's great to try graphing some polar equations with it. It's a new way of looking at graphing, so it helps just to play around. For more information, check out the appendix on graphing calculators.

Polar Equations & Functions

Fill in the table of values for the polar function below.
r(θ) = 3+cos(2θ)                   
θ
r
0
π

4
π

2

4
π

4

2

4
  • Filling in a table of values for a polar function is the same as doing it for a normal function. The input value is θ for the function, so we plug in some value for θ, then we see what output value we get for r.
  • For example, to find r(0), we plug in like normal:
    r(0) = 3 + cos(2 ·0)     =     3 + cos(0)     =     3+1     =     4
  • Repeat the above process for each value of θ in the table:
    θ
    r
    0
    4
    π

    4
    3
    π

    2
    2

    4
    3
    π
    4

    4
    3

    2
    2

    4
    3
    4
    [Notice that the values of r repeat after θ = π. This is because cos(2θ) has a period of π. Still, the values in the table after that are important because the θ values are distinctly different. When it comes time to draw polar graphs, remember that the θ value is just as key to plotting a point as the r value.]
θ
r
0
4
[(π)/4]
3
[(π)/2]
2
[(3π)/4]
3
π
4
[(5π)/4]
3
[(3π)/2]
2
[(7π)/4]
3
4
Below is the graph of r(θ) = 3cos(5 θ). Find an interval of θ where the graph is only drawn out once. (That is, the graph does not get re-traced as later values for θ wind up creating matching locations to those already drawn in by earlier values for θ.)
  • In polar coordinates, it's possible (and common) for numerically different points to give the same location. For example, all of the below points are equivalent:
    (7, π)   ≡     (7, 3π)   ≡     (−7, 0)
    [If this idea is unfamiliar to you, make sure to check out the previous lesson on the idea of polar coordinates. Being comfortable with this idea and how polar coordinates are plotted is critical to understanding polar equations/functions.]
  • We are looking for when the equation r=3 cos(5θ) begins to repeat locations. While it will never repeat the same coordinate numerically, it will eventually create equivalent coordinates, and the equation will re-trace the locations the graph has covered in previous points. Let us arbitrarily choose a starting angle of θ = 0. We could start before that, but 0 is a nice, easy place to start. With that in mind, notice that we will be pointing back in the same direction at θ = 2π. Thus, if we get the same r value for θ = 0 and θ = 2π, we see that the graph will be repeating then. Plug in to check:
    r(0) = 3cos(5 ·0)    =   3       
           r(2π) = 3cos(5 ·2π)    =   3
    Furthermore, if we continue onward from θ = 2π, all the values will be the exact same as if we had started at θ = 0 because of the periodic nature of the trig function. Thus, the graph is repeating after 2π. However, there is one more thing to keep in mind...
  • It is also possible to have duplicate locations from different coordinates if the r is negative in one of them while the θ points in the opposite direction. For example, (7, π)  ≡  (−7,  2π). This means that, assuming we start at θ = 0, there is another, earlier opportunity for repetition: θ = π. If that causes the function to spit out a negative version of the r, they will be giving equivalent locations:
    r(0) = 3cos(5 ·0)    =   3       
           r(π) = 3cos(5 ·π)    =   −3
    Thus we see that, yes, r(0) and r(π) graph to the exact same location. Furthermore, if we look at how r(θ) will behave for angles starting at θ = π, we see that it will produce the exact opposite (negative) to what it produced when starting at θ = 0. [This happens because θ = π catches cos(5θ) in the exact middle of a period interval, so everything is flipped to the opposite from there on out.] Thus, we see that when we start at θ = 0, the first repetition of the function begins at θ = π. This means we can create the entire graph of the function by just using the interval  θ: [0,  π).
  • Finally, if you found all the above a little difficult to follow, don't worry! Parametric coordinates are confusing when they're new to you, and parametric equations/functions are even tougher the first time around. A great way to help yourself understand what's happening is to use a graphing calculator. Enter in the function to be graphed (don't forget to give it an interval of θ to graph, most graphing calculators will require you to enter one), then try out the trace function. Trace the graph using the calculator and get a better understanding of how it's working. Feel free to play around with the functions you're looking at too! Adjust some numbers, try other trig functions, and just generally explore. The easiest way to get comfortable with polar equations is by exploring them, and a graphing calculator makes it possible to see a lot of different things very quickly.
θ: [0,  π)
Graph the polar equation:   r = sin(θ).
  • Unless you're already extremely familiar with a certain type of equation, it's almost always necessary to create a table of values to help see how the function graphs. Make sure to get enough values so that you understand what's going on in the equation and can comfortably graph it. If you're ever unsure about what it will look like, just plot more points until it becomes clear to you.
  • We might create a table of values like the one below:
    θ
    r
    0
    0
    π

    4
    0.71
    π

    2
    1

    4
    0.71
    π
    0

    4
    −0.71

    2
    −1

    4
    −0.71
    0
  • Once you have enough points to feel comfortable with how the equation works, you're ready to start plotting. With polar equations, it's very important to keep in mind that the r value changes based on the θ involved. Thus, as the angle you are drawing on rotates, the distance from the origin varies. Pay careful attention to this as you connect the points with curves: don't just go directly from point to point, make sure to fulfill the changing θ-direction as r grows or shrinks (or goes negative!). For this problem in specific, notice how sin(θ) hits its maximum at θ = [(π)/2]. It starts at 0, then increases to 1 as it rotates, then comes back down to 0. The graph is then re-traced by r becoming negative, telling us to go in the opposite direction to the θ angle, causing us to cover the same graph again.
Graph the polar equation θ = − [(π)/6].
  • Notice that this polar equation has no r term in it whatsoever. The only thing that has a restriction on it is the θ term.
  • This means that θ is restricted to only being −[(π)/6], but that r can be anything at all (since r does not appear anywhere, the equation puts no restrictions on it). Thus, we set the angle of θ = −[(π)/6] while r is allowed to run the interval of (−∞, ∞). This is similar to a rectangular equation like x=5. We set x as 5, then y is allowed to run anywhere, so we get a vertical line out of that equation.
  • When drawing in the graph, don't forget that negative values for r cause them to go "backwards". This means θ sets an angle to point in, and the various possible r values fill out the forward and backward directions for that angle, creating a line.
Graph the polar equation  r=√{θ} with the restriction that θ ≤ 4 π.
  • Begin by noticing that while there is an explicit upper restriction on what θ can be (the problem says θ ≤ 4 π), there is also an implied lower restriction. The equation we're working with is r=√{θ}: if θ < 0, then the equation won't make any sense, because we can't take the square root of negative. Thus, the only domain we have to consider for this problem is θ: [0, 4π].
  • Once we know where we're looking in terms of θ, we approach the problem like other graphing problems. Create a table of values so you have a sense of how r grows and changes and so you can later plot points. Using a calculator to find approximate values for r, we have:
    θ
    r
    0
    0
    π

    2
    1.25
    π
    1.77

    2
    2.17
    2.51

    2
    2.80
    3.07

    2
    3.32
    3.54
    As always, if you're unsure about how the graph works or would like to have more plots to point, just calculate some extra points until you feel comfortable with what you have.
  • Once you have enough points to feel comfortable with how the equation works, you're ready to start plotting. With polar equations, it's very important to keep in mind that the r value changes based on the θ involved. Thus, as the angle you are drawing on rotates, the distance from the origin varies. Pay careful attention to this as you connect the points with curves: don't just go directly from point to point, make sure to fulfill the changing θ-direction as r grows or shrinks (or goes negative!). For this problem in specific, notice that r=√{θ} grows very quickly at first, but slows down the larger it gets. This means that the changes when it first starts out at θ = 0 will be especially quick, so it would probably be useful to get a few extra points for those early θ-values:
    θ
    r
    π

    6
    0.72
    π

    4
    0.88
    π

    3
    1.02
    This goes to show just how quickly r grows for early values of θ. Although r=0 when θ = 0, it shoots out very quickly for the first tiny bit of turning. As θ turns more, the growth rate of r slows down.
Graph the polar function  r(θ) = 1+4sin(2θ).
  • Unless you're already extremely familiar with a certain type of equation, it's almost always necessary to create a table of values to help see how the function graphs. Make sure to get enough values so that you understand what's going on in the equation and can comfortably graph it. If you're ever unsure about what it will look like, just plot more points until it becomes clear to you. When figuring out what values of θ to use in your table, think in terms of which values will be "interesting". For example, the most "interesting" values for cos(x) are x=0, [(π)/2], π, [(3π)/2]. These values are extremely important because they make up the zeros, maximum, and minimum for cos(x). Along these lines, think of what values for θ will be "interesting" for sin(2θ).
  • Because sin(2θ) effectively goes twice as "fast" as sin(θ), this causes all the interesting values to occur on intervals of [(π)/4]. We will see our zeros, maximums, and minimums all fall on the below values:
    θ = 0,   π

    4
    ,   π

    2
    ,   

    4
    ,   …
    Furthermore, notice that sin(2θ) will start repeating outputs after π since π is its period. We still have to care about the θ-values after π because, even though the r-values will repeat, the θ-values will indicate new angles and thus create unique locations.
    θ
    r
    0
    1
    π

    4
    5
    π

    2
    1

    4
    −3
    π
    1

    4
    5

    2
    1

    4
    −3
    1
    As always, if you're unsure about how the graph works or would like to have more points to plot, just calculate some extra points until you feel comfortable with what you have.
  • Once you have enough points to feel comfortable with how the equation works, you're ready to start plotting. With polar equations, it's very important to keep in mind that the r value changes based on the θ involved. Thus, as the angle you are drawing on rotates, the distance from the origin varies. Pay careful attention to this as you connect the points with curves: don't just go directly from point to point, make sure to fulfill the changing θ-direction as r grows or shrinks (or goes negative!). For this problem in specific, notice that we get the exact same r values, in the same order, for θ from [0, π] as we do for [π, 2 π]. This, combined with how the angle is spinning, causes the graph to mirror around the origin. Noticing this mirroring effect can make it easier to draw in the graph. As you get really skilled, you can even notice this sort of mirroring before ever making the table, allowing you to create graphs much more quickly but without losing any quality.
Convert the equation from polar to rectangular.
r = cos(θ)
  • We can convert from polar coordinates to rectangular coordinates through the use of the following identities:
    x = rcosθ,       y = rsinθ,        r2 = x2 + y2,       tanθ = y

    x
    When using them, it's almost always a good idea to swap one entire side of the identity for the other entire side. For example, modifying the first identity to [x/r] = cosθ so you can swap out cosθ for the fraction [x/r] is almost never a good idea-you want to convert to one coordinate type, not use a mix of them.
  • With this idea in mind, we see that it's difficult to directly swap in an identity for the equation r = cos(θ): neither side directly fits one of the identities. However, we see that we're close to an identity. If we had r2 or r cosθ, we could match up to an identity. We can achieve both of these by multiplying each side of the equation by r:
    r = cos(θ)     ⇒     r ·r = r·cos(θ)     ⇒     r2 = r cos(θ)
  • With this new formation of the original equation, it's now quite easy to swap based on identities:
    r2 = r cos(θ)     ⇒     (x2 + y2) = (x)     ⇒     x2 + y2 = x
  • At this point, we're done: we've achieved a rectangular equation. However, if we want to understand it even better, we can go one more step and put this into a form we're used to-a circle:
    x2 + y2 = x     ⇒     x2 − x + y2 = 0     ⇒     x2 − x + 1

    4
    + y2 = 1

    4
        ⇒    
    x− 1

    2

    2

     
    + y2 =
    1

    2

    2

     
    If we're familiar with conic sections, we see that this is the graph of a circle centered at ([1/2], 0) with a radius of [1/2]: exactly what we would have gotten by graphing the original polar equation.
x2 + y2 = x, or, equivalently, (x−[1/2])2 + y2 = ([1/2] )2
Convert the equation from polar to rectangular, then solve for y.
r=−1

1+sinθ
  • We can convert from polar coordinates to rectangular coordinates through the use of the following identities:
    x = rcosθ,       y = rsinθ,        r2 = x2 + y2,       tanθ = y

    x
    When using them, it's almost always a good idea to swap one entire side of the identity for the other entire side. For example, modifying the first identity to [x/r] = cosθ so you can swap out cosθ for the fraction [x/r] is almost never a good idea-you want to convert to one coordinate type, not use a mix of them.
  • Looking at the equation, we see that there's nothing we can currently swap out with an identity. However, having that fraction on the right side isn't helping things. Let's get rid of that by multiplying both sides by the denominator of 1+sinθ:
    r=−1

    1+sinθ
        ⇒     r (1+sinθ) = −1     ⇒     r + r sinθ = −1
    Now we see that we can apply the identity y = rsinθ:
    r + r sinθ = −1     ⇒     r + y = −1
  • At this point, we still need to convert that r into something rectangular. The problem is that the only identity that uses just r is based on r2, which we don't currently have. However, we can get r2 to appear if we move the y to the other side, then square both sides:
    r+ y = −1     ⇒     r = −1 − y     ⇒     r2 = (−1−y)2
    Now that r2 has appeared, we can apply the identity r2 = x2 + y2:
    r2 = (−1−y)2     ⇒     x2 + y2 = (−1−y)2
  • We now have a rectangular equation, but the problem told us to also solve for y, so we need to get y alone on one side. Start off by expanding:
    x2 + y2 = (−1−y)2     ⇒     x2 + y2 = 1 +2y + y2
    We can subtract y2 on both sides, then get the y alone:
    x2 + y2 = 1 +2y + y2     ⇒     x2 = 1+2y     ⇒     x2 − 1 = 2y     ⇒     1

    2
    x2 1

    2
    = y
y = [1/2] x2 − [1/2]
Convert the rectangular equation to polar form.
y = x
  • We can convert from polar coordinates to rectangular coordinates through the use of the following identities:
    x = rcosθ,       y = rsinθ,        r2 = x2 + y2,       tanθ = y

    x
    When using them, it's almost always a good idea to swap one entire side of the identity for the other entire side. For example, modifying the first identity to [x/r] = cosθ so you can swap out cosθ for the fraction [x/r] is almost never a good idea-you want to convert to one coordinate type, not use a mix of them.
  • We can apply the identities right from the start because they have x and y alone in them. Swap out based on the identities:
    y = x     ⇒     r sinθ = r cosθ
  • At this point, we've technically completed the problem, since we have an equation in polar form. Still, it might be nice to simplify it in case we wanted to graph it. In that case, start off by canceling out the r's on each side;
    r sinθ = r cosθ    ⇒     sinθ = cosθ
    Next, we can combine the trig functions into a single tangent function by dividing cosθ on both sides:
    sinθ = cosθ    ⇒     sinθ

    cosθ
    = 1     ⇒     tanθ = 1
    Finally, to make it super easy to graph, we can take the inverse tangent of both sides to just get θ alone:
    tanθ = 1     ⇒     θ = tan−1 ( 1)     ⇒     θ =π

    4
    This makes sense! The rectangular equation y=x is just a straight line that goes up at an angle of 45° (because it has a slope of 1). The polar equation gives us the exact same line, just like it should.
θ = [(π)/4]
Convert the rectangular equation to polar form, then solve for r.
x2−9 = 6y
  • We can convert from polar coordinates to rectangular coordinates through the use of the following identities:
    x = rcosθ,       y = rsinθ,        r2 = x2 + y2,       tanθ = y

    x
    When using them, it's almost always a good idea to swap one entire side of the identity for the other entire side. For example, modifying the first identity to [x/r] = cosθ so you can swap out cosθ for the fraction [x/r] is almost never a good idea-you want to convert to one coordinate type, not use a mix of them.
  • This is a very tricky problem. Our knee-jerk reaction would be to swap out x and y based on the identities, as below:
    x2−9 = 6y     ⇒     (r cosθ)2 − 9 = 6 (rsinθ)     ⇒     r2 cos2 θ− 9 = 6 ·r sinθ
    However, it turns out that it's actually pretty difficult to solve for r from the above equation. Give it a try, and you'll quickly notice how tough it is to pin down r. [It is possible, but you have to use the identity sin2θ+ cos2 θ = 1 to change the cos2 to a sin2, then also do something similar to what we do below.] Instead, it will help us a bit to take a different approach we might not have initially thought of...
  • Notice that we have x2 on the left. One of the identities we have is r2 = x2 + y2, so alternatively, if we had y2 show up, we could use that identity. Let's try that instead:
    x2 − 9 = 6y     ⇒     x2 + y2 − 9 = y2 + 6y     ⇒     r2 − 9 = y2 + 6y
    At this point, we might try moving the 9 to the other side:
    r2 − 9 = y2 + 6y     ⇒     r2 = y2 + 6y + 9
    Which then might cause us to realize we can factor it:
    r2 = y2 + 6y + 9     ⇒     r2 = (y+3)2
  • Our goal is to eventually get r alone, so since we now have something squared on both sides, we might as well take the square root. Remember that taking a square root causes a `±' to show up.
    r2 = (y+3)2     ⇒    

     

    r2
     
    = ±

     

    (y+3)2
     
        ⇒     r = ±(y+3)
    We still need to fully convert to polar, so switch out the y:
    r = ±(y+3)     ⇒     r = ±(rsinθ+ 3)
    Remember, ± means that there is both a + version and − version, so we can split the above into its two versions:
    r = rsinθ+ 3              
        r = −(rsinθ+3)     ⇒     r = −rsinθ− 3
  • Finally, we have an equation that is entirely polar and it's not too difficult to solve for r. To do so, we need to get all the r's on one side, then pull out the r, then put everything else on the other side.
    r=rsinθ+ 3       
           r = −rsinθ− 3

    r − r sinθ = 3       
           r + r sinθ = −3

    r(1−sinθ) = 3       
           r(1+sinθ) = −3

    r = 3

    1−sinθ
          
           r =−3

    1+sinθ
    At this point, we have a polar equation and we've solved for r. There is a little bit of strangeness, though...  How can we have two different solutions? Each of those equations is distinct from the other, but somehow they are both conversions of the same rectangular equation? That's weird.
  • It turns out, there's an explanation. If you graph either of them, you get the exact same graph, which is pictured below. Each of the equations produces the same parabola that the original rectangular equation gave, they just do it from different starting locations. The graph of r = [3/(1−sinθ)] "starts" from the positive x-axis at 3, then works up to the right, until eventually flipping to the left side. The other one starts differently: r = [(−3)/(1+sinθ)] "starts" from the negative x-axis at −3, then works down at first, eventually coming up on the right side, and then also flipping like the other. The fact that we can express the original rectangular equation as two different polar equations is based on how we draw polar coordinates. We can name the same location with different sets of polar coordinates, and these two equations correspond to the different ways we can name the points. We can name with positive ("forward") r or we can name them with negative ("backward") r, which is the two types that come out of the equations.
r = [3/(1−sinθ)]     or     r = [(−3)/(1+sinθ)] [See the last step for a discussion of why the rectangular equation can be turned into two different polar equations.]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Polar Equations & Functions

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

Transcription: Polar Equations & Functions

Hi--welcome back to Educator.com.0000

Today, we are going to talk about polar equations and functions.0002

In the previous lesson, we introduced a new way to look at location in the plane, polar coordinates.0005

By now, we have seen thousands of graphs (probably literally), where x and y are in relationship with each other.0010

As the horizontal changes, the vertical changes somehow.0018

Either x and y are both in an equation, and we graph the equation, and we figure out all of the solutions to that equation;0020

we graph it; or y is a function of x, and we graph the function--as x changes, how will y respond?0025

We can do the exact same thing with polar coordinates: the variables r and θ can be put in some sort of relationship,0031

and then we can graph the resulting polar coordinates.0037

Before you watch the lesson, make sure you watch the previous lesson (before watching this one).0039

You really, really need to understand polar coordinates on their own.0045

You have to be able to understand how polar coordinates make locations on the plane before any of this stuff is going to really make sense.0048

So, if you are having difficulty with working through polar equations, but you don't have a really good understanding0056

of polar coordinates yet, that is the thing--you really want to work on polar coordinates.0060

Make sure you have watched the previous lesson before watching this one,0063

and that you are comfortable with polar coordinates before you try to work on polar equations and functions.0065

And if you can make sense of how polar coordinates work, polar equations and functions probably won't actually be that much harder.0070

All right, let's get started.0074

We can set up equations and functions using r and θ, exactly the same way as we did for x and y.0077

Generally, θ is going to be the independent variable, like x was.0082

Our x was allowed to change and vary around, and then r will be our dependent variable, in the same way that y was.0086

So, x was allowed to change around, and y responded to x's changes.0094

Similarly, here it is going to be θ that will be allowed to change around, and our distance r will respond to the angle that we are at.0098

Here are some examples; we could have an equation r = 1 + 2cos(θ) (we plug in a θ, and it tells us an r),0106

or a function r(θ) = 3sin(2θ), which is the same thing as...we plug in a θ, and it gives us what the r-value will be.0113

So, while r could be the independent, and θ the dependent, such relationships are pretty uncommon in polar equations and functions.0121

We normally think in terms of how length changes based on the angle--0132

if we go to this angle, what length it will be at--and not the opposite way,0136

just like when we are doing a rectangular graph, we normally think in terms of "for this horizontal location,0140

what height will I be at? For this horizontal location, what height will I be at?" and not0145

"For this height, what horizontal location will I be at?"0150

We don't normally think of height and then horizontal; we think of horizontal, and then height,0152

because x is the independent, and y is the dependent, just like here θ is the independent and r is the dependent.0156

We always, always assume that θ is in radians.0164

Whenever we are looking at our θ, it is assumed that θ is going to be in radians.0169

That is numbers like π/2, 7π/4, decimal things, etc.0173

It could be explicitly put in degrees, but that would be extremely rare.0177

Always, always, always use radians, unless you are being told explicitly otherwise, that this thing is in degrees.0182

It is extremely uncommon: you almost never see something like that.0189

I can't even think of one time I have seen it; so just don't expect that to happen--expect to be using radians when you are working with polar equations and graphs.0193

So, always think in terms of radians: you are plugging in radian values, and you are plotting with radian values on your angles.0202

All right, how do we graph in polar coordinates?0210

We graph polar equations and functions pretty much the same way that we graphed normal stuff.0212

You plug in some value for your independent (in this case, θ).0217

You plot the point that gets put out; and then you connect the whole thing with curves to make it into a graph.0220

So, θ is the independent variable; we plug in some value for it, and then we see what distance r would get out.0226

Over here, let's plot some points: if we are looking at the equation r = 1 + 2cos(θ), then r is going to come out once we plug in some θ.0231

So, if we plug in 0, well, 1 + 2cos(0)...cos(0) is 1, so 2 times 1 is 2, and 1 + 2 is 3.0239

So, we now have the point...not (0,3), but (3,0); that is one little confusing thing--0248

the fact that it is not (x,y); it is now (r,θ); so our independent variable is actually the thing that comes second.0255

So, don't let that confuse you.0261

Our distance out is 3, and our θ is going to be an angle of 0.0263

So, we end up getting this point right here; we have an angle of 0; we are at 0 above the starting location.0269

We haven't moved at all, and we are out on the third circle out; so we are at (3,0).0276

Next, π/4: if we plug in π/4, 1 + 2cos(π/4)...cosine of π/4 is √2/2; 2(√2/2) would be √2;0282

so that means that we get 1 + √2 out of this.0293

1 + √2...we figure that out with a calculator, so we can actually plot something down.0296

That is approximately 2.41; so at this point, we are at angle π/4, so we are on this arc sector line right here.0300

Notice, it is broken into eight pieces, so each one of them is going to be π/4, because up here is π/2.0309

So, we are at the line of π/4 angle; and then we go out 2.41: (2.41,π/4) is the point that we get out of this.0315

So, we are at 2.41 out, somewhere that looks around 2.4, a little more than 2, but even more less than 3.0328

So, we get this point right here; OK.0336

The same thing if we plug in π/2: if we plug in π/2, 1 + 2cos(π/2)...cosine of π/2 is 0, so we just get 1.0339

So, we have the point (1,π/2); and that gets us this point here.0348

So now, we think about how these things are going to connect through curves.0354

If we are really confused about it, we could just plot down more points.0357

We could put down π/6 and π/3, as well.0360

And if we want even more points, we could continue to plot down more and more points.0362

But we would probably be able to get a pretty good sense with just π/6 and π/3.0366

But we can even just figure this out by thinking, "Well, how is it curving--what is happening?"0370

So, as the angle goes up, notice that, because it is cosine of θ, as this goes to larger and larger values,0373

0 to π/4 to π/2, cosine shrinks down and down.0381

So, this portion of our equation, the 2cos(θ), will get smaller and smaller as cos(θ) gets smaller and smaller.0384

So, as we get larger and larger, it shrinks down to ultimately this value of 1 right here.0391

We go up, and as it gets farther out, it shrinks down; it shrinks down; it cuts through here; it shrinks down; it shrinks down.0396

And we get something like that curve.0403

We can see it precisely here, now drawn by a computer: r = 1 + 2cos(θ).0405

At this point, we just keep plotting more points.0411

Here is a new, interesting one to think about: if we plotted 3π/4, well, that is 1 + 2cos(3π/4).0413

What does 3π/4 come out to be? That is 1 + 2(-√2/2), so that gets us 1 - √2.0421

We approximate that with a calculator; we get -0.41, which means that we have the point (-0.41,3π/4).0430

So, here is our 3π/4; it is right here; this is the 3π/4 location.0440

We are on this line; but we have -0.41, so we are going in the opposite direction, and we are here for our point.0447

We plug in π; 2 times cosine of π; what is cos(π)? It is -1.0455

So, 2 times -1 get us -2; 1 - 2 gets us -1; so we have gotten the point (-1,π) out of this.0462

So, at the angle of π (here is angle π), we are going to be going opposite the length of 1, and so we are here, as well.0470

So, once again, it is continuing to drop down; this part here is continuing to get smaller and smaller, until eventually it becomes negative.0478

It actually continues out to this curve here, and then it cuts through.0485

It will always have to cut through the origin, because if it gets a 0 in the r, then it has to go through the origin,0488

because distant 0 from the origin, distant 0 from the pole, means at the pole; cut through here.0493

And then, curve up to here; and that is what we have so far.0500

It is computer-drawn there, so it is a little more accurate; but we basically just keep plotting points.0503

If we plug in 5π/4, we see that that is at -0.41; so at 5π/4, that would be this angle here.0508

So, we are in the opposite now; we are over here at 3π/2; we have length 1, so at 3π/2, we are here; length is 1 in that direction.0515

7π/4: we are at a length of approximately 2.41, so at 7π/4...we are at 2.41 from here.0523

And we continue this curving; we can also see at this point, perhaps, that is going to end up being symmetric.0529

If we looked at the entire table of values, we would see that there is some symmetry going on in the way these distances are coming out,0534

and that we are going to end up seeing the top part...this curve so far will happen and just sort of flip over;0539

and we will end up getting it out like this.0547

And finally, here is a computer-drawn version that is better than my slightly-imperfect drawing.0553

And that is what we end up getting out of this.0559

You plot points; you work it out; you graph the whole thing.0560

Just like graphing with rectangular equations, you don't need to plot a huge number of points.0564

It won't hurt; if you plot more points, it is not going to hurt.0568

But you really only have to plot enough so that you can sketch the graph.0571

Which points should you plot--what are the useful points to plot--what are the interesting points to plot?0575

Many polar equations involve trigonometric functions; the interesting points are when the trigonometric function produces a zero,0580

when we plug some θ into that trigonometric function and it puts out 0,0588

or when it puts out an extreme value (for sine and cosine, that is positive or negative 1).0592

Plugging in those values for our θ lets us see what the most extreme values are that our function is going to make,0597

which helps us have an understanding of how the whole thing is behaving.0602

More points will just make it smoother and easier to make the curves.0605

But those are the most important ones of all; so you definitely want to make sure that you are plotting those.0608

If we want to try to plot these extreme values and these 0's, we want to figure out where these interesting values will occur.0613

How are we going to get to it? Trigonometric functions tend to have a pattern.0618

We are used to working with sine and cosine; so we want to think in terms of sine--0621

we have to plug in 0 or π/2 or π; if it is sin(θ), we have to plug in 0 or π/2 or π or 3π/2...0625

any of these would end up getting an interesting number.0633

Sin(0) is 0; sin(π/2) is 1; sin(π) is 0; sin(3π/2) is -1; we are used to working with that from all of our work in trigonometry.0635

If we go to something else, though, like cos(5θ), well, it is not just θ anymore; it is 5 times that.0644

So, if it were cosine of something, we would want that something to be 0, and then π/2, and then π, and then 3π/2, and then 2π, and so on.0651

But in this case, it is 5 times θ; so that has to be taken into account in how our thing works.0658

So, the 0 is still just going to be the same; if we plug in a 0, 5 times 0 still gets a 0.0663

But if we wanted to try to figure out π/2 = 5θ, what does our θ have to go in to get that interesting first value of π/2?0669

We divide both sides by 5, and we would get π/10 = θ.0677

π/10 = θ, because if we plug in π/10, 5 times π/10 is π/2; cos(π/2) gets us 0.0682

So, that is our next interesting thing.0688

Similarly, it is going to continue on this pattern of π/10 being the interval here.0690

π/5 (I said that the wrong way...) 5 times π/5 would get us π; cos(π) is -1, an interesting value.0695

The cosine of 5 times 3π/10 would get us cos(3π/2), which is 0, an interesting value.0704

So, we are thinking in terms of what we have to plug in here, total, to get 0, π/2, π, 3π/2, 2π, the normal, interesting values.0711

But then, we have to pay attention to the fact that it is not just θ; it is something interacting with θ.0720

So, we have to pay attention to what the numbers are going to be.0725

And that helps us figure out what numbers we want to plot in when we are trying to graph.0727

Finally, what if it was 2θ + 1? In this case, θ = 0 won't even show up, because we have to figure out what would make this something.0732

What would make 2θ + 1 turn into 0?0739

Well, if we plug in -1/2 for θ, 2 times -1/2 gets us -1; -1 + 1 gets us 0.0742

So, if we plug in -1/2 for our θ, we get out a 0 from it.0752

What if we plug in θ = π/4 - 1/2?0756

Well, then we have 2θ + 1; so 2(π/4) - 1/2 gets us π/2 - 1 + 1; so we get π/2 out of this.0759

Let's work that one out: if we had 2θ + 1 = π/2, we would have to work out how to get to this.0768

2θ = π/2 - 1...θ...if we are trying to get to this value of π/2,0777

if we want this whole something here to come out to be π/2, the θ will have to end up being π/4 - 1/2.0784

And that is where we are getting it.0792

It is the same thing if we want this whole thing to be π; we end up needing π/2 - 1/2.0793

We want the whole thing to be π/2; we need to plug in 3π/4 - 1/2 for our angle θ.0798

Notice that, in each one of these, we are stepping up by π/4 each time.0804

Over here, we were stepping up by π/5 each time.0808

Once you start to notice the pattern, the pattern will normally continue.0811

But it will depend on the specific circumstances of what is inside of your trigonometric function.0814

Beyond looking for these interesting points, you might notice repetition in trigonometric functions leading to symmetries in the graph.0820

Depending on the way the trigonometric functions are set up, you might notice that all of the same stuff is going to happen here.0825

And because of the way that my angle is going to work, there is going to end up being a symmetry occurring in the graph.0830

If you notice the symmetry, if you see that a symmetry will certainly occur, just use that to make graphing easier.0835

You won't have to plot all of those points, because you will see0839

that it is just going to end up doing effectively the same thing, but reversed or flipped in some way.0842

If at any time you are unsure how the graph will behave, just plot more points; that is the easiest way to be sure of what is going to happen of all.0846

If you are uncertain about what will come next, calculate a bunch of points.0854

Just drop them in and connect the curve through those points.0859

The more points you have down, the easier it will be to see how the curve works.0861

After time, you will develop a sensibility; you will get an intuition for how these curves are going to come out--how they are going to look.0865

You won't have to plot as many points.0871

But when you are just starting out, if you are uncertain about a graph, plot more points, and you will have a good idea of where to go.0873

All right, occasionally you will see an equation that only uses one variable, where there is just one variable there.0879

A lot of students get scared; there is no reason to get scared.0884

It is totally fine to have one variable; it just means that that variable that you said is fixed, and the other one can change freely.0887

For example, if we had r = 2, then that means our distance is always 2.0893

But θ--does θ show up in r = 2? θ doesn't show up at all.0898

So, θ can be anything over here; in our red one over here, θ can be anything.0902

So, if θ can be anything, then that means we set r = 2; so it is going to always be on this length of 2, this distance of 2 from the center.0908

But our θ can end up going to any spin at all--it can be anything.0916

So, as θ is allowed to spin positive or negative, it is going to always end up being stuck on the circle.0921

So, we end up drawing this perfect red circle at a distance of 2 from the origin.0926

A similar thing is going on over here with θ = π/3; if θ = π/3, we never mentioned r in there.0932

We never mentioned the distance, so that means r can be anything.0937

Since r can be anything, we have to be on the angle of π/3.0941

But r could be any positive thing, so it could go out forever this way.0947

And r could be any negative thing, so it could go out in the opposite direction.0950

So, with θ = π/3, with θ at a fixed value, we end up creating a line.0953

With r at a fixed value, like r = 2, we end up creating a circle, because either with r (we fix r) we have fixed a distance,0959

but we are able to go to any angle (so we are making a circle); or if we fix an angle,0966

and we are allowed to go to any distance, we have drawn a line.0970

And that is what is going on if we just fix a single variable.0972

Converting coordinate types: sometimes, we will want to convert an entire equation or function from polar to rectangular, or vice versa.0976

We can do so with the same conversion formulas we figured out and used in the previous lesson.0983

So, when we figured out these formulas, they were based off any x, y, r, and θ.0987

We didn't say what x, y, r, and θ had to be; so this is going to work for converting the variables in equations.0992

They can be used to convert from one variable type to the other variable, to convert from (x,y) to (r,θ) or (r,θ) to (x,y).0998

So, previously we had x = rcos(θ), y = rsin(θ) for one set.1004

And then, the other set of formulas was r2 = x2 + y2; tan(θ) = y/x.1009

Pretty much any way that you can see these working out to allow you to swap variables around,1014

so you can get to the kind of variables you want--go ahead and use it.1018

These are the formulas that will allow you to convert from one type of equation to another type of equation,1021

to switch from polar to rectangular or rectangular to polar.1026

And if you ever forget any of these formulas, if you forget, you can re-derive them by drawing this picture right here,1029

because we know that r is always the hypotenuse, and we know that x and y are the horizontal and vertical;1036

and we know that θ has to be the angle inside of the triangle.1043

So, we can figure out all of these equations: x = rcos(θ); y = rsin(θ);1046

r2 = x2 + y2, tan(θ) = y/x,1051

just through basic trigonometry, because we know that (right down here) it is a right triangle.1054

So, you can just use basic trigonometry and re-derive this if, in the middle of an important exam, they all disappear from your head.1058

You can just draw a picture and do basic trigonometry; it is not too hard.1063

All right, polar equations allow us to make really interesting graphs--that is to say, crazy, bizarre, cool, strange graphs--1068

graphs that look nothing like the kind of graphs that we are used to with rectangular equations.1074

They allow us to make really, really interesting stuff very easily.1080

For example, look at this red one right here: could you imagine figuring out any way to graph that with rectangular x and y,1083

to make an x and y equation that could make this flower-looking thing?1089

That looks so unlike what we are used to graphing, but it only takes 2sin(6θ) + 0.5.1094

We are able to say that incredible thing, where it has these large petals and these small petals,1101

and a bunch of them repeating in this kind-of symmetrical pattern--with very, very little work.1105

Very, very little writing is required to be able to make this incredibly detailed picture.1110

The same thing over here in the blue picture: we are able to get this weird, squished thing that doesn't really look like anything specific.1114

But it is a picture, and it is not very hard to write out.1120

Once again, 2cos(θ) - sin(5θ)--we are able to get this very strange-looking picture,1124

that there would be no easy way for us to create a graph with x and y coordinates that we would be used to using here.1130

But in polar graphs, it is pretty easy to do.1136

We can make really interesting things, things that we were really not used to seeing before, with not that long of an equation.1139

Polar equations and functions are really a new way of thinking about graphing.1148

As such, it is a great time to use a graphing calculator.1151

This is the best time for graphing calculators: plot random equations in there.1155

Alter equations that you already understand, and just get a sense for how polar stuff works by playing around with graphs on a graphing calculator.1159

If you want more information, check out the appendix.1166

There is an appendix to the course entirely on graphing calculators--how to use graphing calculators,1168

what good graphing calculators are, and even if you don't own one, and you are not going to buy one--1172

absolutely, for sure--there are free options out there.1176

So, there is lots of cool stuff where you can go on the Web really quickly.1179

And in five minutes, you can be graphing polar stuff; probably in one minute, you can be graphing polar stuff.1183

And you would be able to get that without having to spend any money on a graphing calculator.1188

And just playing around with this stuff is going to help you understand polar graphs massively.1192

Trying different things, playing around, putting in things, looking at how changing one number here changed the whole thing--1199

just being able to see that incredible speed of responsiveness of a graphing calculator,1205

being able to change immediately when you do something...1210

You don't have to take all of the time to plot it carefully, because that goes so slowly; it is hard to realize what is going on.1212

But if you change one variable in a graphing calculator, and it creates a new graph,1217

you will be able to gain this really beautiful intuition of how polar graphs work--how a polar equation creates the graph associated with it.1220

I really, really, strongly recommend: if you have a graphing calculator, and even if you don't have a graphing calculator,1228

check out the appendix; I will talk about lots of places where you can get free ones,1233

where you can just go and play with them right now, immediately.1236

And you will be able to get a really good understanding of how polar graphs work with not that much effort.1239

Just playing around for 10 or 15 minutes will give you such a better understanding of polar graphs, if you find them difficult, even in the slightest.1244

Also, I want to just point out one really important thing.1250

When you are using a graphing calculator, pay attention to the interval that θ has.1253

Normally, they are going to start your θ being from 0 to 2π.1256

And that is going to often be enough to give you the entire graph; but it won't always.1261

It might not show you everything; so if it doesn't show you everything you need to see--1265

if there is some part missing (you might want to expand it, and you might not even realize that there is something missing)--1270

you might want to just try increasing your interval to -6π, +6π, or -10 to +10.1274

And just see if that changes the graph.1280

If it doesn't change the graph, then you know that a 0 to 2π interval is enough.1282

But if it does change the graph, that tells you that you need to think about how big your interval needs to be, to be able to see the whole graph.1285

And maybe it won't even be possible to graph the entire interval all at once on one graph, like we are going to see in Example 2.1290

All right, we are ready for some examples.1296

First, let's graph the function r(θ) = 3sin(2θ).1297

We plug in θ's here; it gives some value here, and that tells us what our r is going to be.1302

It is just plugging in a number and getting something out of it; and that tells us our r.1307

We plug in some value for θ; we get r.1312

Let's figure out some values for this; let's figure out how often we need to do this.1314

Well, we have 2θ as our thing in here: 2θ...if we were solving for the very first interesting point,1319

which would normally occur at π/2 (0 is an interesting point, but we can be certain that 0 is already going to show up,1324

and that one doesn't tell us as much), then that is going to be θ = π/4.1330

So, we are going to have interesting points occur at an interval of every π/4 that we go out.1334

Let's plug in π/4 to figure this out.1339

We will plug in θ's; we are going to make a whole bunch of them.1342

And here are our r's; and if we plug in 0, 3sin(0)...sin(0) is 0, so we are just going to get 0 in here.1350

If we plug in π/4 (we figured out that our first interesting point was π/4), 3sin(2π/4)...2π/4 is π/2; sin(π/2) is 1; 3 times 1 is 3.1362

So, we get +3 over here for our r.1372

At this point, we can draw a graph; so we can start plotting some things.1374

We know that we are going to have to at least get out to a distance of 3.1378

And I will tell you that it actually is only going to go out to a total distance of 3, at the maximum.1381

So, we will have circles every 3; so here is our first distance circle...1385

Pardon me if my circles aren't absolutely perfect; I am but a human.1392

The second distance circle; and our third distance circle...OK.1396

And then, let's see: since we know that we are going to be based off of π/4, let's cut angles at π/4, as well; cool.1416

So now, we can plot some points: at an angle of 0, we go out 0, so our first point is just at the pole, on what we used to call the origin.1428

At an angle of π/4, this one right here, we are a distance of 3: 1, 2, 3 distance out.1438

Next, let's try π/2, the next π/4 forwards, the next interesting point.1445

We plug in sin(2π/2); 2π/2 is π; sin(π) is 0, so we end up getting 3(0) is 0.1450

By the time we get back to π/2, this angle here, we are back down to 0; what does that look like?1458

As π/4 goes to π/2, it goes down; from 0 up to π/2, we increase to 3, and then we decrease back down to 0.1464

We increase to 3; and then we decrease down to 0; from 0 to π/2, we go up to 3, and then down to 0.1473

So, as we spin counterclockwise, our thing increases; we touch that, and we spin back down.1481

We get smaller and smaller, back down to 0; so that is the first part of our graph.1491

Let's see what else is going to happen here: if we plug in 3π/4, sin(2(3π/4))...2(3π/4) is 3π/2;1495

sin(3π/2) is -1; 3 times -1 is -3; so at an angle of 3π/4, which is here, we are at -3.1506

So, we are going to go in the opposite direction; we are going to go opposite.1515

We were going this way, but we are now going to go opposite, because it is -3.1518

So, we are out 1, 2, 3 here; and then, at π, the next interesting place, sin(2π) is 0;1521

2 times π is 2π, so sine of 2π is 0; so we get 0 once again here.1528

And so, it is going to do the same thing, where, as it goes from π/2 to π, it becomes...1532

here is 0; here is π/2; here is π; so for the first part, it went up to 3, and then it went back down to 0,1538

when it got to π/2; and now it is going to go down to -3, and then it is going to go up to 0.1544

So, we are seeing it go up, and then down, and then negative, and then back to 0.1550

So, for this part, it is going more and more negative; so we end up seeing it curve out like this as it gets to larger and larger angles.1554

We see it spin this way; OK.1561

Next, we have 5π/4; at 5π/4, we plug that in; 2 times 5π/4 is going to be 10π/4, which is the same thing as 2π +...1567

well, let's just write this out, because that way it is a little less confusing.1581

5π/4 times 2 is equal to 10π/4; and look, that is the same thing as 8π/4 + 2π/4.1583

So, that is the same thing as 2π + π/2; everything that we have here is just going to end up being the same thing as this.1599

The π/4 here will match up to the 5π/4 here; so we are going to end up getting 3.1608

With this idea in mind, we could actually realize, really quickly, that π/2 here is going to match to 6π/4, which is 3π/2.1613

So, as we go that extra π forward, because we have this 2 here, it is going to end up repeating everything, as well.1622

Our angles are going to be new and different and interesting, but the r's will end up repeating.1627

We are going to see a repeat of this part here; it is just going to repeat here.1631

So, 3π/2 gets us 0; 7π/4 will get us a -3; and at 2π, we will be right back where we started at 0.1635

So, it will end up getting back to where we started.1644

Let's plot 5π/4; at an angle of 5π/4, we are at this part; we go out a distance of 3, so we are here.1647

So, we just got back to the origin when we were at π; the same thing--it curves out and grows larger when it gets out to it.1653

And then, it gets smaller and smaller at this point; it drops back down.1659

We are starting to see some symmetry; it is like there are petals here.1662

"Petals" is a way of talking about this.1665

At 7π/4, we are here; but once again, it is -3, so we go in the opposite direction.1667

We go out to here, and there we go--they should be perfectly symmetrical, if the graph was absolutely perfect.1675

If it was drawn by a computer, they would end up being perfectly symmetrical petals.1684

But we can get a pretty good sense of what is going on, even drawing it by hand.1687

The second example: Graph the equation r = 1.5θ/2, where θ is between -2π and 2π; it goes from -2π to 2π.1692

At this one, let's start...we will graph...well, we really have no idea of what this is going to end up being.1702

We have 1.5r = 1.5θ/2; we are dividing our θ by 2.1708

So, we know that we are going to want 0 in there, because it is right in between -2π and +2π.1719

And we want to go all the way down to -2π and all the way up to 2π.1724

So, let's do this by π/2; π/2 will be easy to graph, as well, since they are the cross-axis that we normally have in there.1727

A distance of r...if we plug in θ at 0 first, well, 1.5 raised to the 0--that is easy.1734

We always end up getting 1; if you raise any number to the 0, you end up getting 1 out of it.1741

Next, though, this is a little bit more difficult: if we plug in π/2, then what do we get out of this?1746

Well, we will have to use a calculator; so here is how we do the first one.1752

If we were to try to figure out 1.5, raised to the π/2, over 2...we have π/2, over 2...1754

well, that is going to be the same thing as 1.5 to the π/4.1762

We might not be able to figure this out by hand, but we could put this into a calculator: 3.14...1767

1.57 would be half of that; so it is just going to be plugging in those numbers and getting an approximate value.1772

So, this comes out to be approximately...using a calculator, we get 1.37 out of it.1779

Sorry, I meant to say 1.57, because 3.14/2 would be equal to 1.57.1784

But then, this is 3.14/4, so that is 1.57/2; the point is that we could work it out with a calculator1790

and get an approximate decimal value for what that ends up being.1799

That comes out to be 1.37; we do the same thing for each one of these.1802

We plug in π; so it is 1.5 to the π/2 or 1.5 to the approximately 1.57.1806

We raise that...our calculator gives us that that is approximately 1.89.1813

3π/2...our calculator gives us that that gives us approximately 2.60...2π...we get approximately 3.57.1818

What if we went the other way? -π/2...raising that up there...1.5-π/2 divided by 2, so 1.5-π/4...1829

we get approximately 0.72; raising it to the -π/2...we get 0.53; raising it to the -3π/2, over 2, gets us 0.38.1837

And finally, at -2π, we get 0.28.1852

So notice; at 0, if we take 0, and we go up with 0, if we go up, the value gets larger and larger.1857

We have it getting larger and larger faster and faster, because remember: this is an exponential function.1864

So, it will get faster and faster growing as we go to larger and larger values of θ.1869

As we go to negative values of θ, though, it gets smaller and smaller, because it is an exponential function.1874

Once again, we are seeing the tail part get really down close to that x-axis.1877

So, we can draw this out: the most extreme value that we get out to is 3.57.1882

So, I will set our most extreme circle at a distance of 4.1889

These single cross bars will be enough, because we are only concerned1893

if π/2 is the only real reference angle we have going on here; so that will be OK.1896

So, here is a distance of 1 circle; here is a distance of 2 circle; here is a distance of 3 circle; here is a distance of 4 circle.1901

Oops, that got a little bit out of hand; here is a distance of 4 circle; that is better.1916

OK, that is a pretty reasonable setup for our axes.1929

At this point, we can plot some points: at 0, at an angle of 0, we are 1 out; so here is our first point.1932

At an angle of π/2, going straight up, we are at 1.37; so we are almost to halfway out.1940

At π, this angle here, we are at 1.89, getting pretty close to that distance of 2.1949

At 3π/2, we are at 2.60, a little bit over halfway between the 2 and the 3 ring.1956

At 2π, we are at 3.57, a little bit over halfway between the 3 and the 4 ring.1962

We have this...as we go to a larger and larger angle, the distance out increases.1968

We see it spiraling out, the farther out it gets.1973

It continues to spiral out this way; so this is what we see as the angle gets larger and larger--it spirals out.1977

What if the angle goes negative? Well, at -π/2, remember, this is -π/2,1986

because it is talking about going clockwise instead; so at -π/2, we have 0.72.1991

0.72 is around here; at -π, that is here, because remember: it is clockwise now;1997

we are at 0.53; at -3π/2 (that is this one here), we are at 0.38; at -2π, we are at 0.28.2005

So, we end up seeing it continue to spiral in and in and in and in and in.2018

-2π to 2π: that is what is stopping this from continuing out forever.2024

If this was allowed to keep going forever, we would see it spiral off way out forever.2027

If this was allowed to continue forever, we would see it spiral into the center more and more and more and more.2032

So, that -2π to 2π--that is why we have to have an interval set--because sometimes,2036

if we don't set an interval, we could just keep going forever.2041

All right, and that is why we have it set in our graphing calculator, if you are using a graphing calculator.2044

You have to pay attention to the interval, because sometimes it will end up cutting off parts of the graph that you want to see.2047

You wouldn't see that part where it gets to continue to spin in if you didn't have a larger than 0 as the bottom of your interval.2052

You have to go to -2π, -4π, -10π...to really get a sense of just how much that spirals into the center.2057

All right, the third example: convert the equation from polar to rectangular, then solve for y in terms of x.2064

What were all of our equations? We have x = rcos(θ) as one of our formulas for changing.2070

y = rsin(θ) is another formula; r2 = x2 + y2 is another; and tan(θ) = y/x.2076

So in this case, we see right away that 6r2 times cos2(θ)...2086

we are not quite sure about that part, but here is our sin(θ), and here is our sin(θ).2091

So, we can swap them out; that is 3y = 7.2096

What about this part here--what about r squared, cosine squared, θ?2099

Well, we realize that that sounds an awful lot like r cosine θ; so how can we get rcos(θ) to show up there?2103

We maybe think, "Oh, well, there is r2; there is cosine squared; we can pull that squared out, and we could write this as 6rcos(θ)."2109

And then, that whole thing is squared; and then, minus 3y = 7.2117

At this point, we can swap out x = rcos(θ); we have 6 times x2 - 3y = 7.2122

Add 3y; subtract 7 on both sides; so we have 6x2 - 7 = 3y.2131

We can divide by 3 on both sides; we could write this as y = 6x2 - 7, and that divided by 3.2137

We have managed to convert this to a rectangular x and y, and it is now in the form y = stuff involving x.2145

What if we are doing the reverse? We are going from rectangular to polar.2153

Last time, we went from polar to rectangular; now we are going the other way.2157

Solve for r in terms of θ; once again, we have the same x = rcos(θ), y = rsin(θ);2160

r2 = x2 + y2; and tan(θ) = y/x.2169

In this case, we see that that is nice; we have rcos(θ), rsin(θ)...we have y; we have x.2176

So, we can just swap those out directly; we can swap them out for what we have here.2182

We swap them out; y is rsin(θ)...equals 2 times...x is rcos(θ), so 2 times rcos(θ), plus 3.2187

At this point, we were told to solve for r in terms of θ, so we need to get our r's on one side, so we can get just r by itself.2199

We move the 2rcos(θ) over by subtracting it, so we have rsin(θ), now minus 2rcos(θ), from both sides;2207

that equals...+3 is still left over here; now, notice: we have an r here and an r here, so we can use the distributive property in reverse;2214

we pull that r out, and we have sin(θ) - 2cos(θ) = 3.2222

We now divide that out; so we have just r; so r = 3/(sin(θ) - 2cos(θ)).2232

That might be a little surprising; that seems like a fairly complicated thing, if it is just going to give us a line.2243

But some things...polar is better at graphing certain kinds of pictures, and rectangular is graphing other kinds of pictures.2248

So, it depends on things; rectangular is great for graphing lines; polar is not as great at graphing lines.2255

And you might be surprised that that would even end up coming out to be a line.2259

Try plugging it into a graphing calculator, and you will see that that ends up giving us y = 2x + 3.2263

It is just another way of graphing it.2267

You might have a little bit of difficulty seeing why that ends up giving it.2269

If you think about it, this bottom part here is going to sort of work as an asymptote,2272

as it approaches the same angle that this line is based on, which is why it is going to shoot off infinitely in both the top and the bottom.2276

So, think about that for a while; try graphing it.2283

Just in general, try to play around with graphing as many polar functions as you can.2285

It will really give you such a great sense of how the stuff is working if you just play with it for a while on a graphing calculator.2289

All right, we will see you at Educator.com later--goodbye!2294