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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Finding Limits

Slide Duration:

Table of Contents

I. Introduction
Introduction to Precalculus

10m 3s

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

45m 11s

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

35m 31s

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

35m 2s

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

48m 43s

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

56m 31s

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

39m 54s

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

58m 26s

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

48m 49s

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

29m 20s

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

48m 35s

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

33m 24s

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

51m 42s

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

49m 37s

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

28m 49s

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

38m 41s

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

41m 7s

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

39m 43s

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

45m 34s

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

46m 8s

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

45m 36s

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

19m 9s

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

33m 22s

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

34m 16s

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

49m 7s

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

44m 56s

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

35m 17s

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

47m 4s

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

40m 31s

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

42m 33s

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

34m 10s

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

48m 46s

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

39m 5s

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

43m 16s

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

33m 5s

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

52m 3s

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

36m 4s

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

27m 18s

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

32m 58s

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

31m 8s

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

19m 11s

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

23m 16s

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

52m 52s

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

29m 5s

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

43m 55s

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

25m 43s

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

56m 40s

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

49m 5s

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

27m 37s

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

34m 25s

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

55m 40s

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

1h 13s

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

41m 1s

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

1h 9m 31s

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

35m 20s

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

54m 7s

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

47m 12s

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

58m 34s

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

53m 33s

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

48m 7s

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

38m 16s

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

40m 43s

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

57m 37s

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

31m 36s

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

44m 3s

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

36m 58s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

57m 45s

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

40m 27s

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

31m 36s

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

39m 27s

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

49m 53s

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

1h 13m 13s

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

40m 22s

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

57m 11s

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

32m 40s

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

32m 43s

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

32m 49s

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

51m 13s

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

45m 26s

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

10m 41s

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

10m 51s

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

10m 38s

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

9m 45s

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

7m 8s

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

1 answer

Last reply by: Professor Selhorst-Jones
Thu May 9, 2013 4:50 PM

Post by Valentina Gomez on May 7, 2013

the example problems had me cracking up! Thank you for making limits seem easy!

Finding Limits

  • The easiest limits to find are the limits of "normal" functions. Now "normal" is not a technical term, it's just supposed to mean the kind of functions we're used to dealing with. Functions where
    • It does not "break" at the point we're interested in (the function is defined and makes sense);
    • It is not piecewise and/or the point we're interested in is not at the very edge of the domain.
    Assuming the two conditions above are true for a function and the point we're interested in is the value that x approaches in the limit, it will almost always be the case that the limit for the function is the same as the value for the function.
  • The above is true because, in general, most of the functions we're used to working with don't do anything "weird". That is, they're defined everywhere, they don't have holes, and they don't jump around. Therefore the functions we're used to working with go where we expect them to go.
    If f(x) is "normal" around x=c,    then
    lim
     
    f(x)  = f(c).
    This is true even if f(x) has "weird stuff" happening somewhere else. All we care about is x→ c, so as long as the neighborhood around x=c is normal, this works.
  • Often we can't use the above because something "weird" does happen at x=c. A common weird thing is dividing by 0. In this case, we can sometimes find the value of the limit (if it exists) by canceling factors before taking the limit.

    lim
     
     KA

    KB


     
       =   
    lim
     
     A

    B


     
  • For a radical expression (one with a root), the conjugate is the same expression, but flipping the sign on one side:


     

    x2 − 3x
     

     
    − 47x       

     
          

     

    x2 − 3x
     

     
    + 47x
  • Conjugates allow us to rationalize fractions, which can sometimes help us find the value of a limit. By multiplying the top and bottom of a fraction by the appropriate conjugate, we can often find limits that would not otherwise be possible.
  • We'll discuss evaluating the limits of piecewise functions in the next lesson, Continuity and One-Sided Limits. For now though, remember: as long as you're not trying to evaluate a limit on a piecewise "breakover" (where it switches from one function piece to another), the function is probably behaving "normally" on the pieces that contain the point you care about. Thus, if it's not a breakover point, you can approach it like you're evaluating the limit of "normal" function: just plug in the appropriate value and see what comes out.

Finding Limits

Evaluate limx→ −3  2x2 + 4x −5.
  • Notice that the function in the limit (2x2 + 4x −5) is a "normal" function. There's nothing strange or weird about it: it's just a standard polynomial. It doesn't "break" anywhere, it doesn't jump around, and it just generally works as we would expect it to work.
  • When dealing with a "normal" function (or, if the whole function is not "normal", then the neighborhood around the location we're interested in is), we can almost always just plug in the location that limit is approaching. That is, if the function is "normal" at the location being approached, then

    lim
    x→ c 
    f(x)   =  f(c).
  • As we noticed in the first step, f(x) = 2x2 + 4x −5 is a "normal" function. That means we can find the limit by simply plugging in the location it is headed towards: x→ −3, so plug in c=−3:

    lim
    x→ −3 
     2x2 + 4x −5     =  2(−3)2 + 4(−3) −5

        =  2·9 −12 −5

        =  18 −17

        =  1
1
Let f(x) = {
3x+2,
x ≤ 2
x2−8,
2 < x < 4
−2x+7,
4 ≤ x
.    Evaluate limx→ 5  f(x).
  • As we noted in the previous problem and in the lesson, when we're dealing with a "normal" function, we can almost always just plug in the location that the limit is approaching. This can still be true even if the function we're working with is not "normal": all we need is that the neighborhood around the location behaves in a "normal" fashion. Thus, if the function is "normal" at the location being approached, then

    lim
    x→ c 
    f(x)   =  f(c).
  • Notice that f(x) is decidedly not "normal": it is a piecewise function, so it does something "weird"-it jumps whenever it switches pieces. The function has "breakover" points at x=2 and x=4: these make the boundary edges between the different pieces of the function. However, we also must notice that we aren't interested in those "breakover" locations! We're interested in x→ 5, so there is a small neighborhood of "normalcy" around c=5. Thus, even though f(x) is not "normal" in general, we can still plug in because the specific location is in a "normal" section.
  • Because f(x) is "normal" in the neighborhood of c=5 (the location of the limit is not a "breakover"), we can plug in. [When evaluating the function, make sure to use the appropriate piece from the piecewise function.]

    lim
    x→ 5 
     f(x)     =  f(5)

        =  −2(5)+7

        =  −10+7

        =  3
3
Evaluate limx → 4 [(3x−12)/(x−4)].
  • Begin by noticing that we can not simply plug c=4 in to the function. If we did so, we would get a 0 for the denominator, and the function would break down. Thus, the function is not "normal" at the location we're interested in.
  • Once we realize we can't simply plug in the value for the location, we should begin looking for common factors that we can cancel. Factor the top and bottom of the fraction as completely as possible. This problem is fairly simple, and it's not too hard to see that a factor of (x−4) is in the top as well:

    lim
    x → 4 
    3x−12

    x−4
        =    
    lim
    x → 4 
    3(x−4)

    x−4
  • While we can't just plug in the location of the limit, we can now cancel out the common factor:

    lim
    x → 4 
    3x−12

    x−4
        =    
    lim
    x → 4 
    3(x−4)

    x−4
        =    
    lim
    x→ 4 
    3
    After canceling the factor, we see that there is now no issue with plugging in the location of the limit. (After all, at this point there isn't even an x variable remaining, so the limit has no effect: the limit of a constant is just the constant.)

    lim
    x→ 4 
    3     =     3
    Thus, by canceling out the common factor, we were able to make the function "normal" at the limit location and find the limit by plugging in.
3
Evaluate limx → −6 [(12x+72)/(x2−36)].
  • Begin by noticing that we can not simply plug c=−6 in to the function. If we did so, we would get a 0 for the denominator, and the function would break down. Thus, the function is not "normal" at the location we're interested in.
  • Once we realize we can't simply plug in the value for the location, we should begin looking for common factors that we can cancel. Factor the top and bottom of the fraction as completely as possible.

    lim
    x → −6 
    12x+72

    x2−36
        =    
    lim
    x → −6 
    12(x+6)

    (x+6)(x−6)
  • While we can't just plug in the location of the limit, we can now cancel out the common factor:

    lim
    x → −6 
    12x+72

    x2−36
        =    
    lim
    x → −6 
    12(x+6)

    (x+6)(x−6)
        =    
    lim
    x → −6 
    12

    (x−6)
    After canceling the factor, we see that there is now no issue with plugging in the location of the limit. The limit is x → −6, so plug in c=−6:

    lim
    x → −6 
    12

    (x−6)
        =     12

    (−6−6)
        =     12

    −12
        =     −1
    Thus, by canceling out the common factor, we were able to make the function "normal" at the limit location and find the limit by plugging in.
−1
Evaluate lima → 3  [(a4−81)/(a2+2a−15)].
  • First off, don't get freaked out by the fact that the limit is based around a instead of x. The specific variable being used is unimportant: everything still works the same way. Next, notice that we can not simply plug c=3 in to the function. If we did so, we would get a 0 for the denominator, and the function would break down. Thus, the function is not "normal" at the location we're interested in.
  • Once we realize we can't simply plug in the value for the location, we should begin looking for common factors that we can cancel. Factor the top and bottom of the fraction as completely as possible. Factoring the top might be a little challenging, but notice that we can re-write a4 as (a2)2. With this in mind, we get

    lim
    a → 3 
      a4−81

    a2+2a−15
        =    
    lim
    a → 3 
      (a2−9)(a2+9)

    (a−3)(a+5)
        =    
    lim
    a → 3 
      (a−3)(a+3)(a2+9)

    (a−3)(a+5)
  • While we can't just plug in the location of the limit, we can now cancel out the common factor:

    lim
    a → 3 
      a4−81

    a2+2a−15
        =    
    lim
    a → 3 
      (a−3)(a+3)(a2+9)

    (a−3)(a+5)
        =    
    lim
    a → 3 
      (a+3)(a2+9)

    (a+5)
    After canceling the factor, we see that there is now no issue with plugging in the location of the limit. The limit is a → 3, so plug in c=3:

    lim
    a → 3 
      (a+3)(a2+9)

    (a+5)
        =     (3+3)(32+9)

    (3+5)
    Simplify from there:
    (3+3)(32+9)

    (3+5)
        =     (6)(9+9)

    8
        =     (6)(18)

    8
        =     3 ·9

    2
        =     27

    2
[27/2]
Evaluate limx → 0  [(√{2+x}−√2)/x].
  • Begin by noticing that we can not simply plug c=0 in to the function. If we did so, we would get a 0 for the denominator, and the function would break down. Thus, the function is not "normal" at the location we're interested in.
  • Once we realize we can't simply plug in the value for the location, we should begin looking for common factors that we can cancel. However, we see that the function (in its current form) cannot be factored. Notice that the function involves square roots, though: this brings up the idea of rationalization. Hopefully, if we rationalize the numerator of the fraction, we will notice a common factor that can be canceled. Let's try that. [Remember, you rationalize by multiplying top and bottom by the conjugate-the same expression, but with a flipped sign.]

    lim
    x → 0 
     


     

    2+x
     
    −√2

    x
    ·


     

    2+x
     
    +√2



     

    2+x
     
    +√2
        =    
    lim
    x → 0 
      (2+x)−2

    x(

     

    2+x
     
    +√2)
    While we could distribute the x in the denominator, it won't actually help us: our goal is to look for common factors to cancel, so unless it actually simplifies things, we want to keep things factored. Simplify the top, and we finally see the common factor we can cancel:

    lim
    x → 0 
      (2+x)−2

    x(

     

    2+x
     
    +√2)
        =    
    lim
    x → 0 
      x

    x(

     

    2+x
     
    +√2)
  • While we still can't just plug in the location of the limit, we can now cancel out the common factor:

    lim
    x → 0 
      x

    x(

     

    2+x
     
    +√2)
        =    
    lim
    x → 0 
      1

    (

     

    2+x
     
    +√2)
    After canceling the factor, we see that there is now no issue with plugging in the location of the limit. The limit is x → 0, so plug in c=0:

    lim
    x → 0 
      1



     

    2+x
     
    +√2
        =     1



     

    2+0
     
    +√2
        =     1

    √2+√2
        =     1

    2√2
    Thus, by rationalizing to find a common factor and then canceling it, we were able to make the function "normal" at the limit location and find the limit by plugging in.
[1/(2√2)]
Evaluate limk → −4  [(√{k+13}−3)/(k+4)].
  • First off, don't get freaked out by the fact that the limit is based around k instead of x. The specific variable being used is unimportant: everything still works the same way. Next, notice that we can not simply plug c=−4 in to the function. If we did so, we would get a 0 for the denominator, and the function would break down. Thus, the function is not "normal" at the location we're interested in.
  • Once we realize we can't simply plug in the value for the location, we should begin looking for common factors that we can cancel. However, we see that the function (in its current form) cannot be factored. Notice that the function involves square roots, though: this brings up the idea of rationalization. Hopefully, if we rationalize the numerator of the fraction, we will notice a common factor that can be canceled. Let's try that. [Remember, you rationalize by multiplying top and bottom by the conjugate-the same expression, but with a flipped sign.]

    lim
    k → −4 
     


     

    k+13
     
    −3

    k+4
    ·


     

    k+13
     
    +3



     

    k+13
     
    +3
        =    
    lim
    k → −4 
      (k+13) − 9

    (k+4)(

     

    k+13
     
    +3)
    While we could expand (FOIL) the two factors in the denominator, it won't actually help us: our goal is to look for common factors to cancel, so unless it actually simplifies things, we want to keep things factored. Simplify the top, and we finally see the common factor we can cancel:

    lim
    k → −4 
      (k+13) − 9

    (k+4)(

     

    k+13
     
    +3)
        =    
    lim
    k → −4 
      k+4

    (k+4)(

     

    k+13
     
    +3)
  • While we still can't just plug in the location of the limit, we can now cancel out the common factor:

    lim
    k → −4 
      k+4

    (k+4)(

     

    k+13
     
    +3)
        =    
    lim
    k → −4 
      1

    (

     

    k+13
     
    +3)
    After canceling the factor, we see that there is now no issue with plugging in the location of the limit. The limit is k → −4, so plug in c=−4:

    lim
    k → −4 
      1



     

    k+13
     
    +3
        =     1



     

    −4+13
     
    +3
        =     1

    √9 + 3
        =     1

    3+3
        =     1

    6
    Thus, by rationalizing to find a common factor and then canceling it, we were able to make the function "normal" at the limit location and find the limit by plugging in.
[1/6]
Evaluate limx→ 0  [cos(x)/(ex)].
  • Begin by noticing that the function, while unusual, is not actually "weird". While it's not something we're used to working with, it never breaks down. Why? The numerator and denominator are both defined for any x value, and the denominator can never equal 0 (because although ex can get close to 0 as x → −∞, it can never actually reach 0). Thus, we see that the function is actually "normal".
  • When dealing with a "normal" function (or, if the whole function is not "normal", then the neighborhood around the location we're interested in is), we can almost always just plug in the location that limit is approaching. That is, if the function is "normal" at the location being approached, then

    lim
    x→ c 
    f(x)   =  f(c).
  • Thus, since the function we're dealing with is actually a "normal" function (even if it's one we're not used to working with), we can just plug in c=0:

    lim
    x→ 0 
      cos(x)

    ex
        =     cos(0)

    e0
    From here, just simplify. [Remember that cos(0)=1 (from the unit circle) and e0=1 (because anything raised to the 0 equals 1).]
    cos(0)

    e0
        =     1

    1
        =     1
1
Evaluate limv→ 0  [([1/(v−2)] + [1/2])/v].
  • First off, don't get freaked out by the fact that the limit is based around v instead of x. The specific variable being used is unimportant: everything still works the same way. Next, notice that we can not simply plug c=0 in to the function. If we did so, we would get a 0 for the denominator, and the function would break down. Thus, the function is not "normal" at the location we're interested in.
  • Once we realize we can't simply plug in the value for the location, we should begin looking for common factors that we can cancel. However, we see that the function (in its current form) cannot be factored. Notice that the function currently has fractions inside of fractions, though: this brings up the idea of simplifying it so no fractions appear in the numerator or denominator. Hopefully, if we find a way to express the function without all these extra fractions, we will notice a common factor that can be canceled. Let's try that. [Remember, we can knock out these fractions by multiplying the top and bottom by each denominator.]

    lim
    v→ 0 
     
    1

    v−2
    + 1

    2

    v
    · (v−2)(2)

    (v−2)(2)
        =    
    lim
    v→ 0 
      2 + (v−2)

    v(v−2)(2)
    While we could distribute and expand all the factors in the denominator, it won't actually help us: our goal is to look for common factors to cancel, so unless it actually simplifies things, we want to keep things factored. Simplify the top, and we finally see the common factor we can cancel:

    lim
    v→ 0 
      2 + (v−2)

    v(v−2)(2)
        =    
    lim
    v→ 0 
      v

    v(v−2)(2)
  • While we still can't just plug in the location of the limit, we can now cancel out the common factor:

    lim
    v→ 0 
      v

    v(v−2)(2)
        =    
    lim
    v→ 0 
      1

    (v−2)(2)
    After canceling the factor, we see that there is now no issue with plugging in the location of the limit. The limit is v → 0, so plug in c=0:

    lim
    v→ 0 
      1

    (v−2)(2)
        =     1

    (0−2)(2)
        =     1

    (−2)(2)
        =     − 1

    4
    Thus, by "cleaning out" the nested fractions to find a common factor and then canceling it, we were able to make the function "normal" at the limit location and find the limit by plugging in.
−[1/4]
Let f(x) = √x.     Find limh→ 0  [(f(x+h) − f(x))/h].
  • First, notice that while we can't really work with [(f(x+h) − f(x))/h], we can evaluate the function with those inputs to give us something we can work with:

    lim
    h→ 0 
      f(x+h) − f(x)

    h
        =    
    lim
    h→ 0 
     


     

    x+h
     
    − √x

    h
    Second, notice that we have h→ 0, not x → 0. This means we are looking for a way to swap out h for 0. However, if we try to plug in 0 for h right now, it will break down because we'll have a denominator of 0.
  • Once we realize we can't simply plug in the value for the location, we should begin looking for common factors that we can cancel. However, we see that the function (in its current form) cannot be factored. Notice that the function involves square roots, though: this brings up the idea of rationalization. Hopefully, if we rationalize the numerator of the fraction, we will notice a common factor that can be canceled. Let's try that. [Remember, you rationalize by multiplying top and bottom by the conjugate-the same expression, but with a flipped sign.]

    lim
    h→ 0 
     


     

    x+h
     
    − √x

    h
    ·


     

    x+h
     
    +√x



     

    x+h
     
    +√x
        =    
    lim
    h→ 0 
      (x+h) − x

    h(

     

    x+h
     
    +√x)
    While we could distribute and expand the factors in the denominator, it won't actually help us: our goal is to look for common factors to cancel, so unless it actually simplifies things, we want to keep things factored. Simplify the top, and we finally see the common factor we can cancel:

    lim
    h→ 0 
      (x+h) − x

    h(

     

    x+h
     
    +√x)
        =    
    lim
    h→ 0 
      h

    h(

     

    x+h
     
    +√x)
  • While we still can't just plug in the location of the limit, we can now cancel out the common factor:

    lim
    h→ 0 
      h

    h(

     

    x+h
     
    +√x)
        =    
    lim
    h→ 0 
      1

    (

     

    x+h
     
    +√x)
    After canceling the factor, we see that there is now no issue with plugging in the location of the limit. The limit is h → 0, so plug in c=0:

    lim
    h→ 0 
      1



     

    x+h
     
    +√x
        =     1



     

    x+0
     
    +√x
    Finally, simplify the expression as much as you can. It won't come out as a number, but that's okay: we weren't told a value to use for x, so x should appear in the final result.
    1

    √x+√x
        =     1

    2√x
[1/(2√x)] [While the problem makes no mention of it, the limit expression limh→ 0  [(f(x+h) − f(x))/h] is a very special thing in calculus. This problem foreshadows the idea of the derivative, which, while currently meaningless, will turn out to be an incredibly important idea for later mathematics courses (and science, economic, and engineering courses!). You'll get a glimpse of the concept in the second-to-last lesson of the course, Instantaneous Slope & Tangents (Derivatives).]

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

Answer

Finding 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 0:08
  • Method - 'Normal' Functions 2:04
    • The Easiest Limits to Find
    • It Does Not 'Break'
    • It Is Not Piecewise
  • 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
    • A Limit is About Figuring Out Where a Function is 'Headed'
  • Method - Canceling Factors 7:18
    • One Weird Thing That Often Happens is Dividing By 0
    • Method - Canceling Factors, cont.
    • Notice That The Two Functions Are Identical With the Exception of x=0
    • Method - Canceling Factors, cont.
    • Example
  • Method - Rationalization 12:04
    • Rationalizing a Portion of Some Fraction
    • Conjugate
    • Method - Rationalization, cont.
    • Example
  • Method - Piecewise 16:28
    • The Limits of Piecewise Functions
  • Example 1 17:42
  • Example 2 18:44
  • Example 3 20:20
  • Example 4 22:24
  • Example 5 24:24
  • Example 6 27:12

Transcription: Finding Limits

Hi--welcome back to Educator.com.0000

Today, we are going to talk about finding limits.0002

We often need to find the precise value that a limit will produce.0004

However, the methods we saw when we first introduced limits (that is, graphing it or a table of values for the function) are not precise.0007

They can give us a good idea of what the limit will be, but they don't give us certainty.0015

They don't let us know that it will be exactly something.0020

Likewise, the formal (ε,δ) definition of a limit that we talked about in the last lesson--0022

and it is totally fine if you do not know it; that was a completely optional lesson, only if you are really interested in math,0027

and wanted to find out more about stuff that is going to come later in a few years if you keep studying math;0033

it is totally fine if you didn't do it--but if you did, that is still not really going to help us find limits.0038

It allows us to formally prove that this limit has to be here.0043

And it is the deeper mechanics of what is going on "under the hood" for how a limit works.0047

But it doesn't let us find limits; it doesn't make finding them easier; it is just about proving limits.0051

In this lesson, we will see various methods to find the precise value; that is what this lesson will be about.0056

The basic idea that we are going to see with all of these methods is to transform the function into something0061

that works pretty much the exact same way, that we will just be able to plug in a value for the x0066

in this equivalent version, and we will be able to churn out some value for what the limit will come out to be.0072

Now, before you watch this, make sure that you are already familiar with the concept of a limit.0077

You really want to have a good understanding of how a limit works.0083

We will be working out how to get numbers in this lesson.0086

But if you don't actually understand what this stuff means, it is all going to fall apart really quickly.0088

So, it is really important that you understand how a limit works before you watch this.0092

If you don't already have a good understanding of how limits work, check out the lesson two lessons back,0096

Idea of a Limit, where we will explain and get an idea of what a limit is about.0101

And that way, we will have some meaning for how we actually get precise values.0105

You can figure out the precise values without really understanding what is going on.0110

But that will fall apart really, really quickly; and you might as well just have a nice foundation to work from there.0114

It won't take that long to get an idea of what is going on.0119

All right, first, the easiest limits to find are limits for "normal functions."0121

And now, that is in quotes, because normal is not a technical term.0127

What I just mean here is...it is supposed to mean the kind of functions that we are used to dealing with,0131

the sort of thing that we use most often, functions where it does not break at the point we are interested in--0135

that is to say, the function is defined and makes sense; it doesn't suddenly break down when we get to the place that we really care about.0142

And it is not piecewise, and/or the point that we are interested in is not at the very edge of the domain.0148

The point that we care about, whatever x we are going to (we are going to some x going to c)...0154

whatever c we are going towards, it is not going to be at the very edge of the domain, or where we split on some piecewise function.0159

So, as long as that location makes sense--everything in that area makes sense--0166

we know how to use the function in that area around that place, and it isn't piecewise--0169

there aren't different parts of it, and it is not the very edge of the domain,0174

the very starting or very last value for it--as long as those aren't the case, we will be able to do it really easily.0177

If these two conditions are met, where it isn't breaking down and it is not piecewise,0184

and the point isn't at the edge of the domain, then it is really easy to figure out what the value is going to be.0189

So, if the point we are interested in is the value that x approaches in the limit--0195

that is to say, x goes to c, and c is the place where things aren't breaking down,0200

and c is not at a piecewise break-over, and it is not at the very edge of the domain,0204

then it is almost always going to be the case that the limit as x goes to c of f(x)0208

is as simple as just plugging c in for x and getting f(c).0212

So, let's get an example first, to see how we could use this.0218

If we looked at the limit as x goes to 2 of 1/x2, what would that end up being?0220

Well, first notice: while 1/x2 breaks down (it isn't defined, that is to say) at x = 0, that is here.0225

But we don't care about x = 0; that is not the area that we are interested in.0232

We care about x going to 2; so if x is going to 2, if we look in that area over here, that region, that is totally fine; it makes perfect sense.0236

1/x2 works fine in the region around x going to 2.0245

As x goes to 2, well, all of this stuff makes sense; we can see it.0250

It clearly just maps to values, and it is perfectly reasonable.0254

Second, 1/x2 is not a piecewise function; we don't have to worry about that.0258

And x = 2 is not at the edge of the domain (and the domain for 1/x2 is every x, with the exception of x equaling 0; it is all x not equal to 0).0261

So, we are not at the edge of a domain; it is not piecewise; and it makes sense in the area we are looking for.0270

It makes total sense in here; so since it does all of those, it means that we can just plug in the value that we are going to.0275

Because of these two conditions, limit as x goes to 2, we just plug it in for x,0282

and we have 1/22, which simplifies to 1/4, and there is our limit.0286

Why can we do this--what is the reason that we can get away with doing this?0291

Well, in general, most of the functions we are used to working with don't do anything weird.0294

They aren't strange in any way; and what I mean by not being weird is that they are defined everywhere; they don't have holes; they don't jump around.0299

They are defined anywhere that we might be interested in looking; they don't have any holes in them; and they don't jump around.0311

They work in a pretty reasonable way; they work normally--they are not weird.0317

What this all means is that the functions we are used to working with, the functions that we normally work with, go where we expect them to go.0322

Since a limit is about figuring out where a function is headed (a limit is about what our expectation is0330

for this function), and a normal function has our expectations fulfilled (what we expect from the function0336

is what we get out of the function), that means that we can evaluate normal functions at the location the limit approaches.0343

Our expectation, the limit as x goes to c of f(x), what we expect to end up landing on in our journey,0349

what we expect as we come in, ends up being what it actually is--what it is at the location.0356

So, if it is normal, if f(x) isn't doing something weird, then what we expect ends up being what we actually get.0363

So, the limit as x goes to c...well, we can just plug that in for f(x),0371

and we have that f(c) will end up being the limit, any time that we are dealing with a normal function.0374

In fact, this is true even if f(x) does have weird stuff, but as long as it happens somewhere else.0381

All we care about is x going to c; so as long as the neighborhood around x going to c is normal,0387

and the weird stuff happens off somewhere else--it isn't happening directly on top of that c--0392

then the weird stuff...we don't care about it, because the region we care about being normal is the region around x = c.0397

As long as the neighborhood around x = c is normal, as long as around x = c does this,0403

then we will end up being able to just plug into that, just fine.0409

So, if there is weird stuff, that can be OK, as long as it is far enough away.0413

When we looked at 1/x2, there was weird stuff at x = 0, but we didn't care about x = 0.0416

We cared about x going to 2; so as long as the weird stuff isn't right on top of where we are going to,0421

we can just plug in our c, and that will tell us what the limit will come out to be,0426

if it is this fairly normal function that we have been working with for years and years.0431

Of course, sometimes x goes to c, but something weird does happen at x going to c.0436

When we are at x = c, it is a weird place.0441

One weird thing that often happens is dividing by 0; you are not defined when you divide by 0.0445

So, consider the function and limit below: f(x) = 2x/x.0450

Well, we can see it graphed here; it makes perfect sense, up until we try to plug in x = 0, at which point the function breaks down.0455

But the limit is pretty clear; it is going to 2 the entire time, so what is it headed towards?0462

It is headed towards 2; that is what we get out of the limit.0466

So, of course, we can clearly see that the limit in this case exists, and it is 2.0469

However, let's also notice that with the exception of x = 0, everywhere other than actually at x = 0,0474

where the weird thing happens, the function f(x) = 2x/x is just equivalent to if we had canceled out those x's and gotten g(x) = 2.0481

Notice: f(x) = 2x/x and g(x) = 2...these two functions are identical, with the exception at x = 0.0492

Everywhere other than x = 0, these two are totally the same.0503

f(x) = 2x/x and g(x) = 2 behave exactly the same, with an exception at x = 0.0509

However, since we are looking at the limit as x goes to 0, we don't care about x = 0.0516

It is about the journey, not the destination; so if it is x to 0, the destination we don't care about is at x = 0.0523

So, the weird thing here at x = 0...we might as well forget about it.0529

That means, since we don't care about what happens at the weird place, and g(x) = 2 is exactly the same0533

everywhere but the weird place, that means that we can use g(x) to evaluate the limit.0540

And we can just plug in g(0) to get 2.0545

So, g in general works just the same as f; g is just the same as f.0548

It works the same everywhere, with the exception of this one weird little point.0555

However, since we are looking at a limit going to that one weird little point, we don't actually care about the weird little point.0559

A single point doesn't matter, because the limit is about the journey towards that point.0564

So, g(x) = 2 behaves the exact same for the journey portion.0568

The journey towards behaves the exact same, whether we are using f or g, which means that g(x) is what we can use for figuring out the limit.0574

And then, by the same logic we were just talking about previously, since g(x) is totally normal at 2,0584

we can just plug in there, and we can get the answer from g(x).0588

Now, how did we find g(x)?--by canceling common factors.0591

This logic works in general; if we have some function given as a fraction, we can cancel out factors between top and bottom,0598

because a single point...if we cancel a factor, the only thing that can possibly happen0607

that would be bad is that we would cause one point to change around, like with f(x), x = 0 did not exist;0611

but with g(x), x = 0 did exist; so we caused a problem for specifically one point.0617

But a single point has no effect on a limit, because with a limit, it is about the journey, not a single point that is our destination.0622

So, since a journey is made up of a multitude of points, taking out a single point,0634

changing a single point, doesn't actually have an effect on where we end up landing for our limit.0638

That means, any time we have common factors for a limit, we can cancel out common factors0643

and just get what it would be without those common factors for the limit.0647

We will have changed the function that we are using, but the limits will be equivalent.0651

So, here is an example: we have limit as x goes to 3 of (x - 3)(x + 2)/(x - 3).0656

Well, that means that, if we were to plug x = 3 into this, we would have 0/0; 3 - 3 turns to 0; 3 - 3 on the bottom turns to 0; so we would get 0/0.0661

And so, we can't do that; it goes crazy; it is weird there.0671

But we can cancel out x - 3 and cancel out x - 3, just like we canceled out the k's here; and we get some other something over something.0674

We get what remained, a and b; in this case, we have x + 2 divided by 1 now.0682

So now it is the limit as x goes to 3 of x + 2; that is effectively going to work the same.0687

x + 2 is pretty much equivalent to x - 3 times x + 2 over x - 3, with the exception of x = 3.0692

But since we don't care about that, since that is where we are headed towards, we can end up using that limit instead.0700

So, we now plug in x going to 3 into x + 2.0704

Well, x + 2 is perfectly normal at x = 3; nothing weird happens there.0707

So, since nothing weird happens there, it works normally; we can just plug our value in there.0712

We plug in 3; 3 + 2...we get 5; our answer is 5 for the limit.0716

A similar idea to canceling out factors is rationalization: rationalizing a portion of some fraction.0722

If we have a radical that is in our way, and it is part of a fraction, we can change it into a non-radical0728

by multiplying the numerator and denominator by the same thing.0735

What do we get rid of a radical with--how do we rationalize an expression?0738

We rationalize an expression that contains a radical by multiplying by its conjugate.0743

That is the same expression, but now with a negative on one side.0748

For example, if we have √x + 2, well, that is a radical and some non-radical thing.0752

If we go through the conjugate process, we get √ - 2; plus switches to negative.0756

If we have √(x2 - 3x) - 47x, some radical minus something that is not in a radical,0762

then its conjugate is √(x2 - 3x) + 47x.0768

So notice: plus, if we are going through a conjugate, becomes minus; and minus becomes plus.0773

We just swap the sign on one of the portions, and that is how we get the conjugate.0781

Now, if we multiply a radical expression by its conjugate, we will end up canceling out the radicals.0785

And we will see how that works in just a moment.0790

Since multiplying a fraction on the top and bottom by the same thing gives an equivalent expression--0793

if I have a fraction, I can multiply it by 5/5, because 5/5 is just the same thing as 1, so it is still equivalent--0798

we can figure out limits by doing this thing.0804

And we can trust that this works, that this doesn't introduce any issues, by the exact same logic that we use to cancel out factors.0806

If we were to multiply by something over something, it could introduce one slight issue;0813

but it would only introduce a slight change to the function at one point.0818

But because it is a limit, we don't care about single points on their own; we only care about continuums of points.0821

So, that single point being changed is not really an issue; the limit will still work the same.0826

So, if we have the limit as √(x + 4) - 2, over x, as x goes to 0,0831

well, we would want to multiply this by the conjugate, √(x + 4)...but it was a minus previously,0838

so now it swaps to a + 2, and it will have to be divided by the same thing,0844

because we can only multiply by 1 effectively, which means the same thing on top and bottom: + 2.0848

And we multiply by that, and we start working things out.0854

What do we get out of that? Well, limit as x goes to 0 of √(x + 4) - 2, over x, times √(x + 4) + 2,0857

over √(x + 4) + 2...well, the top here is now going to become x + 4 - 4.0864

And if we are not quite sure how we see that, let's look at √(x + 4) - 2 times √(x + 4) - 2.0869

Well, what is...oops, I should not have put that radical over the whole thing; the radical ends there.0878

What is √(x + 4) times √(x + 4)?0883

Well, the square root of thing times the square root of same thing always just lets out the thing on its own.0885

The square root of smiley-face times the square root of smiley-face becomes smiley-face.0890

So, √(x + 4) times √(x + 4) becomes x + 4.0894

So, x + 4 is what we get out of that; and now we have √(x + 4) times -2...0898

oops, that shouldn't be -2; it should be plus, because it is a conjugate; I'm sorry about that.0905

√(x + 4) times positive 2 is 2√(x + 4): and then -2 times √(x + 4) is -2√(x + 4).0909

So, we have positive 2 √(x + 4) and -2√(x + 4); those two things cancel each other out.0916

Positive 2 and negative 2 cancel each other out; so the middle part disappears.0922

And now, it is -2 times +2; that becomes -4; so that is where we get x + 4 - 4.0927

On the bottom, it is x times this thing over here; so we just put in the quantity, because we will have some convenient canceling happening very soon.0934

On our top, we have x + 4 - 4; so plus 4 and minus 4 cancel each other out; we are left with just x on top;0943

it is still x times √(x + 4) + 2 on the bottom.0948

But now, we say, "We have x on top and x on bottom," and now we have the limit as x goes to 0 of 1/(radic;(x + 4) + 2).0952

At this point, we see that if we plug in 0...does anything weird happen?0959

Well, 1/(√(0 + 4) + 2)...we aren't dividing by 0 anymore, so it effectively works as a normal function.0964

We don't have any weird thing happening, so we plug in for our x at this point.0971

The square root of 0 + 4, plus 2 is in our denominator: 1 over...√4 is 2, so we have 1 over 2 + 2, which gets us 1/4.0975

And that is how we get to our answer for that limit.0985

We will discuss evaluating the limits of piecewise functions.0988

We haven't talked about piecewise functions yet, because we will be talking about that in the next lesson, Continuity and One-Sided Limits.0991

We will want a couple of little new ideas before we talk about piecewise functions.0997

That is why we are saving them for the next lesson.1001

For now, though, remember: as long as you are not trying to evaluate--as long as it is not trying1002

to evaluate the limit at some piecewise break-over, where it switches from one piece to another piece,1007

the function is probably going to be behaving normally on the pieces that contain the point you care about.1014

It might have a piecewise here and a piecewise here, and then it suddenly switches over.1018

But as long as we are over here in the normal area, or we are over here in the normal area, nothing weird happens.1021

So, if it is not at a break-over point, then that means that, since it is behaving normally,1027

you can approach it the same way as you do when just dealing with the limit of a normal function.1032

Just plug in the appropriate value and see what comes out.1037

Plug in the value that you are going towards and see what comes out.1040

However, if you do need to evaluate a break-over point, where you have to be talking1043

about where it switches from one to the next, check out the next lesson,1048

because we will see how that idea works specifically in the next lesson.1051

All right, we are ready for some examples.1055

The first one: Evaluate the limit as x goes to 2 of x4 - 3x2 + 4x - 10.1057

Our first question that we want to ask ourselves is if x4 - 3x2 + 4x - 10 is normal.1062

Yes, there is nothing weird that happened in that; that is just a polynomial--we are used to that sort of thing.1066

So, it is normal; if it is normal, then that means that we can just plug in our value for each of the x.1071

So, we plug in our 2, because we know that the limit of what it gets is...well, we don't have to plug x into 10...1078

we know that the limit as what we are going to get out of this is just the same thing as what the function would be there.1084

What we expect is what we get when we are dealing with a normal function.1089

So, we plug in 2; we now have 24 - 3(2)2 + 4(2) - 10.1094

24 is 16; minus...3(2)2...22 is 4; 3 times 4 is 12,1103

plus...4 times 2 is 8; minus 10; 16 + 8 gets us 24; minus 12 minus 10 gets us -22; we put those together, and we now have 2--done!1109

The next one: let f(x) equal x2 - 3 when x is less than or equal to 2 and 5x + 2 when x is greater than 2.1120

First, let's just see, really quickly, what this looks like.1127

Here is a rough picture of what it looks like; I will do that with blue here; so x2 - 3...that is basically like a parabola.1131

It has just been lowered by 3 from a normal standard parabola.1139

And it goes until x ≤ 2, at which point it stops here, at 2.1143

And then, after that, we are at x > 2, so we switch over to 5x + 2 when x is greater than 2.1147

It starts here, and then it goes off like this.1153

And that is what we end up seeing.1155

The question here is: if we are going to evaluate the limit as x goes to 1 of f(x)...oh, no, it is a piecewise function!1157

Well, yes; but we are clearly contained within x ≤ 2.1165

The area we care about is this area here; that is far enough from something weird happening.1169

Something weird does end up happening over a little bit further to the right.1174

But we don't care about that, because in the neighborhood we are interested in (that is x going to 1),1178

we are definitely less than or equal to 2, if we are close enough to 1.1183

So, since being close enough to 1 means nothing weird happens, all we are dealing with is x2 - 3.1187

So, we are effectively normal, because we only have to consider the part of the piecewise function that we are inside of.1194

The part of the piecewise function that we are inside of is x2 - 3.1201

So, we can just plug our 1 into x2 - 3; we plug in our 1; 12 - 3; 1 - 3 gets us -2, and there is our limit.1204

The next example: Evaluate the limit as x goes to -3 of (x2 + 3x)/(x2 - 9).1216

Our first question is, "Is it normal?" Well, if we plug in -3, what do we get on the top?1221

Well, we will get 9 - 9; so that is 0; and on the bottom, we will get x2 - 9,1226

so that will be positive 9, minus 9; oh, we get 0/0; so that means it is not normal.1231

Oh, no! But what do we do as soon as we are not normal?1237

We start thinking, "All right, well, what else can we do?"1241

We have the possibility of canceling factors; so what we want to do now is think, "Is there a way for us to cancel out factors?"1243

Can we cancel out factors? Well, we have x2 + 3x...1251

so I will write limit as x goes to -3...technically, we really should have the limit at every step.1254

If you end up really not feeling like writing out the whole thing, at least write limit, so that we know1259

that we are still dealing with the limit; and we will plug in something later.1263

(x2 + 3x)/(x2 - 9): well, limit as x goes to -3...how can we change the top?1266

How can we factor the top? Well, we could pull out an x, and we would have x(x + 3) on top.1277

How can we factor the bottom, x2 - 9? Oh, that is just the same thing as (x + 3)(x - 3).1282

Great; we can cancel factors; we cancel x + 3 and cancel x + 3.1289

And that is fine, because we are just working with a limit.1295

Our limit is now the same thing as x goes to -3 of x on top, divided by x - 3 on the bottom.1298

Now, we ask ourselves, "If we plug in -3, does anything weird and disastrous happen?"1307

-3 on top; -3 on the bottom; we don't get 0/0; we don't even get dividing by 0 once.1311

We are totally fine; so we plug in, because now it is effectively a normal function, since something weird isn't happening.1317

So, we have -3/(-3 - 3); that gets us -3/-6; the negatives cancel; 3/6 is 1/2, and so we get 1/2 as the answer to our limit.1324

Great; the next one: Evaluate the limit as x goes to 0 of tan(x)/sin(x).1337

The first question that we ask ourselves is, "Is this normal?"1342

Well, if we plug in 0, tan(0) is 0; sin(0) is 0; oh, that means it is 0/0; so it is not normal.1347

But if it is not normal, the first thing we try is asking ourselves, "Can we cancel factors?"1357

So, if we can cancel factors, we are good; how can we cancel factors?1363

Limit as x goes to 0...how can we change tan(x)?1367

Well, tan(x)...remember, that is just the same thing as sin(x)/cos(x).1372

And any time we don't have just sine and cosine, and we are dealing with trigonometric stuff,1377

it normally helps to put it just in terms of sine and cosine; so we have sin(x)/cos(x), all divided by sin(x).1380

Oh, OK; well, now we see that we can start canceling stuff; limit as x goes to 0...1390

well, we could rewrite this as sin(x)/cos(x); we could just cancel out the sin(x) if we see that directly.1395

But if we find it difficult to divide fractions and fractions, we could think of this as divided by sin(x),1401

which means that that is the same thing as limit as x goes to 0 of sin(x)/cos(x), times 1/sin(x).1407

If you divide, it flips to a fraction, which we could have also done by just breaking out the fraction of the sin(x) on the bottom to the side.1416

So now, we see that there is sin(x) on the bottom and sin(x) on the top.1422

We cancel some stuff out; limit as x goes to 0...now we have 1/cos(x).1426

Since we managed to cancel some stuff out, let's ask ourselves, "If we plug in 0, is it weird? Is it normal? What happens?"1432

Well, x goes into 0 for cos(x); cos(0) is 1; 1/1 is totally not weird anymore.1438

So, that means we can now swap in our 0; 1/cos(0)...cos(0) is 1; we have 1/1, so our limit comes out to be 1.1445

OK, the next one: Evaluate the limit as x goes to 4 of (2 - √x)/(4 - x).1460

The first thing we ask ourselves is, "Is it normal?"1466

Well, if we plug in 4 on the top, we will get 2 - √4, so that will come out to be 0,1468

divided by 4 - 4...even worse...dividing by 0...so 0/0 is not normal--no! No!1472

It is not normal, so the next thing we ask ourselves is if we can cancel factors.1480

Well, 2 - √x...4 - x...we might be able to figure out a way to cancel factors, but not easily.1484

Canceling factors...we might be able to figure out a way; we could figure out a way;1490

but let's say that we don't want to figure out canceling factors; canceling factors is not easy.1496

So, there is a radical; what was the trick we learned for dealing with radicals?1501

We rationalize; we move on to the next trick in our selection; we rationalize.1505

What do we do? We have the limit as x goes to 4 of 2 - √x, over 4 - x.1514

How do we rationalize? We multiply by the conjugate on the top and the bottom.1523

2 - √x: its conjugate is 2 + √x; we could also put a negative on the 2, but it doesn't really matter which side gets the negative.1526

2 + √x; 2 + √x; great--multiply our tops together, and multiply our bottoms together.1534

2 - √x times 2 + √x: these are now factors with parentheses around them, because we have to have distribution going on.1542

So, 2 times 2 is 4; 2 times √x is 2√x; -√x times 2 is +2√x and -2√x; they cancel each other out;1550

-√x times +√x becomes -x (√x times √x always comes out to be x;1559

√smiley-face times √smiley-face always comes out to be smiley-face; root(root) cancels the roots,1565

as long as it is the same thing underneath it).1569

4 - x times 2 + √x: well, we could multiply them together--but 4 - x is what we have on the top right now.1571

We are basically working towards canceling out factors; so 2 + √x...1577

at this point, we are now going to be able to cancel out factors.1582

We have the 4 - x on the bottom and the 4 - x on the top; they cancel each other out.1587

And now, I have the limit as x goes to 4 of 1/(2 + √x).1592

Great; so, if we were to plug 4 into this, would something weird happen?1601

1/(2 + √4)...no: we don't have 0 showing up; we aren't dividing by 0.1605

It basically works like a normal function; it does work like a normal function.1609

Nothing weird is going on there; so that means we can just plug in our value.1613

So, 1 over 2 plus...plugging in 4 for our x...1 over 2 + 2 (the square root of 4 is 2)...and 1/4 is our answer; nice.1617

The final example: Evaluate the limit as x goes to 0 of 1/(x + 3) - 1/3, all divided by x.1632

The first question we ask ourselves is, "Is it a normal function?"1638

So, is it normal? Well...you can guess by my hint: no.1641

If we plugged in 0, it would be 1/3 - 1/3, so 0 on top, divided by 0 on bottom...no, it is definitely not normal.1646

So, it is not normal; oh, no, what are we going to do?1654

Well, the next thing we ask ourselves is, "Can we cancel factors?"1658

Can we cancel? Not easily: 1/(x + 3) - 1/3...I really don't see any easy ways to make cancellation show up there.1663

So, we are probably not going to be able to cancel, at least not easily.1672

So, our next question is...last time we asked ourselves, "Can we rationalize?"1675

Well, there are no radicals here, so we can't rationalize.1678

But we can take a hint from the idea of rationalization.1681

The idea of rationalization was to multiply the top and the bottom by something that makes some part not weird anymore,1684

not as strange to deal with, so that hopefully, we can get cancellation to appear later.1689

Well, what would make the top...the thing that is really strange about this is that we have fraction over fraction.1693

We don't like fractions and fractions; so how can we get rid of some of those fractions by multiplying?1699

Well, the easiest way to get rid of the denominator in the top is to just multiply by the denominators in the top.1706

If we multiply...I will rewrite the thing out...limit as x goes to 0 of 1/(x + 3) - 1/3, all over x:1712

well, what would get rid of the x + 3? x + 3 would get rid of the denominator of x + 3.1723

What would get rid of 1/3? Well, multiply by 3.1730

We can get rid of both of those denominators by multiplying that whole top by (x + 3) times 3.1733

That will cancel out each of the denominators as we work through it.1738

And remember: it is always going to multiply the whole thing; when we multiply, we multiply the quantity, because we have to have distribution showing up.1740

And on the bottom, we will have to have the same thing, because otherwise we are changing the expression; we can't change the expression.1747

x + 3, times 3, over x + 3, times 3; great; our limit continues...limit as x goes to 0...1754

What do we get on the top? Well, (x + 3) times 1/(x + 3)...those cancel out, and we are left with just the 3 left over.1760

So, (x + 3) times 3, on 1/(x + 3)...the (x + 3)s cancel out; we are left with just 3.1767

Minus...when (x + 3) times 3 hits 1/3, well, the (x + 3) doesn't do anything; but the times 3 cancels out, so we are left with minus (x + 3).1774

All right, now we could expand the bottom, but that won't actually help us.1785

One of the ideas that we are going to hopefully manage to get to is to figure out a way to cancel things.1789

We couldn't cancel things easily by factoring; but hopefully we will still manage to cancel something at some point later on.1793

We don't want to expand factors; we want to actually keep up this process of keeping things in factors.1799

So, let's not put anything together; we will have it as x times x + 3 times 3.1805

At this point, we see x + 3 on top and x + 3 on the bottom, but we have to be careful; don't cancel stuff.1810

We can't cancel, because there is still a subtraction sign on the top; we have to have the whole factor.1814

We keep working to simplify: limit as x goes to 0...3 - (x + 3)...well, the 3's will cancel out,1818

and we will be left with just -x on the top, divided by x times (x + 3) times 3; great.1825

Limit as x goes to 0 of -x/x(x + 3)(3); at this point, we say, "We can cancel some stuff!"1838

This x and this x cancel, and we are left with the limit as x goes to 0 of -1 now (because it just canceled out the x,1844

not also the negative), times (x + 3), times 3.1852

Now, we ask ourselves, "Now that we have managed to cancel something, if we were to plug in a number, would we have something weird happen?"1856

Would it be normal now? If we plug in 0, we get -1/(0 + 3)(3); it doesn't look like we are going to be having dividing-by-zero issues anymore.1862

It isn't weird anymore, so we can just plug in; this is equal to the -1, over (0 + 3) times 3.1869

Once it is not weird, we can plug in, because now it is effectively normal.1879

And when it is effectively a normal function, you can just plug into it with your limit.1882

-1/3(3) gets us -1/9; and there is our answer--great.1887

All right, at this point, we have a really good understanding for how to figure out how limits work.1896

The basic idea is, "All right, I have a limit that is normal and doesn't have anything weird happen."1900

That is easy--just plug in something and crank it out--see what number you get out.1905

That is what the limit is, because "normal" means that your expectations will be met.1908

If it is not normal, if there is something weird happening, you try to manipulate things.1912

You either pull out factors or multiply the top and the bottom; you do something where you are allowed to cancel factors later on.1916

And then, you check and see, "OK, now that I have canceled out the factors, is it possible for me to plug things in and have it be normal, effectively?"1922

Can I now plug in (now that there isn't, hopefully, a weird thing happening)?1929

Sometimes there will still be weird things happening; in all of the examples we saw here, we canceled out anything that would cause weird stuff to happen.1932

But sometimes, you will end up still having weird stuff, no matter what you manage to cancel out.1939

And in that case, it can help to check a graph and think, "Oh, I see: it is going to go out to infinity," or something like that.1943

And you will see that it is never going to work.1947

But a lot of the time, you can cancel stuff out, and you can say, "Oh, OK, now it is effectively behaving like a normal function;1949

so I can plug in the x-value that I am going towards and just crank out an answer."1954

All right, we will see you at Educator.com later--goodbye!1958