Vincent Selhorst-Jones

Piecewise Functions

Slide Duration:

Section 1: Introduction
Introduction to Precalculus

10m 3s

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

45m 11s

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

35m 31s

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

35m 2s

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

48m 43s

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

56m 31s

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

39m 54s

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

58m 26s

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

48m 49s

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

29m 20s

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

48m 35s

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

33m 24s

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

51m 42s

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

49m 37s

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

28m 49s

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

38m 41s

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

41m 7s

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

39m 43s

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

45m 34s

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

46m 8s

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

45m 36s

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

19m 9s

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

33m 22s

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

34m 16s

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

49m 7s

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

44m 56s

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

35m 17s

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

47m 4s

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

40m 31s

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

42m 33s

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

34m 10s

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

48m 46s

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

39m 5s

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

43m 16s

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

33m 5s

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

52m 3s

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

36m 4s

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

27m 18s

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

32m 58s

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

31m 8s

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

19m 11s

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

23m 16s

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

52m 52s

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

29m 5s

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

43m 55s

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

25m 43s

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

56m 40s

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

49m 5s

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

27m 37s

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

34m 25s

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

55m 40s

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

1h 13s

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

41m 1s

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

1h 9m 31s

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

35m 20s

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

54m 7s

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

47m 12s

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

58m 34s

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

53m 33s

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

48m 7s

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

38m 16s

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

40m 43s

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

57m 37s

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

31m 36s

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

44m 3s

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

36m 58s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

57m 45s

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

40m 27s

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

31m 36s

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

39m 27s

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

49m 53s

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

1h 13m 13s

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

40m 22s

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

57m 11s

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

32m 40s

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

32m 43s

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

32m 49s

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

51m 13s

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

45m 26s

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

10m 41s

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

10m 51s

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

10m 38s

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

9m 45s

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

7m 8s

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

### Piecewise Functions

• Up until now, all the functions we've seen used a single "rule" over their entire domain. No matter what goes into the function (as long as it's in the domain), the same process happens to the input. In this lesson, though, we'll examine functions where the process changes depending on what goes into the function. This is the idea behind piecewise functions.
• As a non-math example, we could imagine cooking potatoes in different ways depending on the size of the potato. If it's small, you boil and mash the potato. But if it's over a certain size, you cut it up and turn it into fries. In both cases, something happens to the potato. But the transformation is different depending on the type.
• We use the following notation for piecewise functions:
f(x) =

 Transformation Rule 1,
 x is in Category 1
 Transformation Rule 2,
 x is in Category 2
 Transformation Rule 3,
 x is in Category 3
 :
 :
That is, given some input, we first check which category it belongs to, then use the corresponding transformation. [Check out the video for some concrete examples of this.]
• We graph piecewise functions the same way as other functions: a series of points ( x, f(x) ). The difference is that the rule determining where x maps to can change depending on which x we're looking at. Often it looks like the graph "changes" at switchovers between rules: when x switches from one category to another, the shape/location of the curve can change.
• An important graphical note is that we show inclusion with a solid circle. We show exclusion with an empty circle. That way, when we have two categories like x < 1 and 1 ≤ x, we can see which curve in the graph "owns" x=1.
• This is a great time to bring up the idea of a continuous function. We'll occasionally refer to this idea in this course, and it will come up often in Calculus. It's hard to formally define continuous with symbols and numbers right now, but we can understand what it means graphically. All the below mean the same thing:
• All the parts of its graph are connected.
• Its graph could be drawn without ever having to lift your pencil off the paper.
• There are no "breaks" in the graph.

### Piecewise Functions

f(x) =

 3x−3,
 x ≤ 2
 x2 + 7,
 x > 2
What is f(−1)? What is f(2)? What is f(4)?
• This function is a piecewise function. Depending on the input that goes in, different transformations are applied. To figure out the transformation that should be applied, look at the category attached to each transformation rule. For each of the inputs, check which category the input falls in, then use the associated transformation.
• For f(−1), we have −1 ≤ 2, so it is in the first category: x ≤ 2. Thus,
 f(−1) = 3 (−1) − 3.
• For f(2), we have 2 ≤ 2, so it is in the first category: x ≤ 2. Thus,
 f(2) = 3 (2) − 3.
• For f(4), we have 4 > 2, so it is in the second category: x > 2. Thus,
 f(4) = (4)2 +7.
f(−1) = −6        f(2) = 3        f(4) = 23

g(t) =

 √ 3−t + 2,
 t ≤ −1
 − 20 t+2 ,
 −1 < t ≤ 3
 47,
 t > 3
What is g(−6)? What is g(3)? What is g(8)?
• This function is a piecewise function. Depending on the input that goes in, different transformations are applied. To figure out the transformation that should be applied, look at the category attached to each transformation rule. For each of the inputs, check which category the input falls in, then use the associated transformation.
• For g(−6), we have −6 ≤ −1, so it is in the first category: t ≤ −1. Thus,
 g(−6) = √ 3 − (−6) + 2.
• For g(3), we have −1 < 3 ≤ 3, so it is in the second category: −1 < t ≤ 3. Thus,
 g(3) = − 20 (3)+2 .
• For g(8), we have 8 > 3, so it is in the third category: t > 3. Thus,
 g(8) = 47.
[Notice that the third transformation rule is just a constant function. As long as t > 3, the function will always output 47.]
g(−6) = 5        g(3) = −4        g(8) = 47
Graph the function
f(x) =

 −x−3,
 x < −1
 x2−4,
 x ≥ −1
• This function is a piecewise function. Depending on the input that goes in, different transformations are applied. To figure out the transformation that should be applied, look at the category attached to each transformation rule. As you draw the function's graph, notice that each transformation rule will only apply as long as the x comes from its associated category.
• For x < −1, draw the graph of y=−x−3. Remember that this only applies for x < −1, so the graph of y=−x−3 will only exist over that portion of the x-axis. Once we go past x=−1, the graph will change to something else.
• If you're not sure how to draw the graph of y = −x−3, create a table of values and plot points until you see how to graph it. But remember, it only graphs for x < −1.
• For x ≥ −1, draw the graph of y = x2 −4. Remember that this only applies for x ≥ −1, so the graph of y=x2−4 will only exist over that portion of the x-axis. If we look to the left of x=−1, the graph will change to something else.
• If you're not sure how to draw the graph of y = x2−4, create a table of values and plot points until you see how to graph it. But remember, it only graphs for x ≥ −1.
• The only thing left to do is to designate what happens at any "switchover" locations, where it changes between transformation rules. We show inclusion with a solid circle and exclusion with an empty circle. Since x < −1 does not include x=−1, its associated graph (y = −x−3) has an empty circle there. For the other portion, because x ≥ −1 does include x=−1, its associated graph (y = x2 −4) has a solid circle there.
Graph the function
g(x) =

 |x+5|,
 x < −3
 7,
 x=−3
 √ x+3 ,
 x > −3
• This function is a piecewise function. Depending on the input that goes in, different transformations are applied. To figure out the transformation that should be applied, look at the category attached to each transformation rule. As you draw the function's graph, notice that each transformation rule will only apply as long as the x comes from its associated category.
• For x < −3, draw the graph of y=|x+5|. Remember that this only applies for x < −3, so the graph of y=|x+5| will only exist over that portion of the x-axis. Once we go past x=−3, the graph will change to something else.
• If you're not sure how to draw the graph of y = |x+5|, create a table of values and plot points until you see how to graph it. But remember, it only graphs for x < −3.
• For x=−3, the function has a value of g(−3)=7, so plot a point at (−3,  7).
• For x >−3, draw the graph of y = √{x+3}. Remember that this only applies for x >−3, so the graph of y=√{x+3} will only exist over that portion of the x-axis. If we look to the left of x=−3, the graph will change to something else.
• If you're not sure how to draw the graph of y = √{x+3}, create a table of values and plot points until you see how to graph it. But remember, it only graphs for x > −3.
• The only thing left to do is to designate what happens at any "switchover" locations, where it changes between transformation rules. We show inclusion with a solid circle and exclusion with an empty circle. Since x < −3 does not include x=−3, its associated graph (y = |x+5|) has an empty circle there. For x=−3, it clearly includes that x-location, so it has a solid circle. Because x > −3 does not include x=−3, its associated graph (y = √{x+3}) has an empty circle there.
Graph the function
 h(x) = 3· ⎡⎣ [0.5x ] ⎤⎦
• [ [t ] ] is the step function: it produces the greatest integer less than or equal to t.
• For h(x), we have more than just x inside the brackets, so we round down based on the entirety of what is inside the brackets. For example:
 h(3) = 3· ⎡⎣ [0.5 (3) ] ⎤⎦ = 3 ⎡⎣ [1.5 ] ⎤⎦ = 3 ·1 = 3
• Using the above understanding, we can create a table of values that will allow us to graph the function. As you create the table of values, notice that the graph "steps up" at intervals of length 2.
• For each of the steps, we need to appropriately show what happens at the "switchover" location with either an empty circle (exclusion) or a solid circle (inclusion). The left side of each chunk will get the solid circle, because the left side is included each time. For example h(2) = 3, but for any value less than x=2, the function will output less than 3. Thus, x=2 is included on the line chunk at height 3. By this same logic, the right side of each chunk will get an empty circle, because it must be excluded each time.
The graph of f(x) is below. Fill in the question marks in the function by using the graph.
f(x) =

 2x+4,
 ??????????
 −x3+1,
 ??????????
 −7,
 ??????????
• We already know what each of the transformation rules are, we just need to figure out where the "switchovers" occur. Look at the graph to figure out what portion of the graph "belongs" to each of the functions.
• The first transformation rule switches over at x=−1. Since y=2x+4 does not have the end point at x=−1 (which we see because of the empty circle), the category for the first transformation rule is x <−1.
• The second transformation rule starts at x=−1. Since y = −x3+1 has the end point at x=−1 (which we see because of the solid circle), we have −1 ≤ x  ???? so far for the second category.
• By looking at the graph, we see that it switches from y = −x3+1 to y = −7 at the horizontal location x = 2. However, there is no jump in the graph there. Both y = −x3+1 and y=−7 output the same value for x=2, so they "agree" on the location. Since either transformation rule can be considered to "own" the point, there are two possibilities. Either the second category contains it (−1 ≤ x ≤ 2), or the third category contains it (x ≥ 2). Either of these two possibilities is correct, but we can't have both simultaneously, because it doesn't make sense to allow both categories to contain the location.

f(x) =

 2x+4,
 x < −1
 −x3+1,
 −1 ≤ x < 2
 −7,
 x ≥ 2
or, equivalently,    f(x) =

 2x+4,
 x < −1
 −x3+1,
 −1 ≤ x ≤ 2
 −7,
 x > 2
Graph the function
f(x) =

 2x+5,
 x ≤ 2
 − 1 2 x + 3,
 x > 2
.
Is f(x) continuous?
• Graph each portion of f(x) over the appropriate x-locations.
• Don't forget to put a solid/empty circle at the end of each portion depending on if it is included/excluded.
• A function is continuous when all the parts of its graph are connected and it can be drawn without any "breaks".

No, f(x) is not continuous because it has a break at x=2.
Graph the function
g(x) =

 2x+5,
 x ≤ 2
 − 1 2 x + 10,
 x > 2
.
Is g(x) continuous?
• Graph each portion of g(x) over the appropriate x-locations.
• If there is no "break" or "jump" in the graph when it switches transformation rules, then there is no need to put a solid/empty circle, because the value of the function is clearly defined at that x-location.
• A function is continuous when all the parts of its graph are connected and it can be drawn without any "breaks".

Yes, g(x) is continuous because it has no breaks anywhere.
At Fermat's Little Bookstore, each book costs $9. However, there's a discount if you buy in bulk. If you buy more than 20 books, each book after the twentieth costs only$5.
Give a piecewise function p(b) that describes the price of purchasing b books.
• For the first twenty books, the price is simply the number of books (b) multiplied by the cost of each book ($9). • After 20 books though, things change. At first, we might be tempted to simply multiply the number of books (b) by the new price ($5), but notice that will not take into account the fact that the first 20 books cost $9. • We have to account for the change in cost by still charging$9 for the first 20 books, then change the price only for the remaining books.
• We have b−20 remaining books which will get the discounted price. Thus, if we buy more than 20 books, the cost comes out to
 9 ·20 + 5(b−20)
• To create the piecewise function, notice that the first pricing scheme occurs only when 0 ≤ b ≤ 20, while the second pricing scheme only happens for b > 20.

p(b) =

 9 ·b,
 0 ≤ b ≤ 20
 5(b−20) + 180,
 b > 20
At Hooke's Gym (which specializes in resistance training on springs), they are currently offering a discount on membership: for the first year of membership, it only costs $10 per month. If you decide to keep your membership longer than a year, the cost then rises to$35 per month.
Give a piecewise function p(t) that describes the price of being a member at Hooke's gym for t months.
• For the first year (12 months), the price is simply the number of months (t) multiplied by the cost of each month ($10). • After 12 months though, things change. At first, we might be tempted to simply multiply the number of months (t) by the new price ($35), but notice that will not take into account the fact that the first 12 months cost $10. • We have to account for the change in cost by still charging$10 for the first 12 months, then change the price only for the remaining months.
• We have t−12 remaining months which will get the increased price. Thus, if we buy more than 12 months, the cost comes out to
 10 ·12 + 35(t−12)
• To create the piecewise function, notice that the first pricing scheme occurs only when 0 ≤ t ≤ 12, while the second pricing scheme only happens for t > 12.

p(t) =

 10 ·t,
 0 ≤ t ≤ 12
 35(t−12) + 120,
 t > 12

*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.

### Piecewise Functions

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

• Intro 0:00
• Introduction 0:04
• Analogies to a Piecewise Function 1:16
• Different Potatoes
• Factory Production
• Notations for Piecewise Functions 3:39
• Notation Examples from Analogies
• 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
• Example of Continuous and Non-Continuous Graphs
• Interesting Functions: the Step Function 22:00
• Notation for the Step Function
• How the Step Function Works
• Graph of the Step Function
• Example 1 26:22
• Example 2 28:49
• Example 3 36:50
• Example 4 46:11

### Transcription: Piecewise Functions

Hi--welcome back to Educator.com.0000

Today, we are going to talk about piecewise functions.0002

So far, all of the functions we have seen use a single rule over their entire domain.0005

No matter what goes into the function, as long as it is part of the domain, the same process happens to that input.0009

For example, consider x2 + 3; the same process happens, no matter what input goes in.0015

While f(1), f(27), f(-473) all produce different outputs, the function is doing the exact same thing to each.0022

It squares the input; then it adds 2; f(1) is 12 + 3; f(27) is 272 + 3; f(-473) is (-473)2 + 3.0034

Different inputs will make different outputs, but ultimately the process we are going through is the same for each one of these.0046

What if the rule wasn't always the same, though--what if the rule we used, the process we used, changed depending on the input?0053

Instead of always using the same process, the transformation could vary based on what goes into the function.0061

This is the idea behind piecewise functions--functions that do different things over different pieces of their domain.0066

Let's start by looking at two analogies, to get this idea really into our heads.0073

The first analogy: imagine you are cooking potatoes, and you are going to turn them either into mashed potatoes or French fries.0078

Now, since the large potatoes make better French fries, you decide to turn your big potatoes into fries, and to mash up the small potatoes.0085

You just bought a big pile of potatoes, so you have some big potatoes and some small potatoes.0092

So, some of them are going to make better fries, and some of them would probably make better mashed potatoes.0097

So, the big ones become fries, and the small ones will become mashed potatoes--OK, that makes sense.0101

When you start cooking, you begin to work through your pile of potatoes.0106

You pick up one potato at a time, and you start by deciding if it is small or if it is big.0109

If it is a small potato, you boil it and mash the potato; if it is a big potato, you cut it up and turn it into fries--it seems pretty simple.0114

But we are getting sort of that idea of piecewise functions here, because in both cases, something will happen to the potato.0122

The potato will go through a process, but the transformation is different, depending on the type.0128

Depending on what kind of potato we have--if it is big potato or if it is a small potato--different processes are going to occur next.0134

We have to pay attention to what we are putting into our cooking process before we know what steps to take next.0141

Another one--here is another analogy: let's consider a factory.0148

Imagine this factory where, depending on what materials you bring to the factory, it produces different things.0151

If you bring along two kilograms of wood, they make a chair.0157

If you bring in 400 kilograms of metal, they make a car.0160

2 kilograms of wood--the factory makes a chair; if you bring in 400 kilograms of metal to the factory, it makes a car.0165

Now, notice that it is not enough to simply say how much material you bring.0172

If you bring in a mass of 400 kilograms, that is not enough information.0176

We need to know if it is wood or metal; if it is wood, then 400 kilograms of wood at 2 kilograms for a chair means we will make 200 chairs.0181

But if it is 400 kilograms of metal, well, it was 400 kilograms of metal to make 1 car; so you would get one car.0191

400 kilograms isn't enough information; we need to know 400 kilograms and what type it is.0198

We need to know what category it belongs to.0202

So, something is going to happen to the material, no matter what; but we have to know the category that the material belongs to.0205

We have to know what we are putting in--not just a specific number, but what sort of group it is from, before we can tell what is going to happen.0210

This is now a vague sense of piecewise functions; we have this idea that a piecewise function0219

is something where a process will happen to the input, but different things will happen, depending on the specific nature of the input.0225

That is a really good sense for what a piecewise function is.0232

So now, let's consider the notation that is used for piecewise functions.0234

Generally, it is f(x) = [...and this bracket just says that it breaks into multiple different things.0237

There are different possible paths that we can take.0243

So, transformation rule 1 is our normal rule, like x2 + 7; and then it says x is in category 1.0246

So, what that means is that we look at our input; and then we go and we look at the various categories we have.0253

Is x wood? Is x metal? Is the potato big? Is the potato small?0262

We look for which category it belongs to; once we have found that it belongs to category 2,0268

then we go ahead and use transformation rule 2 on that input.0275

We look at the input that is going into the function; we then see which one of the categories it belongs to.0281

And that tells us which of the rules to use.0287

When we want to talk about the function, we have this bracket, so we can see all of the possible rules at once,0289

and all of the categories that go along with the possible rules.0294

Which set of circumstances do we use each rule under?0297

Given some input, we first check which category it belongs to.0301

Then, we use the corresponding transformation; so there are all of these transformation rules; we first check the category; then we use it.0306

Also, notice that since f is a function, two categories cannot overlap.0313

So, categories cannot overlap--why? because, if they overlapped, and they had different rules,0320

we would get two different outputs from using the same input.0326

We would get two different outputs from using two different rules, if the categories overlapped.0330

Remember, if we put x into a function, it has to only produce one output.0335

If we put in x into a function, it can't put out a and put out b; it is not allowed to produce two different things.0341

So, if we put in x, and x belongs to two different categories, each of those rules would have to either be the same thing,0347

or we would have to make sure that the categories don't overlap.0353

We are allowed to have categories overlap; but if that is the case, we have to make sure that the transformation rules0356

produce the same output during that overlapping space; otherwise, we have broken the nature of being a function.0361

All right, let's start looking at some examples to help us get a sense of how to use this notation.0367

So, this isn't really formal mathematics; but we can get an idea of how this notation works by seeing how it would work on those previous analogies.0372

First, our potatoes analogy: potatoes of input...how does our potato function work on input?0378

If we plug in x, the first thing we do is see, "Is x small?"0385

If x is small, then we turn it into mashed potatoes; if x is big, then we turn it into French fries.0389

So, we plug in our potato x, and then we see which category x belongs to.0396

The same sort of thing is going on at the factory.0403

If we plug x, our input, into the factory function, some sort of quantity--some number of kilograms--of a material,0405

we then say, "OK, is x wood? If x is wood, it is x/2," because remember, it took 2 kilograms of wood to make one chair.0414

So, it is x/2 chairs; or if x happens to be metal, it is x/400 cars, because it was 400 kilograms of metal to make one car.0422

We plug in the x; we take the x; we see which category it belongs to; and then we plug it into the appropriate rule, based on the category.0435

Great; all right, let's see an example of the piecewise function, actually working through with numbers.0443

Here is a table; this is the most extreme sort of table we can use.0449

When we are actually doing this, we probably won't want to use a table that has this much possible information in it.0453

But we will get the idea of how piecewise functions work from this table.0459

So, to start with, let's look at what would happen to -4.0463

Well, actually, first let's look...which one would -4 belong to?0467

-4 would be in x < -1; so it is going to belong in x2 - 1, and it is not going to belong in the 2 rule,0471

because -4 is not between -1 and 1, and -4 is not greater than 1.0480

So, it is going to knock out these two rules.0488

Next, -3: -3 is still less than -1, so once again, that knocks out the second rule and the third rule.0491

What about -2? Well, once again, -2 is still less than -2, so that knocks out the second rule and the third rule again.0499

What about if we plug in -1? Well, -1 is not less than -1; -1 equals -1, so we have this less than or equal right here.0506

-1 is less than or equal; it knocks out that first rule, but -1 is still not greater than 1 from our third category.0514

So, it knocks out the third rule, as well.0523

What about plugging in 0? Well, 0 is not less than -1; and 0 is not greater than 1;0525

so our first and third categories just got knocked out--the first and third rules are out.0532

What if we plug in 1? Well, once again, 1 is equal to 1, so it is part of this second category.0535

1 is not less than -1, so it knocks out the first rule; and 1 is not greater than 1, so it knocks out the third rule.0541

We get to 2, and finally 1 is less than 2; 2 is greater than 1; we have 1 being less than 2,0550

so we are using the third rule, which means that our first rule and our second rule are knocked out.0558

What about 3? 3 is still greater than 1; 3 is not less than -1, and 3 is not between -1 and 1, so those rules and categories are out.0565

What about 4? 4 is not less than -1, and 4 is not between -1 and 1; but 4 is greater than 1, so only the third rule gets used there.0575

So now, we have a sense of how this table comes together.0585

So now, let's actually start plugging in numbers.0588

-4 goes into x2 - 1: (-4)2 - 1 gets us +16 - 1, so we get 15.0590

What about -3? We plug in (-3)2 - 1; that gets 9 - 1, so we get 8.0602

What about -2? We plug in (-2)2 - 1; 4 - 1 gets us 3.0611

All right, now we switch rules; for this one, we plug in -1, but -1 doesn't really do anything.0620

All the function says is that, if you are between -1 and 1 as your input, it outputs 2.0627

It doesn't care what you are putting in as an input, as long as it is between -1 and 1.0632

It is going to be constant; it is going to always give the same thing in there; so it is going to just be 2, 2, 2; 2 for all of those.0636

-1, 0, and 1; it is 2...it is going to be a constant value of 2 in that interval.0644

Now, we switch rules once again, and we are at 2, -2(2) + 4; -2(2) gets us -4; -4 + 4 gets us 0.0649

What about 3? -2...plug in our 3...+ 4; -2(3) gets us -6; -6 + 4 gets us -2.0660

Plug in 4: -2(4) + 4...-2 times 4 gets us -8; -8 + 4 gets us -4.0670

We have managed to fill out this table.0679

The important thing is to start by figuring out which one of these inputs is going to go to which category.0681

Where are my inputs going to go? You have to figure out an input and its connection to which of the possible categories it can be connected to.0688

All right, let's also see an example of a non-numerical piecewise.0694

Many lessons ago, when we first introduced the idea of a function, we talked about a non-numerical initial function.0698

It took in names spelled with the Roman alphabet, and it output the first letter of the name.0704

For example, if we gave it the name Robert, the initial function would come along,0708

and it would say, "Your first letter is R, so we put out the letter R; done!"0715

So, it is just going and saying, "Let's grab the first initial and do that."0722

That was our idea of the initial function when we first introduced it.0725

We can have functions operating on non-numerical things.0728

But we can also have piecewise functions on non-numerical things.0730

We can modify that and make a piecewise function; we will have f(x) is equal to two categories.0734

Our first rule will be the first letter of x, the first letter of the name, if the name starts with A to M (x is just a placeholder for a name here).0740

It is the first letter of the name, if x starts with A - M.0749

And then, it is the last letter of the name, if the name starts with N - Z.0752

And notice that that covers all of the possible letters that names could start with: A to M, N to Z;0758

A, B, C, D, E, F, G, H, I, J, K, L, M; N, O, P, Q, R, S, T, U, V, W, X, Y, Z; great.0763

Albert: we plug in Albert, and Albert belongs to the red category; it belongs to starting with A to M.0771

That one is pretty easy; we use the first letter, so it gets A as the letter out of it.0778

What about Isabella? Well, Isabella is between A and M (A, B, C, D, E, F, G, H, I, J, K, L, M);0784

so Isabella also belongs to the red category; so it is going to return an I.0794

What about Nicole? Nicole is an N, so it is using the blue category; so it uses the last letter of the name.0800

The last letter of Nicole is E.0808

Vincent begins with a V, which is between N and Z (M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z).0812

So, we are going to be using the blue category, the second category where you use that last letter, T.0822

What about Zach? If we have Zach, Zach is pretty clearly going to be starting with a Z.0827

That means we are in the second category; we are going to cut off that last letter.0832

We are going to get the last letter, and it will be H.0836

We can have a piecewise function operating on non-numerical things.0838

The idea of a piecewise function is just that we take in things; we see which category the thing belongs to;0841

and then we apply a rule based on the category it belongs to.0848

We figure out the category; then we apply a rule based on it; great.0851

Graphing piecewise functions: how do we graph these things?0857

It is actually really similar to graphing a normal function.0859

A series of points, (x,f(x)): when we graph x2, the reason why (0,0) is a point0862

is because, if we plug in 0, we get 0; (0,f(0)); for x2, we get a (0,0).0868

Then we plug in (1,f(1)), because we plug in 1, and it gets 12; so it is (1,1).0876

We plug in 2, and we get f(2); so that would be 2; and then 22 is 4, so we would be at (2,4).0883

We are moving to a height of 4; and that is how it is working for x2.0890

And that is how it works for graphing any function; we put in the input, and then we see what output that gets to.0893

The difference with a piecewise function is that the rule determining where x is going to map will change, depending on which x we are looking at.0898

So, often it will look to use like the graph is changing at switchovers--that we are breaking from one thing to another.0905

And in a way, we are: we are switching between rules.0911

So, when x switches from one category to another, the shape or location of the curve can change,0914

because all of a sudden, we are doing a new way of outputting things.0919

It is not that nice, smooth connection anymore, because we are working all through x2.0922

All of a sudden, we are jumping from x2 to maybe x3 - 1.0926

And we are going to see, suddenly, a new, totally different kind of output when we change categories.0929

An important graphical note is that, to show inclusion, we use a solid circle.0934

Solid circles say "this point is here--this point is actually being included."0940

We show exclusion, excluding it, saying it is not there, with an empty circle.0946

Empty circles give us exclusion; exclusion is an empty circle; inclusion is a filled-in circle.0952

That way, when we have two categories, like x < 1 and 1 ≤ x, we want to be able to know which curve owns x = 1.0961

And because it is less than or equal to, we would have 1 ≤ x, so it would get the dot at that point.0970

And x < 1 gets to go right up to 1, but it doesn't actually get to include 1; so it uses the exclusion, the hollow, empty circle.0977

Let's see an example: if we graph f(x) = x2 - 1 when x < -1, 2 when -1 ≤ x ≤ 1,0985

and -2x + 4, which is when 1 < x, we would see this graph right here.0996

And I also just want to point out that that is what we just did in our table--our really big table,1001

where we figured out all of the possibilities--that is what we just did a few slides back.1004

So, x2 - 1; we see this portion (I will make it in blue, actually) of the graph right here,1009

because x2 - 1 is a parabola; we are seeing the left portion of the parabola, because it is only1018

the portion of the parabola when x is less than -1.1023

We plug in those values, and we see where it gets mapped.1026

However, because x is less than -1, we have exclusion right here; we are not allowed to actually use that point.1028

It gets right up to it, but it can't actually touch x = -1, because it has exclusion on x < -1.1036

And it is strictly less than; it is not actually allowed to equal -1.1044

Similarly, we have to flip to inclusion on the rule of being 2; so it gets to include it on this.1048

And then, it is just a constant from -1 up until 1; so we have this nice straight line here.1055

And then, in green, -2x + 4: 1 < x; we have this thing right here.1062

Now, why don't we see a dot at this junction right here?1069

Well, it is because it is actually agreed on; this point shows up here, and it would show up here, if it were able to hold it.1076

-2x + 4, evaluated at 2...sorry, not 2, but 1, because we are at the point 1; 1 is our changeover, right here;1084

1 < x and x ≤ 1; so -2...plug in 1; plus 4; we get -2 + 4; so we get 2.1095

If -2x + 4 was allowed to use x = 1, it would end up agreeing with it; so it is actually going to the exact same point.1107

It is sort of a hollow for -2x + 4; but then, it immediately gets filled in for 2.1116

So, ultimately, we don't see a break there.1123

There is no loss, because while one end is excluding it, the other one is including it.1125

And they are all in the same place, so we end up only seeing the inclusion; it is held there; they are together in that place.1129

And that is one example of graphing a piecewise function.1135

All right, this is a great time to bring up the idea of a continuous function.1138

We are going to occasionally refer to the idea in this course, and it is going to come up a whole bunch in calculus.1142

So, it is important to get used to it now.1146

It is hard to formally define continuous right now; right now, we don't have enough symbolic technology.1148

We aren't used to using symbols in the way we would need to to talk about continuous; we can't really talk about it with numbers right now.1154

But we can understand what it means graphically; it makes great intuitive sense in pictures.1160

All of these different things--all of these three different ways of talking about it--all mean the exact same thing.1165

So, if a function is continuous, all of the parts of its graph are connected.1170

Its graph could be drawn without ever having to lift your pencil off the paper.1176

And there are no breaks in the graph.1181

These three things all mean the exact same thing.1183

The parts of the graph are connected; that means that there are no breaks.1185

And if there are no breaks, then you could just put your pencil down and draw the whole thing, without ever having to lift your pencil off the paper.1188

They are three different ways of thinking about it, but they all mean the same thing: the whole thing is connected.1194

It is a continuous flow--this nice, connected piece of information.1198

Great; let's look at some examples to help us understand this.1203

For a function to be continuous, it must be any one of the below statements (we just said them; let's say them again).1206

The parts of its graph are connected; the graph could be drawn1211

without ever lifting your pencil or pen off the paper; and there are no breaks in the graph.1213

So, here is one example of something being continuous.1218

Even though it has this sort of "juke" (it suddenly changes direction there) in this corner,1221

it is still continuous, because the graph connects in this corner.1228

The two ends touch; you can draw it in one smooth thing without ever having to take your hand off of drawing.1232

You could draw it in one smooth thing, without ever having to lift up.1239

So, it is a continuous graph, at least in that viewing window--what we can see.1243

Now, here is an example of something being not continuous.1247

This one right here is not continuous, because we have this break; all of a sudden, we jump locations.1251

We have an empty circle here, and we are now, all of a sudden, in a totally different place.1257

If we were to try to draw this, we could get up to here; but to go any farther,1261

we would have to lift our pencil up, move down, and now start down here.1265

So, we would be in a totally different place; there is a break in the graph.1269

The parts of the graph are not connected; it is not just one nice, connected curve; it is not a continuous function.1272

Finally, another one that not continuous: this one is pretty close to being continuous.1278

We can draw and draw and draw and draw and draw, but there is an empty point here.1284

We have one point; we have this single discontinuity, this single point that has been moved off of the line.1288

We have to move down here to draw in this single point; and then we go back to normal.1294

So, it is really, really, really close to being continuous; but it is not perfectly continuous,1298

because this point here has been moved down here; it is in a different place; it is not where it needs to be, to be continuous.1302

So, a continuous function--all of the parts of its graph are connected.1309

And that means even one point could be out, and it would break the continuity.1312

It would break being continuous; it would no longer be connected.1316

Great; that is an interesting function.1320

At this point, it is probably becoming clear that you can have some kind of weird-looking functions.1322

So far, over the course of algebra and geometry, you have seen pretty reasonable-looking things, like x2, √x, x3...1327

Even the weirdest things you have seen have been pretty reasonable--just sort of smooth curves.1334

But we are starting to see, with piecewise functions, that things can be a little odd.1338

Let's look at another one, a little rule--a rule that is a little more complex than x2,1344

something that is a little more interesting than the ones we have been used to so far.1349

So, here is an example, the step function: sometimes it is also called the greatest integer function (and it will make sense why in just a second).1353

f(x) = double bracket x on both ends; it is just make a bracket; make another bracket; put x inside; and then close both of those brackets.1360

The greatest integer less than or equal to x--what does that mean?1370

The greatest integer less than or equal to x--let's try it out.1375

What would happen if we put in 3? Well, the greatest integer that is less than or equal to 3 is 3, because 3 is an integer.1377

And there are other integers out there, like 2; but 2 is not the greatest possible integer that is less than or equal to x,1387

since there is -1, 0, 1, 2, 3; 3 is the greatest one that is less than or equal to 3.1393

What if we went higher, if we said 4? Well, 4 would be greater than 3; so it is not in the running--it doesn't have a possibility.1402

So, that would be 3; but what if we tried something that wasn't just already a straight integer, like, say, 4.7?1410

If we plugged in 4.7, well, what is the greatest integer that is less than or equal to x?1417

3 is a possibility; 4 is a possibility; 5 is a possibility; and it would keep going in either direction.1423

Well, if we went from the left, let's start like this; we could say, "1--1 is a possible thing! Let's go with 1!"1429

Oh, well, if we look at 2, it turns out that is even bigger than 1, and it is still less than 4.7; so 2 is our best option.1444

Oh, what about 3? Well, 3 is still less than or equal to 4.7, and it is bigger than 2; so it is the best option.1450

Oh, what about 4? 4 is bigger than 3, and 4 is less than or equal to 4.7; so it is the best option so far.1456

What about 5? Oh, wait, 5 is greater than 4.7, so it is not in the running, because it has to be the greatest integer less than or equal to x.1465

So, that means 5 is out of the running, and also anything larger than 5.1478

So, everything less than 1, 2, 3...those are not going to work, because we have found 4, and it is the best so far.1481

And everything 5 or greater isn't going to work; so that means our answer is 4.7.1487

It is basically always rounding down; 4.7 would become 4; 3.5 would become 3; -2.5 would become -3,1492

because we have to round down, and what is below -2.5? -3.1501

All right, and finally, we can also sometimes call this the int(x), the integer function on x.1507

Sometimes you will see it denoted as that; it will be written as int(x), as opposed to [[x]].1514

It is the same idea though--this greatest integer thing, this step function where we are breaking.1520

Now, why is it called a step function? We will look at a picture, and that will help explain it a lot.1525

So, the graph of f(x) = [[x]] looks like this.1529

Why does it look like this? Well, remember, at -3, where do we get placed?1533

Well, -3 is an integer, so it just goes right here.1536

Well, what about anything in the middle? Anything in the middle would get placed onto -3,1540

because they would have to be rounded down to the greatest integer they are connected to.1545

So, that is what we get there.1548

As soon as we get to -2, though, we are going to jump up, because -2 is an integer, so it gets to be used here.1549

And so, it is going to have the same sort of thing; anything in the middle would end up getting placed onto -2.1555

But once we get to -1, we make it to this one; and so on and so forth.1560

And so, we just keep stepping along and stepping along and stepping along.1563

And every time we hit an integer, we jump up to the next height, and so on and so forth.1567

And so, we have the greatest integer less than or equal to x, which ends up looking like a staircase,1572

in terms of its steps, because we keep stepping up every time we hit a new integer.1577

Cool; all right, we are ready for some examples.1582

So, the first one just to get started: let's evaluate this function at four different points:1584

f(x) = 3x + 10 when x < -2, 8 when x = -1, and x2 - 10 when x > -1.1589

All right, at f(-3), first what we have to do is say, "Which category do you belong to?"1597

Well, -3 is less than -1, so it belongs to the 3x + 10 rule.1603

So, we use 3x + 10; we plug in the -3; we have 3(-3) + 10; -9 + 10; so we have 1; f(-3) = 1.1610

Great; what about f(-1)--what does that belong to?1624

Well, -1 = -1, so it is using this category right here, so that means we have 8; so we have 8.1631

There is nothing else that we have to do; it is already as simple as it can be; f(-1) = 8.1638

And that is our answer, right there.1642

What about f(-0.9)? This one is really close to -1; but remember, this thing was x = -1, and only happens on precisely -1.1645

-0.9 is, in fact, slightly greater than -1; -0.9 is greater than -1, so we use the rule x2 - 10.1655

We plug in that -0.9; we have (-0.9)2 - 10; -0.9 squared is 0.81; it becomes positive.1666

Anything squared becomes positive, as long as it is a real number.1676

0.81 - 10 becomes negative, because the 10 is bigger: -9.19.1679

So, f(-0.9) is equal to -9.9.1688

Finally, one more example, f(5); which category does this belong to?1694

It pretty clearly belongs to x > -1; 5 > -1, so we use the x2 - 10 rule once again; we use that process.1699

Plug in the 5; 52 - 10 is 25 - 10, is 15; so f(5) = 15.1707

And there we are; and that is how you evaluate a piecewise function.1720

You see which category it belongs to; then you plug it into the appropriate rule, and you just plug it in and work,1722

like you are doing a normal function at that point; great.1727

The next one: all right, in this one, we will graph a piecewise function; so our function this time is:1729

f(x) = x + 6 when x ≤ -3 and -x2 - 2x + 1 when x > -3.1734

So, first, let's make a table to help us graph this thing.1742

x and f(x); what would be a good place to start out?1746

Well, we have -3 showing up here and here; so that is probably going to be the midpoint, mid-"zone" in our graph.1755

So, let's start by plugging in -3; and we will go more negative as we go up: -4, -5, -6, and we will go more positive as we go down: -2, -1, 0, 1; great.1762

Let's try plugging in...which rule will we end up using?1778

Well, when x is less than or equal to -3, we will end up using the things that are above -3 or equal to -3.1781

So, this rule up here gets the x + 6 portion; and down here, when we are below the line, we get -x2 - 2x + 1,1791

because then x is greater than -3; -2 is greater than -3; 0 is greater than -3; etc.1805

All right, so let's try doing some of these.1811

If we plug in -3, -3 + 6 is going to equal positive 3; -4 + 6, -5 + 6, -6 + 6; what do these all come out to be?1813

-4 + 6 gets us 2; -5 + 6 gets us 1; -6 + 6 gets us 0; so we have a pretty good idea of how to graph the x + 6, the portion of the graph where x ≤ -3.1830

Now, what about going the other way?1844

Well, if we plug in -2 into -x2 - 2x + 1, we have -(-2)2 - 2(-2) + 1.1845

Let's plug in all of them, and then we will just do them at once.1855

-12 - 2(-1) + 1; -02 - 2(0) + 1; -12 - 2(1) + 1; what do these all come out to be?1857

Well, first, -2 squared becomes positive 4; so we hit that with another negative, and we have -4 right here.1874

-2 times -2 gets us +4, + 1, so -4 + 4 gets canceled; and then + 1...we get 1.1882

-1 squared gets us positive 1, but then, hit with another negative, we get -1; -2 times -1 gets us + 2, + 1; so -1 + 2 + 1...we get 2.1892

0 squared gets us 0; -2 times 0 gets us 0, plus 1--we get 1.1903

-1 squared gets us -1; -2 times 1 gets us -2, plus 1; so we get -2 here.1909

Great; all right, at this point, we can start graphing this thing.1920

We are graphing from -6 to 1; and our extreme y-values are...we have from 0, 1, 2, 3, so we will make it 1, 2, 3, 4, -1, -2, -3, -4, -5.1924

That is probably enough information; I have to do down there...1, 2, 3, 4, -1, -2, -3, -4, -5.1947

And there doesn't seem to be any reason why we shouldn't do this on a square axis.1957

So, the tick mark length, the length of our vertical tick marks, can be the same as our horizontal tick marks.1961

And of course, I am just doing this by hand, so it is approximate.1966

But this isn't too bad: 1, 2, 3, 4, 5, 6, and it would keep going out that way, as well.1968

All right, -1, -2, -3, -4, -5, and -6; positive 1, positive 2; great.1977

So, at this point, we plot down our points, just like we are doing a normal thing.1985

-6 goes to 0, right here; -5 goes to positive 1 here; -4 goes to positive 2 here; -3 goes to positive 3 here.1988

And at this point, we have the line portion.2003

Does the line keep going to the right, though? No, because it stops once it goes greater than -3.2005

It only works, the rule only happens, when x is less than or equal to -3.2013

But it would keep going off to the left; so it stops right here, but it does include that point, because of the "less than or equal."2016

Now, what about the parabola part of it?2024

Well, we plug in -2; -2 gets us 1; -1 gets us 2; 0 gets us 1; 1 gets us -2; and 2 would continue down.2026

So, we have a pretty clear parabolic arc going on here.2039

We are used to this; and it is going to keep going straight off forever to the right,2045

because it is x > -3; so as long as we are continuing to go to the right, it will continue on.2049

What happens to the left, though? We know what is happening--it is going to be in a parabolic arc.2054

But we are not quite sure where it is going to land, because it has to stop somewhere.2058

But we don't know what height it will stop at.2062

We know it will stop just before -3; so it will stop at -2.9999999999999...forever, continuing forever and ever.2065

It can't actually touch -3, but it can get infinitely close; it can get right up next to it.2074

So, let's figure out where it would be going if it got right up next to it.2078

What we do is: let's see what would happen if we plugged in -2.99999; now, I need even more nines, right? -2.99999999...nines forever.2082

Now...well, not quite forever, because then it would turn into 3.2093

But the point is -2.lots-of-nines; now, I don't know about you, but I don't want to have to plug in -2.9999999 into a calculator,2096

because it is going to end up getting me these ugly numbers, and I will end up having decimals.2104

And really, when you get right down to it, isn't -2.99999999 going to behave a lot like we plugged in -3?2106

It is so close to -3 that we could probably just plug it in as if we had plugged in -3; and indeed, we can do that.2113

We will just know that it will be an empty circle of that, because it has exclusion; it has strictly greater than.2120

So, we will plug in -2.99999999 and 9, which still belongs to the -x2 - 2x + 1,2126

because -2.999999999 is greater than -3, if only by a little tiny bit.2132

And it is going to behave pretty much the same as if we had plugged in -3, so we can calculate it more easily by plugging in -3.2137

So, -(-3)2 - 2(-3) + 1--what does that come out to be?2144

-(-3)2 becomes -9; -2(-3) becomes +6; plus 1, so we have -9 + 7 = -2.2151

So, we know that this is going to go out to -2 when it gets to -3; but it is not actually going to be at -3.2163

It is going to be hollow there, because we are excluding it--it is not actually allowed to go to that point,2170

because the exclusion was already put on that first category on that first rule.2175

This will curve down in a parabolic arc into the exclusion hole, and then just stop right there.2180

It doesn't actually get to touch -3, but we can basically calculate it as if it had gotten to -3,2185

because -2.999999999999 is so close to being just like -3, we can calculate it as if it had gotten there.2190

But then, we just have to remember that we have to make sure that we put it in this circle here,2198

because we are actually excluding it at -3, because x isn't equal to -3 if x > -3.2202

Great; all those ideas that we just talked about are going to come up a lot with this.2209

Let's go just a little bit off and pretend that we are using the real number, and then see what it is like.2214

And we are going to do that here; so if what I do here doesn't quite make sense, look back at the explanation of that 2.999999 thing.2219

And we will get an idea of "Oh, that is why we can do this sort of thing."2226

So, once again, we will set this up in the same way: (x,f(x)): now what values...2229

Clearly, -2 is kind of important; it shows up in a lot of places.2239

What are we going to do? Well, the first thing that we are told to do in this problem is to give the domain of f; then graph it.2244

First, let's do the domain; how do we come up with the domain?2251

Well, remember, domain is all of the inputs that are allowed to go into a function.2256

x2 - 5 never breaks down; 3 never breaks down; -2x + 1 never breaks down; so none of the rules break down.2261

So, none of the processes, none of the rules, break; they are always defined.2272

However, are the categories always defined?2281

x < -2 means we can just keep on going; we can keep on going.2286

So, it is really negative infinity less than x; so we can go all the way down to negative infinity.2290

What about to the right, though? Is there anything that we are not allowed to get to?2295

Well, negative infinity up to -2; and then -2 is here at equal; and then -2 is less than...2298

so we have covered all of our bases, from negative infinity up to -2; and keep going, up until 1.2302

Are there any rules for what happens if x is greater than 1?2309

No, we don't have any rules; we have x < -2, x = -2, and -2 < x ≤ 1; but we don't have any rules for when x > 1.2313

So, no rules for x > 1 means that f doesn't tell us what to do if we are plugging something in.2324

The f fails to tell us what to do to this input if we plug in something that is greater than 1.2334

If we plug in, say, 500, we look at this, and we say, "Oh, this doesn't belong to any categories."2340

So, f is undefined at 500; it doesn't work; it is not in the domain.2345

So, the domain fails to contain everything in x > 1; so that means our domain is not going to be x > 1.2350

That is the things of failure, because we don't have rules; the domain is everything from negative infinity2362

(we use a parenthesis for negative infinity, and infinity), and we go up until 1; and we include the 1,2367

because we have less than or equal to, but we can't go past it; we have no more rules to go up past it.2373

So, f has a domain from negative infinity up until 1, including 1.2378

Great; now, let's build up that table.2383

-2 seems like a good place to make our middle; and if we are above -2, we will use which rule?2385

We use the x2 - 5 rule.2394

Sorry, by "above," I meant to say more negative than -2.2399

And if we are below on this table, which is to say more positive, closer to 0, we are going to use -2x + 1.2402

So, -2...we will have -3 and -4, -1, 0, 1...but just like we did in the last thing, it will be useful to know2411

where it is going if it had been allowed to get to -2.2423

So, for the above part, we will say -2.0001; and -1.9999; these things are because -.1999999 is greater than -2,2426

and -2.000001 is less than -2, but they are going to behave effectively as if we had plugged in -2.2440

So, when we are actually figuring out the numbers, we can pretend as if we had plugged in -2, just to make it easier on us to do the calculation.2448

All right, the first one, -2: what are our f(x)'s?2454

-2's rule just says to give out 3; it doesn't matter what your input is, even though we have to use the category of -2.2459

So, it automatically gives out 3 at -2.2466

What about -2.00001, which would use the x2 - 5 rule? Well, that is about the same thing as plugging in -2.2471

We have (-2)2 - 5; keep going--let's just keep going up to get them all written out.2477

(-3)2 - 5, and (-4)2 - 5--what do those all equal?2484

Well, (-2)2 becomes 4; 4 - 5 is -1; (-3)2 is 9; 9 - 5 is not -4, but +4; 9 isn't bigger than -5.2491

And (-4)2 is +16; 16 - 5 is 11; OK.2502

What about the other way, if we go to the -2x + 1 rule?2506

Well, if we had plugged in -2, we would get -2 times -2; we aren't literally plugging it in.2510

We are just saying, "What if we had gone all the way up to it? Let's see what would have happened,"2514

even though ultimately we will have to exclude it, because we have these strictly less than and strictly greater than signs.2519

So, -2 times -2 plus 1; -2 times -1 plus 1; -2 times 0 plus 1; 1 times 0...sorry, not 1; sorry about that...2531

-2 times positive 1 (I got that confused with the one above it); -2 times 1 plus 1; what do those all equal?2543

The thing that is effectively going to be like -2...-2 times -2 is positive 4; 4 plus 1 is 5.2551

-2 times -1 is positive 2, plus 1 is 3; -2 times 0 is 0, plus 1 is 1; -2 times 1 is -2, plus 1 is -1.2557

Great; so now we are in a position to be able to graph it.2567

Our extremes...vertically we can get up to really high things when we are in the x2 - 5; so we won't worry about the 11 part.2569

But we are going between extremes of 5, maybe a little lower; so we will graph this...2575

We never get to very low values, it seems; so we will put our corner down here.2585

And we also never get past 1; remember, our domain is only -∞ up until 1.2589

So, we also don't have to have a whole lot of stuff on the right.2593

So, we have positive 1 here, positive 2 here, positive 1, positive 2, positive 3, positive 4, positive 5, positive 6; -1, -2, -1, -2, -3, -4, -5.2597

Great, and that is plenty of room, because we only get up to -4; and we know that x2 - 5 is going to blow out.2622

So, 1, 2, 3, 4, 5, 6, -1, -2, 1, 2, 3, -1, -2, -3, -4, -5; making tables...making axes.2627

So, let's plug in some things and see what happens.2643

We plug in -4, and it goes out to 11; so we can't even see it; it is so high up.2646

-3 gets to 4, though; we can definitely plot that; so -3 goes to 4, (-3,4).2651

-2, if it had a -2...it doesn't actually have it, but we know that -2.0001 would practically be going to -1.2659

So, we are putting an exclusion hole down here, just below the -2.2670

At -2, though, we actually end up being at 3; so we have this single point right there.2676

If we had -2 for the -2x + 1 process, we would be at 5; but we can't actually go there, so once again, we have an exclusionary hole there.2683

-1 is at 3; 0 is at 1; 1 is at -1; and it stops right there, because we stop at 1.2693

We can't go any higher than positive 1; our domain caps out there.2704

So, our straight line is just a straight line, up until where it stops at that exclusionary hole.2709

We have this point in the middle, the 3 point, but it is just x = -2; and then we have a parabola curving up.2715

It is already moving pretty fast by this point, so it is not going to be a nice, smooth parabola like this.2722

It is already moving fast up, because it is pretty far up.2727

It manages to jump from -1 to 4, and then from 4 to 11; so it is not the bottom part of the parabola.2731

It is already in process, in a way; so curve this parabola up, and it just zooms way, way off really quickly.2735

All right, that is basically what we are seeing here for our graph.2745

We have a parabola on the left-hand side, which drops to the single point--there is just a single point in the middle.2747

And then, we switch to -2x + 1, which goes to 1, and then stops at 1,2754

because the category just stops at 1; so it stops right here, and we don't have anything farther to the right.2760

There is nothing further off to the right, because the categories don't include anything further right.2765

All right, the final example: A certain phone company charges $20 for using its service, along with 10 cents for each minute under 200 minutes.2771 After 200 minutes, they charge 5 cents for each additional minute.2780 Let's give a piecewise function, p(t), price in terms of t, that will describe the price in terms of t, the minutes spoken.2784 So, t is the minutes spoken.2791 It is pretty easy for us to figure out what the first part is.2793 The first portion: when we are under 200 minutes, which is to say when t ≤ 200, the price of t is not too hard for that.2796 p(t) =...well, a$20 flat rate--they charge us $20, and then they charge us 10 cents for each additional minute.2812 So,$20 plus that additional 10 cents...how many minutes did we have? We had t.2820

So, 0.1t--that is what it is; let's do a really quick test--let's say if we had talked 100 minutes.2825

Then 100 minutes times 10 cents would be $10, so we would have a$30 total,2834

which, if we plug that into our new function that we just made, p(t), p(100) would be 20 + .1(100), which would come out to be 30.2839

Great, so the first part of it checks out.2846

What about the second portion, though?--that is where things start to get a little complicated.2848

So, in the second portion, when we are over 200, which is to say t > 200...2853

and actually can be greater than or equal, because we know that they are going to have to agree;2861

there is not going to be a sudden jump there; and we will talk about that more later.2865

It is a way of checking this function, actually.2867

We know that it is going to be 5 cents for each additional minute.2870

Our first thought might be, "Oh, great, easy; it is going to be 0.05t."2875

Not true--this is not going to be the case.2880

Why not? Because it is for each additional minute, over 200; so after 200, you get charged at 5 cents per minute.2884

Before that, you still get charged at the 10 cents; so how many minutes over 200?2894

Well, that is not too hard; we know that we have t minutes total.2904

We know that we are already over 200, so it is going to be the number of minutes we have talked, minus 200.2908

So, t - 200 is the number of minutes we talked; so it is 0.05 cents, times t - 200.2914

Now, that is the amount of additional money that will be on top of some lump.2922

How much is it to even make it to 200 minutes in the first place?2927

Well, 200 minutes in the first place: let's see what it is from our first one.2930

p(200) would be equal to 20 + 0.1(200); just move the decimal place over one, so it becomes 20; so 20 + 20 is 40.2934

So, it costs $40 to get up to 200 minutes; so it costs$40 at 200, and then it is plus the additional amount per minute.2947

So, for the second portion, our function is going to be p(t) = $40, the lump sum that we have to pay at first2967 to have even made it to the 200-minute mark, plus 5 cents for the number of minutes over 200 minutes.2975 So, our function has been broken into 2 pieces; so we have a piecewise function here, p(t) = 20 + 0.1t when t ≤ 200,2984 and 40 + 0.05 times the minutes over 200 when t ≥ 200.3001 Now, we know that the two have to agree; otherwise, people would make sure to make that jump or not make that jump,3014 because otherwise with the sudden change, or the switch--it wouldn't make sense for the phone company3019 to have it suddenly leap more on your bill or cut off a portion of your bill if you were to hit the 200 mark.3024 It is going to just continue in a continuous function, we would expect.3029 So, we can check this; and we can check and make sure that, indeed, p(200) = 20 + 0.1(200).3034 We already did this before; it was$40; and let's check and make sure that the second portion, p(200), would agree.3044

40 + 0.05 at the minutes over 200, so that is 200; 200 - 200 is just 0, so that cancels out the 0.05; so we get 40.3053

So, those two things check, and our function, price in respect to time, makes perfect sense.3068

So, p(t) = 20 + 10 cents per minutes when minutes are less than or equal to 200,3075

or 40 + 5 cents per minutes over 200, when the number of minutes is greater than or equal to 200.3080

Great; I hope piecewise functions are making a lot more sense now.3087

Remember: it is an idea about putting into the category, then applying the rule based on the category.3090

That is the prime, the major idea in piecewise functions; if you can hang onto that, you will be able to make sense of them.3095

All right, we will see you at Educator.com later--goodbye!3100