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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Parametric Equations

Slide Duration:

Table of Contents

I. Introduction
Introduction to Precalculus

10m 3s

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

45m 11s

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

35m 31s

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

35m 2s

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

48m 43s

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

56m 31s

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

39m 54s

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

58m 26s

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

48m 49s

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

29m 20s

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

48m 35s

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

33m 24s

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

51m 42s

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

49m 37s

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

28m 49s

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

38m 41s

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

41m 7s

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

39m 43s

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

45m 34s

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

46m 8s

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

45m 36s

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

19m 9s

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

33m 22s

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

34m 16s

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

49m 7s

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

44m 56s

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

35m 17s

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

47m 4s

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

40m 31s

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

42m 33s

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

34m 10s

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

48m 46s

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

39m 5s

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

43m 16s

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

33m 5s

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

52m 3s

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

36m 4s

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

27m 18s

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

32m 58s

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

31m 8s

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

19m 11s

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

23m 16s

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

52m 52s

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

29m 5s

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

43m 55s

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

25m 43s

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

56m 40s

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

49m 5s

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

27m 37s

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

34m 25s

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

55m 40s

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

1h 13s

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

41m 1s

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

1h 9m 31s

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

35m 20s

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

54m 7s

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

47m 12s

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

58m 34s

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

53m 33s

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

48m 7s

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

38m 16s

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

40m 43s

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

57m 37s

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

31m 36s

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

44m 3s

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

36m 58s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

57m 45s

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

40m 27s

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

31m 36s

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

39m 27s

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

49m 53s

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

1h 13m 13s

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

40m 22s

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

57m 11s

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

32m 40s

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

32m 43s

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

32m 49s

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

51m 13s

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

45m 26s

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

10m 41s

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

10m 51s

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

10m 38s

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

9m 45s

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

7m 8s

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

1 answer

Last reply by: Mohammed Jaweed
Tue Sep 22, 2015 12:35 AM

Post by Mohammed Jaweed on September 22, 2015

You did the math wrong on the first problem. when you plugged in 2 for x you got -1, but it's supposed to be 1.

1 answer

Last reply by: Professor Selhorst-Jones
Sun May 31, 2015 11:11 AM

Post by Datevig Daghlian on May 26, 2015

Thank you for the lecture! Great explanation!

Parametric Equations

  • Parametric equations are a new way to look at graphing. Instead of graphing input versus output, we'll base both x and y on a third, new variable: a parameter.
  • Using this new idea for graphing, we can describe a set of points in the plane (a graph) as a plane curve. A plane curve is created by two functions f(t) and g(t) defined on some interval of the real numbers. The curve is the set of points (x,y) = ( f(t),  g(t) ). The equations
    x = f(t)        and        y=g(t)
    are called parametric equations and t is the parameter.
  • Graphing with parametric equations is very similar to "normal" graphing. You plug in a value, then see what point you get. Just instead of plugging in x to get y, you plug in t to get x and y.
  • If we want to show the direction of motion in the plane curve, we can draw arrows along the curve to show which way it moves.
  • Sometimes it's useful to turn a pair of parametric equations into an old fashioned rectangular equation (one using just x and y). To do that, we must eliminate the parameter t from the equations. We do this by solving for t in one equation, then plugging it into the other. POOF! No more parameter. [Caution: Be careful when eliminating parameters. We will sometimes need to alter domains to keep the same graph. Furthermore, it is not always possible to solve for t directly, so occasionally we'll have to be clever.]
  • If you have access to a graphing calculator, it's great to try graphing some parametric 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.

Parametric Equations

Fill in the table of values for the below parametric equations.
x = 2t2+1,       y=−t3+5                    
t
x
y
−3
−2
−1
0
1
2
3
  • Parametric equations are based on some outside parameter. As this parameter changes, it causes the x and y values to change. Unlike a normal equation, where the change in one variable causes a change in the other, we have the change in some third parameter variable (t) causing changes in both x and y. A table of values is very similar for parametric equations. Plug in the given input value to get the outputs. The only difference is that instead of one input producing one output, we have one input (t) producing two different outputs (x and y).
  • For example, to find the values of x and y at t=−3, simply plug in to the x- and y-equations:
    t=−3        ⇒        x = 2(−3)2+1     =     19

    t=−3        ⇒        y = −(−3)3+5     =     32
  • Repeat the above process for each of the t values given in the table, then fill in the corresponding x and y values.
t
x
y
−3
19
32
−2
9
13
−1
3
6
0
1
5
1
3
4
2
9
−3
3
19
−22
Given the parametric equations below, find the domain of t.
x = 3
√
 
t+7
 
              y = ln(42−t) + 4
  • The domain is the allowed set of input values. If a problem does not specify or imply a specific set of allowed inputs, then we assume the domain is the set of all inputs that do not "break" the function(s)/equation(s). Thus, for this problem, the domain of t is the set of all input values that do not cause the equations for x or y to "break" (be undefined).
  • Looking at x = 3 √{t+7}, we see the only part that can "break" is the square root function. Square root works as long as what is underneath the square root is not negative, so we have
    t+7 ≥ 0     ⇒     t ≥ −7
    Looking at y = ln(42−t) + 4, w see the only part that can "break" is the natural log function. The natural log works as long as its argument is positive (0 is not allowed), so we have
    42−t > 0     ⇒     42 > t
  • Since a value for t must work in both the x-equation and the y-equation, both restrictions on the domain of t apply simultaneously. Combining these restrictions, we have
    −7 ≤ t < 42.
    We can write this in interval notation as  [−7, 42).
Domain of t:  [−7, 42)
Graph the plane curve described by the parametric equations below.
x = 2−t               y = −t2+6
  • Graphing a set of parametric equations is very similar to graphing a normal function/equation. Simply plug in various input values, take down the outputs, then plot the points. The only difference is that instead of plotting input vs. output, we plot the (x, y) values created for various values of t.
  • Create a table of values, filling it out as necessary to get a good sense for how the graph will behave. If you already feel comfortable with the equations and think you have a good idea for what the graph will look like, calculate just a few points. If you're unsure about what it will look like, calculate more until the graph becomes clear to you. We create a table along the lines of the one below:
    t
    x
    y
    −4
    6
    −10
    −3
    5
    −3
    −2
    4
    2
    −1
    3
    5
    0
    2
    6
    1
    1
    5
    2
    0
    2
    3
    −1
    −3
    4
    −2
    −10
  • Finally, once you have enough points to understand how the graph works, plot the points and connect them with curves. When graphing parametric equations, we often also put little arrows along the graph to show which direction the graph "moves" as the t-values move to larger and larger (positive) values.
Eliminate the parameter from the below parametric equations to obtain a rectangular equation that describes the same curve.
x = 2−t               y = −t2+6
  • Sometimes we want to turn a pair of parametric equations into a rectangular equation: an equation using just x and y, like we normally use. To do this, we must remove t from the equations and "merge" the x- and y-equations together.
  • We can do this through substitution. Solve for t in one of the equations, then plug that in to the other equation. In so doing, we will have eliminated t from existing in either equation.
    x = 2−t     ⇒     t = 2−x
    Now that we have t solved for in one equation, plug that in to the other.
    y = −t2 + 6     ⇒     y = − (2−x)2 + 6
  • The above equation of y = − (2−x)2 + 6 is perfectly acceptable as an answer. It describes the same parabolic curve that the original parametric equations gave, but does so without using t. Still, depending on the problem, we might be expected to give an alternative, equivalent form. For example, we might want to be able to easily identify the vertex of the parabola:
    y = − (2−x)2 + 6     ⇒     y = −⎛
    ⎝    		− (x−2)⎞
    ⎠    		2
     
    		 + 6     ⇒     y = − (x−2)2 + 6
    Or we might be expected to expand the right side into a clear polynomial:
    y = − (2−x)2 + 6     ⇒     y = −(4−4x+x2) + 6     ⇒     y = −x2 + 4x + 2
    All of these are equivalent equations, so they are all acceptable answers for the problem.
y = − (x−2)2 + 6 [Note: The above is one possible answer. There are many equivalent equations that would satisfy the problem. See the final step for a more detailed explanation.]
Graph the plane curve described by the parametric equations below.
x = 3cos(t)              y = sin(t)
  • Graphing a set of parametric equations is very similar to graphing a normal function/equation. Simply plug in various input values, take down the outputs, then plot the points. The only difference is that this time instead of plotting input vs. output, we plot the (x, y) values created for various values of t.
  • Create a table of values, filling it out as necessary to get a good sense for how the graph will behave. If you already feel comfortable with the equations and think you have a good idea for what the graph will look like, calculate just a few points. If you're unsure about what it will look like, calculate more until the graph becomes clear to you. For the specific case of this problem, notice that since x and y are based on trigonometric functions, the "interesting" places will occur at the "interesting" locations on the unit circle. Use those locations to help choose which t-values to use in the table. [Remember, unless indicated otherwise, assume you are working with radians whenever dealing with trig functions.]
    t
    x
    y
    0
    3
    0
    π
    4
    2.12
    0.71
    π
    2
    0
    1
    3π
    4
    −2.12
    0.71
    π
    −3
    0
    5π
    4
    −2.12
    −0.71
    3π
    2
    0
    −1
    7π
    4
    2.12
    −0.71
    2π
    3
    0
    Furthermore, notice that the graph will repeat this set of values for t-values above or below this interval because of the periodic nature of trig functions.
  • Finally, once you have enough points to understand how the graph works, plot the points and connect them with curves. When graphing parametric equations, we often also put little arrows along the graph to show which direction the graph "moves" as the t-values move to larger and larger (positive) values.
Eliminate the parameter from the below parametric equations to obtain a rectangular equation that describes the same curve.
x = 3cos(t)              y = sin(t)
  • Sometimes we want to turn a pair of parametric equations into a rectangular equation: an equation using just x and y, like we normally use. To do this, we must remove t from the equations and "merge" the x- and y-equations together.
  • Normally, we do this by substitution: solve for t in one equation, then plug it in to the other. However, for this problem, doing so would not work out very well. For example, if we solved for t in the y-equation we would get
    y = sin(t)     ⇒     t = sin−1 (y)
    This is tough, because inverse sine (sin−1) is not very friendly. We could plug it in, then carefully think through the trigonometry involved, and eventually solve for a rectangular equation. But it would be a little bit ugly. Instead of substituting, it can be really useful to remember the most common identity from trigonometry:
    sin2 (θ) + cos2 (θ) = 1
  • The above identity is true for any θ at all, so it works for t as the variable too:
    sin2 (t) + cos2 (t) = 1
    Thus, if we can get the x- and y-equations to give us sin(t) and cos(t), we can plug into the above instead. Working towards that, we have
    x = 3cos(t)     ⇒     x
    3
    		 = cos(t)       ⎢
    ⎢    		       y = sin(t)
    Plugging in to the above, we get
    sin2 (t) + cos2 (t) = 1     ⇒     (y)2 +⎛
    ⎝    		 x
    3
    		⎞
    ⎠    		2

     
    		 = 1     ⇒     x2
    9
    		 + y2 = 1
    [If you're familiar with conic sections, notice that the above equation gives an ellipse that is exactly the same as the one we graphed in the previous problem.]
[(x2)/9] + y2 = 1
Graph the plane curve described by the parametric equations below.
x = et              y = 2et − 3
  • Graphing a set of parametric equations is very similar to graphing a normal function/equation. Simply plug in various input values, take down the outputs, then plot the points. The only difference is that this time instead of plotting input vs. output, we plot the (x, y) values created for various values of t.
  • Create a table of values, filling it out as necessary to get a good sense for how the graph will behave. If you already feel comfortable with the equations and think you have a good idea for what the graph will look like, calculate just a few points. If you're unsure about what it will look like, calculate more until the graph becomes clear to you. We create a table along the lines of the one below:
    t
    x
    y
    −4
    0.02
    −2.96
    −3
    0.05
    −2.90
    −2
    0.14
    −2.73
    −1
    0.37
    −2.26
    0
    1
    −1
    1
    2.72
    2.44
    2
    7.39
    11.78
    3
    20.09
    37.17
  • Notice that no matter how negative t gets, x will never reach 0, and y will never reach −3. This is because et only has a range of (0, ∞), so it can never reach 0, causing the previously mentioned effects on x and y. We can show that we approach the point (0, −3) but never actually reach it by using an empty circle at that location on the graph. Other than that, plot the points as normal, then connect them appropriately to create the graph. Since it is a graph of parametric equations, we can also place little arrows to indicate the direction of motion as t grows from negative to positive values.
Eliminate the parameter from the below parametric equations to obtain a rectangular equation that describes the same curve. [Make sure that the resulting equation describes the same curve as the original parametric equations. You may need to restrict the domain to do so.]
x = et              y = 2et − 3
  • Sometimes we want to turn a pair of parametric equations into a rectangular equation: an equation using just x and y, like we normally use. To do this, we must remove t from the equations and "merge" the x- and y-equations together. Normally, we solve for t in one equation, then plug that in to the other equation. In this case though, we have x=et and since et is all that appears in the other equation, we simply substitute based on that. Do so:
    y = 2et − 3     ⇒     y = 2(x) − 3     ⇒     y = 2x−3
  • However, there is an issue with this. The equation y=2x−3 does not describe the same graph as the original parametric equations. Compare the graph of y=2x−3 to the graph of the parametric equations, which we found in the previous problem. Clearly, they're different. The difference is that y=2x−3 goes to the left and right forever, while the original parametric graph "starts" at x=0 and only goes to the right from there. The reason for this is that, no matter what value we choose for t, et can never be equal to 0 or a negative number. Thus, since x=et, it must be that x cannot be equal to 0 or a negative number either.
  • This means we need to put a restriction on what numbers x is allowed to be. We restrict x to the interval of (0, ∞). From the parametric equation, there was a similar restriction on y: because y=2et−3, the value for y can never equal −3 or lower. However, once we put a restriction on the domain allowed for x, this issue is resolved as well. If we can never plug x=0 or lower in to y=2x−3, then we don't have to worry about y ever going into its "forbidden" values either.
y=2x−3,    Domain of x: (0, ∞)
Find a pair of parametric equations that describe the below rectangular equation given that t=x3−7.
y = −4x+5
  • Our pair of parametric equations must be in the below structure
    x = stuff involving t           ⎢
    ⎢    		           y = stuff involving t
    Thus, using that we were given t = x3−7, we must somehow solve for x and y as above.
  • From the equation t = x3−7, we can solve for x:
    t = x3−7     ⇒     t + 7 = x3     ⇒     x =
    3
    √
     
    t+7
     
    We have now found our x-equation. Next, we must find y.
  • We know y = −4x+5, and from what we just solved for, we also have x = 3√{t+7}. Therefore we can solve for y in terms of t by swapping out x based on the equation we just found:
    y = −4x+5     ⇒     y = −4⎛
    ⎝
    3
    √
     
    t+7
     
    		⎞
    ⎠    		 + 5     ⇒     y = −4·
    3
    √
     
    t+7
     
    		 + 5
    At this point, we now have an x-equation and a y-equation that are both based on t as the independent variable. We have found a pair of parametric equations.
x =
3
√{t+7},       y = −4·3√{t+7} + 5
A projectile that is launched from some starting location (d, h) with an initial velocity of v0 and an angle of θ above horizontal can have its motion described by the parametric equations
x =⎛
⎝		 v0 cosθ⎞
⎠		 t + d              y = − 1
2
 g t2 +⎛
⎝		 v0sinθ⎞
⎠		 t + h,
where the constant g is the acceleration due to gravity. On Earth, it is 9.8 [(m/s)/s].

A sailing ship is attacking a citadel at the top of some sea cliffs. The sailing ship is 1000 m from the base of the cliffs, and the cliffs are 100 m tall. The citadel is at the top of the cliffs, situated right next to the edge, and has a height of 20 m. The sailing ship fires a cannonball from a height of 10 m above sea level with a speed of 250 [m/s] and an angle of 10° above horizontal. Does the cannonball hit the citadel? If not, does it strike the cliffs or fly over the citadel? If it does hit the citadel, where does the cannonball make contact relative to the top of the cliffs?
  • Figuring out whether or not the cannonball hits the citadel is a matter of knowing the ball's location at the appropriate moment in time. We can set the bottom of the ship as the origin (0, 0). With that in mind, since the cannonball starts 10 m above sea level, the cannonball starts at the location (0, 10). The bottom of the citadel wall starts at (1000, 100) and the top of the wall is at (1000,  120). Thus, we need to know if the cannonball passes through the little line segment defined by the bottom and top of the citadel walls.
  • We can model the motion of the cannonball by using the parametric equations above. From the problem, we have
    v0 = 250,       θ = 10°,        d=0,        h = 10
    Plugging in to the parametric equations, we get
    x = 250 cos(10°)·t               y = −4.9 t2 + 250sin(10°) ·t + 10
  • Earlier, we realized that the cannonball hits the citadel if it passes through the line segment defined by the points (1000, 100) and (1000,  120). Therefore we want to know the cannonball's height when it makes it to x=1000 (the horizontal edge of the citadel/cliff). However, we can not directly plug x=1000 in to the y-equation to find the ball's height at that x-location. Instead, we must find the time that the ball reaches that horizontal location, then use that to find the height there.
  • Let t1000 represent the time it takes for the cannonball to reach a horizontal distance of x=1000. Plugging in to the x-equation, we can solve for t1000:
    x = 250 cos(10°)·t     ⇒     1000 = 250 cos(10°) ·t1000     ⇒     t1000 = 1000
    250cos(10°)
    Using a calculator, we find that t1000≈ 4.062.
  • Now that we know it takes 4.062 seconds for the cannonball to horizontally reach the citadel/cliff, we need to know what height it has when it gets there. We plug t1000=4.062 in to the y-equation to find the height of the cannonball at that time:
    y = −4.9 t2 + 250sin(10°) ·t + 10     ⇒     y = −4.9 (4.062)2 + 250sin(10°) ·4.062 + 10
    Using a calculator, we work out that the height is 105.49 meters above sea level. Therefore, since the bottom of the citadel is at 100 meters above sea level and its top is at 120 meters, the citadel is struck by the cannonball. Furthermore, we see that the citadel is struck at a height of 105.49−100=  5.49 meters above the top of the cliffs.
The cannonball hits the citadel. It makes contact 5.49 m above the top of the cliffs.

*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

Parametric Equations

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

  • Intro 0:00
  • Introduction 0:06
  • Definition 1:10
    • Plane Curve
    • The Key Idea
  • Graphing with Parametric Equations 2:52
  • Same Graph, Different Equations 5:04
    • How Is That Possible?
    • Same Graph, Different Equations, cont.
    • Here's Another to Consider
    • Same Plane Curve, But Still Different
  • A Metaphor for Parametric Equations 9:36
    • Think of Parametric Equations As a Way to Describe the Motion of An Object
    • Graph Shows Where It Went, But Not Speed
  • Eliminating Parameters 12:14
    • Rectangular Equation
    • Caution
  • 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

Transcription: Parametric Equations

Hi--welcome back to Educator.com.0000

Today, we are going to talk about parametric equations.0002

Up until this point, whenever we have looked at graphing an equation or a function, we thought in terms of input and output.0005

If we put in x for this, what will y come out to be?0010

If we plug in x at 10, y is going to be some number; if we plug in x at 2, y is going to be some number.0014

We always think of x going in and y coming out, and that is how we graph it.0020

We have the x-axis and the y-axis, so we plug in this x, and that goes to a y; we plug in this x; that goes to a y; we plug in this x; it goes to a different y.0024

And that is how we have thought of graphing so far.0031

Parametric equations are a new way to look at graphing.0033

Instead of graphing input versus output, we will base both x and y on a third new variable, a parameter.0037

Parametric equations give us a great new way to look at how time, or some other changing quantity--0044

some changing parameter--some outside thing--affects an object, such as letting us look at motion over time.0049

There are lots of other uses, but motion over time is a really commonly-used one.0055

This way of graphing has a variety of applications in science, calculus, and advanced math.0059

So, let's look at what it is: using this new idea for graphing, we can describe a set of points in the plane, a graph, as a plane curve.0065

As opposed to thinking of a graph as the x's going to y's, we can now just think of it as a bunch of points on the plane.0072

So, how do we get that bunch of points that we call a plane curve onto the plane?0078

A plane curve is created by two functions, a function f(t) and g(t), that are defined on some interval of the real numbers.0083

t is allowed to go from some set of numbers--has some set of numbers like, say, 0 up until 10.0091

And the curve is the set of points, (x,y), equal to (f(t),g(t));0096

that is, as you plug in all of the various t's that we are allowed to have, it creates a bunch of points on the graph.0102

And that is our curve--that is what we get there.0108

The equations x = f(t) and y = g(t) are called parametric equations, and t is the parameter.0110

While we use t most often for parametric equations, since they usually involve time (t for time), we can use any symbol.0119

We could use whatever symbol made sense for the specific parametric equations we were working with.0127

You might see θ show up, or it might be some other letter, depending on what we are working with.0132

The key idea of a parametric graph is that, instead of having x and y be based on each other, we base them on some outside parameter.0137

For example, as time goes on, how does the object move in the x and the y?0145

As this outside thing goes forward or goes to negative values--as this outside thing changes around--how will x and y be affected by this outside thing changing?0151

x and y are no longer directly linked; they are linked to a parameter now.0162

How is that parameter changing, affecting x and y?0166

How do we graph parametric equations?0169

Graphing a parametric equation is very similar to what we think of as normal graphing.0171

You plug in a value; then you see what point you get.0176

It is just that, instead of plugging in x to get y, like we used to, we now plug in t, and we get x and y.0178

We plug in one t, and it gives us (x,y) out of it.0185

So, if we have x = t + 1, y = 2t - 1, and these are our parametric equations,0190

we can just plug in a variety of different t-values, then plot the points that come out for those t-values; and we will have a graph.0195

So, let's start at 0: we plug in 0; then that would give us the point...x is 1; y comes out to be -1.0203

If we plug in 0 for t, we get 0 + 1 (1); it is 2t - 1 for y, so 2(0) - 1 gets us -1.0210

So, that would come out to be: at t = 0, we have (1,-1) as a point.0220

We go and we plot that: (1,-1)--we plot that point.0226

We do the same thing at time = 1: we have 1 + 1, so 2; and 2(1) - 1 comes out to be 1; so we have the point (2,1).0231

So, we go and we plot (2,1), and that is on it.0243

And we have already drawn this; so if t is allowed to vary, just have complete varying, going over everything (that red line that we see there).0246

But we could just keep plotting points like this: (2,3)...3...we would get...2 + 1 comes out to be 3; 2 times 2 minus 1 is 4 - 1, so 3; we would get (3,3).0253

If we went to negative values of t (we aren't required to only have positive values of t--0266

we weren't given any limitations on what t could be, so we have to allow for t to be anything when we are graphing it)--0270

if we plug it t = -1, -1 + 1 comes out to be 0; 2(-1) - 1 comes out to be -3; so we have (0,-3).0277

So, at this point, it becomes pretty clear that we are graphing a line,0288

although in this picture specifically, we had already seen the line.0291

But it would become clear to us that we are graphing a line; and we could graph points in between them,0294

as well, if we wanted to get an even finer idea of what is going on.0298

But it is pretty obviously a line.0301

The same graph with different equations: it is possible to create the same plane curve with different parametric equations.0304

This is kind of surprising, because we think of the equation changing automatically meaning0310

that the graph will change, that our picture, our plane curve, will change.0313

On the previous slide, we graphed x = t + 1; y = 2t - 1.0317

That was what we had on the previous one.0323

On this slide, we have the exact same graph, but these new equations, x = 3t + 1, y = 6t - 1.0324

But the plane curve, the graph that we get out of it, looks exactly the same as the one we just had on the previous slide.0332

How is this possible--what is going on? Let's investigate.0338

We have x = t + 1; y = 2t - 1 as our left side, and x = 3t + 1, y = 6t - 1 as our right side.0342

Both pairs of equations will produce the same plane curve, will produce the same set of points for our graph.0353

But there is a difference: the second pair is moving faster--this set of equations here is moving faster, in a way.0360

It will move three times as far for the same change in t.0368

Let's try some values here: for our red equations here, if we plug in 0 for t, then we end up getting (1,-1), this point right here.0372

If we plug in 1 for t, we will get (2,1) out of it, this point right here.0385

Compare that to the blue one: if we plugged in 0 in the blue one, we will have 3(0) + 1, so we will be at 1 in our x;0392

and then, 6(0) - 1...so -1; we will have the same starting location.0400

At time = 0, at t = 0, we will be at the same place as in the red one.0405

However, when we plug in time = 1, t = 1, we end up getting 3(1) + 1 is 4, and our y-value is 6(1) - 1; it comes out to be 5.0410

We have managed to go much farther than we did; look at how much farther the blue graph has gone than the red graph.0421

So, the blue graph has managed to go three times as far.0428

We would have to go out to t = 3 to be able to get the same set of points: we would have 3 + 1 = 4 and y...2(3) is 6, minus 1 is 5.0432

So, if we had plugged in t = 3, we would be at the same point, (4,5).0442

So effectively, the blue graph is moving three times faster.0448

It does the amount that would take one time interval in blue--that would take three time intervals in red.0451

So, three time intervals in red is one time interval in blue.0459

That is one way of looking at it: they have the same picture, when we just look at it;0463

but if we think about how fast the point is moving in regards to t changing around, they are totally different in regards to that.0466

All right, here is another one to consider: x = -t + 1, and y = -2t - 1.0474

We get, once again, the exact same graph; what is going on here--how is this possible?0482

Well, we actually have the same plane curve, but once again, the motion, how the t change shows which point we are at, is completely different.0487

At 0, we are at (1,-1), just like before.0495

However, as we go to positive numbers...if we plug in 1, we get (0,-3), because -1 + 1 gets us to 0, and then -2t - 1 will get us to -3.0499

So, we will be down here.0510

If we plug in +2, we will be at (-1,-5).0511

Previously, as t went up, as our t increased, the graph went up to the top right; we saw it moving up.0517

And it went down to the bottom left as t became negative.0525

But now, if we plug in a negative value, we get (2,1) for plugging in -1; so we see that it is the opposite.0528

Negative points go to the top right, but positive points go down to the bottom left; so it is the opposite of what it was before.0535

If we want to show the direction of motion--if we want to show which way it is moving--we can draw arrows along the curve.0542

In this case, it would be useful to distinguish this graph from the previous one by showing it with arrows.0549

We just place little arrowheads along the curve occasionally, so that we can see which way it is pointing.0554

That is just two things like that, the arrowhead, just on part of the line...0559

And we wouldn't make it that large, unless we were trying to make a point.0564

We just make little arrowheads, so we can see which way the motion is going.0568

All right, a way that we can think about a parametric equations (and plane curves, and all of these ideas):0572

we can think of parametric equations as a way to describe the motion of an object.0578

As the object moves, it leaves a glowing trail behind where it moved.0582

So, the places that it moves through--what we see is the graph; the plane curve is the places that it has been through.0586

The glowing trail is the plane curve graphed by the equations.0592

So, the equations tell us its motion, its location at a specific time.0596

As it passes through various times, it moves through different locations.0600

And what we see as a graph, what we see as the plane curve, is just where it has been.0603

However, this graph can show us where it went (it shows us where it has been); and it can show us the direction of motion by those little arrows on it.0609

But it can't show us the speed.0616

Consider these three graphs: briefly, really quickly: I am human; I won't make exactly the same graph each of the three times I am going to draw this.0618

But it will be pretty close; the idea is just that, if you drew the exact same graph, we could draw it in three different ways.0625

In our first one, we draw it like this--fairly slow-moving; it makes a loop down here, and then goes up like this.0631

So, if we saw this as a picture, we would be able to see its motion, and we would be able to see where it had gone.0643

But we wouldn't have any idea that it went pretty slowly.0649

But we could have the same thing go through the exact same set of locations;0652

and it would show us the same motion, but we would have no idea that that one went so much faster.0660

That red one went so much faster than the blue.0665

My picture isn't exactly the same between the two; but let's pretend it is.0667

So, the red one has the same motion and the same picture as the blue one, but only by having watched it move were we able to see that it went faster.0672

Finally, we could have one that is different than both of those,0681

where it starts slow, and then it speeds up, and then it slows back down, and then it speeds up,0683

and then it slows back down, and then it speeds up...so it is changing around as it goes through it.0690

Once again, my picture is not quite perfect; it is a little bit more jerky than I would like.0696

But what we are seeing is that we can only show where it is being and the direction it went.0700

But once we are looking at a still, 2D picture, we can't see the speed of motion.0706

That is the difference, in many ways, between what we are seeing with the 3t + 1 and 6t - 1, versus the set of parametric equations.0711

We are just seeing how fast it is going.0721

They give us the same picture, but the picture isn't quite everything with parametric equations.0723

There is also this question of how fast it managed to move through that picture.0727

All right, sometimes it is useful to turn a pair of parametric equations into an old-fashioned rectangular equation.0731

A rectangular equation is just a fancy way to say an equation that only uses x and y,0737

where it is the two things related to each other, like we were used to before in math.0742

To do that, we must eliminate the parameter t from the equation.0746

How do we go about eliminating parameters from parametric equations?0749

Well, we do this by solving for t in one equation; and then, since we have t--we know what t is, as a value--we can plug it into the other.0753

We plug that into the other; there is no more parameter--it is gone.0761

For example, the ones that we were working with were x = t + 1, y = 2t - 1.0764

Well, what we do is: we would solve for the t here, and so we get x - 1 = t.0769

It is not too difficult; at that point, we can take this value for t, and we can plug it in here;0774

so we have x - 1 taking the spot of the t over here in the y equation.0779

We have y = 2(x - 1) - 1; we simplify that, and we get y = 2x - 3, at which point we have managed to solve this;0784

and we have a rectangular equation--we have eliminated the parameter from it, because we no longer have t.0794

Notice: if we were to graph y = 2x - 3, it would give us the exact same picture as the x = t + 1, y = 2t -1.0800

But because it no longer has the parameter, it no longer has a direction, and it no longer has this idea of speed.0808

So, when we turn it into a rectangular equation, it is just a question of what its graph looks like.0813

These ideas of speed and direction disappear once we get rid of the parameter,0818

because it is the parameter changing that allows us to have the idea of speed;0822

it is the parameter changing that allows us to have the idea of which way we are moving.0826

Are we moving from left to right? Are we moving from right to left?0829

I want you to be careful when you are eliminating parameters.0835

You will sometimes need to alter domains to keep the graph the same.0837

Furthermore, it is not always possible to solve for t directly, so occasionally we will have to be clever.0841

Sometimes, you will have to come up with an identity or some other relationship that is going on.0847

You won't be able to get this nice, clean t = something involving a variable.0850

It will be something a little bit more thoughtful, a little bit more difficult.0855

Sometimes, it will be as easy as just solving for t and plugging it into the other one; but other times, it will actually take a little bit of thought and creativity.0858

We will see a little bit of this in the examples.0864

We can also do the reverse of this, where we start with a rectangular equation, and then we want to parameterize it.0867

We want to get a parameter into that, so that we can express this rectangular equation, instead, using parametric equations.0873

This is really easy to do if the original graph is in this form: y = f(x).0880

That is to say, y is simply a function of x; pretty much all of the things that we are used to0886

of y = stuff involving x...stuff involving x...stuff involving x.0890

If that is the case, you just set x = t; say x is equal to your parameter t, and you are done--it is as easy as that.0894

So, for example, if we had y = x3 - 2x + 3, that is a function of x; x is the only thing that shows up in there, so it is purely a function of x.0900

At that point, we say, "All right, let's say x is equal to t; let's just set x equal to the parameter t."0910

And then, t...x...they are the same thing, so we go back here, and we swap out t's for x, and we get t3 - 2t + 3 for y.0917

So, at this point, we have x = t, y = t3 - 2t + 3; we have parametric equations.0927

And it is really, really easy, if it is in this nice, clean form, y = f(x).0935

Also, if it was in the form x = function of y, where it is x = stuff involving y, then you just set y = t.0940

You can do the same thing in the other direction.0946

However, it won't always be that easy.0949

Sometimes, the original rectangular equation can be a little more complicated, where it has x2 + y2 = 1, or something like that.0951

In those cases, it is going to take more thought.0957

If you want to try to come up with identities or relationships that will help you create the appropriate parametric equations,0959

how can you get these two things to connect to each other in a way where you can bring a parameter to bear?0965

There is no one easy way to do this; it will depend on the specific thing you are working on.0970

So, just try to think of things that look similar or that might be connected to this.0973

Just try to play around and use whatever you know to be able to come up with some way to turn it into something0977

where you can get a parameter to show up, at which point you can easily turn it into parametric equations.0982

But it can require some playing around and cleverness that won't just be immediately apparent.0986

So, just think of things that look similar, and see if there is any way to use them.0990

Parametric equations allow us to make really interesting graphs very easily.0995

And by "interesting," I mean things that are crazy, bizarre, cool, strange...they are pretty unlike any other graph that we are used to.0999

Let's start with this red one first, because it is considerable more tame.1007

If x is equal to 3cos(πt) and y = t, well, as t increases, our y is just going to increase, as well.1010

As t goes up, y goes up; however, as t goes up for 3cos(πt), well, at t = 0, cos(π0) would be cos(0), or 1.1017

So, we would be out here at 3; as t goes to 1, we are going to get cos(π) eventually.1027

The cosine of π is -1, so we will end up going out to -3.1035

So, 3cos(πt) is just going to oscillate back and forth; it will end up oscillating faster than normal cos(t), because it has this π factor in there.1038

But other than that, it is just going to end up oscillating back and forth.1048

That is why we see this curve: as it goes up, it oscillates back and forth; let me get that so that it goes your way.1051

That is to the left: so it starts out on the right, and then as it goes up, it bounces back and forth, and we see this oscillation, like that.1058

If we want to draw it in, we can see that from the first two points that we plot about, it is just going to be going back and forth, like this.1067

And we can keep that going down this way, as well.1075

All right, there we go; now let's talk about the blue one.1079

This one is considerably more crazy: x = t times cos(2t); y = 5sin(3t).1082

And we are restricting t to only go from 0 until 6, so it is not allowed to go to anything except 0 until 6.1089

So, this starts...is t is 0 for both of them, then we are going to start out at (0,0), because cosine of 2t...1095

cosine of 0 would be 1, but it is multiplied by 0 times cosine of stuff.1101

So, we start out at (0,0) for the beginning, and then, from there, we work our way out.1105

It goes up and around, and around some more, like that.1110

It is a pretty unusual thing; there is no way that we would be able to write that easily with rectangular coordinates.1123

There is no way that we could create a function--that clearly fails the function test.1130

But as a parametric equation, creating it parametrically is totally fine, and not very difficult to do.1134

We can express this really weird-looking figure in very, very few symbols: t times cos(2t) and 5 times sin(3t); t goes from 0 to 6.1139

We can create these really strange-looking things.1147

Parametric equations give us a lot of power to make really interesting-looking stuff without that much difficulty.1149

This brings us to the idea that graphing calculators are nice to have; why?1155

because working with parametric equations is a great time to use a graphing calculator.1159

It is a totally new way of looking at graphing.1163

Even if you have seen it just a little bit before in previous classes, parametric equations takes a little while to understand--1166

this idea of something that...you are not seeing t on the graph; t doesn't ever show up on the graph.1171

You see what its effects are through x and y, but t itself never shows up on the graph.1177

So, it is this new way of thinking: if t moved, how would it cause x to move--how would it cause y to move?1182

You are thinking in terms of this thing that never shows up.1187

It is a totally new way of thinking about graphing.1190

It really helps to just play around; if you have a graphing calculator, just plot random things.1192

Plot down some equation that you think might be interesting.1198

And then, once you have an understanding of an equation, alter that equation if you already understand it;1201

and see how you can get it to move it in some different way--how you can get the whole thing to move up to 3;1206

how you can get it to move left by 2; how you can get the thing to squish down.1211

What can you do to it to get different stuff to happen--how can you play with the thing?1215

Just do weird stuff to it; play with your graphing calculator, and just get a sense for how parametric equations work.1219

There is pretty much no better way to learn this sort of thing than just playing around for a while with it.1225

You just do weird things, and eventually you realize, "Oh, this all makes sense!"1230

And it will make sense eventually, but you have to get experience with it.1235

And the easiest way to get experience quickly with graphing things isn't by graphing it by hand.1238

That takes a long time; but if you use a graphing calculator, you can get really the best of both worlds.1243

You can see your graphs quickly, but you can also think about what is going on.1246

If you want more information on graphing calculators, there is an appendix on graphing calculators at the end of the course.1249

So, just go and check that out; there is a lot more information there1254

if you don't know much about graphing calculators and if you are interested in getting one.1256

Even if you don't own one, and you know for sure that you are not going to buy one, there are lots of free options out there.1260

In the very first lesson on graphing calculators, I talk about some of the free options that you can have1265

for ways that you can have the function of a graphing calculator without actually needing to go out and buy one.1269

If you are watching this video right now, there are graphing calculators that you can use for free right now on the Internet.1274

And if you just go and play around for a little while on one of these free things, it is going to help massively for understanding how parametric equations work.1280

There is really nothing better that you can do for understanding this stuff than just getting the chance to play around.1287

Also, when you are using a graphing calculator, pay attention to the interval that the parameter is using.1292

Most will only start with t going from 0 to 2π or t going from -10 to 10.1298

But because the interval is limited at the beginning, it might end up cutting off some of your graph.1303

So, you want to pay attention to what interval it starts by giving your t.1308

Set the interval as you need for whatever you are plotting.1313

If you have absolutely no idea what kind of interval you want, you might want to just start with a really, really big interval, like -20 to 20.1315

Or maybe go crazy, like -100 to 100; and that will very, very likely catch anything that you would end up wanting to graph.1321

But it is going to take your graphing calculator longer to work through all of that interval, than if it had a small interval to work through.1328

So, that is something to think about as you are working with the graphing calculator.1334

Also, if you don't quite understand how to get a graphing calculator to work with parametric equations,1337

you can check out the lesson on graphing parametric and polar stuff in the appendix.1342

And we will talk a little bit more about what is actually going to be involved in getting a calculator to be able to work with graphing a parametric equation.1348

All right, let's look at some examples: the first one: Graph x = t2 - 3; y = t - 2; then go on to eliminate the parameter.1355

First, let's see what this thing graphs out as.1362

What we do is make our normal table of values; we plug in some t-values, and that is going to end up giving out x-values and giving out y-values.1365

Instead of giving out one value, it now gives out a pair of values, which is our point.1373

Let's consider if we plugged in 0: if we plugged in 0 into x, 02 - 3, we would have -3.1378

And plug in 0 into y; 0 - 2 gets us -2.1385

If we plug in +1 into x, that gets us 12 - 3, so that will be -2.1390

1 into y...that is 1 - 2, which is -1.1394

2: 22 - 3, 4 - 3, -1; 2 - 2 is 0; 3: 33 is 9 - 3 is positive 6; 3 - 2 is positive 1.1397

If we went in the other direction and we plugged in -1, (-1)2 - 3 is positive 1 minus 3; that gets us -2.1409

-1 - 2 is -3; -2 into t2 - 3...(-2)2 becomes positive 4, minus 3 becomes positive 1; -2 - 2 is -4.1416

-3 squared is positive 9; 9 - 3 is positive 6; -3 - 2 is -5.1428

That gives us a pretty good set to plot--let's plot this out; OK.1435

Let's do markings of length 1: 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3; OK.1447

So, let's start plotting some of these points.1468

We see, at 0, when we plug in time = 0, notice: 0 is not going to show up at all on our graph.1471

But when we plug in at time = 0, we have the point (-3,-2); we go to -3: 1, 2, 3; down 2: 1, 2; we plot our first point.1476

Let's go to positive 1 for time; that is (-2,-1), so it will be here.1490

We plug in time = 2; then we are at -1 for x and...oops, that won't end up being the case.1497

Let's go back a little bit: 22 is 4, minus 3, so that gets us positive 1, not -1; I'm sorry about that mistake that I made a while back there.1510

It makes sense, because that matches up to the -2 value up here; I'm sorry about that.1521

So, plug in time = 2; we are at (1,0) (the 0 is for our y-value).1525

Plug in time = 3; we are at 6: 1, 2, 3, 4, 5, 6; and 1 height.1533

So, we could draw in a curve; let's also think about it a bit.1539

x is basically behaving as a parabola, compared to time; the speed of motion in x, our horizontal motion,1542

is going to speed up as time increases, because it has that t2 factor.1551

So, it will start at -3; but as time gets bigger and bigger, it is going to move faster and faster and faster.1556

What about y? y, on the other hand, is t - 2; it is linear, so it just maintains the same constant rate of increase.1561

It never gets faster; it never gets slower.1567

That is why we end up seeing this top curving out like this, because as time increases more and more,1569

our y-value stays at the same amount of increase, but our x-value moves more and more to the right.1575

So, we move faster horizontally, and that is why we are seeing it curve out like that.1581

If we went to the negatives, we would plug those in as well.1586

At time = -1, we are at (-2,-3); at time = -2, we are at (1,-4); at time = -3, we are at (6,-5).1590

So, we are going to end up seeing the same thing, because it is curving out like this.1604

It basically curves a lot like a parabola.1614

If we want to know which direction...out here at -3, it is here; and then, as it goes to larger times, it curves in this way; so we can see its direction, like this.1617

All right, so now that we have the graph, let's see about eliminating that parameter.1627

We have the graph part done; how can we eliminate that parameter?1632

Well, notice: we have y = t - 2; so if y equals t - 2, then we have y + 2 = t--that is nice.1635

So, at this point, we can plug that in for x = t2 - 3; we have x = y + 2,1645

what we are swapping out for our t; and then we just go back to what it had been, squared, minus 3.1654

So, that this point, we have x = (y + 2)2 - 3, which we could expand if we wanted.1659

But that doesn't really help us understand what is going on any better, so that is a fine answer, right there.1664

And if you wanted to, if you had to, you could expand (y + 2)2 - 3.1670

But for my purposes, that actually makes it easier to understand, because then,1674

it is in a fairly normal form for a sideways parabola, and so we can see that it has been shifted left by that -3.1677

So, it has been shifted left by three, and it has been shifted down by 2, because it has y + 2.1684

So, if you are used to reading this sort of stuff (conics), then you can see that this ends up coming out to make this picture, just the same.1690

All right, let's graph x = 2cos(θ) and y = sin(2θ).1701

The first thing to do here is to think about where the interesting stuff happens.1707

θ is our parameter here, so θ is allowed to change freely; so where would we want to start?1711

Where do we want to stop? How often do we need to have points?1718

Well, the really interesting, the absolutely most interesting, points to look at on a trigonometry function are 0, π/2, π, 3π/2, and 2π.1721

Those are the absolute most interesting parts to look at on a trigonometric function.1731

Notice, though, that we have 2θ; 2θ isn't going to have its most interesting stuff happen at 0 and then π/2,1736

because if you plug in π/2 into 2θ, it is going to put out π.1743

So, we will have missed the π/2; so the most interesting thing for it, the first interesting thing for it after 0, would be π/4.1746

If we have θ = 0, then we will have a sine of 0; if we plug in θ = π/4, then we will have the sine of 2 times π/4, so the sine of 2π/2.1753

That is a really interesting thing to look at, so we will start at 0, and then work our way down with π/4 chunks.1762

We have θ, x, and y; let's plot a bunch of points, so that we can see what is going on here.1771

The first θ is 0; we plug that into here, into our x: 2 times cos(0)...cos(0) is 1, so we just get 2.1785

Plug 0 into sin(2θ); sin(0) is just 0; so that is our first point.1794

Next, we want to do π/4, because we just talked about how the most interesting things are going to be on the π/4 interval.1800

And cos(θ)...well, it has to pay attention to what 2θ's most interesting stuff is.1806

So, π/4 into 2cos(θ): well, 2cos(π/4)...we use a calculator to come up with what that is approximately.1810

That comes out to be around 1.41.1817

We plug in π/4 into sin(2θ); well, that is going to come out to be sin(π/2): sin(π/2) is 1.1820

Continue on with this process: we plug in π/2; π/2 into cos(θ) is going to come out as 0.1827

π/2 into sin(2θ) is going to come out as sin(π); so we get 0 here, as well.1833

Next, 3π/4: plug in 3π/4 in for 2cos(θ); that is now going to be getting us a negative value--it is going to come out as around -1.41.1841

For 3π/4 into 2θ, that gets us 3π/2; 3π/2 is now on the bottom of the unit circle,1853

so it is pointing down at the bottom of the unit circle; so that gets us a -1 in here.1862

Plug in π; π into cos(θ) gets us -1; 2 times that will be -2.1869

The sine of 2π is just going to come out to be 0; notice that we are starting to see this pattern with how this sin(2θ) is working.1875

5π/4...at this point, sine has managed to make one entire arc of the unit circle, because it is doubled up.1883

It has 2θ, so it moves twice as fast.1891

So normally, it is 0 to π for cos(θ), but cos(2θ) does double that, because it is 2θ.1894

So, it has already hit one entire course around the unit circle; so we are just going to end up seeing a full repeat at this point.1900

5π/4: plug that in for cos(θ); then we are down to slightly negative: we end up being in -1.41.1906

5π/4 times 2 gets us 5π/2, which is equivalent to π/2, so we have sin(π/2), or 1.1917

Next, 3π/2; put that in for cosine; you get 0; 3π/2 into sin(2θ) is sin(3π), effectively, which is the same as sin(π), which is 0.1925

We are repeating there, remember.1936

7π/4...when we plug that in, that comes out to be around 1.41; 7π/4 into sin(2θ) comes out to be -1.1938

7π over...let's just write it out...4, times 2, becomes 7π/2, which is the same thing as 3π/2,1948

because we can subtract by 4π/2, and it will still be the same, because that is just one whole unit circle rotation.1961

That is the exact same thing, which explains why we get -1 out of there.1966

And finally, 2π: now our cosine has managed to make one entire wrap, and it is back to 2, and we are back to 1.1970

Notice...oops, we are not back to 1; we are back to 0; sine of 4π is sine of 2π is sine of 0, which is 0, not 1.1978

At this point, we have managed to make an entire wrap on our cos(θ) and an entire wrap, twice now, on our sin(2θ).1987

If we were to keep going up with θ, we would just end up seeing completely repeating values.1993

If we were to have gone down with θ, we would see repeating going in the other direction.1996

So, this is actually enough for us to have.2000

All right, let's draw some axes here: OK, let's make a unit of 1, 2, 1, 2, 1, 1.2002

Here is 1, 2, 1, -1, -2, -1; OK.2023

We plot these points; let's do just the first three, so we can understand what is going on there.2034

The first θ = 0: we have x at 2 and y at 0.2039

At π/4, we have x at around 1.41, so a little bit under halfway to the 1; and then, at a height of 1...2044

And then, at θ = π/2, we have managed to get to (0,0).2053

Let's try to think about what is going on here.2058

2 times cos(θ) is just going to be 2 multiplied on cos(θ)--that is kind of obvious.2060

But cos(θ)--how is cos(θ) going to move from 0 to π/2?2066

Well, that is a question of how the x changes as we spin up to the top.2071

It is going to move faster, the closer θ gets to π/2; that is what we might be used to from how trigonometric things work.2075

It is going to move faster as it gets closer to being at the top.2082

It will move a little slower at the first, which is why it hasn't gotten very far by π/4.2086

But then, it manages to jump all the rest of the way down to 0 by the time it gets to π/2, only another π/4 forward.2090

What about sin(2θ)? Well, sin(2θ) is doubling the speed.2096

So, it manages, by the time it gets to π/4...π/4 has managed to have the sine effectively feel like it is going to π/2.2099

So, it manages to flip up to here and then flip down to here.2107

Sine is a question of how high we are, like this; that is what sine is measuring--the height of the angle for the unit circle.2110

What we end up seeing is it going like this: it cuts through and cuts down like this.2123

So, if you end up having difficulty understanding how we are figuring out that the curve looks like that,2132

and it is not just straight lines going together, just try plotting more points.2136

Any time you have confusion about how to plot something, just plot more points, and that will tell you the story of what is going on.2140

If you are not sure how all of the curves connect, just plot down more points, and the things will start to make sense.2146

The same basic structure is going to go on with the rest of these, so I will move a little faster now.2150

The next one: at angle 3π/4, we are at (-1.41,-1), and then at π, we are at -2 and 0.2153

At 5π/4, we are at -1.41 and positive 1; at 3π/2, we are back to (0,0).2165

At 7π/4, we are at 1.41 and -1; and then we are back to (2,0), and from there, it just wraps.2173

But what we end up seeing is that it does the same sort of curving thing, like this.2182

So, we get this hourglass figure on its side, and it looks kind of like an infinity.2198

And we can see that the direction it is moving is this way; cool.2207

All right, any time you have one of these, and you are really not sure how these things work,2217

in the worst-case scenario, you just plot a bunch of points.2221

Plot really small things for your angle θ; break it into even smaller chunks, and just try it.2224

You could always have just plotted in θ = 0, θ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6...2229

and just use a calculator to get decimal approximations for your x and y.2237

And then, just plot all of those, and that will help you see the curve.2240

And then, once you get a sense for how the curve is moving, you will be able to use that to not have to plot as many θ values,2242

as many time values, for later parts of the curve.2248

Always just plot more points if you are not sure how the curve looks.2252

All right, the third example: Eliminate the parameter, but make sure that the resulting rectangular equation gives the same graph.2256

The hint here is to control the domain in the rectangular equation.2262

Before we try to eliminate this, let's look at what x = et, y = e2t + 7 would look like.2264

It would actually look a lot like a line, surprisingly...2271

Actually, no: it won't look a lot like a line--it will look a lot like a parabola, surprisingly.2274

But there is this strange thing that is going on.2278

Think about what values et can become; if you plug in really large values for t, you get huge values out of et.2280

But if you plug in really, really negative values for t, you can only get close to 0.2288

e to the -100 is 1 divided by e to the 100, which is really, really small, but it is not actually 0.2294

You can never get actually to 0; so that means the range for our x = et is everything from 02301

(but not including 0, because it can never actually get there) up until positive infinity.2310

Similarly, the range for y = e2t + 7...well, that +7 will always be added in;2315

so the question is how small e2t can get.2322

Once again, it can only get close to 0; it can't even actually make it up to 0.2325

So, we can only make it up to 7, but can't actually get to 7; 7, but not including 7, is the bottom of the range.2328

But e2t can become arbitrarily large, so we can get all the way up to positive infinity.2336

With that idea in mind, let's get rid of this parameter--let's eliminate this parameter.2340

We know that x equals et, so we can rewrite y = e2t as et times 2, plus 7.2344

We are still not quite sure how to plug into that; we can write that as (et)2,2353

which we remember from what we learned about exponents: (et)2 + 7 =...2357

at this point, we see x = et, so we swap out, and we have x2 + 7.2362

So now, we have y = x2 + 7.2371

However, what we figured out about range, right from the beginning, was that x can never get below 0.2375

Our x can't actually get to 0; they can't go more to the left of 0, so we can't actually get to the 0 for x.2382

Similarly, the y-value can't actually drop below 7; it can't even get quite to 7.2391

So, y = x2 + 7 is kind of a problem, because while that will never get below 7 for its range,2396

we will certainly be allowed to put in x-values that are different than 0 to infinity.2403

We will be able to put negative infinity into this; so we have to restrict the domain.2407

We restrict the domain, and we say that the domain for x is going to be from 0 up until infinity.2411

If we allow 0 up until infinity, well, then, we have satisfied the range that was allowed for x.2419

What our x-values are allowed to be is only between 0 and infinity.2423

And we have also satisfied from 7 to infinity, because if we can't ever actually plug in 0 for x2,2427

y (which is 02 + 7) will never come out to be...we will never be able to get y = 7 out of it.2432

We will only get really close to it, which is going to satisfy the range for our y, as well.2436

So, that also will have to have this domain before our answer is truly right.2441

The parameter, eliminated, gives us the rectangular equation y = x2 + 7.2445

But we have to have this restriction, that the domain has to be x going from 0 up until infinity, not including 0.2452

The fourth example: Notice that x = cos(θ), y = sin(θ), gives a circle centered at (0,0) with radius 1.2459

We won't show this precisely; if you are not sure about this, try graphing it--it is pretty cool.2465

It comes out to be pretty clear, if you just go through a few points.2468

So, x = cos(θ); y = sin(θ) gives us (0,0) with radius 1; we end up getting a circle with radius 1.2471

All right, let's think about if we wanted to give a parametric equation for a circle centered at (3,-2) with radius 5.2481

How could we do each one of those things?2488

Well, we could move the center by adding things to x and y.2490

If we have x = cos(θ), y = sin(θ), well, if we just add 3 to x (it is 3 + cos(θ)),2496

well, then, all of our x-values will have shifted the entire thing horizontally to the right by 3.2503

We will have shifted the entire thing horizontally, because we just added the 3 to x.2509

So, every x point just moved over 3.2513

If all of the points move over 3 at once, we have just moved the center 3 horizontally.2516

If we want to move the center vertically, we just add or subtract the amount to our y.2520

We can move the center to (3,-2) with x = 3 + cos(θ), y = -2 + sin(θ).2524

And that will give us a circle that is moved over 3 and down 2, and so we will be down here.2538

If we want to expand to a radius of 5, if we want to increase our radius, then we are going to end up just multiplying the x and the y by 5.2547

If we make them bigger by 5 on the whole thing, then that means every point of it just went out by 5.2558

So, we can multiply 5 on cos(θ) and 5 on sin(θ), and we will end up having expanded the entire circle by 5.2563

We can change the radius to 5 by having it be x = 5cos(θ), y = 5sin(θ).2569

And that will end up giving us a larger circle that now has a radius of 5.2580

Notice: if you wanted to make an ellipse, where, instead of having a constant radius everywhere,2590

you wanted to maybe make the radius (no longer technically a radius) smaller on the top,2594

but then expand out, and then smaller again--if you wanted to have a major axis and a minor axis--2599

you could not use the same number multiplied on your cos(θ) and your sin(θ),2604

so that one of them will end up being larger out, and then it will shrink down for the other one.2609

That is one trick if you want to be able to make an ellipse.2613

OK, we take these two ideas, and we put them together; we combine moving the center and expanding the radius out.2619

So now, we have a radius of 5 that is 5cos(θ) and 5sin(θ), x and y respectively.2626

So, if we want to move that, we just add 3 and -2; we put these two ideas together, and we get x = 3 + 5cos(θ) and y = -2 + 5sin(θ).2631

That would end up getting us a circle that starts at (3,-2) and has a radius of 5; cool.2647

All right, one little idea, before we get to our very last example: projectile motion.2664

This is a really good use of parametric equations.2670

A projectile that is launched from some starting location, whether that means thrown,2673

shot from an arrow, shot from a gun, thrown out of a catapult--whatever it is--2677

any projectile--anything that is moving--a human cannonball--whatever it is--anything that is moving from some starting location, (d,h),2681

where d is the horizontal location and h is the vertical height, with an initial velocity of "v naught," v0,2688

pronounced "v naught," N-A-U-G-H-T, like "all for naught," and an angle of θ above the horizontal2696

(how much above the horizontal--if we wanted to draw that in here, then this here would be our angle θ,2705

and this is our v0, how fast we started out before gravity started to affect us and pull us back down to the earth),2713

if we have these ideas here, it can have its motion described by the parametric equations2719

x = v0cos(θ)t + d (our starting horizontal location),2725

and y = -1/2gt2 + v0sin(θ)t + height.2733

In the equation for y, we are probably wondering what this g is.2741

Well, this g is the constant of acceleration for gravity.2744

So, on earth, we have an acceleration of gravity of 9.8 meters per second per second.2749

Equivalently, in the imperial system, it is 32 feet per second per second.2756

We have this acceleration; if we went somewhere else, like, say, the moon or Jupiter, we would end up getting a different value of g.2761

But most of the problems we end up ever looking at are on earth, so you will probably end up seeing 9.8 meters per second per second a lot.2767

Let's understand just how this is working.2776

We have this initial location that it gets shot out of.2777

Then, this velocity ends up going out, and then gravity pulls the thing down; gravity is always pulling down on the object.2780

It gets pulled down more and more and more and more and more, until eventually it lands and hits the ground.2788

If you take any object, and you toss it up, if you were to carefully graph out what it looked like--2793

if you were to see what it was, you would see it as an arc of a parabola; this is true for any thrown object.2802

Any object ends up having a parabolic arc to it.2807

And so, that is what we are seeing, because we are seeing this parabolic arc as a parametric equation.2811

All right, we are ready for this final example.2817

We have a reminder of these formulas on the top, and our problem is: A marauding horseback archer fires an arrow at a castle.2818

And he is on a height of 2 meters, because he is on top of a horse.2825

So, he fires, and he starts at a height of 2 meters; and the arrow comes out2828

with a speed of 50 meters per second and an angle of 25 degrees above the horizontal.2831

He starts at a height of 2 meters with a speed of 50 meters per second and an angle of 25 degrees above the horizontal, all for the arrow.2837

The castle walls are 80 meters away; he is firing at a castle, and the walls of the castle are 80 meters away, and they are 10 meters tall.2844

How far above the wall is the arrow when it flies over?2851

Let's draw a little picture: we see our man--he is on horseback, but I will not draw the horse.2855

And there are castle walls out here in the distance.2860

And so, there is a distance of 80 meters between him and them, and the castle walls are 10 meters tall.2866

He fires an arrow, and it flies through the air.2874

And the question we want to know is: Just as it gets above this wall, what is the extra height above that wall?2882

How high is the arrow above the wall?2892

And then, it will end up coming and landing on the other side; hopefully it won't hurt anybody.2894

Well, he is a marauder.2898

OK, let's see how we can figure this out.2900

Our first question is when the arrow is above the wall.2903

We have this great formula here; we can figure out what the height of the arrow is if we know the time,2907

because we know what v0 is (it is 50 meters per second); we know v0 = 50;2913

we know θ = 25 (he fired it at an angle of 25 degrees above the horizontal);2919

we know that its starting height was h = 2, and we were given g; so that is everything that we need for y, except for the time.2927

But we don't know what time it is before it manages to make it over those walls.2935

What we need to do is: we first need to figure out when it makes it to the walls--at what point is it at the walls?2938

x = v0cos(θ)t + d: we know what v0 is--it was 50; we know what θ is--it is 25.2945

We know how far they are, so we know what our x-value is going to be: it is 80 meters away.2953

So, finally, what is our d? That is the one thing we don't know there.2959

What is the d? Well, let's just say that where the horseback rider starts is 0.2963

We might as well make his horizontal location 0; so he starts at 0 horizontally, so 0 = x here, and then this is 80 = x here.2969

The castle walls are at 80; he starts at 0 in terms of horizontal x location.2979

At this point, we are ready to solve this thing.2984

In general, we have that, for any horizontal location, x is equal to 50 (our initial speed, v0),2986

times cosine of 25 degrees, times the amount of time that the arrow has flown, plus our initial location (our initial location was 0).2994

At this point, we want to solve: so at x = 80, our time is equal to what?3004

We plug in 80 = 50cosine of 25 degrees, all times time.3011

We divide by 50 times cosine of 25 degrees, so we get t = 80/50, times cosine of 25 degrees.3019

We plug that into a calculator, and we get that t is approximately equal to 1.765 seconds.3029

So, after 1.765 seconds of flight time, the arrow is now at the horizontal location of the walls.3038

So, after 1.765 seconds, we are at the walls; so now we can plug that in; and we can figure out,3045

once it makes it to the walls horizontally, how high up it is--what the arrow's height is once we are at the walls horizontally.3051

So now, we use y = -1/2 times 9.8 (our acceleration due to gravity) t2 + v0...3058

50 times sin(θ), sine of 25 degrees, all times time, plus our initial starting height;3072

our initial starting height was 2, because he fires the arrow--he is on top of a horse,3079

so he is firing it from above the ground; he is not firing it from actually the level of the ground.3086

So, we will start working through that: we want to plug in at time = 1.765 seconds,3090

because that is the time that we are interested in knowing the height.3098

We plug that in here; we plug that in, and we get y =...-1/2 times 9.8 is -4.9, times 1.765 squared, plus 50sin(25) degrees, times 1.765 + 2.3100

It is kind of a lot there; but at this point, we can work this all out.3123

We work it out with a calculator, and we get that it is at 24.03 meters high.3126

So, the arrow is 24.03 meters high when it gets to the walls.3131

However, that is not our answer; we were asked how far above the walls when it gets to it.3137

So, how far above the wall is it?3144

At this point, we take 24.03 minus...the height of the walls is 10 meters tall, so minus 10.3147

That will give us the amount that it is above the wall, so that comes out to be 14.03 meters above the castle walls when it flies over them.3156

All right, that finishes up for parametric equations.3169

The important part is to think about it as describing the motion of an object in terms of its time.3172

Try to think about it as how time would change x, how time would change y...3177

Try to think of both of those together, and you will start to slowly build up a sense of how parametric equations work without even having to graph them.3181

Mainly, experience is a great way to learn how to do these things.3188

But you can really speed up the process of learning and understanding parametric equations3191

by just playing around, honestly, for five or ten minutes with a graphing calculator--just playing around,3194

plugging in random things, and seeing how one thing affects another thing--3199

how changing one constant causes things to move around.3202

Just playing around for five or ten minutes will help you so much more than trying to do 10 graphing problems.3205

All right, we will see you at Educator.com later--goodbye!3210