Vincent Selhorst-Jones

Continuity & One-Sided Limits

Slide Duration:

Section 1: Introduction
Introduction to Precalculus

10m 3s

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

45m 11s

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

35m 31s

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

35m 2s

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

48m 43s

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

56m 31s

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

39m 54s

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

58m 26s

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

48m 49s

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

29m 20s

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

48m 35s

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

33m 24s

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

51m 42s

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

49m 37s

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

28m 49s

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

38m 41s

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

41m 7s

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

39m 43s

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

45m 34s

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

46m 8s

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

45m 36s

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

19m 9s

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

33m 22s

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

34m 16s

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

49m 7s

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

44m 56s

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

35m 17s

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

47m 4s

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

40m 31s

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

42m 33s

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

34m 10s

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

48m 46s

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

39m 5s

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

43m 16s

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

33m 5s

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

52m 3s

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

36m 4s

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

27m 18s

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

32m 58s

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

31m 8s

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

19m 11s

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

23m 16s

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

52m 52s

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

29m 5s

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

43m 55s

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

25m 43s

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

56m 40s

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

49m 5s

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

27m 37s

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

34m 25s

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

55m 40s

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

1h 13s

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

41m 1s

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

1h 9m 31s

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

35m 20s

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

54m 7s

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

47m 12s

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

58m 34s

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

53m 33s

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

48m 7s

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

38m 16s

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

40m 43s

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

57m 37s

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

31m 36s

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

44m 3s

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

36m 58s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

57m 45s

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

40m 27s

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

31m 36s

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

39m 27s

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

49m 53s

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

1h 13m 13s

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

40m 22s

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

57m 11s

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

32m 40s

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

32m 43s

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

32m 49s

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

51m 13s

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

45m 26s

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

10m 41s

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

10m 51s

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

10m 38s

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

9m 45s

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

7m 8s

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

• Long ago in this course, we learned about continuous functions. At the time, we lacked the formal ideas to precisely define continuity, so we intuitively defined it as being any of these three equivalent things:
• All the parts of the function are connected;
• The function's graph can be drawn without ever having to lift your pencil from the paper.
• There are no "breaks"/"holes" in the graph.
• Now we have the the formal background to expand on this intuitive definition and define continuity precisely. Formally, a function f(x) is continuous at c if f(c) exists and
 lim f(x)   =  f(c).
This is the same as the above intuitive ideas, because when a limit matches up with its function, it means that the function goes where we "expect"-there are no jumps or weird stuff.
• If f(x) is not continuous at c, we say it is discontinuous at c. We call such a location a discontinuity. If f(x) is continuous at every point in an interval (a,b), we say f is continuous on (a,b). If f(x) is continuous for every real number, we simply say that f is a continuous function.
• Notice that the vast majority of the functions we're used to working with are continuous. This is why "normal" functions allow us to simply plug in the value we're approaching. Because we generally have
 "Normal"    ⇒     Continuous     ⇒ lim f(x) = f(c)
Furthermore, this also explains why a function that is "weird" in one place still allows us to plug in the value we're approaching if it's not the "weird" value. Because, other than the "weird" value, the rest of the function is continuous, so we can apply the same logic.
• If a limit does not exist because the two sides do not agree on a single value, we can consider a one-sided limit: the limit of a function as it approaches from only one side.
 lim f(x)
denotes the limit as x approaches c from the left or negative side. It is the limit as x→ c while x < c.
 lim f(x)
denotes the limit as x approaches c from the right or positive side. It is the limit as x→ c while x > c.
• For one-sided limits, be careful to keep track of which symbol goes with which side; they can be easy to get confused at first. Remember, −' means look at the negative' side, while +' means look at the positive' side.
• If both sides of a function go to the same value as a location is approached, then the normal limit exists there. Furthermore, if a normal limit exists, both sides must go to the same value as they approach it.
 lim f(x) = L = lim f(x)     ⇔ lim f(x)   =  L
Similarly, if both one-sided limits do not go to the same value (or one side doesn't exist), then a normal limit does not exist there.
 lim f(x) ≠ lim f(x)    ⇒ lim f(x) does not exist
• The above idea allows us to find the limits of piecewise functions at "breakover" points. (If it's not a breakover, we can usually just plug the location in, as discussed in the previous lesson.) We find the limit of a piecewise function by checking if the left- and right-side limits agree with each other.
• If they agree, then the limit exists and equals them.
• If they do not agree, then the limit does not exist.

### Continuity & One-Sided Limits

Let f(x) = {
 x+4,
 x ≤ 3
 2x−8,
 x > 3
.     Find limx → 3 f(x)  and   limx → 3+ f(x).
• The limits that we see in the problem are one-sided limits. A one-sided limit works basically the same as a normal limit, except it only considers one side of the function. Here is how we should interpret the extra symbol of − or + after the location the limit is approaching:
 limx→ c− f(x)
denotes the limit as x approaches c from the left (negative) side. It is the limit as x→ c while x < c.
 limx→ c+ f(x)
denotes the limit as x approaches c from the right (positive) side. It is the limit as x→ c while x > c.
• Thus, for this problem, we are looking at the limit as f(x) approaches 3 from the left (x → 3) and the limit as f(x) approaches 3 from the right (x→ 3+). Notice that, although f(x) is a piecewise function, the only place that is not continuous is at x=3. In other words, while f(x) is "weird" at the "breakover" of x=3, it behaves normally everywhere else. Since the function tells us how it behaves on each side of the breakover and these function pieces are continuous ("normal"), we can use them to find the left and right side limits.
• Use the appropriate piece from the piecewise function to find the value as it approaches from each side. x→ 3, so use the function piece for x ≤ 3:
 limx → 3− f(x)   =  (3) +4

 =  7
x→ 3+, so use the function piece for x > 3:
 limx → 3+ f(x)   =  2(3) − 8

 =  6−8

 =  −2
limx → 3 f(x) = 7,     limx → 3+ f(x)=−2
Let g(x) = {
 3x+1,
 x < −4
 x2−2,
 −4 ≤ x <−3
 √{x+3},
 −3 ≤ x
.     Find limx → −4 g(x)  and   limx → −4+ g(x).
• The limits that we see in the problem are one-sided limits. A one-sided limit works basically the same as a normal limit, except it only considers one side of the function. Here is how we should interpret the extra symbol of − or + after the location the limit is approaching:
 limx→ c− f(x)
denotes the limit as x approaches c from the left (negative) side. It is the limit as x→ c while x < c.
 limx→ c+ f(x)
denotes the limit as x approaches c from the right (positive) side. It is the limit as x→ c while x > c.
• Thus, for this problem, we are looking at the limit as g(x) approaches −4 from the left (x → −4) and the limit as g(x) approaches −4 from the right (x→ −4+). Notice that, although g(x) is a piecewise function, the only places that are not continuous are at x=−4 and x=−3. In other words, while g(x) is "weird" at those "breakovers", it behaves normally everywhere else. Since the function tells us how it behaves on each portion and these function pieces are continuous ("normal"), we can use them to find the left and right side limits.
• Use the appropriate piece from the piecewise function to find the value as it approaches from each side. x→ −4, so use the function piece for x < −4:
 limx → −4− g(x)   =  3(−4) +1

 =  −12+1

 =  −11
x→ −4+, so use the function piece for −4 ≤ x <−3:
 limx → −4+ g(x)   =  (−4)2 −2

 =  16−2

 =  14
[Remark: Notice that for x→ −4+, we use the function piece based on −4 ≤ x <−3 but not the piece for −3 ≤ x. This is because the right side limit is based on the part of the function that is just slightly to the right of −4. While the piece for −3 ≤ x is to the right of −4, it is not adjacent to the location, so we do not use it. On the other hand, −4 ≤ x <−3 is "touching" the limit location on the right side, so we use that piece of the function.]
limx → −4 g(x)=−11,     limx → −4+ g(x)=14
Find   limx → 2 x3−x2+3   and    limx → 2+ x3−x2+3.
• Begin by noticing that the function x3−x2+3 is continuous ("normal"). It is connected everywhere and does not behave "weirdly" at any location. This means that the normal limit of the function exists everywhere and equals the value of the function:
 limx→ c f(x) = f(c)
• From the lesson, we learned that if the normal ("two-sided") limit of a function exists at some location, then both one-sided limits exist at the location and are equal to each other.
 limx→ c f(x)   =  L    ⇔ limx→ c− f(x) = L = limx→ c+ f(x)
• Therefore to find both of the one-sided limits in the problem, we just need to find the limit of the function for x→ 2. Furthermore, since the function is continuous ("normal") we can just plug in the limit location of c=2 to find the value of the limit:
 limx → 2 x3−x2+3   =  (2)3 − (2)2 + 3

 =  8 − 4 + 3

 =  7
limx → 2 x3−x2+3   =    7    =   limx → 2 x3−x2+3
Let g(x) = {
 2x+5,
 x ≤ 1
 √{x2+7},
 x > 1
.     Find limx → −1 g(x)  and   limx → −1+ g(x).
• Start off by noticing a small but very important fact: while the piecewise function has its "breakover" location at x=1, the one-sided limits are x → −1 and x → −1+, so they are about the location −1. This means that, while f(x) does have two different pieces, we are only interested in one of those pieces. Since the breakover location is not the location that the limit is focused on, we do not have to care about the function piece that does not contain the limit location.
• Since we're only interested in the function piece that contains the limit location of −1, we only need to work with 2x+5. Next, notice that 2x+5 is continuous ("normal"). This means that the normal limit of the function exists and equals the value of the function:
 limx→ c f(x) = f(c)
Furthermore, because the normal ("two-sided") limit exists then we know both one-sided limits exist at the location and must equal each other.
 limx→ c f(x)   =  L    ⇔ limx→ c− f(x) = L = limx→ c+ f(x)
• Therefore to find both of the one-sided limits in the problem, we just need to find the limit of the function for x→ −1. Also, since the function is continuous ("normal") at the location since it is not a piecewise breakover, we can just plug in the limit location of c=−1 to find the value of the limit:
 limx → −1 g(x)   =  2(−1)+5

 =  −2+5

 =  3
limx → −1 g(x)   =    3    =   limx → −1 g(x)
Find the location of any and all discontinuities: f(x) = [(x−2)/(x2−x−2)].
• A function is continuous at c when f(c) exists and
 limx→ c f(x)   =  f(c)
A discontinuity is the opposite of the above. A discontinuity can happen because the function does not exist at the location, the limit does not exist at the location, or the limit does not match the value of the function at the location.
• To do this, figure out all the locations where f(x) is not defined. In other words, find where the function "breaks". Notice that [(x−2)/(x2−x−2)] can "break" because of dividing by 0. Factor the denominator to find where this can happen:
 x−2 x2−x−2 = x−2 (x+1)(x−2)
Thus f(x) is not defined at x=−1 or x=2, so there is a discontinuity at each of those locations.

[Remark: Notice that, although we can simplify the function as
 x−2 x2−x−2 = x−2 (x+1)(x−2) ⇒ 1 x+1 ,
this does not get rid of the function "breaking" at x=2. While we can simplify functions by canceling factors to find limits, this does not change the domain of the original function. The domain is based on the function before simplification, so x=2 is not a part of the domain even if we can later "fix" the hole with simplification.]
Discontinuities: x=−1,  2
For what values is the function f(x) = √{|x| −1}  continuous? [Give your answer in interval notation. For example, (5, 7], (−∞, 0), etc.]
• A function is continuous at c when f(c) exists and
 limx→ c f(x)   =  f(c)
Notice that f(x) can "break": taking the square root of a negative number is not defined, so the function will not exist whenever that happens.
• Figure out when a negative value would occur beneath the square root. There are two possibilities for the absolute value |x|: either x is positive (or 0) and it has no effect, or x is negative and it flips the sign. Deal with each of these cases separately. If x is positive, then the value under the square root is negative when
 x−1 < 0     ⇒     x < 1
Therefore the value under the square root is negative when 0 ≤ x < 1. If x is negative, then the value under the square root is negative when
 −x−1 < 0     ⇒     −1 < x
Therefore the value under the square root is negative when −1 < x ≤ 0. Since either one of these conditions will cause the value under the square root to be negative, that means the function is not defined ("breaks") when we have −1 < x < 1.
• Notice that for any other value, the function will be defined. That is, for any x where x ≤ −1 or x ≥ 1, the function will work "normally". How does this happen? Because for x ≤ −1, the function behaves as if it were just √{−x−1}, while for x ≥ 1, the function behaves as if it were just √{x−1}. With this in mind, let us graph the function to better understand it:
• Based on this graph and our understanding of how the function works, we see that for any x where x < −1 or x > 1, the function works "normally": the function exists at the location, the limit exists at the location, and the two match each other. Thus for x < −1 and x > 1, the function is continuous. From earlier, we saw that the function does not exist for −1 < x < 1, so it is not continuous at any of those values. This leaves us with two points to investigate: what happens at x=−1 and x=1? Clearly, from the graph and by using the function, we see that these locations do not "break" the function and f(−1) = f(1) = 0. However, for a point to be continuous, it must exist and have a matching limit at that location. Let us now figure out if limits exist at these locations. For ease, let's just consider x=1 at first. Notice that while the function exists for x ≥ 1, for x < 1 the function does not exist. But to have a limit, the function must be approaching the same value from both sides. We see that while the function exists to the right of x=1, there is nothing adjacent to x=1 on its left. Thus, the limit as x1 does not exist. (Notice that the right-side limit of x→ 1+ would exist, but that's not enough: continuity is based on the normal, two-sided limit.) We see that the exact same thing happens for x=−1, but in the opposite way. Thus the limit as x→ −1 does not exist either. Therefore, because the limits do not exist at these locations, they cannot be continuous. Combining that with our earlier observations, we now see that f(x) is discontinuous for −1 ≤ x ≤ 1. Thus the continuous points must be the opposite of the above, which is x < −1 and x > 1. In interval notation, we write that as (−∞,  −1) and (1,  ∞).
f(x) is continuous on (−∞,  −1) and (1,  ∞). [Be sure to notice that the function is not continuous at x=−1 or x=1. For a detailed explanation of why that is the case, check out the last step to the problem.]
Is this function continuous?     f(x) = {
 3x−7,
 x < 2
 −2x+3,
 x ≥ 2
• Begin by noticing that each piece of the piecewise function is "normal". That is, each piece of the function is continuous on its own. The issue comes at the "breakover" point where we switch from the first function piece to the second function piece. We need to know whether or not these two function pieces are headed towards the same location. In other words, we need to know if the left- and right-side limits of x→ 2 and x→ 2+ match up with each other.
• One way we could do this problem is by making a high-quality graph to see if the function pieces "connect" with each other at the breakover location of x=2. However, making a good graph takes a lot of effort and it is impossible to make an absolutely perfect graph. Instead, an easier way is to just figure out what the values being approached by the left and right sides are. If the limit as x→ 2 matches the limit as x→ 2+, then the pieces "connect" and the function is continuous. If they do not, then it is not continuous.
• Find the values being approached by each side. For x→ 2, we use the function piece based on x < 2 because we're considering the values to the left of x=2:
 limx → 2− f(x)     =     3(2) −7     =     6−7     =     −1
For x→ 2+, we use the function piece based on x ≥ 2 because we're considering the values to the right of x=2:
 limx → 2+ f(x)     =     −2(2) + 3     =     −4 + 3     =     −1
Because the left and right sides approach the same value, we see that the normal, two-sided limit exists there. This means that the function "connects" its two pieces, so the function as a whole is continuous.
Yes, the function is continuous.
Is this function continuous?     g(x) = {
 −x2−4x+2,
 x ≤ −3
 2x−1,
 x >−3
• Begin by noticing that each piece of the piecewise function is "normal". That is, each piece of the function is continuous on its own. The issue comes at the "breakover" point where we switch from the first function piece to the second function piece. We need to know whether or not these two function pieces are headed towards the same location. In other words, we need to know if the left- and right-side limits of x→ −3 and x→ −3+ match up with each other.
• One way we could do this problem is by making a high-quality graph to see if the function pieces "connect" with each other at the breakover location of x=−3. However, making a good graph takes a lot of effort and it is impossible to make an absolutely perfect graph. Instead, an easier way is to just figure out what the values being approached by the left and right sides are. If the limit as x→ −3 matches the limit as x→ −3+, then the pieces "connect" and the function is continuous. If they do not, then it is not continuous.
• Find the values being approached by each side. For x→ −3, we use the function piece based on x ≤ −3 because we're considering the values to the left of x=−3:
 limx → −3− g(x)     =     −(−3)2−4(−3)+2     =     −(9) +12 +2     =     5
For x→ −3+, we use the function piece based on x >−3 because we're considering the values to the right of x=−3:
 limx → −3+ g(x)     =     2(−3) − 1     =     −6−1     =     −7
Because the left and right sides do not approach the same value (5 ≠ −7), we see that the normal, two-sided limit does not exist there. This means that the function fails to "connect" its two pieces, so the function as a whole is not continuous. [Remark: Notice that each piece on its own is still continuous, the only issue is the breakover point. This means that g(x) is continuous on (−∞,  −3) and (−3,  ∞). It's just that since g(x) is not continuous at x=−3, it is not continuous everywhere, so we do not call it a continuous function.]
No, the function is not continuous.
What value must k be for the function to be continuous?
f(x) =

 −2x+4,
 x < 1
 x+k,
 x ≥ 1
• Begin by noticing that each piece of the piecewise function is "normal". That is, each piece of the function is continuous on its own. [While the value we choose for k will affect the second piece of the function, it will wind up being "normal"/continuous no matter what value we choose.] The issue comes at the "breakover" point where we switch from the first function piece to the second function piece. For the function to be continuous, we need these two pieces to "connect" together. To help us understand what's going on, we can imagine the problem like this: the first function piece is fixed in its location, but we are allowed to adjust the "height" of the second function piece (because we can change k, the y-intercept). We can visualize this with the below graph: the red part on the left is the first piece, while the other colored pieces represent some possible heights to place the second piece at by adjusting k.
• What we need to do is figure out the value for k that will "connect" these two pieces together (in the graph above, this would be how the red and blue pieces connect at the breakover). Formally, this means we need the left- and right-side limits to give the same value. If these sides match up to the same value, then the pieces "connect" and we will have a continuous function. Therefore we need to figure out what value for k will make the below equation true:
 limx→ 1− f(x)   = limx→ 1+ f(x)
• Notice that the limit as x → 1 is based on the left side, so it uses the function piece where x < 1. Similarly, x→ 1+ is based on the right side, so it uses the piece where x ≥ 1:
 limx→ 1− f(x)   = limx→ 1+ f(x)

 −2(1) + 4   =  (1) + k

 2   =  1+k
We can now solve the resulting equation for k to get that k=1. Therefore, to make the two sides "connect" and cause the function to be continuous, we must have k=1.
k=1
What value must c be for the function to be continuous?
g(x) =

 x+8,
 x < c
 −3x−2,
 x ≥ c
• Begin by noticing that each piece of the piecewise function is "normal". That is, each piece of the function is continuous on its own. [While the value we choose for c will affect where each function piece is used, both pieces will still be "normal"/continuous no matter what we choose for c.] The issue comes at the "breakover" location (which we are choosing since we are figuring out c). For the function to be continuous, we must choose c such that the two pieces will "connect" at the breakover. To help us understand what's going on, we can imagine both function pieces being graphed as if there were no conditions on either of them. Our job is then to choose the value of c that matches up to where they "connect". This will give us a breakover point that is continuous.
• More formally, we are trying to make the left- and right-side limits match up to each other at the location of x=c. While we do not know the value for c yet, we do know that if they match up, we will have
 limx→ c− g(x)   = limx→ c+ g(x)
Therefore, if we can figure out a value for c that makes the above equation true, we will have a continuous function.
• Even if we don't know what the value of c is yet, we can still consider how the limit interacts with the function. Notice that if we are looking at the limit as x→ c, then we are looking from the left-side, so x < c. Similarly, x→ c+ uses the right-side, so x > c. Using the appropriate pieces from our piecewise function, we have
 limx→ c− g(x)   = limx→ c+ g(x)
c+8   =   -3(c) -2\] We can now solve this equation to find c:
 c+8    =   −3c −2

 4c    =   −10

 c    =   − 5 2
Therefore, if we set c = −[5/2], then the breakover will be in the right location to cause the two pieces of the function to "connect", and therefore the function will be continuous.
c = −[5/2]

*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.

### Continuity & One-Sided Limits

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

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

### Transcription: Continuity & One-Sided Limits

Hi--welcome back to Educator.com.0000

Today, we are going to talk about continuity and one-sided limits.0002

By now, we have a pretty good intuitive understanding of how limits work, and a reasonable sense of how we can evaluate them precisely.0005

However, we haven't really worked with piecewise functions yet.0011

In this lesson, we will not only learn how to apply limits to piecewise functions, but in doing so,0015

we will learn about some new ideas involving limits.0019

Our exploration will deepen our understanding of what it means for a function to be continuous, and also reveal the idea of a one-sided limit.0022

And if you don't quite remember how piecewise functions work, or you find them confusing,0028

make sure that you check out the lesson on piecewise functions that we did a long, long, long time ago,0032

almost 40 lessons back--maybe even more; we did a lesson on piecewise functions.0037

So, if you don't have an understanding of how piecewise functions work, you probably want to check that out first,0042

because the idea of piecewise functions will end up showing up a lot when we are working with limits, especially in this lesson.0045

All right, let's go: first we will set up a motivating example, so that we can have some realizations come out of this motivating example.0050

This piecewise function example will help lead us to some ideas.0058

We have -1 when x is less than 0; that is the portion right here.0062

We have 0 when x equals 0 (that is the portion right there); and we have positive 1 when x is greater than 0.0067

And notice how there are holes at actually equal to 0 on the top and actually equal to 0 on the bottom,0075

because it jumps to the middle portion, right here, when it is actually at x = 0.0084

OK, notice that, while f(0) exists (we get a value--we get f(0) = 0), the limit as x goes to 0 does not exist.0091

As we show up from the two different sides, as we come in, they don't agree with each other.0100

So, the point in the middle, f(0), does exist; but the limit does not exist.0107

The issue is not so much that f(x) is a weird thing; it is not particularly weird.0110

It is more an issue of f(x) all of a sudden doing this jump, where it is doing this giant leaping around.0116

It is jumping from one place to another...wait a second...jumping?0122

That sounds familiar--the idea of moving around suddenly...breaking the graph...0127

Long, long ago, in this course, when we first learned about piecewise functions, about 40 lessons ago, we mentioned the idea of a continuous function.0132

At the time, we lacked the formal ideas to precisely define continuity.0142

So, we intuitively defined it as being any of these three equivalent things:0146

all of the parts of the function are connected--we have some function, and everything in the function connects together;0150

alternatively, we can think of this as that the function can be drawn without ever having to lift your pencil from the paper,0156

because if you have to lift your pencil, if we draw part of it, but then you have to stop here,0164

and lift your pencil to go somewhere else to keep going, well, we have this break here;0169

and because of that break, it means that it is not continuous.0174

And finally, there are no breaks or holes in the graph.0177

We see a break pretty clearly in this, where we go one way, and then all of a sudden it switches to something else.0180

But we can also it just having a hole as being the same issue, where it goes up here,0186

but then for some reason this point is not defined, and then it keeps going.0191

It is continuous, except for the part where it has this little break in the middle, which breaks continuity.0194

At this point, we actually now have the formal background to expand on this intuitive definition of it not being connected,0200

it not being continuous, and actually define continuity in a precise term, using limits.0207

To help us see this, we will compare a piecewise example with a continuous function.0213

Here is our piecewise example that we have been working with so far, and here is our continuous function friend0217

for any time we need to pull out a random function, g(x) = x2, a parabola.0224

Over here in our piecewise break, we can interpret the break in continuity, this part where it jumps, as it jumping.0230

The issue with it being continuous isn't that something breaks in the function.0237

The function is perfectly defined at every number; any number that you put in...it knows what to go out to.0242

But the issue is that it jumps all of a sudden; that is, at x = 0, it doesn't go where we expect; it does not go to the place that we expect.0248

Either direction we come at this thing, no matter how we look at it, we can't build an expectation.0256

We can't build an expectation for where the next place is, because there is no agreement; we can't agree on where we are going.0261

Formally, when we can't have an expectation--there is no agreement on where we are going0267

as we come from the two sides--that means that there is no limit as it goes to x = 0.0271

As we get towards x = 0, limit as x goes to 0 does not exist, because there is no agreement on the value that they are going to.0276

Those two sides are going to two totally different values.0283

On the other hand, in g(x), there are no jumps; the nice thing about g(x), the nice thing about the parabola,0285

is that the function goes where we expect; it always ends up showing up where we expect.0291

Every point is exactly where we expect that point to be.0297

That is to say, every point always has a limit, and that point matches with that limit.0300

All of g(x) is exactly where we expect it to be.0306

The limits and the points in the function are exactly the same thing.0308

This is the idea that we use for continuity: we see that a break in continuity, the break in a function,0312

is caused by the limit...our expectation not matching up with what the function does.0318

Formally, we say that a function f(x) is continuous at some location c if f(c) exists and the limit as x goes to c of f(x) equals f(c).0323

In other words, the function gives the value we expect when we evaluate it at c.0335

The value that we expect to come out as we get close to c is exactly what the value ends up being.0341

We expect some value, and then that value turns out to actually be what it is.0346

That idea of the limit matching what the function actually is there is what it means for something to be continuous at a point.0350

If f(x) is not continuous at c, we say that it is discontinuous at c; the opposite of continuous is discontinuous.0361

If we want to talk about some specific location where it breaks from being continuous, we say that it is a discontinuity.0369

A single discontinuous point is a discontinuity.0375

If f(x) is continuous at every point in an interval, over an entire interval--0380

it is always continuous throughout that interval--we say f is continuous on (a,b), on some interval.0385

If f(x) is continuous for every real number--it is always continuous--we simply say that f is a continuous function.0391

The function is continuous if it always ends up working out continuously; great.0399

Notice that the vast majority of the functions that we are used to working with are continuous functions.0405

When we think of functions, we usually think of these nice, smooth, curved things that all fit together quite cleanly.0409

So, when we think "normal," what we are really thinking is "continuous."0416

This is why normal functions allow us to simply plug in the value we are approaching--0420

because we generally have...a "normal" function, normally, in our minds, means that it is a continuous function.0424

And by definition, a continuous function is one where the limit as x goes to c of f(x) is the same thing as f at c--0430

that we can just plug in the c for our x, and we will end up getting the same thing as if we looked at the limit.0437

The expectation is where we actually come out to be.0442

Furthermore, this idea also explains why a function that is weird in one place still allows us to plug in the value we are approaching if it is not the weird value.0445

That is because, if it is not the weird value, then that means that it is normal in that area;0453

or more accurately, it is continuous in that place, if it is not the weird place.0458

And because it is continuous in that place, that means we still have the same thing: that the limit as x goes to c of f(x) will match with f(c).0462

That means that we can find out what the limit is by simply plugging in c into our function.0470

So, the same logic is what we can apply here.0474

And we end up seeing that, as long as it is normal in this region around c, as long as there is0477

some tiny little neighborhood around c that behaves normally (which is to say continuously),0481

we can just plug c in, and we will end up being able to figure out, "Oh, that is what the limit comes out to be."0486

Let us return to our motivating example, now that we have the notion of continuity learned.0492

Back to our motivating example: we have -1 when x is less than 0, 0 when x equals 0, and 1 when x is greater than 0.0497

On the one hand, as we approach 0, we totally don't have a limit.0503

As x goes to 0, it does not produce a limit, because they don't agree.0508

If they don't agree, it is not a limit; both sides have to agree.0512

However, if we were to imagine only approaching from a single side, all of a sudden, it starts to work; we could find limits.0516

This realization leads us to the creation of a new type of limit, the one-sided limit.0527

If we approach only from the negative side, it is clear that we are headed to -1.0531

If we were to approach only from the positive side, it is clear that we would be headed to positive 1.0535

So, we can have this idea of a one-sided limit, where we just look at approaching from one side or the other.0540

We don't worry about what the other side is doing; we just worry about what this side is doing;0545

where is it headed towards, as we stay coming from this one direction?0550

A one-sided limit basically works just the same as a normal limit, except that it only considers one side; it is only looking at one side.0555

The limit as x goes to c with a negative symbol in the top right--our c that we are going to, and then a negative symbol in the top right--0563

denotes the limit as x approaches c from the left, or the negative side, equivalently.0570

It is the limit as x goes to c, while x is less than c.0576

So, as x remains less than c--that is, as we come from the left side, the negative side, that is the limit that we end up getting:0579

limit as x goes to c, with that little negative sign in the corner.0587

Similarly, if we have limit as x goes to c with a positive sign in the corner, that denotes the limit as x approaches c from the right, or positive, side.0590

As we come from the right side, the positive side--that is going to be all of the x's where x is greater than c.0598

So, as our x stays above c, what are we going towards?0604

Be careful to keep track of which symbol goes with which side.0608

Remember: the negative symbol means to look at the negative side; and positive means to look at the positive side.0616

If we say that x goes to c with a negative symbol in that top right corner,0626

that means that we are saying to look as we come from the left side, the negative side.0631

If it is x goes to c with a positive sign in that top right corner, we are saying,0636

"Look at what happens as we come from the right side, as we come from the positive side."0640

All right, let's see this idea applied to our example.0646

We have -1 when x is less than 0; so as we go in from the left side, limit as x goes to 0 with a negative sign,0650

we get -1, because that is the value that we appear to be approaching.0658

If we come from the positive side, the limit as x goes to 0 from the positive side, we end up approaching positive 1.0662

If we just simply evaluate what f at 0 is, we get just simply 0.0670

Now, notice: there is not necessarily any connection between the left and right side or the actual value of what the function is.0677

-1 is totally different than 0, which is totally different than positive 1.0688

The limits, the two side limits, don't agree, and the actual value that comes out of the function doesn't agree with either of them.0692

So, each thing can behave totally independently.0698

However, one connection is that the two different side limits don't agree.0700

And because they don't agree, that means that there can't be a normal limit, the limit of it coming from both sides.0705

For there to be a normal limit, they have to match up; if they don't match up, then that means that there is no normal limit.0710

So, matching up: if the left and right side, the two sides, match up together, that means that we do have a normal limit.0716

If, on the other hand, they don't match up, then that means that we don't have a normal limit.0723

So, normal limits and one-sided limits: if both sides of a function go to the same value0727

as some location is approached, then the normal limit exists there.0735

Furthermore, if a normal limit exists, then both sides must go to the same value as they approach that location.0739

Symbolically, we can write this as the left-side limit equals l, and the right-side limit equals l,0748

implies that the normal limit is equal to l, as well.0756

If our left side goes to the same value as our right side, then that means that the sides match up.0760

So, we have a normal limit; and because they both went to l, then it comes out as an l limit, a limit going to l.0768

Similarly, if we went the other direction, if we know that there is a limit here at l;0774

then that means that, if we went off to the left side, or we went off to the right side,0778

we are going to end up having it be l in both of those directions, as well.0782

So, if the normal limit exists, then we know that the left-side limit and the right-side limit,0787

the negative limit and the positive limit, have to both be equal to the same thing.0795

They have to be equal to this same l.0799

All right, continuing with the same idea: if both one-sided limits do not go to the same value--0801

that is, they don't match up, or one of them simply doesn't exist at all--then a normal limit does not exist there.0808

That is to say, if the left-side limit is not the same as the right-side limit, then that means that the normal limit does not exist.0814

And if the normal limit does not exist, that tells us that one of them has to not exist, or that the sides don't match up to each other.0825

So, there is this connection between normal limits and one-sided limits.0834

If the one-sided limits meet up together, there is a normal limit at that place.0837

If there is a normal limit at that place, then that means that the one-sided limits must match up together.0840

And the same thing happens if the one-sided limits don't match up: then that means that there is no normal limit.0844

If there is no normal limit, then we can't have matching up; great.0848

All right, limits of piecewise functions: we are now ready to actually talk about how to apply all of this stuff to a piecewise function.0851

It is this idea about "Do they match up? Do they not match up?" that lets us find the limits of a piecewise function at a break-over point.0857

A break-over point is switching from one function to another function.0867

We are going here, and then, all of a sudden, we come out as some other thing.0871

Let's see what we are talking about here: it is some function here, and then all of a sudden, it swaps over to being some other function, like this.0874

If it is a break-over point, and we want to evaluate this, well, we can say, "Where were you going from the left side?0892

Where were you going from the right side?" and we can ask questions about each of the one-sided limits and how they relate to each other.0897

And if it is not a break-over point, then we don't have to worry about that, because that normally means0902

that we are just in the middle of a normal, nice, continuous part of a piecewise function.0907

And that means that we can usually just plug the location, as we discussed in the previous lesson, and as we sort of discussed earlier in this lesson.0912

If you are in the middle of a nice continuous chunk of a function, you just plug in the location that you are going to,0920

because "continuous" means that the limit matches up with what it evaluates to be there.0926

So, if we do have a piecewise function, though, and we are looking at a break-over,0931

where we switch from one track to another track, then we can find the limit of a piecewise function0934

by checking if left and right side agree; if the left side and the right side agree with each other,0943

then that means that we have a limit there; so if they agree, the limit exists.0952

And it is equal to what they both are, because they have to match up to the same value.0957

If they do not agree (one goes here and the other one goes here), then there is a jump between them,0961

which means that there can't be a normal limit, because we have to agree from both sides; then, the limit does not exist.0966

Since piecewise functions are usually made up of fairly normal functions0972

(you don't really normally see any very weird functions when we are dealing with piecewise functions),0975

since each side is normally fairly normal, it is normally pretty easy to find the left and right side limits,0979

when we want to compare them, because we can just say, "OK, if we plug in for the left side over here,0984

and plug in for the right side over here, we are able to figure out that that is what the left-side limit is;0988

that is what the right-side limit has to be; now decide whether or not they actually match up."0992

So, you can normally pretty easily figure out what they are.0996

All right, let's see some examples: the first example: Find both of the one-sided limits below0998

for f(x) = -3x + 3 when x < 2, and x2 + 2 when x ≥ 2.1004

First, let's just see a quick sketch to get some idea of how this thing works.1012

-3x + 3 when x is less than 2: we are going to end up being at a fairly steep incline like this.1017

And it pops out of existence here, because now we hit x < 2.1029

And then, we switch over to some x2 + 2; x is greater than or equal to 2;1034

so, at x2 + 2, we would pop up here; and then we would continue on our way,1039

with whatever the parabola is, already in motion.1043

All right, that is what we are seeing from this piecewise function.1047

When it says "the limits below for x going to 2 from the negative side," well, if it is from the negative side,1050

we are looking for what is happening as we come in here.1056

So, if we are looking at what is coming in here, then that is x less than 2, which means we are just concerned about -3x + 3.1059

So, if that is the case, then that means limit as x goes to 2 from the left side is just...1067

if we plugged in -3x + 3, and we plug in 2 (it is 2 from the left side; we plug in 2), -3(2) + 3 is -6 + 3, so we get -3 there.1072

Over here, if we want to talk about the limit as x goes to 2 from the positive side,1092

then what we are concerned about is as we come in from this side; as we come in from this side,1096

we belong to x being greater than or equal to 2; we don't actually have to worry about the x = 2,1101

because they are both limits, so it is only the journey towards that location.1106

We use x2 + 2 if we want to talk about what it is going to be.1110

x2 + 2 was nice and normal until that flip-over; so we can just plug into that.1113

x2 + 2 means it is going to be 22 + 2, 4 + 2, or 6.1117

And that is what we get for our two different sides for the limits here.1127

Notice that they don't match up.1130

All right, the second example: Evaluate the limit as x goes to 3 from the positive side of √(x - 3) + 4.1132

First, let's just quickly draw a quick sketch, so that we can see what is going on here.1137

So, √(x - 3) + 4: where would that start?1142

Well, the first value that would make sense in there...if we plug in anything less than 3,1148

we are going to be taking the square root of a negative number, so it can't be any less than 3.1152

So, the very first point we could plot is at +3; and we would have 4 come out of it.1155

And then, it would curve out like a square root function, like this.1160

OK, if we are concerned about x going to 3 from the positive side, then what we care about is what happens on our way there.1163

On our way there, it ends up just behaving like √(x - 3) + 4.1170

It is totally normal, up until the moment where it stops existing completely, when we try to take square roots of negatives.1174

The limit as x goes to 3 from the positive side, of √(x - 3) + 4...1180

well, since it behaves totally normally up until the point where it stops existing,1189

but we don't care about that side, because we are on the existing side, the positive side--we can just plug in our value for 3.1193

√(3 - 3) + 4 is √0 + 4, or positive 4; and that totally makes sense.1202

The point that we expect to go to is this one right here, where it starts.1211

There are no weird jumps; there is nothing weird going on as we go in.1216

As we come in from the positive side, it is very clearly headed towards 4, so it makes total sense that it has a limit.1219

Now, what about explaining why the limit as x goes to 3 of √(x - 3) + 4 does not exist?1230

Well, the real thing here is: does it come into the same thing from both sides?--no, because there is nothing over here at all.1236

If we try to come in from the left side, what is going on?1243

We have no idea what is going on; that is why it does not exist.1245

If we want to get even more formal, we can say that, for the limit as x goes to 3 to exist,1248

that must mean that the limit as x goes to 3 from the positive side is the same thing as limit as x goes to 3 from the negative side.1256

But clearly, the limit as x goes to 3 from the negative side of f(x) (this being f(x) in this case) does not exist.1267

You can't say that it is something that we are headed towards, because it just doesn't exist in that area.1279

If it doesn't exist, as we try to come in from the left side, there is no limit to say that it is going to.1284

So, if the right side simply does not exist, then that means that the normal limit can't exist, because we only have one-half of a normal limit.1291

Therefore, the limit as x goes to 3 does not exist; and that is why we see that.1298

But more informally, it is just the fact that we can't see it coming to the same thing from both sides.1309

We have to have it from both sides; so if one side simply just isn't there, then it doesn't exist.1313

The third example: Determine if the limit exists; if so, evaluate it.1319

We have a piecewise function here: x3 + 4 for x < -1; 7 when x = -1; and (x - 1)2 - 1 when x > -1.1324

So, our first question is, "Where are we headed towards?"1336

We are headed to x going to -1; so we don't just have it being in a nice, convenient, normal section.1338

We are going exactly in the break-over; all right, well, if that is the case, how do we check to make sure it exists?1345

Well, it only exists if from the right side and the left side it meets up to the same value.1352

The two different sided limits have to agree with each other.1358

If that is the case, what we are looking for is the limit as x goes to -1 of f(x); it is going to be based on the limit,1362

as x goes to -1 from the negative side (and don't get confused about the -1 and then the negative;1374

the negative in the top right tells us that we are coming from the negative side; the negative in the normal place1381

just means that it is a negative number), is going to be the same thing as the limit as x goes to -1 from the positive side.1385

If they end up being the same value for the limit, then we end up getting that it does exist as an actual limit, and it is that location.1394

The first thing to notice at this point is...do we care about 7 when x = -1?1402

No, we don't care about it, because it is x going to -1.1408

And remember: it is about the journey, not the destination; it is about where you are headed,1411

but you don't actually care about where you show up, when you are looking at a limit.1415

You just care about where it seemed like you were going to.1418

So, where we actually end up going doesn't matter; we don't have to worry about 7 at all.1420

It is just there as a distraction to get us confused.1425

The only things that we really have to care about are the x3 + 4, the part that we are coming from on the left side,1427

x less than -1, and the (x - 1)2 - 1, the part that we are coming from on the positive side, because it is x greater than -1.1433

So, if that is the case, let's work this out; we know that our left side limit will be based off of x3 + 4,1442

so that the limit as x goes to -1 from the negative side of f(x) will be the same as if we had simply plugged in -1 for our x here,1452

since we are coming from the left side, and x3 + 4 is the way that the thing behaves as long as it is on the left side of -1.1464

So, we plug into there; we have (-1)3 + 4; -1 cubed is just -1, plus 4...so we get +3 from the left side.1472

Switch to looking at our right side now; the question is, "Do the two sides agree, or do they disagree?"1483

If they go to different places, then the limit does not exist; if they go to the same place, then they do exist, and the limit is where they meet up.1489

So, in this case, we are now looking at...the right-side function is (x - 1)2 - 1.1495

We are looking at the limit, as x comes in from the right side (that is, the positive side), of f(x).1502

Once again, all we are concerned with is the right side of this, and the right side is entirely determined by this function,1510

because how that is how the right side is behaving that whole time, just as the left side was behaving the whole time for x3 + 4.1516

So, it is just a question of how that will fit into there, because (x - 1)2 - 1 is a nice continuous function.1521

So, we can just plug into it, because we don't have to worry about anything weird happening.1527

We plug into that, and we have...-1 swaps out for our x, minus 1, squared, minus 1; -2 inside of there, squared, minus 1;1531

so that gets us 4 - 1, which gets us 3; look at that: 3 and 3 check out.1541

So, that means that indeed the limit does exist.1548

Thus, we can combine these two things, and we know that, since the left- and right-side limits agreed with each other--1551

they both came out to be 3--that means that the limit as x approaches -1 from both sides of f(x) is equal to 3.1556

I do want to point out to you, though, that f(x) is not continuous.1567

Why? Because at x = -1, it jumps; so the left side expects to go to 3; the right side expects to go to 3.1570

But when we actually get to 3, it jumps up to 7.1578

So, because it jumps to somewhere else, it is not continuous.1580

The limit exists, but it is not actually a continuous function.1583

It has to have a limit and agree with what that point actually comes out to be, to be continuous.1586

All right, our final one, where we do actually talk about continuity: Determine the value of a that makes f(x) continuous.1592

What does it mean to be continuous? (I will write that out as cts, just because I am lazy).1600

To be continuous: that means that the limit, as x goes to c, of f(x), is equal to f(c).1605

Remember: we talked about this as the expectation for the function being the same thing as what we actually get out of the function.1612

That is what it means for something to be continuous.1618

The expectation, the limit, is the same thing as what we actually get out, f(c).1621

So, to be continuous, the limit as x goes to c of f(x) must be equal to f(c).1625

However, in this case, we don't just have to worry about limits; we also have to worry about the fact that this is a piecewise function.1629

Since it is a piecewise function, we need to make sure that both of the pieces end up agreeing.1635

The first question is, "Does the limit exist?"1641

Well, our question here--the function will be continuous if the limit as x goes to 1 of f(x) is equal to f at 1.1645

All right, that is the case; so what is f at 1?1658

We will come back to the left side and right side in just a second.1661

So, what is f at 1? Well, f at 1 is equal to 5 minus...1663

Oh, which one do we have to use? We use x less than or equal to 1, so we are using this right here.1668

So, 5 - 2: we swap out the x for a 1, minus...swap out the x for a 1...squared; 5 - 2 - 1 comes out to be 2.1674

So, f(1) comes out to be 2; great.1687

All right, now what we need to figure out is if the limit exists there.1693

If our limit is going to exist, we have two different sides that we are coming from.1698

We are coming from the left side, and we are coming from the right side.1702

We are coming from the less than or equal and the greater than.1706

We have both a left and a right side that we have to make sure match up.1709

Since we have two different possibilities, we have to make sure that the limit as x goes to 1 of f(x) is the same thing as saying1713

(because we have to check that both sides match) that the limit, as x goes to 1 from the negative side,1724

is equal to the limit as x goes to 1 from the positive side.1733

Now, we have to make sure that they are the same; and they should, of course, both be f(x) in here.1741

We have been talking about f(x) this whole time.1747

The limit as x goes to 1 from the negative side: well, actually, we already figured it out.1749

What is the limit as x goes to 1 from the negative side?1753

Well, that is going to be based off of how this works, because it is the left side.1756

x ≤ 1 behaves like 5 - 2x - x2; so if you are at the less than, you are on the negative side, so it behaves just like 5 - 2x - x2.1761

So, we already figured out what happens there; that comes out to be 2.1769

Since we are behaving just like the left side, that is just like it is going towards 2.1774

So, we know that this is going to come out to be 2.1779

Really, our only question is, "Does this part here equal the limit as x goes to 1 from the positive side?"1783

We are allowed to determine ax - 1; we can't change x, because that is just the variable;1791

but a--we are supposed to determine the value of a that will make this whole thing continuous.1796

We know 2, because that is what the limit is as x goes to 1 from the negative side, has to be equal to ax - 1.1802

What x are we going to? We are going to 1 from the positive side, so we just use ax - 1; we plug in 1 for our x; minus 1.1811

Start working that out: we have 2 = a - 1; we add 1 to both sides; we have 3 = a, so 3 must be equal to a.1818

And if 3 is equal to a, then that means that the limit on the right side is equal to the limit on the left side,1826

which means that the limit does exist, and that the limit will come out to be 2.1833

And since the limit comes out to be 2, and we know that f(1) = 2, we now see that, yes, it is, indeed, continuous.1838

It might be a little bit hard to understand what is going on, so it can really help to see this graphically.1845

Let's draw a quick picture, just to cement our understanding.1849

That is how you do it technically; but it is really useful to understand what is going on intuitively, as well.1852

Our 5 - 2x - x2 would graph basically like this.1858

And when we get to x ≤ 1, we jump over to the other one.1867

Now, since it is ax - 1, a is taking the position of the slope: mx + b is how we normally graph a line.1871

So, mx - 1 means that we are shifted down 1; we are definitely going to have a point there, down one on the y-axis.1881

But our a--we are allowed to choose our value of a.1889

So, what we are doing is effectively choosing the slope that we are going to have.1892

So, that means that we get to choose some possibility for our slope.1895

We have any possible rotation of this line; all of the different lines that could go through here with various different slopes are all of the different possibilities.1899

The one that we have to choose to make this thing come out to be continuous1910

is where that line matches up to where we have this hand-off, where we have this break-over point.1915

So, we choose the one that matches up; we choose that slope, and it continues out from here.1922

And that way, we end up having that the break-over ends up changing to a new track, but that track starts in the same place.1926

We have chosen the slope so that the line matches up with where the other one finished off; and it goes through; great.1935

All right, that finishes this; we now have an understanding of how to deal with piecewise functions.1941

It is basically a function of if the left side and right side, the one-sided limits, match up to each other.1946

And if we are talking about continuity, it is a question of if the limit matches what comes out of it.1952

And sometimes, it will become a question of if the two sides match what comes out of it.1956

All right, that finishes this lesson; we will see you at Educator.com later--goodbye!1960