  Vincent Selhorst-Jones

Properties of 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
Bookmark & Share Embed

## Copy & Paste this embed code into your website’s HTML

Please ensure that your website editor is in text mode when you paste the code.
(In Wordpress, the mode button is on the top right corner.)
×
• - Allow users to view the embedded video in full-size.
Since this lesson is not free, only the preview will appear on your website.

• ## Related Books 1 answer Last reply by: Professor Selhorst-JonesFri Aug 23, 2019 10:12 PMPost by Pedro Galvao on August 23, 2019Hello Jones,I'm not a math person but I'm really enjoying your course and how easily and clearly you speak about precalculus.  By the way, what do you think about the book Precalculus by Sullivan 10th edition? 1 answer Last reply by: Professor Selhorst-JonesSun Aug 11, 2013 11:03 PMPost by Tami Cummins on August 11, 2013In the second part of example 2 what about the negative? 1 answer Last reply by: Professor Selhorst-JonesThu Jul 11, 2013 1:36 PMPost by Montgomery Childs on June 26, 2013Dear Mr. Jones,I really appreciate the time you spend on "definitions" of math terms - i have come to realize this is one of the biggest issues i have had over the years - not the math. This helps so much in my understanding of relationships! Very cool!!!

### Properties of Functions

• Over an interval of x-values, a given function can be increasing, decreasing, or constant. That is, always going up, always going down, or not changing, respectively. This idea is easiest to understand visually, so look at a graph to find where these things occur.
• We talk about increasing, decreasing, and constant in terms of intervals: that is, sections of the horizontal axis. Whenever you talk about one of the above as an interval, you always give it in parentheses.
• An (absolute/global) maximum is where a function achieves its highest value. An (absolute/global) minimum is where a function achieves its lowest value.
• A relative maximum (or local maximum) is where a function achieves its highest value in some "neighborhood". A relative minimum (or local minimum) is where a function achieves its lowest value in some "neighborhood". [Notice that these aren't necessarily the highest/lowest locations for the entire function (although they might be), just an extreme location in some interval.]
• We can refer to all the maximums and minimums of a function (both absolute and relative) with the word extrema: the extreme values of a function.
• We can calculate the average rate of change for a function between two locations x1 and x2 with the formula
 f(x2) − f(x1) x2 −x1 .
• It is often very important to know what x values for a function cause it to output 0, that is to say, f(x) = 0. This idea is so important, it goes by many names: the zeros of a function, the roots, the x-intercepts. But these all mean the same thing: all x such that f(x) = 0.
• An even function (totally different from being an even number) is one where
 f(−x) = f(x).
In other words, plugging in the negative or positive version of a number gives the same output. Graphically, this means that even functions are symmetric around the y-axis (mirror left-right).
• An odd function (totally different from being an odd number) is one where
 f(−x) = − f(x).
In other words, plugging in the negative version of a number gives the same thing as the positive number did, but the output has an additional negative sign. Graphically, this means that odd functions are symmetric around the origin (mirror left-right and up-down).

### Properties of Functions

Assume that what we can see on the graph is the entirety of the function f(x). What are the intervals that f is increasing, decreasing, and constant on? • Increasing, decreasing, and constant behave like they sound. Assuming you are "reading" the graph from left to right, an increasing interval means it is going up, decreasing means it is going down, and constant means the height does not change.
• To find the increasing intervals, we look on the graph to see where the function is going up. To find decreasing, we find where it is going down. To find constant, we find where it does not change.
• Write each of these intervals in interval notation, such as (−5, −3). With increasing/decreasing/constant intervals, we always use parentheses, never brackets, because we can't include where the function "turns" in the interval.
Increasing: (−2, 2) and (3,5);    Decreasing: (−5, −3) and (2, 3);    Constant: (−3, −2)
Assume that what we can see on the graph is the entirety of the function f(x). What are the intervals that f is increasing, decreasing, and constant on?
At what x-values do the absolute maximum and minimum occur? What are the values for the maximum and minimum? • Increasing, decreasing, and constant behave like they sound. Assuming you are "reading" the graph from left to right, an increasing interval means it is going up, decreasing means it is going down, and constant means the height does not change.
• To find the increasing intervals, we look on the graph to see where the function is going up. To find decreasing, we find where it is going down. To find constant, we find where it does not change. Write these intervals in interval notation and only using parentheses ( ). [Notice that this function has no constant intervals.]
• The absolute maximum is the highest value achieved by the function. The absolute minimum is the lowest value achieved by the function.
• To find the max/min, look on the graph for the highest/lowest points. The horizontal location of these points gives the x-value, and the height gives the value for the max/min.
Increasing: (−1, 3);    Decreasing: (−3, −1);
Absolute Maximum: at x=3⇒ f(3) = 4;    Absolute Minimum: x=−1 ⇒ f(−1) = −4
Consider the function f(x) = x3 + 3x2 −1, whose graph is shown. What are the intervals that f is increasing, decreasing, and constant on?
At what x-value do we find a relative maximum? At what x-value do we find a relative minimum? What are the values of those maximums and minimums?
Explain why the function does NOT have an absolute maximum or minimum. • Increasing, decreasing, and constant behave like they sound. Assuming you are "reading" the graph from left to right, an increasing interval means it is going up, decreasing means it is going down, and constant means the height does not change.
• The graph helps us find the increasing/decreasing/constant intervals, but it doesn't tell the whole story. Remember, we were told that it is a graph of f(x) = x3 + 3x2 −1. Unlike previous problems, f(x) is not confined to this graphing window. It keeps going beyond the edges of this graph.
• The graph will go down forever as it goes off to the left, while it will go up forever as it goes to the right. This means we will need to use ∞ in our interval notations.
• The relative maximum is the highest value achieved by the function in some neighborhood. The relative minimum is the lowest value achieved by the function in some neighborhood. Relative max/min don't have to to be the highest or lowest for the whole function, just the highest or lowest in some local "zone" of the function.
• Looking at the graph, we can see an upward bump and a downward bump. The points at the tips of these bumps give us the relative maximum and minimum. The horizontal location of these points gives the x-value, and the height gives the value for the max/min.
• To have an absolute max/min, the function must achieve a value that is higher/lower than all the other values. On this function though, can that ever happen? If you were to name a highest value, would be able to find another value that surpasses it?
Increasing: (−∞, −2) and (0, ∞);    Decreasing: (−2, 0);
Relative Maximum: at x=−2⇒ f(−2) = 3;    Relative Minimum: x=0 ⇒ f(0) = −1;
The function cannot have an absolute max/min because it continues to travel up and down forever. There is no single highest/lowest location, because f(x) will always go above/below any value.
Using the graph of the function f(x), find the roots of f(x). [Assume that there are no roots beyond those we can see in the graph.] • The roots of a function (also called the zeros or x-intercepts) are all the x-values such that f(x) = 0. In other words, they are the locations that cause the function to output 0.
• Graphically, we can see this as where the function crosses the x-axis, because that height is equivalent to f(x)=0.
x=−3,  0,  1,  4
Find all the zeros of the function g(x) = √{2x+5}−5.
• The zeros of a function (also called the roots or x-intercepts) are all the x-values such that g(x) = 0. In other words, they are the locations that cause the function to output 0.
• Previously, we found the zeros of a function by looking at its graph and seeing where it crossed the x axis. Potentially, we could do that here by carefully graphing g(x), but that would be difficult and time-consuming. Instead, we can do it algebraically.
• Since a zero (root) is an x-value where g(x) = 0, we just set g(x) to 0 in the function, then solve the resulting equation for x:
 0 = √ 2x+5 −5
x=10
For the function h(x) = 3x2+2, what is the average rate of change from x=0 to x=2? What about from x=2 to x=4?
• The average rate of change is the slope of an imaginary line between two points on the graph of the function (this line is called the secant line). These two points are determined by the two x locations we are finding the average rate of change between.
• As a formula, the average rate of change is
 f(x2) − f(x1) x2 −x1 .
• For the first pair of x-values, we plug them into the average rate of change formula to get
 f(2) − f(0) 2 −0 = (3·22 +2) − (3·02 + 2) 2 .
From there, simplify to find the average rate of change.
• For the second pair, we do the same thing:
 f(4) − f(2) 4 −2 = (3·42 +2) − (3·22 + 2) 2 .
From there, simplify to find the average rate of change.
From x=0 →2: 6;        From x=2 → 4: 18
Let f(x) = x2−3. Find an equation for the secant line passing through (1, f(1)) and (4, f(4)).
• The secant line is an imaginary line that passes through two points on a function. The slope of the secant line is equal to the average rate of change over that interval of the function.
• Begin by finding the slope of the secant line. This will be the same whether we use the formula for the average rate of change or just find the slope between the two points:
 f(4) − f(1) 4−1 = (42−3)−(12−3) 3
Simplify this to find the slope of the secant line.
• Once you know the slope of the secant line is m=5, you can find the rest of the equation for the secant line. A useful and common formula for a line is y=mx +b, where m is the slope and b is the y-intercept. At this point we just need to find b.
• We can plug m=5 into y=mx+b to get y = 5x+b. Now if we know any other point on the line, we can plug it in to the equation, and that will allow us to find b. We know both (1, f(1)) and (4, f(4)) are on the secant line, so let's use one of them. Let's choose (1, f(1)) = (1, −2). Plugging that into our equation, we have
 −2 = 5 (1) + b,
which we can solve for b.
y=5x−7
Show that f(x) = 3x4 − x2 is an even function.
• An even function is one where f(−x) = f(x). In other words, plugging in the negative version of a number gives the same result as the positive version of a number.
• To show that f(x) is even, we need to show that this property is true for any number. We can do that by considering the arbitrary number x and its negative version −x. Thus, if f(−x) = f(x), we have shown that f(x) is even.
• Set up f(−x) by plugging in −x for x from the original function given:
 f(−x) = 3(−x)4 − (−x)2
• Simplify the above. Remember that if you raise a negative number to an even power, it is turned positive. Thus
 f(−x) = 3(−x)4 − (−x)2 = 3x4 − x2 = f(x),
which shows that f(x) is even.
[Answer is given by showing that f(−x) = f(x). See the steps for a detailed explanation.]
Show that f(x) = [x/(x2+3)] is an odd function.
• An odd function is one where f(−x) = −f(x). In other words, plugging in the negative version of a number gives the opposite result as the positive version of a number.
• To show that f(x) is odd, we need to show that this property is true for any number. We can do that by considering the arbitrary number x and its negative version −x. Thus, if f(−x) = − f(x), we have shown that f(x) is odd.
• Set up f(−x) by plugging in −x for x from the original function given:
 f(−x) = (−x) (−x)2+3 .
• Simplify the above to show that f(−x) = −f(x):
 f(−x) = (−x) (−x)2+3 = −x x2+3 = − x x2+3 = − f(x).
Thus f(x) is odd.
[Answer is given by showing that f(−x) = −f(x). See the steps for a detailed explanation.]
Show that f(x) = x2 − 4x is neither even nor odd.
• A function f(x) is even if f(−x) = f(x). A function f(x) is odd if f(−x) = − f(x).
• To show that f(x) = x2 − 4x has neither of these qualities, we must find out what f(−x) is.
• Looking at f(−x) and simplifying, we get
 f(−x) = (−x)2 − 4(−x) = x2 + 4x.
• To show that f(x) is not even, compare f(−x) = x2 + 4x to f(x) = x2−4x. Since f(−x) ≠ f(x), we have that f(x) is not even.
• To show that f(x) is not odd, compare f(−x) = x2 + 4x to − f(x) = −(x2−4x) = −x2 + 4x. Since f(−x) ≠ − f(x), we have that f(x) is not odd.
[Answer is given by showing that f(−x) ≠ f(x) and f(−x) ≠ −f(x). See the steps for a detailed explanation.]

*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.

### Properties of 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 0:05
• Increasing Decreasing Constant 0:43
• Looking at a Specific Graph
• Increasing Interval
• Constant Function
• Decreasing Interval
• 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
• Formal Definition of Relative Minimum
• Max/Min, More Terms 14:18
• Definition of Extrema
• Average Rate of Change 16:11
• Drawing a Line for the Average Rate
• Using the Slope of the Secant Line
• Slope in Function Notation
• Zeros/Roots/x-intercepts 19:45
• What Zeros in a Function Mean
• Even Functions 22:30
• Odd Functions 24:36
• Even/Odd Functions and Graphs 26:28
• Example of an Even Function
• Example of an Odd Function
• Example 1 29:35
• Example 2 33:07
• Example 3 40:32
• Example 4 42:34

### Transcription: Properties of Functions

Hi--welcome back to Educator.com.0000

Today, we are going to talk about properties of functions.0002

Functions are extremely important to math; we keep talking about them, because we are going to use them a lot; they are really, really useful.0005

To help us investigate and describe behaviors of functions, we can talk about properties that a function has.0012

There are a wide variety of various properties that a function can or cannot have.0017

This lesson is going to go over some of the most important ones.0022

While there are many possible properties out there that we won't be talking about in this lesson,0024

this lesson is still going to give us a great start for being able to describe functions,0028

being able to talk about how they behave and how they work.0032

So, this is going to give us the foundation for being able to talk about other functions in a more rigorous way,0035

where we can describe exactly what they are doing and really understand what is going on; great.0039

All right, the first one: increasing/decreasing/constant: over an interval of x-values,0045

a given function can be increasing, decreasing, or constant--that is, always going up, always going down, or not changing.0050

Its number will always be increasing; its number will always be decreasing; or its number will be not changing.0060

And by number, I mean to say the output from the inputs as we move through those x-values.0066

This is really much easier to understand visually, so let's look at it that way.0072

So, let's consider a function whose graph is this one right here: this function is increasing on -3 to -1.0076

We have -3 to -1, because from -3 to -1, it is going up; but it stops right around here.0083

So, it stops increasing on -3; it stops increasing after -1, but from -3 to -1, we see that it is increasing.0091

It is probably increasing before -3, but since all we have been given is this specific viewing window to look through,0099

all we can be guaranteed of is that, from -3 to -1, it is increasing.0104

Then, it is constant on -1 to 1; it doesn't change as we go from -1 to 1--it stays the exact same, so it is constant on -1 to 1.0108

However, it is increasing before -1, and it is decreasing after 1; so it is decreasing on 1 to 3, because we are now going down.0121

So, it continues to go down from 1 on to 3, because we can only be guaranteed up until 3.0131

It might do something right after the edge of the viewing window, so we can only be sure of what is there.0138

It is decreasing from 1 to 3--great; that is what we are seeing visually.0143

It is either going up, straight, or down; it is either horizontal, it is going up, or it is down.0147

Increasing means going up; constant means flat; decreasing means down.0153

Formally, we say a function is increasing on an interval if for any a and b in the interval where a < b, then f(a) < f(b).0160

Now, that seems kind of confusing; so let's see it in a picture version.0167

Let's say we have an interval a to b, and this graph is above it.0172

If we have some interval--it is any interval, so let's just say we have some interval--that is what was between those two bars--0179

and then we decide to grab two random points: we choose here as a and here as b;0188

a is less than b, so that means a is always on the left side; a is on the left, because a is less than b.0195

That is not to say b over c; that is because--I will just rewrite that--we might get that confused in math.0204

a is on the left because a is less than b; if we then look at what they evaluate to, this is the height at f(a), and then this is the height at f(b).0211

And notice: f(b) is above f(a); f(b) is greater than f(a).0224

So, it is saying that any point on the left is going to end up being lower than points on the right, in the interval when it is increasing.0231

In other words, the graph is going up in the interval, as we read from left to right.0240

Remember, we always read graphs from left to right.0245

So, during the interval, we are going from left to right; we are going up as we go from left to right.0248

We have a similar thing for decreasing; we have some interval, some chunk, and we have some decreasing graph.0255

And if we pull up two points, a and b (a has to be on the left of b, because we have a < b), then f(a) > f(b).0265

f(a) > f(b); so decreasing means we are going down--the graph is going down from left to right.0277

We don't want to get too caught up in this formal idea; there is some interval, some place,0287

where if we were to pull out any two points, the one on the left will either be below the one on the right,0293

if it is increasing; or if it is decreasing, it will be above the one on the right.0299

We don't want to get too caught up in this; we want to think more in terms of going up and going down, in terms of reading from left to right.0304

Finally, constant: if we have some interval, then within that interval, our function is nice and flat,0310

because if we choose any a and b, they end up being at the exact same height.0318

There is no difference: f(a) = f(b); the graph's height does not change in that interval.0324

While the definitions on the previous slide give us formal definitions--they give us something that we can really understand0332

if we want to talk really analytically--we don't really need to talk analytically that often in this course.0338

It is going to be easiest to find these intervals by analyzing the graph of the function.0343

We just look at the graph and say, "Well, when is it going up? When is it going down? And when is it flat?"0347

That is how we will figure out our intervals.0352

We won't necessarily be able to find precise intervals; since we are looking at a graph, we might be off by a decimal place or two.0354

But mostly, we are going to be pretty close; so we can get a really good idea of what these things are--0360

what these intervals of increasing, decreasing, or constant are.0366

So, we get a pretty good approximation by looking at a graph.0370

And if you go on to study calculus, one of the things you will learn is how to find increasing, decreasing, and constant intervals precisely.0373

That is one of the major fields, one of the major uses of calculus.0380

You won't even need to look at a graph; you will be able to do it all from just knowing what the function is.0383

Knowing the function, you will be able to turn that into figuring out when it is increasing, when it is decreasing, and when it is flat.0387

You will even be able to know how fast it is increasing and how fast it is decreasing.0392

So, there is pretty cool stuff in calculus.0395

All right, intervals show x-values: for our intervals of increasing, decreasing, and constant,0399

remember, we are giving intervals in terms of the x-values; it is not, not the points.0403

We describe a function's behavior by saying how it acts within two horizontal locations.0411

We are saying between -5 and -3, horizontally; it is not the point (-5,-3); it is between the locations -5 and -3.0416

And don't forget, we always read from left to right; it is reading from left to right, as we read from -5 up until -3.0425

So, the other thing that we need to be able to do is: we need to always put it in parentheses.0432

Parentheses is how we always talk about increasing, decreasing, and constant intervals.0437

Why do we use parentheses instead of brackets?0442

a parenthesis indicates that we are dropping that point, not including that in the interval.0449

But the places where we change over, the very end of an interval, is where we are flipping0453

from either increasing to decreasing or increasing to constant; we are changing from one type of interval to another.0457

So, those end points are going to be changes; they are going to be places where we are changing from one type of interval to another.0463

So, we can't actually include them, because they are switchover points.0470

We want to only have the things that are actually doing what we are talking about.0473

The switchovers will be switching into something new; so we end up using parentheses.0476

All right, a really quick example: if we have f(x) = x2 - 2x, that graph on the left,0480

then we see that it is decreasing until it bottoms out here; where does it bottom out?0485

It bottoms out at 1, the horizontal location 1; and it is decreasing all the way from negative infinity, out until it bottoms out at 1.0489

And then, it is increasing after that 1; it just keeps increasing forever and ever and ever.0499

So, it will continue to increase out until infinity.0503

So, parenthesis; -∞ to 1 decreasing; and increasing is (1,∞).0506

We don't actually include the 1, because it is a switchover.0513

At that very instant of the 1, what is it--is it increasing? Is it decreasing?0515

It is flat technically; but we can't really talk about that yet, until we talk about calculus.0520

So for now, we are just not going to talk about those switchovers.0524

All right, the next idea is maximums and minimums.0527

Sometimes we want to talk about the maximum or the minimum of a function, the place where a function attains its highest or lowest value.0530

We call c a maximum if, for all of the x (all of the possible x that can go into the function), f(x) ≤ f(c).0538

That is to say, when we plug in c, it is always going to be bigger than everything else that can come out of that function,0545

or at the very least equal to everything else that can come out of the function.0553

A minimum is the flip of that idea; a minimum is f(c) is going to be smaller or equal to everything else that can be coming out of that function.0556

So, a maximum is the highest location a function can attain, and a minimum is the lowest location a function can attain.0565

On this graph, the function achieves its maximum at x = -2; notice, it has no minimum.0575

So, if we go to -2 and we bring this up, look: the highest point it manages to hit is right here at -2.0580

Why does it not have a minimum? Well, if we were to say any point was its minimum--look, there is another point that goes below it.0589

So, since every point has some point that is even farther below it, there is no actual minimum,0594

because the minimum has to be lower than everything else.0600

There is a maximum, because from this height of 3, we never manage to get any higher than 3, so we have achieved a maximum.0603

And that occurs at x = -2; great.0610

We can also talk about something else; first, let's consider this graph,0615

this monster of a function, -x4 + 2x3 + 5x2 - 5x.0619

Technically, this function only has one maximum; you can only have one maximum,0624

and it is going to be here, because it is the highest point it manages to achieve; it would be x = 2.0630

But it actually has no minimum; why does it not have any minimum?0636

Well, it kind of looks like this is the low point; but over here, we managed to get even lower.0638

Over here, we managed to get even lower; and because it is just going to keep dropping off to the sides,0643

forever and ever and ever, we are going to end up having no minimums whatsoever in this function,0647

because it can always go lower; there is no lowest point it hits; it always keeps digging farther down.0651

But nonetheless, even though there is technically only one maximum and no minimums at all,0659

we can look at this and say, "Well, yes...but even if that is true, that there isn't anything else,0663

this point here is kind of interesting; and this point here is kind of interesting,0669

in that they are high locations and low locations for that area."0674

This is the idea of the relative minimum and maximum; we call such places--these places--0679

the highest or lowest location (I will switch colors...blue...oh my, with yellow, it has managed...blue here; green here)...0686

relative minimums are the ones in green, and the relative maximums are the ones in blue.0701

And sometimes the word "local" is also used instead; so you might hear somebody flip between relative or local, or local or relative.0708

These places are not necessarily a maximum or a minimum for the entire function, for every single place.0715

But they are such a maximum or minimum in their neighborhood; there is some little place around them where they are "king of their hill."0721

So, this one is the maximum in this interval, and this one is the minimum in this interval; and this one is the maximum in this interval.0730

But if we were to look at a different interval, there would be no maximum or minimum in this interval,0737

because it just keeps going down and down and down.0740

And if we were to look at even in here, it is clearly right next to them--if we were to put a neighborhood around this, it would keep going down.0743

It is not the shortest one around; it is not the highest one around.0749

But these places are the highest or lowest in their place.0754

OK, so this gives us the idea of a relative maximum or a relative minimum.0761

Formally, a location, c, on the x-axis is a relative maximum if there is some interval,0765

some little place around that, some ball around that, that will contain c, such that,0770

for all x in that interval, f(x) ≤ f(c)--in its neighborhood, c is the highest thing around.0775

It is greater than all of the other ones.0783

Similarly, for a relative minimum, there is some interval such that f(x) is going to be less than or equal to f(c).0785

In its neighborhood, it is the lowest one around; lowest one around makes you a minimum--highest one around makes you a maximum--0792

that is to say, a relative maximum or a relative minimum.0799

Once again, this is sort of like what we talked about before with the previous formal definition for maximum and minimum,0804

and also for the formal definition of intervals of increasing, decreasing, and constant.0809

Don't get too caught up on what this definition means precisely.0813

The important thing is that we have this graphical picture in our mind that relative maximum just means the high point in that area.0817

And relative minimum just means the low point in that area; that is enough for us to really understand what is going on here.0823

Getting caught up in these precise things is really something for a late, high-level college course to really get worried about.0829

For now, it is enough to just get an idea of "it is the high place" or "it is the low place."0835

Don't forget: the terms relative and local mean basically the same thing--actually, they mean exactly the same thing.0841

They can be used totally interchangeably; and some people prefer to use one; some people prefer to use another.0846

Some people will flip between the two; so don't get confused if you hear one or you hear another one; they just mean the same thing.0851

To distinguish relative local maximums and minimums from a maximum and minimum over the entire function,0859

we can use the terms "absolute" or "global" to denote the latter.0864

If we want to say it is the maximum over the entire function, we could call it the absolute maximum or the global maximum.0868

So, an absolute, global maximum/minimum is where the function is highest/lowest over the entire function,0875

which is exactly how we defined maximum/minimum at first, before we started to talk about the idea of relative maximum/relative minimum.0881

So, absolute or global maximum/minimum is over the entire function--the function's highest/lowest over everywhere in the domain.0886

If we want to talk about all of the relative or absolute maximums/minimums in the functions, we can call the them the extrema (or the "ex-tray-ma").0899

Why? Because they are the function's extreme values: they are the extreme high points0906

and the extreme low points that the function manages to go through, so we can call them the extrema.0911

So, there we are; there is just something for us: extrema.0916

If we want to talk about relative or absolute maximums/minimums in general, we use this word to do it.0919

And absolute or global talks about the single highest or single lowest;0924

relative just talks about one that is high or low in its neighborhood, in the area around that point.0928

Just like find increasing/decreasing/constant intervals, we want to do this from the graph.0936

We don't want to really get too worried or too caught up on these very specific definitions,0941

the formal definitions we were talking about on the previous slide.0945

We just want to say, "OK, yes, we see that that is a high point on the graph; that is a low point on the graph."0948

So, find your minimums; find your maximums by looking at the graph.0953

And once again, if you go on to study calculus, you will learn how to find extrema precisely, without even needing to look at a graph.0957

You will be able to find them exactly; you won't have to be doing approximations because you are looking at a graph.0963

And you won't even have to look at a graph to find them.0966

So once again, calculus is pretty cool stuff.0968

Average rate of change: this also can be called average slope.0972

When we talked about slope in the introductory lessons, we discussed0976

how it can be interpreted as the rate of change, how fast up or down the line is moving.0978

If we have a line like this, it is not moving very fast up; but if we have another line like this, it is moving pretty quickly up.0984

So, it is a rate of change; the slope is how fast it is changing--the rate of change; how fast are we going up?0993

Now, most of the functions we are going to work with aren't lines; but we can still use this idea.0999

We can discuss a function's average rate of change between two points.1003

So, if an imaginary line is drawn between two points on a graph, its slope is the average rate of change.1009

Say we take two points, this point here and this point here; and we draw an imaginary line between them.1015

Then, the slope of that imaginary line is the average rate of change,1022

because what it took to get from this point to the second point is that we had to travel along this way.1025

And while we actually went through this curve here--we actually went through this curve,1031

but on the whole, what we managed to do, on average, is: we really just kind of went along on that line.1037

We could forget about everything we went through, and we could just ask, "Well, what is the average thing that happened between these two points?"1045

And that would be our average rate of change--how fast we were moving up from our first point to our second point.1050

So, if we want to find the average rate of change, how do we do this?1057

Let's say we have two locations, x1 and x2,1061

and we want to find the slope of that imaginary line between those two points on the function graph.1065

So, that line is sometimes called the secant line; for the most part, you probably won't hear that word too often.1072

But in case it comes up, you know it now.1078

Remember, if we want to find what the slope of this imaginary line is, the slope of this secant line, we know what slope is.1080

How do we find slope? Remember, slope is the rise over the run, so it is the difference between our heights1088

y2 and y1, our second height and our first height--what did our height change by,1094

and what did our horizontal location change by--our second location minus our first location?1099

So, our horizontal distance is x2 - x1; and our vertical distance is y2 - y1.1104

So, y1 is the height over here; y2 is the height over here.1115

y2 - y1 over x2 - x1 is the rise, divided by the run.1119

But what are y1 and y2?--if we want to figure out what y1 and y2 are,1127

well, we just need to look at what x1 and x2 are.1132

So, since x1 and x2 are coming to get placed by the function,1134

then y2's height is really just f(x2), because that is how the graph gets built.1141

The input gets dropped to an output; we map an input to an output.1146

And y1 over here is from f(x1).1151

So, since our original slope formula is y2 - y1 over x2 - x1,1156

and we know that y1 is the same thing as f(x1) and y2 is the same thing as f(x2),1160

we can just plug those in, and we get the change in our function outputs, f(x2) - f(x1),1164

divided by our horizontal distance, x2 - x1.1171

For our average rate of change, we just look at how much our function changed by between those horizontal locations.1174

How much did its output change by? Divide that by how much our distance changed by.1180

It is often really useful and important to find what inputs cause a function to output 0.1186

So, if we have some function f, we might want to know what we can put into f that will give out 0.1191

That is the values of x such that f(x) = 0.1197

Graphically, since f(x)...remember, f(x) is always the vertical component; the outputs come to the vertical;1200

so, if our outputs are coming from the vertical, then 0 is going to be the x-axis.1208

We have a height of 0 here; so graphically, we see that this is where the function crosses the x-axis.1213

Our crossing of the x-axis is where f(x) = 0.1219

This idea of f(x) = 0 is so important that it is going to go by a bunch of different names.1224

It can be called the zeroes of a function; it can be called the roots of a function; and it can be called the x-intercepts.1229

x-intercepts--that makes sense, because it is where it crosses the x-axis.1236

Zeroes make sense, because it is where we have the zeroes showing up.1241

But how can we remember roots--why is roots coming out?1245

Well, one way to think about it--and actually where this word's origin is coming from--1249

is because it is the roots that the function is planted in.1254

The function we can think of as being planted in the ground (not literally the ground, but we can think of it as being the ground of the x-axis).1258

So, it is like the function has put down roots in the soil.1265

It is not exactly perfect, but that is one good mnemonic to help us remember.1269

"Roots" means where we are stuck in the soil; it is where we are stuck in the x-axis;1273

it is where we have f(x) equal to 0, or where we have an equation equal to 0.1277

But all of these things--zeroes, roots, x-intercepts--they all mean the same thing.1282

They are the x such that f(x) = 0; we can also use these for equations--1287

we might hear it as the zeroes of an equation, the roots of an equation, or the x-intercepts of an equation.1291

There is no one way to find zeroes for all functions.1297

We are going to learn, for some functions, foolproof formulas to find zeroes, to tell us if there are zeroes and what those zeroes are.1301

But for other functions, it can be very difficult--very, very difficult, in fact--to find the zeroes.1308

And although we are going to learn some techniques to help us on the harder ones, there are some that we won't even see1312

in this course, because they are so hard to figure out.1316

But right now, the important thing isn't being able to find them, but just knowing1319

that, when we say zeroes, roots, x-intercepts of a function, or an equation, we are just talking about where f(x) = 0.1323

So, don't get too caught up right now in being able to figure out how to get those x-values such that f(x) = 0.1331

Just really focus on the fact that when we say zeroes, roots, or x-intercepts, all of these equivalent terms,1337

we are just saying where the function is equal to 0--what are the places that will output 0?1344

Even functions: this is a slightly odd idea (that was an accidental joke).1351

Even functions: some functions behave the same whether you look left or right of the y-axis.1357

For example, let's consider f(x) = x2: it is symmetric around the y-axis.1363

What do I mean by this? Well, if we plug in f(-3), that is going to end up being (-3)2, so we get 9.1368

But we could also plug in the opposite version to -3; if we flip to the positive side, -3 flips to positive 3.1375

If we plugged in positive 3, then f(3) is 32, so we get 9, as well.1383

It turns out that plugging in the negative version of a number or the positive version of a number,1390

-3 or 3, we end up getting the same thing; for -2 and 2, we end up getting the same thing.1395

For -47 and 47, we end up getting the same thing.1400

So, whatever we plug in, as long as they are exact opposites horizontally--1405

they are the same distance from the y-axis--the points are symmetric around the y-axis--1409

they are going to come out to the same height; they are going to have the same output.1414

We call this property even; and I want to point out that it is totally different from being an even number.1419

It is different from an even number--not the same thing as that.1423

But we call this property even for a function.1427

A function is even if all of the x for its domain, for any x that we plug in...1432

if we plug in the negative x, that is the same thing as the positive x.1437

Plugging in f(-x) is equal to plugging in f(x); so we plug in -x into the function, and we get the same thing as if we had plugged in positive x.1442

We can flip the signs, and it won't matter, as long as it is just negative versus positive.1451

Why do we call it even? It has something to do with the fact that all polynomials where all of the exponents1457

end up being even exponents--they end up exhibiting this property.1462

But then, this property can be used on other things; so don't worry too much about where the name is coming from.1465

But just know what the property is: f(-x) = f(x).1470

Odd functions are the reverse of this idea: other functions will behave in the exact reverse.1476

The left side is the exact opposite of the right side; for example, f(x) = x3 behaves like this.1481

If we plug in -3, we get -3 cubed, so we get -27; but if we had plugged in positive 3, we would get positive 3 cubed, so we would get positive 27.1488

So, you see, you plug in the negative version of a number, and you plug in the positive version of a number;1500

and you are going to get totally opposite answers.1508

However, they are only flipped by sign; -27 and 27 are still somewhat related.1510

They are very different from one another--they are opposites, in a way; but we can also think of them as being perfect opposites.1515

-27's opposite is positive 27; so an odd function is one that behaves like this everywhere.1520

We call this property odd; it is totally different, once again, from being an odd number.1528

A function is odd if, for all x in its domain, f(-x) is equal to -f(x).1532

And that is a little confusing to read; but what that means is that, if we plug in -x,1538

then that is going to give us the negative version of if we had plugged in positive x.1543

So, if we plug in a negative number, and then we plug in a positive number, the outputs1549

that come out of them will be positive-negative opposites.1555

One of them will be positive; the other one will be negative.1559

So, negative on one side and positive on one side means that the outputs will also be negative on one side and positive on the other side.1561

It is not necessarily going to be the case that the negative side will always put out negative outputs.1568

But it will be the case that it will be flipped if it is odd.1572

This will make a little more sense when we look at some examples.1575

And once again, why are we calling it odd?1577

Once again, don't worry too much about it, but it because it is connected to polynomials where all of the exponents are odd numbers.1579

But don't really worry about it; just know what the property is.1585

Even/odd functions and graphs: we can see these properties in the graphs of functions.1589

An even function is symmetric around the y-axis: it mirrors left/right, because when we plug in a positive number,1593

and we plug in a negative number, as long as they are the same number, they end up getting put to the same location.1600

They get output to the same place.1605

An odd function, on the other hand, is symmetrical around the origin, which means we mirror left/right and up/down,1607

because when we plug in the positive version of a number, it gets flipped to the negative side, but also shows up on the opposite side.1613

It flips to the negative height or the positive height; it flips the positive/negative in terms of height.1622

So, let's look at some examples visually; that will help clear this up.1627

An even one: f(-x) = f(x); let's see how this shows up; if we plug in 0.5, we get here; if we plug in -0.5, we get here.1630

And look, beyond the fact that I am not perfect at drawing, they came out to be the same height.1642

If we plug in 2.0, and we plug in -2.0, they came out to be the same height.1648

You plug in the negative number and the positive number, and they end up coming out to be the same height.1661

That is what it means to be even; and since all of the positives will be the same as the negatives,1666

we end up getting this nice symmetry across the y-axis; it is just a perfect flip.1670

If we took the two halves and folded them up onto each other, they would be exact perfect matches; it is just mirroring the two sides.1676

Odd is sort of the reverse of this: f(-x) = -f(x).1684

For example, let's plug in -1: we plug in -1, and it ends up being at this height, just a little under 2.1690

Let's see what happens when we plug in positive 1; when we plug in positive 1, it ends up being just a little under -2.1699

So, we flip the horizontal location; that causes our vertical location to flip.1708

Let's try another one: we plug in 2.0, and we are practically past it; so we should be just a little bit before 2.0.1713

And we plug in -2.0, once again, just a little past it; so we are just a little before -2.0.1722

And look: we end up being at the same distance from the x-axis, but in totally opposite directions.1727

2.0, positive 2.0, causes us to go to positive 4 in terms of height; but -2.0 causes us to go to -4 in terms of height.1735

So, they are going to flip; if you flip horizontally, you also flip vertically; and that is why we mirror left/right and mirror up/down.1744

We are not just flipping around the y-axis; we are flipping around the origin,1752

because we are flipping the right/left and the up/down; flipping around the origin is flipping the right/left and the up/down.1759

We mirror left/right; we mirror up/down; that is what is happening with an odd function.1771

All right, we are finally ready for some examples.1776

There are a bunch of different properties that we covered; now, let's see them in use.1778

The first example: Using this graph, estimate the intervals where f is increasing and decreasing.1781

Find the locations of any extrema/relative maximums/minimums.1787

And our function is -1.5x4 + x3 + 4x2 + 3.1790

Now, that is just so we can have an idea that that is what that function looks like.1794

But we are not really going to use this thing right here; it is not really going to be that helpful for us figuring it out.1798

So first, let's figure out intervals where f is increasing or decreasing.1802

First, it is increasing from all the way down (and it sounds like we can probably trust1807

that it is going to keep going down, because we have -1.5x4); it is increasing up until...it looks like just after -1.0.1812

It is increasing from negative infinity (because it is going all the way to the left--it is going up1824

as long as we are coming from negative infinity, because it goes down as we go to the left, but we read from left to right),1835

so it is increasing from negative infinity up until...let's say that is -0.9, because it is just after -1.0.1839

And then, it is also going to be increasing from here...let's say it starts there...up until about this point.1846

So, where is that? It is probably about 1.4; so it is increasing from 0 up until 1.4.1852

Where is it decreasing? It is decreasing from this point until this point.1864

That was -0.9 that we said before; so we will go from -0.9 up until 0.1875

And then, it increased up until 1.4; so now it is going to be decreasing from 1.4.1881

And it looks like it is going to just keep going down forever, and it does indeed.1887

So, it is going to be all the way out until infinity; it is going to continue decreasing; great.1890

Now, let's take a look at the extrema; where are the relative maximums/minimums?1895

We have relative maximums/minimums at all of these flipovers that we have talked about, here, here, and here.1902

So, our relative maximum/minimum, our high location, the absolute maximum/minimum, is going to be up here.1909

Relative maximums: we have x =...we said that was 1.4, and that point is going to be 1.4.1916

Let's take a look, according to this...and it looks like it is just a little bit under 8; let's say 7.9.1931

And then, the other one, the lesser of them, but still a relative maximum--it is occurring at x = -0.9.1940

So, its point would be -0.9; and we look on the graph, and it looks like it is somewhere between 4 and 5.1949

It looks a hair closer to 5, so let's say 4.6; great.1958

Relative minimum--our low place: well, we can be absolutely sure of what the x is there--it is pretty clear that that is x = 0.1962

And what is the height that it is at right there? It looks like it is exactly on top of the 3, so it is (0,3).1972

We have all of the intervals of increasing and decreasing.1978

And we also have all of our extrema, all of our relative maximums and minimums; great.1981

Example 2: A ball is thrown up in the air, and its position in meters is described by location of t.1987

Distance of t is equal to -4.9t2 + 10t, where t is in seconds.1993

OK, so we have some function that describes the height of the ball--where the ball is.1998

What is the ball's average velocity (speed) between 0 seconds and 1 second,2004

between 0 and 0.01 seconds, and between 0 and 2.041 seconds?2007

OK, at first, we have some idea...if we were to figure out what this function looks like, it is a parabola.2012

It has a negative here, so it is ultimately going to go down.2017

And it has the 10t here; if we were to graph it, it would look something like this.2020

And that makes a lot of sense, because if we throw a ball up, with time, the ball is going to go up and them come back down.2024

So, that seems pretty reasonable: a ball is thrown up in the air, and its position is given by this.2031

But how does speed connect to position? Well, we think, "What is the definition of speed?"2036

We don't exactly know what velocity is, necessarily; maybe we haven't taken a physics course.2040

But we probably know what speed is from before in various things.2044

Speed is distance divided by time, so distance over time equals speed.2047

It seems pretty reasonable that velocity is going to be the same thing.2058

That is not exactly true, if you have actually taken a physics course; but that is actually going to work on this problem.2063

We are going to have a good idea of what is going on with saying that that is true.2068

All right, so what is the ball's average velocity?2072

The average velocity is going to be the difference in its height, divided by the time that it took to make that difference in height.2074

So, we are going to be looking for distance.2080

If we have 2 times, time t1 to time t2, it is going to be the location at time t2,2084

minus the location at time t1, over the difference in the time, t2 - t1.2095

Oh, and that makes a lot of sense; it is going to be connected, probably, to what we learned in this lesson,2102

since with student logic, they normally try to give us problems that are going to be based off of what we just learned.2107

So, t2 - t1...this looks just like average rate of change.2112

The average rate of something's position--that would make sense, that how fast it is going is the rate of change; the thing is changing its location.2116

The rate at which you are changing your location is the velocity that you have; perfect.2124

Great; so we need to figure out what it is at 0 seconds and what it is at 1 second right away.2128

So, the location at 0 seconds; we plug that in...-4.9(0)2, plus 10(0)...that is just 0, which makes a lot of sense.2133

If we throw a ball up, at the very beginning it is going to be right at the height of the ground.2140

Distance at time 1 is going to be -4.9 times 12, plus 10 times 1; so we get 5.1.2146

If we want to figure out what is its average velocity between 0 seconds and 1 second, then we have d(1) - d(0)/(1 - 0), equals 5.1 - 0/1, which equals 5.1.2159

What are our units? Well, we had distance in meters, and time in seconds; so meters divided by seconds...we get meters per second.2180

That makes sense as a thing to measure velocity and speed.2189

All right, next let's look at between 0 and 0.1 seconds.2192

If we want to find 0.01 seconds, the location at 0.01 equals -4.9(0.01)2 + 10(0.01).2196

Plug that into a calculator, and that is going to end up coming out to be 0.09951; so let's just round that up2211

to the much-more-reasonable-to-work-with 0.01.2218

OK, so it rounds approximately to 0.01; so let's see what is the average rate of change.2222

The average rate of change, then, between 0 and 0.01 seconds, is going to be d(0.01) - d(0) over 0.01 - 0.2228

That equals...oops, sorry, my mistake: 0.01 is not actually what it came out to be when we put it in the calculator.2245

I mis-rounded that just now; it was 0.09951, so if it is 0.09951, if we are going to round that2253

to the much-more reasonable-to-work-with thing, we actually get approximately 0.1.2265

So, it is not 0.01; 0.01 is still on the bottom, but the top is going to end up coming out to be 0.1 - 0, divided by 0.01; sorry about that.2271

It is important to be careful with your rounding.2288

That comes out to be 0.1 over 0.01, which comes out to be 10 meters per second.2290

And now, you probably haven't taken physics by this point; but if you had, you would actually know that -4.9t2...2298

that is the thing that says the amount that gravity affects where its location is.2307

The 10t is the amount of the starting velocity of the ball.2312

The ball gets thrown up at 10 meters per second, so it makes sense that its average speed2315

between 0 and 0.01--hardly any time to have changed its speed--is going to be pretty much what its speed started at.2320

That 10 meters per second is actually showing up there.2327

So, there is a connection here between understanding what the physics going on is and the math that is connecting to it.2329

All right, finally, between 0 and 2.041 seconds...let's plug in d(2.041) = -4.9(2.041)2 + 10(2.041).2335

So, that is going to come out to be -0.0018; so it seems pretty reasonable to just round that to a simple 0.2354

Now, what does that mean? That means, at the moment, 2.041 seconds--that is when the ball hits the ground.2365

It goes up at 0, and then it comes back down.2371

And at 2.041 seconds after having been thrown up, it hits the ground precisely at 2.041 seconds.2374

So, 2.041 seconds--then it has a 0 height; so what is its average velocity between 0 and 2.041 seconds?2380

Location at 2.041 minus location at 0, divided by 2.041 - 0, equals 0 minus 0, over 2.041, which equals 0 meters per second,2388

which makes sense: if we throw the ball up, and then we look at the time when it hits the ground again,2404

well, on average, since it went up and it went down, it had no velocity,2410

because the amount of time that it has positive velocity going up and the amount of time that it has negative velocity going down--2415

it has cancelled itself out, because on average, between the time of its starting on the ground2420

and ending on the ground, it didn't go anywhere.2425

So, on average, its velocity is 0, because it didn't make any change in its location; great.2427

The next example--Example 3: Find the zeroes of f(x) = 3 - |x + 3|.2433

Remember: zeroes just mean when f(x) = 0; so we can just plug in 0 = 3 - |x + 3|.2439

So, we have |x + 3| = 3; we just add the absolute value of x + 3 to both sides.2451

We have |x + 3| = 3; that is what we want to know to figure out when the zeroes are.2458

When is this true? Remember, absolute value of -2 is equal to 2, which is also equal to the absolute value of positive 2.2462

So, the absolute value of x + 3...we know that, inside of it, since there is a 3 over here...2474

there could be a 3, or there could be a -3.2480

So, inside of that absolute value, because we know it is equal to 3, we know that there has to currently be a 3, or there has to be a -3.2484

We aren't sure which one, though; so we split it into two different worlds.2498

We split it into the world where there is a positive on the inside, and we split it into the world where there is a negative on the inside.2502

In the positive world, we know that what is inside, the x + 3, is equal to a positive 3.2508

In the negative world, we know that the x + 3 is equal to a negative 3.2518

Now, it could be either one of these; either one of these would be true; either one of these would produce a 0 for the function.2523

So, let's solve both of them: we subtract by 3 on both sides over here; we get x = 0.2529

We subtract by 3 on both sides over here; we get x = -6.2535

So, the two answers for the roots are going to be -6 and 0; that is when the zeroes of f(x) show up.2538

The zeroes of f(x) are going to be at x = 0 and x = -6.2546

And if we plug either one of those into that function, we will get 0 out of the function.2549

The final example: Show that x6 - 4x2 + 7 is even;2556

show that -x5 + 2x3 - x is odd; and show that x + 2 is neither.2559

All right, the first thing we want to do is remind ourselves of what it means to be even.2564

To be even means that when we plug in the negative version of a number, a -x is the same thing as if we had plugged in the positive x.2572

It doesn't have any effect.2581

And the odd version...actually, let's put it in a different color, so we can see how all of the problems match up to each other.2583

If we do with the odd version, then if we plug in the negative of a number,2590

it comes out to be the negative of if we had plugged in the positive version of the number.2595

All right, so the first one: Show that x6 - 4x2 + 7 is even.2599

So, that was really seeing that expression as if it were a function; so let's show this2605

by showing that if we plug in -x, it is the same thing as if we plug in positive x.2609

On the left, we will plug in -x; -x gets plugged in; it becomes (-x)6 - 4(-x)2 + 7 =...2613

if we plugged in just plain x, we would have plain x6 - 4x2 + 7; great.2622

(-x)6...remember, a negative times a negative cancels out to a positive.2630

We have a 6 up here; we are raising it to the sixth power, so we have an even number of negatives.2634

Negative and negative cancel; negative and negative cancel; negative and negative cancel.2642

That is a total of 6 negatives; they all cancel each other out; so we actually have (-x)6 being the same thing as if we just said x6.2645

Minus 4...the same thing here: -x times -x cancels and just becomes plain x2...plus 7 equals2652

x6 - 4x2 + 7; it turns out that it has no effect.2660

If we plug in a negative x, we get the same thing as if we had plugged in the positive x.2666

Plugging in a negative version of a number is the same thing as plugging in the positive version of the number.2669

So, it checks out; it is even; great.2672

The next one: let's look at odd: -x5 + 2x3 - x is odd.2677

We will do the same sort of thing: we will plug -x's in on the left side.2682

-(-x)5 + 2(-x)3 - (-x); what is going to go on the right side?2686

Well, remember: if we plug in the negative version of the number, then it is the negative of if we plugged in the positive version of the number.2696

So, it is the negative of if we had plugged in the positive version of the number.2703

Plugging in the positive version of the number is just if we have the normal x going in: -x5 + 2x3 - x.2706

All right, so -(-x)5: well, what happens when we have (-x)5--what happens to that negative?2714

Negative and negative cancel; negative and negative cancel; negative--that fifth one, because it is odd, gets left over.2720

So, we have negative; and we just pull that negative out--it is the same thing as -x5.2727

Plus 2...once again, it is odd; a negative and a negative cancel; we are left with one more negative, for a total of 3 negatives; we are left with a negative.2732

So, we get 2(-x)3 minus...we can pull that negative out, as well...-x...equals...2739

let's distribute this negative; so we get...distribute...cancellation...a negative shows up here...cancel;2747

we get positive x5 minus 2x3 + x.2754

So, let's finish up this left side and do cancellations over here as well; positive, positive; this stays negative.2761

Positive, positive; so we get x5 - 2x3 + x equals the exact same thing over here on the right side.2767

It checks out; yes, it is odd; great.2778

Finally, let's show that x + 2 is neither; so, to be neither, we have to fail at being this and fail at being this.2782

So, to be neither, it needs to fail being odd and being even; it needs to fail even and odd.2791

Let's just try plugging in a number; let's try plugging in, say, -2.2809

If we look at x = -2, then that would get us -2 + 2, which equals 0.2814

Now, what if we plugged in the flip of -2--we plugged in positive 2?2821

x = positive 2...we plug that into x + 2, and we will get 2 + 2, which equals 4.2825

Now, notice: 0 is not equal to 4; we just failed being even up here,2832

because the negative number and the positive version of that number don't produce the same output.2841

Plug in -2; you get 0; plug in +2; you get 4; those are totally different things, so we just failed to be even; great.2846

Next, we want to show that it is not odd.2854

Odd was the property that, if we plug in the negative, it is going to be equal to the negative of the positive one.2857

So, 0 is not equal to -4 either, right? If we plug in -2, we get 0, and if we plug in positive 2,2861

it turns out that that is not -0, or just 0; it turns out that that is 4.2869

So, we fail to be odd as well, because it isn't the case that if we plug in opposite positive/negative numbers,2873

we don't get opposite positive/negative results, because 0 is not the opposite of -4; it is just the opposite of 0, so it fails there.2881

So, it checks out: that one is neither; great.2890

All right, we just learned a whole bunch of different properties; and they will each come up in different places at different times.2896

Just remember these: keep them in the back of your mind.2901

If you ever need a reminder, come back to this lesson and just refresh what that one meant,2903

because they will show up in random places; but they are all really useful.2907

And we will see them a lot more as we start getting into calculus.2910

Once you actually get to calculus, this stuff, especially the stuff at the beginning of this,2913

where we talked about increasing and decreasing and relative maximums and minimums--2915

that stuff is going to become so important if you are going to understand why we are talking about it so much right now in this course.2919

All right, I hope you understood everything; I hope you enjoyed it; and we will see you at Educator.com later--goodbye!2924