Vincent Selhorst-Jones

Exponential Functions

Slide Duration:

Section 1: Introduction
Introduction to Precalculus

10m 3s

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

45m 11s

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

35m 31s

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

35m 2s

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

48m 43s

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

56m 31s

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

39m 54s

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

58m 26s

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

48m 49s

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

29m 20s

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

48m 35s

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

33m 24s

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

51m 42s

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

49m 37s

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

28m 49s

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

38m 41s

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

41m 7s

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

39m 43s

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

45m 34s

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

46m 8s

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

45m 36s

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

19m 9s

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

33m 22s

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

34m 16s

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

49m 7s

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

44m 56s

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

35m 17s

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

47m 4s

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

40m 31s

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

42m 33s

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

34m 10s

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

48m 46s

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

39m 5s

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

43m 16s

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

33m 5s

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

52m 3s

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

36m 4s

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

27m 18s

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

32m 58s

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

31m 8s

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

19m 11s

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

23m 16s

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

52m 52s

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

29m 5s

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

43m 55s

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

25m 43s

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

56m 40s

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

49m 5s

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

27m 37s

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

34m 25s

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

55m 40s

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

1h 13s

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

41m 1s

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

1h 9m 31s

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

35m 20s

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

54m 7s

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

47m 12s

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

58m 34s

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

53m 33s

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

48m 7s

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

38m 16s

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

40m 43s

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

57m 37s

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

31m 36s

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

44m 3s

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

36m 58s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

57m 45s

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

40m 27s

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

31m 36s

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

39m 27s

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

49m 53s

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

1h 13m 13s

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

40m 22s

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

57m 11s

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

32m 40s

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

32m 43s

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

32m 49s

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

51m 13s

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

45m 26s

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

10m 41s

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

10m 51s

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

10m 38s

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

9m 45s

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

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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 1 answerLast reply by: Professor Selhorst-JonesFri Mar 25, 2016 5:12 PMPost by Jay Lee on March 21, 2016Hi,How would you simplify (x^(5/4)y^(-5/4))/(x^(-1/2)y^(7/4)x^(2)y^(2))^(5/3)? The question requires the answer to contain only positive exponents with no fractional exponents in the denominator. I tried solving the question and ended up with a fraction in the denominator, so I changed those to the "root" form, but the teacher wouldn't accept that answer either. I solved until: (y^(1/2)x^(3/4))/(y^(8)x^(2)), but I don't know how to simplify it further. Thank you so much:)

### Exponential Functions

• An exponential function is a function of the form
 f(x)   =   ax,
where x is any real number and a is a real number such that a ≠ 1 and a > 0. We call a the base. [The base is the thing being raised to some exponent.]
• From the previous lesson, Understanding Exponents, we can compute the value of a given base raised to any exponent. In practice, we can find these expressions (or a very good approximation) by using a calculator. Any scientific or graphing calculator can do such calculations.
• Exponential functions grow really, really, REALLY fast. There's a fictional story in the lesson to get across just how incredibly fast these things grow. Multiplying by a factor repeatedly can get to extremely large values in a short period of time.
• Compound interest is a form of interest on an investment where the interest gained off the principal (the amount of money initially invested) also gains interest. Thus, compound interest returns more and more money the longer the investment is left in.
• We can describe the amount of money, A, in a compound interest account with an exponential function:
 A(t)   =  P ⎛⎝ 1 + r n ⎞⎠ ,
• P is the principal in the account-the amount originally placed in the account,
• r is the annual rate of interest (given as a decimal: 6% ⇒ 0.06),
• n is the number of times a year that the interest compounds (n=1⇒ yearly; n=4⇒ quarterly; n=12 ⇒ monthly;  n=365 ⇒ daily),
• t is the number of years elapsed.
• In the above equation, we can see that the more often the interest compounds, the more money we make. This motivates us to create a new number: the natural base, e. (See the video for more information on how we find e.) The number e goes on forever:
 e = 2.  718  281  828  459  045  235  360 …
Like π, the number e is an irrational number: its decimal expansion continues forever, never repeating. Also like π, the number e is deeply connected to some fundamental things in math and the nature of the universe. Just as π is fundamentally connected to how circles work, the number e is fundamentally connected to how things that are continuously growing work.
• One application of e is to see how an account would grow if it was being compounded every single instant- being continuously compounded:
 A(t) = Pert,
• P is the principal (starting amount),
• r is the annual rate of interest (given as a decimal),
• t is the number of years elapsed.
The above equation can also be used for a wide variety of things that grow (or decay) continuously. You'll see it show up a lot in math and science.
• We can see exponential decay if 0 < a < 1 in f(x) = ax. The exponential function will quickly become very small as it repeatedly "loses" its value because of the fraction (a) being compounded over and over.

### Exponential Functions

Find the value of e3 to three decimal places.
• The symbol e represents a number in the same way that π does. Also, just like π, e is irrational: its decimal expansion never ends and never repeats.
 e = 2.  718  281  828…
• Clearly, we can't use all the digits of e because there are infinitely many of them. However, the more of them we use, the more accurate our calculation will be. Since we need three decimal places of accuracy, let's begin by using at least six decimals from e: 2.  718  281. [This won't be perfectly accurate, but it will be accurate enough to find e3 to three decimal places. If we want more accuracy, we can use more decimals from e.]
• Plug the approximation into a calculator:
 e3    ≈     2.7182813 = 20.085  518  558  …
Thus, we round to three places, saying e3≈ 20.086. [Notice that the above approximation is not perfect. As we use more decimals, we get a more accurate result. Using a massive number of digits, we could find that
 e3 = 20.085  536  923  …
See how the highly accurate value for e3 above starts to give different values after the fourth digit. While we can get good accuracy with a modest approximation of e, if we really need a massive amount of accuracy, we have to start with a lot of accuracy in the value we use for e.]
20.086 [If your answer is close, but off by the last couple digits, you probably need to use more digits of e. Look at the steps for a detailed explanation.]
Complete the table below for f(x) = 3x.
 x
 f(x)
 −1
 0
 1
 2
 3
 4
 5
• To fill in the table, plug in each value to the function.
• For 3−1, remember that raising anything to the negative exponent causes it to "flip" to its reciprocal.
• For 30, remember that raising anything to the 0 causes it to become 1.
• As you fill out the table, notice how very quickly the function grows. Exponential functions become massively huge very, very quickly.

 x
 f(x)
 −1
 1 3
 0
 1
 1
 3
 2
 9
 3
 27
 4
 81
 5
 243
The function f(x) is an exponential function. Using the table of values below, state the function f(x).
 x
 f(x)
 0
 1
 2
 22.09
 3
 103.823
 4
 487.968
• An exponential function is a function of the form f(x) = ax, where a is some base. Thus, to answer the question, we need to figure out what a is.
• The fact that f(0)=1 is not much help to us. As long as the function is strictly in the form ax, then any value of a will still produce 1. This is because raising any number to the 0 causes it to become 1. So we can't get much useful information from the first entry in the table.
• The second and third entries are much more useful, though. We know f(2) = a2 = 22.09 and f(3) = a3 = 103.823. Notice that a2 ·a = a3. Thus,
 a2 ·a = a3     ⇒     22.09 ·a = 103.823     ⇒     a = 4.7
• Finally, now that we know a=4.7, we can double-check our answer. Currently, we have f(x) = 4.7x. From the table, we know f(4) should be 487.9681. Check using a calculator that 4.74 gives the same value. Indeed, it does, so our answer is correct.
f(x) = 4.7x
Graph the function f(x) = 4x−2−17.
• Like graphing any function, we can always use brute force: calculate and plot a bunch of points, then draw a graph from those points. Approaching it that way, we might create a table such as the below:
 x
 f(x)
 −5
 −16.9999
 −4
 −16.9998
 −3
 −16.9990
 −2
 −16.996
 −1
 −16.984
 0
 −16.938
 1
 −16.75
 2
 −16
 3
 −13
 4
 −1
 5
 47
• Alternatively, we could examine the structure of the function. Notice that f(x) = 4x−2−17 is similar to the expression 4x. From looking at graphs of other exponential functions, it might be very easy for us to graph 4x: it would be 1 at x=0, 4 at x=1, and shoot up rapidly as we go farther to the right. If we examine 4x as it goes to the left, it would approach the x-axis asymptotically. Now notice that 4x−2 would have the same graph as 4x, except it would be shifted 2 units to the right. Next, 4x−2−17 would be shifted a further 17 units down. [If you're not sure how to see this, check out the lesson on function transformations.] Thus, f(x) = 4x−2 −17 will have the same shape as 4x, but shifted 2 units right and 17 units down.
• Either way that we approach understanding the shape of f(x), we now need to graph it. Notice that f(x) moves very quickly vertically for a small amount horizontally. Thus, we might want to choose graphing axes that are very tall for a small width.
Graph the function g(x) = ( [2/3] )x.
• Because the base of the exponent is smaller than 1, the value of the function will become very small for positive x. As the fractions multiply each other more and more, the result will be tiny. On the other hand, when we have a negative x, the negative exponent will cause the fraction to flip, and thus allow the function to grow.
• To help us see how the function graphs, calculate various points:
 x
 g(x)
 −6
 11.39
 −4
 5.06
 −2
 2.25
 0
 1
 2
 0.44
 4
 0.2
 6
 0.09
• Plot points and connect with curves. Notice that as x moves very far to the right, the function will asymptotically approach a value of 0. On the other hand, as x moves to the left, the function will grow very rapidly.
A bank account is opened and $10 000 is placed in the account. If the account has an interest rate of 4.7%, how much money is in the account after 25 years if the account compounds annually? What if it had been compounded semiannually? Daily? • We can describe the amount of money, A, in such an account with an exponential function:  A(t) = P ⎛⎝ 1 + r n ⎞⎠ nt , where P is the principal in the account, r is the interest rate (as a decimal), n is the number of times the account compounds per year, and t is the number of years elapsed. • For this problem, we have  P=10 000, r = 0.047, t = 25. The only thing left is how many times the account compounds per year, and the question asks us to figure that out for three different compounding schedules. • Those compounding schedules are as follows: annually-once a year ⇒ n=1; semiannually-twice a year ⇒ n=2; daily-every day, so 365 times in a year ⇒ n=365. Use this along with the information above to figure out how much money is in the account for each situation. •  Annually: 10000 ⎛⎝ 1 + 0.047 1 ⎞⎠ 1·25 = 10000 (1.047)25 = 31 525.87  Semiannually: 10000 ⎛⎝ 1 + 0.047 2 ⎞⎠ 2·25 = 10000 (1.0235)50 = 31 944.22  Daily: 10000 ⎛⎝ 1 + 0.047 365 ⎞⎠ 365·25 = 10000 (1.000128767…)9125 = 32 378.98 Annually:$31 525.87,    Semiannually: $31 944.22, Daily:$32 378.98
Sally is currently 45 years old and is planning to retire at age 65. She has $25 000 in savings that she is going to put into an investment that has an interest rate of 7% and compounds continuously. How much money will she have when she retires? How much money would she have at retirement if she had made the same investment, but had instead done it at age 35? What about age 25? • If something is being compounded continuously, the amount (A) of money after t years is  A(t) = Pert, where P is the principal, r is the interest rate (in decimal), and e is the natural base. • The principal investment is P=25 000 and the rate is r=0.07. If she puts the money into the investment at age 45, then retires at age 65, that means the investment has t=20 years to mature. Thus, its value is  25000 e0.07 ·20 = 25000 e1.4 = 101 380.00 • For the other ages, we just need to figure out new values for t. If she invests at age 35, then that means t=30 years until retirement:  25000 e0.07 ·30 = 25000 e2.1 = 204 154.25 If she makes the investment at 25, then she has t=40 years:  25000 e0.07 ·40 = 25000 e2.8 = 411 116.17 [Notice the massive increase in the value of the investment. Because compounding investments are based on exponential functions, the earlier the investment can be made, the greater its final value will wind up being.] Age 45:$101 380.00,    Age 35: $204 154.25, Age 25:$411  116.17
Bryan buys a new car for $26 650. Because of depreciation, the value of the car goes down with a time. The value of the car after t years is  p(t) = 26 650 (0.87)t. What is the value of the car after one year of use? If Bryan sells the car after using it for a total of seven years, what will have been his net cost in owning the car? [Hint: The net cost is the original cost of the car, minus the amount he gets when selling the car.] • We have a formula to determine the value of the car after t years of use. To find the value of the car after one year of use, simply plug in t=1:  p(1) = 26650 (0.87)1 = 23 185.50 • To find his net cost upon selling the car at seven years, we first need to know how much he sells the car for. Figure out the value of the car after seven (t=7) years:  p(7) = 26650 (0.87)7 = 10 053.84 • Now that we know Bryan is able to sell the car for$10 053.84, we can calculate his net cost in owning the car. While it cost him $26 650 to purchase the car in the first place, he ultimately managed to sell it for some money, thus lowering the net cost of owning the car. To find the net cost, simply subtract the selling price from the original cost:  26650−10053.84 = 16 596.16 Value of car after one year:$23 185.50,    Net cost of owning for seven years: $16 596.16 A population of bacteria doubles every 26 minutes. If the population starts with 50 cells, how many will there be in 12 hours? (Assume that none of the cells die off.) • Notice that we are told the population doubles after a certain time interval. This is equivalent to multiplying the starting population by 2 after a time interval passes. After the next time interval, it will multiply by 2 again, and so on. Thus, if we say the number of time intervals is n, and the starting population is P, the amount A of bacteria will be  A = P·2n. • Now we need to figure out how many time intervals have elapsed. We know that one interval occurs every 26 minutes. The total time is 12 hours, which is 720 minutes. Thus, the number of intervals that have occurred is  n = 720 26 . • We can plug that in to the above to figure out the amount of bacteria after 12 hours:  A = 50 ·2[720/26] = 10 843 894 130 10 843 894 130 bacteria The radioactive isotope Carbon-14 has a half-life of 5 730 years. (Half-life is the time it takes for [1/2] of the isotope to decay and break down.) If we start with a quantity of 2.3 μg (micrograms) of C-14, how much will have not decayed after 30 000 years? • Notice that we are told the quantity halves after a certain time interval. This is equivalent to multiplying the starting amount by [1/2] after a time interval passes. After the next time interval passes, it will multiply by [1/2] again, and so on. Thus, if we say the number of time intervals is n, and the starting quantity is P, the amount A of quantity in the end will be  A = P· ⎛⎝ 1 2 ⎞⎠ n . • Now we need to figure out how many time intervals have elapsed. We know that one interval occurs every 5 730 years. The total time is 30 000 years. Thus, the number of intervals that have occurred is  n = 30 000 5 730 . • We can plug that in to the above to figure out the quantity of not decayed C-14 after the passage of 30 000 years:  A = 2.3 · ⎛⎝ 1 2 ⎞⎠ [30000/5730] = 0.061 0.061 μg *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 ### Exponential Functions Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. • Intro 0:00 • Introduction 0:05 • Definition of an Exponential Function 0:48 • Definition of the Base • Restrictions on the Base • Computing Exponential Functions 2:29 • Harder Computations • When to Use a Calculator • Graphing Exponential Functions: a>1 6:02 • Three Examples • What to Notice on the Graph • A Story 8:27 • Story Diagram • Increasing Exponentials • Story Morals • Application: Compound Interest 15:15 • Compounding Year after Year • Function for Compounding Interest • A Special Number: e 20:55 • Expression for e • Where e stabilizes • Application: Continuously Compounded Interest 24:07 • Equation for Continuous Compounding • Exponential Decay 0<a<1 25:50 • Three Examples • Why they 'lose' value • Example 1 27:47 • Example 2 33:11 • Example 3 36:34 • Example 4 41:28 ### Transcription: Exponential Functions Hi--welcome back to Educator.com.0000 Today, we are going to talk about exponential functions.0002 Previously, we spent quite a while looking at functions that are based around a variable raised to a number--0005 things like x2 or x47; this is basically the idea of all of those polynomials we have worked with for so long.0010 But what if we took that idea and flipped it?0017 We could consider functions that are a number raised to a variable, things like 2x or 47t,0019 where we have some base number that has a variable as its exponent.0027 We call functions of this form exponential functions, and we will explore them in this lesson.0032 Now, make sure that you have a strong grasp on how exponents work before watching this.0036 If you need a refresher on how exponents work, check out the previous lesson, Understanding Exponents, to get a good grounding in how exponents work.0040 All right, an exponential function is a function in the form f(x) = ax,0048 where x is any real number, and a is a real number such that a is not equal to 1, and a is greater than 0.0054 We call a the base: base is just the name for the thing that is being raised to some exponent.0060 So, whatever is being exponentiated--whatever is going through this process of having an exponent--that is called the base,0066 because it forms the base, because it is below the exponent.0072 We might wonder why there are all of these restrictions on what a can be; well, there are good reasons for each one.0076 If a equals 1, we just have this boring constant function, because we would have 1x, which is just equal to 1 all of the time.0081 So, something that is just equal to 1 all of the time is not really interesting, and it is not really going to be an exponential function.0091 So, we are not going to consider that case.0097 If a equals 0, the function wouldn't be defined for negative values of x.0100 If we try to consider what 0-1 is, well, then we would get 1/0, but we can't do that--we can't divide by 0.0104 So, that is not allowed, so that means a = 0--once again, we are not going to allow that one.0111 And if we had a < 0, then the function wouldn't be defined for various x-values, like x1/2.0116 For example, if we had -4, and we raised that to the 1/2, well, we know that raising it to the 1/2 is the same thing as taking the square root.0121 So, the square root of -4...we can't take the square root of a negative, because that produces imaginary numbers;0129 and we are only dealing with real numbers--we are not dealing with the complex numbers right now.0134 So, we are going to have to ban anything that is less than 0.0137 And that is why we have this restrictions: our base has to be greater than 0, and is not allowed to be 1,0140 because otherwise things break down for the exponential function.0145 All right, notice that, from the previous lesson, we can compute the value of a given base raised to any exponent.0149 We know how exponents work when they are a little more complex (not complex numbers, but just more interesting).0154 And so, we can raise things like...43/2 = (√4)3,0160 which would be equal to...√4 is 2; 23, 2 times 2 times 2, gives us 8; great.0165 If we had 7-2, well, then that would become 1/7, because we have the negative, so the negative flips it to (1/7)2.0171 So, 1 squared is 1; 7 squared is 49; we get 1/49.0179 So, we can do these things that are a little more difficult than just straight positive integers.0184 But we might still find some calculations difficult, like if we had 1.76.2--that would probably be pretty hard to do.0190 Or (√2)π--these would be really difficult for us to do.0196 So, how do we do them? In practice, we just find these expressions, or a very good approximation, by using a calculator.0200 We can end up getting as many digits in our decimal expansion as we want.0206 We can just find as many as we need for whatever our application is--whatever the problem asks for--by just using a calculator.0214 Any scientific or graphing calculator can do these sorts of calculations.0220 There will be some little button that will say xy, or some sort of _ to the _--some way to raise to some other thing--something random.0223 They might have a carat, which says...if I have 36 (not with an a--I accidentally drew that in...oh, I drew it in again),0235 then that would be equivalent to us saying 36.0250 The carat is saying "go up," so the calculator would interpret 36 as 36.0253 There are various ways, depending on if you are using a scientific calculator,0259 or if you are using a graphing calculator, to put these things into a calculator and get a number out.0261 So, we are able to figure these things out, just by being able to say "use a calculator."0266 Now, from a mathematical point of view, that is a terrible statement.0270 We don't want to say, "We can deal with this because we have calculators!" because how did you figure it out before you had calculators?0273 Calculators didn't just spawn into existence and give us the answers.0279 We can't rely on our calculators to do our thinking for us; we have to be able to understand what is going on.0283 Otherwise, we don't really have a clue how it works.0287 But as you will see as you get into more advanced math classes, there are methods to figure out these values.0290 There are ways to do this by hand, because there are various algorithms that give us step-by-step ways to get a few decimals at a time.0296 Now, doing it by hand is long, slow, and tedious; it would be hard to get this sort of thing, just because it would be so much calculation to do.0303 We could do it, but that is what calculators are for; they are to do lots of calculations very quickly.0310 They are to help us get through tedious arithmetic.0316 So, since these sorts of calculations take all of this arithmetic, we designed calculators that can do this method for us.0319 And that is why we can appeal to a calculator--not because the calculator knows more than us,0326 but because, at some point, humans figured out a method to get as many decimals as we wanted to;0331 and then, we just built a machine that is able to go through it quickly and rapidly,0336 so we can get to the thing that we want to look at, which is more interesting, using this.0340 The calculator is a tool; but it is important to realize that we are not just relying on it because it has the knowledge.0345 We are relying on it because, at some point, we built it and put these methods into it.0350 And if you keep going in mathematics, you will eventually see that these are where the methods come from--there is some pretty interesting stuff in calculus.0354 All right, now, if we can evaluate at any place--if we can compute what these values of exponential functions are--0361 then we can make a graph, because we can plot as many points as we want; we can draw a smooth curve.0368 So, let's look at some graphs where the base is greater than 1.0372 If we have 2x, that would be the one in red; 5x is the one in blue, and 10x is the one in green.0375 Now, notice: 2x, 5x, and 10x--all of these end up going through 1, right here,0384 because what is happening there is that 20, 50, 100...anything raised to the 0--they all end up being 1.0392 Remember, that is one of the basic properties of exponents.0401 If you raise something to the 0, it just becomes 1; so that is why we see all of them going through the same point.0403 And notice that they get very large very quickly.0409 By the time 2 is to the fourth, it is already off; and 10 is off by the time it gets to the 1.0411 10x grows very quickly, because it is multiplying by 10, each step it goes forward.0417 Notice also: as we go far to the left, it shrinks very quickly.0421 Let's consider 10-3; 10-3 would be the same thing as 1/103, which would be 1/1000.0425 That is why we end up seeing that this green line is so low.0436 It looks like it is almost touching the x-axis; it isn't quite--there is this thin sliver between it.0440 But it is being crushed down very, very quickly, because of this negative exponent effect,0444 where it gets flipped over, and then it has a really, really large denominator very quickly.0448 So, we see, as we go to the left side with these things, that it will crush down to 0.0454 And as we go to the right, it becomes very, very big.0460 We can change the viewing window, so that we can get a sense for just how big these things get.0464 And look at how big: we have gotten up to the size of 1000 by the time we are only out to 10.0468 And that is on 2x; if we look at 10x, 10x has already hit 1000 at 103.0475 At x = 3, it has managed to hit 1000 as its height.0483 This stuff grows really quickly; this idea of massive growth is so central to the idea of exponential functions.0488 We are going to have a story: there is this story that often gets told with exponential functions,0494 because it is a great way to get people to understand just how big this stuff gets.0499 So, let's check it out: All right, long ago, in a far-off land, there was a mathematician who invented the game of chess.0505 The king of the land loved the game of chess so much that he offered the mathematician any reward that the mathematician desired.0513 The mathematician was clever, and told the king humbly, "Your Highness, I thank you;0519 all I ask for is a meager gift of rice, given day by day on a chessboard."0523 "Tomorrow, I would like a single grain of rice give on the first square;0528 on the next day, two grains of rice given on the second square; then on the following day, the third day,0533 four grains of rice; and so on and so forth, doubling the amount every day until all 64 squares are filled."0541 So, the mathematician is asking for the first square, doubled, doubled, doubled, doubled...0550 The mathematician drew the king a diagram to help make his request clear.0556 On the first day of his gift, he would end up having one grain of rice on the first square.0559 On the second day, there would be a total of two grains of rice (1 times 2 becomes 2).0567 On the third day, there would be a total of 4 grains of rice (2 times 2 becomes 4).0572 On the next day, there will be 8 (4 times 2 becomes 8), and then 16, and then 32, and so on and so on and so on,0578 going all the way out to the 64th day, doubling each time we go forward a square on the board.0586 The king was delighted by the humble request and agreed to it immediately.0593 Grains of rice? You can't get a lot of grains of rice on a single chessboard; "It will be very easy," he thought.0597 He ordered that the mathematician would have his daily reward of rice delivered from the royal treasury every day.0603 A week later, the king marveled at how the mathematician had squandered his reward.0609 After all, he only had to send him 26 = 64 grains of rice that day.0612 Notice: on the seventh day, we are at 26--let's see why that is.0617 On the first day, we have 1 grain; on the second day, we have 2 grains; on the third day, we have 4 grains.0624 On the fourth day, we have 8 grains; on the fifth day, we have 16 grains; on the sixth day, we have 32 grains.0631 And thus, on the seventh day, we have 64 grains.0638 So, notice: we can express this as 20, 21, 22, 23, 24, 25, and then finally 26 on the seventh day.0642 Why is this? Because on the very first day, he just got one grain.0655 Every following day, it multiplies by 2--it doubles.0659 So, that means we multiply it by 2; so we count all of the days after the first day, which is why, on the seventh day, we see an exponent of 6.0662 So, in general, it is going to be 2 to the (number of day minus 1).0671 We will subtract one to figure out the grains on some number of day.0678 So now, we have an idea of how we can calculate this pretty quickly and be able to get these things figured out.0683 Another week later, on the fourteenth day, the king sent him 213 (remember, it is the fourteenth day,0689 so we go back one, because it has been multiplied 13 times) grains of rice, which is 8,192 grains.0694 And 8,192 grains is just about a very large bowl of rice.0702 The king was still amazed at the fantastic deal he was getting.0706 But he was glad that the mathematician was at least seeing some small reward.0709 He loved the game of chess, after all; and if he ended up feeding the mathematician for a year, that was great.0713 It seemed like a wonderful deal; he was willing to give him palaces, jewels, and massive amounts of money.0719 He can give him a little bit of rice for the great game of chess.0724 At the end of the third week, on the twenty-first day, the king had to send the mathematician a full bag of rice,0728 because in the kingdom, a full bag of rice contained precisely 1 million grains.0734 So, on the 21st day, we have 220 grains of rice, which ends up being 1 million, 48 thousand, 576 grains.0739 So we see here: after we jump these first six digits, we have one million plus grains of rice.0747 So, he has managed to get one million grains of rice (which is one bag of rice), plus an extra 48000 in change, in grains.0753 So, perhaps the mathematician was not as foolish as the king had first thought.0761 At the end of the fourth week, the king was starting to get worried.0766 On the twenty-eighth day, he had to send him more than 134 bags of rice, because 227 is more than 134 million grains of rice.0769 So, we are starting to get to some pretty large amounts here.0780 Now, the royal treasury has a lot of rice; he is not worried--he has hundreds of thousands of bags of rice.0782 So, he is not too worried about it; but he sees that this is starting to grow quite a bit.0788 At that moment, the royal accountant bursts into the throne room and says, "Your Highness, I have grave news!0793 The mathematician will deplete the royal treasury! On the forty-first day alone, we would have to give one million bags of rice!"0799 because 240 is here, so we have one million million grains of rice; so we have one million bags of rice,0806 which is more than the entirety that the treasury has in rice.0816 "And if we kept going--if we let it run all the way to the sixty-fourth day,0819 we would have to send him more rice than the total that the world has ever produced,0823 because we would be at 263, which would come out to be 9 trillion bags of rice."0827 Look; we have ones here; we have thousands here; we have millions here, billions here, trillions here, quadrillions here;0835 it would be 9 quintillion grains of rice; if we knock off these first ones, we see that we are still at 9 trillion bags of rice.0842 That is a lot of rice, and the world doesn't have that much by far.0851 So, the mathematician's greed has enraged the king, and the king immediately orders all shipments of rice stopped.0856 The mathematician is not getting any more rice, and the mathematician is to be executed!0862 Now, the mathematician, being a clever fellow, hears the soldiers coming down the road, and he escapes.0866 He fled the kingdom with the few bags of rice that he could manage to carry on his back,0871 and he had to find a new place to live, far, far away from the kingdom.0875 So, the moral of the story is twofold: first, don't be overly greedy--don't try to trick kings.0880 But more importantly than that, exponential functions grow really, really large in a short period of time.0886 They get big fast; even if they start at a seemingly very, very small, miniscule amount, they will grow massive if given enough time.0895 So, that is the real take-away here from this story.0904 Exponential functions get big; they can start small, but given some time, they get really, really big.0907 All right, let's see an application of this stuff.0916 When you put money in a bank, they will usually give you interest on your money.0918 For example, if you had an annual interest rate of 10% (annual just means yearly) on a$100 principal investment0922

(the amount that you put in the bank), the following year you would have that $100 still0931 (they don't take it away from you), plus$100 times 10%.0935

Now, 10% as a decimal is .10; so it is $100 times .10, so you would get that$100 that you originally started with, and you would have $10 in interest.0940 Great; but you could leave that interest in the account, and then your interest would also gain interest.0948 The interest is going to get interest on top of it; so we would say that the interest is compounded, because we are putting on thing on top of the other.0954 So, you have$110 in your bank account now, because you had $110 total at the end last time.0961$110 gets hit by that 10% again; so you still have the $110, plus...now 10% of$110 is $11.0968 Notice that$11 is bigger than 10--your interest is growing.0978

Over time, you are getting more and more interest as you keep letting it stay in there.0985

You continue to gain larger and larger amounts with each interest.0990

Compound interest is a common and excellent way to invest money, because over time,0993

your interest gains interest, and gains interest, and gains interest.0998

And eventually, it can manage to get large enough to be even larger than the principal investment,1002

and be the thing that is really earning you money--the time that you have spent letting it compound.1006

We can describe the amount of money, A, in such an account with an exponential function: A(t) = P[(1 + r)/n]nt.1011

Let's unpack that: P is the principal in the account--the amount that is originally placed in the account.1024

So, in our example, that would be $100 put in; so our principal would be 100 in that last example.1032 r is the annual rate of interest, and we give that as a decimal: here is our r, right up here.1038 In the last one, that was 10%, so it was expressed as .10.1045 n is the number of times a year that the interest compounds; n is the number of times that we see compounding.1050 So, n = 1 would be yearly; n = 4 would be quarterly; n = 12 would be monthly; n = 365 would be daily.1057 In our last one, it compounded annually, every year; so it compounded just once a year, so n was equal to 1.1066 Notice that n also shows up up here; it is n times t.1072 And then finally, t is just the number of years that we have gone through; so it is times t.1077 So, let's understand why this is the case.1084 Well, if we looked at 10%, just on the$100, we would have $100 times 1 + 10%.1086 So,$100 times 1.1 equals $110.1096 Now, if we wanted to have this multiple times, well, the next time it is$110 times 1.1, again.1102

We would get another number out of it; and then, if we wanted to keep hitting it...1109

we can just think of it as (1001.1)t, and that will just give us the amount of times1112

that the interest has hit, over and over and over--our principal times the 11119

(because the bank lets you keep what you started with), plus the interest in decimal form,1125

all raised to the t--the number of years that have elapsed.1131

Now, what about that "divide by n" part?1134

Well, let's say that we compounded it twice a year; so they didn't just give you your interest1136

in a lump sum at the end of the year--they gave it to you in bits and pieces.1140

So, the first time it compounds, if they did it twice a year (let's say they did it semiannually, two times a year),1144

then it would be 1 (because they let you keep the amount of money), plus .1/2 (because they are going to do it twice in a year).1151

So, the first time in the year, we would get 100 times (1 + 0.05), 100 times 1.05.1158

The first time in the year it gets hit, you would get $105 out of that.1169 Now, they could do it again, and we would have$105 get hit with another one of 1.05, and then we could calculate that again.1173

And that would be the total amount that you would have over the year.1181

Now, notice: 105 times 1.05 is going to be a little bit extra, because we are getting that 5 times 1.05, in addition to what we would have ended up having.1183

We would have 105 times 1.05; 5.25...so we will end up getting 5.25 out of this.1192

So, we will have a total of 110 dollars and 25 cents.1200

So, by compounding twice in a year, we end up getting 25 cents more than we did by compounding just once in a year.1208

So, the more times we compound, we get more chances to earn interest on interest on interest.1214

1 + .1, divided by 2...it is going to happen twice in a year; so since it happens twice in a year,1219

we have to have the number of times that it is happening in a year, times the number of years.1225

So, at the twice-in-a-year scale, we would see 1.05 to the 2 times number of years, because it happens twice every year.1230

And this method continues the whole time; so that is why we have the divide by n, because the rate has to be split up that many times.1240

But then, it also has to get multiplied that many times extra, because it happens that many times extra in the year.1246

So, that is where we see this whole thing coming from.1252

Now, we noticed, over the course of doing that, that the more times it compounded, the better.1255

We earn more interest if it is calculated more often; the more often our account compounds,1260

the more interest we earn, because we have more chances to earn interest on top of interest.1267

So, we would prefer if it compounded as often as possible--every minute--every second--every instant--1271

if we had it happening continuously--absolutely constantly.1278

This idea of having it happen more and more often leads to the idea of the natural base, which we denote with the letter e.1281

The number e comes from evaluating 1 + 1/n to the n as n approaches infinity--as this becomes larger and larger--1288

because remember: the structure last time was 1 plus this rate, divided by n to the n times t.1295

So, if we forget about the times of the year that it is occurring, and forget about the rate, we get just down to (1 + 1/n) to the n.1302

So, we can see what happens as n goes out to infinity--what number does this become?1310

It does stabilize to a number, as you can see from this graph here.1314

So, by the time it has gotten to 40, it starts to look pretty stable; it has this asymptote that it is approaching, so it is starting to become pretty stable.1317

We can look at some numbers as we plug in various values of n.1325

At 1, we get 2; at 10, we have 2.594; at 100, we have 2.705; at 1000, 2.717; at 10000, 2.718; at 100000, it is still at 2.718.1328

And there are other decimals there; but we see that it ends up stabilizing.1342

As we put more and more decimal digits, as n becomes larger and larger and larger, we see more and more decimal digits that e is going towards.1347

e is stabilizing to a single value, and we see more and more of its digits, every time we keep going with this decimal expansion.1356

So, as we continue this pattern, e stabilizes to a single number.1364

Now, it doesn't stabilize to a single number where we have finished figuring it; we keep finding new decimals.1367

But we see that decimals we have found so far aren't going to change.1372

e is 2.718281828...and that decimal expansion will keep going forever.1376

Just like π, the number e is an irrational number; its decimal expansion continues forever, never repeating.1382

So, that decimal expansion just keeps going forever, just like π isn't 3.14 (it is 3.141...it just keeps going forever and ever and ever).1389

So, e is the same thing, where we can find many of the decimals, but we can't find all of the decimals, because it goes on infinitely long.1399

Now, also, just like π, the number e is deeply connected to some fundamental things in math and the nature of the universe.1406

e is connected to the very fabric of the way that the universe, and just things, work.1414

So, π is fundamentally connected to how circles work; circles show up a lot in nature, in the universe.1418

π is connected to circles, and e is connected to things that are continuously growing--1423

things that are always growing, that don't take this break between growth spurts, but that are just always, always, always growing.1430

e gives us things that are doing this continual growth; e has this deep connection;1438

and if you continue on in math, you will see e a lot (and also if you continue on in science).1443

One application of e is to see how an account would grow if it was being compounded every single instant.1448

That idea, that we are not just doing it every year; not just every day; not just every minute; not just every second;1454

but every single instant--that gives us P (our principal amount) times ert.1459

The amount in our account is P times ert; we can also just remember this as "Pert"; Pert is the mnemonic for remembering this.1466

P is the principal, or we can just think of it as the starting amount--however much we started with.1474

r is the annual rate of interest, and it can even be used for things that aren't just annual rates,1478

but r is the annual rate of interest; and remember: we give that as a decimal.1483

If we give it as a percent, things will not end up working out.1487

And t is the number of years elapsed.1489

Now, this above equation, this one right here--this "Pert" thing--this can be used for a wide variety of things1493

that grow or decay continuously--things that are constantly growing or constantly decaying.1499

You will see it show up a lot in math and science as you go further and further into it.1503

It is very, very important--this idea of some principal amount, times e to the rate times the amount of time elapsed.1508

You can use it for a lot of things; and while we will end up, in these next few examples, using some other things1516

than just ert (with the exception of the examples that involve continuously compounded interest),1521

you can actually bend a lot of stuff that you have in exponents into using e.1525

So, it is easiest to end up just remembering this one, and then changing how you base your r around it.1530

Now, don't get too confused about that right now; we will see it more as we get into other things and logarithms,1535

and also just as you get further and further into math.1540

You will see how Pert is a really fundamental thing that gives us all of the stuff that is doing the growth.1542

Finally, exponential decay: so far, we have only seen exponential functions that grow as we go forward--1550

f(x) = ax, where our base, a, is greater than 1; so it gets bigger and bigger as we march forward.1556

But we can also see decay, if we look at 0 < a < 1--1562

if a is between 0 and 1--it is a fraction--it is smaller than 1.1566

Here are some examples: if we have 4/5x, we see that one in red;1571

1/2x--we see that one in blue; 1/10x--we see that one in green.1576

Notice how quickly the functions become very small as they repeatedly lose value because of the fraction compounding on them.1583

1/10 becomes very small by the time it has gotten to just 2; we have 1/102, which is equal to 1/100.1590

So, it becomes very, very small: by the time we are at (1/10)10, we are absolutely tiny.1598

Once again, it looks like it touches the x-axis, but that is just because it is a picture.1604

It never actually quite gets there; there is always a thin sliver of numbers between it.1608

But it gets very, very, very close; they will all become very, very small as the fraction on fraction on fraction compounds over and over.1611

Bits get eaten away each time the fraction hits, so it gets smaller and smaller and smaller.1621

But notice: if we go the other direction, we end up getting very large, just like normal exponential functions that grow, where a was greater than 1.1626

They got small when they went negative; they grew when they went positive,1635

because when they went negative, they flipped; we have that same idea of flipping.1639

If we have 1/10 to the -2, well, that is going to be 10/1 squared, which is equal to 100.1642

And that is why we see it blow up so quickly--it becomes very, very large, because we go negative for decay things.1651

But we will normally be looking at it as we go forward in time, which is why we talk about decay,1657

and things that are greater than 1 being growth, because we are normally looking at it1661

as we go forward--as we go to the right on our horizontal axis.1664

All right, let's look at some examples.1668

A bank account is opened with a principal of $5000; the account has an interest rate of 4.5%, compounded semiannually (which is twice a year).1670 How much money is in the account after 20 years?1678 So, what do we need? We go back and figure out the function we are using.1680 The formula is the one for interest compounded; so it is our principal, times 1 plus the rate, but divided by the number of times it occurs,1685 and then also raised to the number of times it occurs in the year, times the number of years that pass.1696 So, what are the numbers we are dealing with here?1701 We have a principal of$5000; we have a percentage rate of 4.5%, but we need that in decimal, so we have 0.045.1703

And what is the amount of time? The amount of time is 20 years.1717

If we do this with it going semiannually, twice a year, when we look at that, it will be n = 2.1722

a at 20 =...what is our principal? $5000, times 1 + the rate, 0.045 divided by the number of times it occurs in the year;1730 it occurred twice; n = 2, so divide by 2; raise it to the 2, times how many years? 20 years.1743 We go through that with a calculator; it comes out to 12175 dollars and 94 cents.1751 Now, what if we wanted to compound more often--what if it had been compounded quarterly or monthly or daily or continuously?1763 If it was compounded quarterly, it would occur four times in the year--every quarter of the year, every season--so n = 4.1770 So, we have 5000 times 1 + 0.045/4; that will be 4 times 20; we use a calculator to figure this out.1779 It comes out to 12236 dollars and 37 cents.1792 So, notice that we end up making a reasonable amount more than we did when it was compounded just twice in the year.1800 We are making about 50 dollars more--a little bit more than 50 dollars.1805 What if we have it do it monthly? How many months are there in a year?1810 There are 12 months in a year, so that would be n = 12.1813 5000 is our initial principal, times 1 + our rate, over 12 (I am losing room)...12 to the t...12 times t; so what is our t?1817 Our t was 20; sorry about that...12 times 20.1829 That will come out to be 12277 dollars and 33 cents.1836 What if we have it at daily--how many days are there in the year?1845 There are 365 days in a year, so that will be an n of 365.1849 So, at 365, we have 5000 times 1 + 0.045/365 (the number of times it occurs--365--the number of times it occurs in the year);1855 we had 20 years total; we simplify that out; we get 12297 dollars and 33 cents.1871 And what if we managed to do it every single instant--we actually had it compounding continuously?1881 Well, if n is equal to infinity, we are no longer using this formula here.1886 We change away from this formula, and we switch to the Pert formula, because that is what we do for compounded continuously.1890 That is going to be 5000 times e; what is our rate? 0.045; how many years? 20 years.1898 Once again, we punch that into a calculator: there will be an e key on the calculator--1906 you don't have to worry about memorizing that number that we saw earlier, because there is always an e key.1909 5000 times...oops, let's just get a number here; we are not going to end up doing this number, because it would be hard to do.1915 We will use a calculator; so let's just hop right to our answer.1926 We get 12298 dollars and 2 cents.1928 Finally, I would like to point out: notice that we ended up seeing reasonable amounts of growth1935 when we jumped from going only semiannually (twice a year) to four times a year.1940 And we also saw an appreciable amount of increase when we went from four times a year to twelve times a year-- when we went to monthly.1945 We got a jump of a little over 40 dollars.1953 When we managed to make it up to daily, we got a jump of about 20 dollars.1956 But going from daily to every single instant forever only got us a dollar.1960 So, we get better returns the more often it happens; but they end up eventually coming to an asymptote.1964 It increases asymptotically to this horizontal...it eventually stabilizes at a single value.1970 So, you won't see much difference between an account that compounds every single day and an account that compounds every single instant.1976 There won't be a whole lot of difference.1984 It is much better to have daily versus yearly, but daily versus continuously is not really that noticeable.1986 The second example: The day a child is born, a trust fund is opened.1992 The fund has an interest rate of 6% and is compounded continuously.1996 It is opened with a principal of$14000; what is the fund worth on the child's eighteenth birthday?1999

What formula will we be using? We will be using Pert.2004

The amount that we have in the end is equal to the principal that we started with, times e to the rate that we are at times t.2007

What is our principal? Our principal is 14000 dollars. What is our rate? Our rate was 6%.2014

We can't just use it as a 6; we have to change it to a decimal form, because 6 percent says to divide by 100; so we get 0.06.2023

Finally, what is the amount of time that we have?2030

In our first one, we are looking at a time of the eighteenth birthday--so 18 years; t = 18.2032

A principal of $14000 times e to our rate, 0.06, times the amount of years, 18 years--2039 we plug that into a calculator, and we see that, on his eighteenth birthday, the child has managed to get 41225 dollars and 51 cents.2050 So, that is pretty good; but what if the child managed to not need the money--didn't really want the money--2062 wanted to save it and maybe use it to buy a house when he was 30 (or put down a good down payment on a house when he was 30)?2067 At that point, if he was 30 before he took out the money, the child would have 14000;2073 it is the same setup, but we are going to have a different number of years--times 30.2079 That would end up coming out to 84000; it has more than doubled since he was 18--pretty good.2083 So, it has more than doubled; he has managed to make$84000 there.2092

That is not bad--he could get a good down payment on a house with that, so it is pretty useful.2097

But if he really didn't need the money--if he managed to not spend that money,2102

and he said, "I will use it as a retirement fund; that way I won't have to invest for my retirement at all--I already have it set up."2105

How much would he end up having at the age of 65?2110

We have 14000--the same setup as before--times e to our rate, 0.06, times our new number of years we are doing--it is 65 years.2114

And you would manage to have a huge 691634 dollars and 29 cents.2124

So, this points out just how powerful compound interest was.2134

We managed to start at 14000 dollars; but if we can avoid touching that money,2137

if we can just leave it for a very long time, we can get to very large values as the interest compounds on itself over and over again.2142

In 65 years, which is a very long time, we managed to grow from 14000 dollars to 691634 dollars--a lot of money.2148

And this gives us an appreciation for how important it is to make investments for retirement at an early age.2162

It is difficult when you are young; but if you manage to invest when you are young--2168

if you can wait on spending that money now--it can grow to very large amounts by the time you want to spend it to retire.2172

So, that is the benefit of investing early--being able to do that.2177

Also, it shows just how great, how useful, an interest rate is.2181

If that 6% was bumped up to 8% or 10%, we would see massive increases.2185

You can get a lot of increase if you can just get that percentage rate up another point or two--it is pretty impressive.2189

All right, the third example: The population of yeast cells doubles every 14 hours.2194

If the population starts with 100 cells, how many cells will there be left in two weeks?2199

So, this isn't compound interest, and it isn't continual growth, like we had before.2205

We might want to build our own here.2210

The population is doubling, so let's say n is the number of cells after some time.2212

We will set it up as a function--that makes sense; we are in "Exponential Function Land" right now.2217

So, n(t) is equal to...well, how many cells did we start with?2221

We started out with 100 cells, and we were told that it doubles.2226

So, we are going to have some "times 2," because we multiply it by 2.2231

How often does it do that? It does it every 14 hours.2234

So, if we have our number of hours, t = number of hours, t divided by 14 will be how many times it has managed to double.2237

After 14 hours, we have multiplied by 2 once.2249

After 28 hours, we have multiplied by 2 twice; we have 2 times 2 at 28 hours.2252

So, let's do a quick check and make sure that this is working out so far.2258

So, if we had n at 14 hours, we would have 100 times 214/14, which would simplify to 100 times 21.2262

So, we would get 200, so that part checks out.2273

Let's try one more, just to be sure: n(28)...if we had double double, then we know that we should be at 400, so we can see what is coming there.2276

So, 2 times 28/14...that simplifies to 100 times 22, which is equal to 100 times 4, or 400.2285

So, that checks out, as well; it passes muster--this makes sense as a way of looking at things.2295

So, as long as we have the amount of time we spent and the number of hours,2300

then we can see how many cells we have after that number of hours.2304

Now, we were told to figure out how many there will be in 2 weeks.2308

And we can assume that none of the cells die off, so the number just keeps increasing.2313

It is a question of how many times the population has gotten to double.2316

If that is the case, what number are we plugging in--it is n of how many hours?2320

Is it 2? No, no, it is not 2! Well, how many weeks...oh, 14 days? No, it is not 14.2324

What were we setting this up in? t was set in number of hours.2330

So, the question is how many hours we have on hand.2334

Let's first see how many hours 2 weeks is: how many days is that?2337

Well, that is going to be 2 times...how many days in a week?...7, so that is 2 times 7 days.2344

How many hours is that? 2 times 7 times 24, or 14 times 24 hours, which we could then figure out with a calculator, and get a number of hours.2349

But we can actually just leave it like that, which (we will see in just a few moments) is a useful thing to do,2361

because we notice that there is a divide by 14 coming up; maybe it would be useful to just leave it as 14 times 24--a little less work for us.2365

So, 14 times 24...now notice: 14 times 24 is the number of hours in 2 weeks.2373

That is why we are plugging that in, because once again, the function we built,2384

our n(t) function that we built, was based on hours going into it.2388

We can't use any other time format.2392

100 times 2 to the 14 times 24 (is the number for t), divided by 14; look at that--the 14's cancel out.2395

We can be a little bit lazier--that is nice.2406

100 times 2 to the 24: we plug that into a calculator, and we get a huge 1677721600 cells.2408

That is more than one and a half billion cells: ones, thousands, millions, billions.2425

So we are at 1.6 billion cells--actually, closer to 1.7 billion cells.2432

This gives us a sense of just how fast small populations are able to grow.2440

And that is how populations grow: they grow exponentially, because each cell splits in half.2443

So, if we have one cell split in half to 2, and then each of those splits in half to 4,2449

and each of those splits in half to 8, this is going to do this process of exponentiation.2453

We are doing this through doubling, so we are going to see very, very fast growth.2458

And we actually see this in the real world.2461

We could also write this, for ease, as...we have 1, 2, 3, 4, 5, 6, 7, 8, 9...so that is the same thing...2464

we could write it as approximately 1.67x109 cells,2474

so that we can encapsulate that information without having to write all of those digits.2482

That is scientific notation for us; all right.2486

The fourth and final example: The radioactive isotope uranium-237 has a half-life of 6.75 days.2488

Now, what is half-life? We would have to go figure that out, but luckily, they gave it to us right here.2495

Half-life is the time that it takes for one-half of the material of our isotope to decay and break down--to go through a process of decay.2499

If you start with one kilogram of U-237, how much will have not decayed after a year?2508

So, we are saying that, after 6.75 days, we will have half of a kilogram.2515

We start with one kilogram, and we know that, after every 6.75 days, we will have lost half of our starting material.2522

So, we will go down from one kilogram to half of a kilogram that has not decayed.2529

So, let's see if we can figure out a way to turn this into another function.2533

The...let's make it amount...the amount of our isotope that has not decayed, based on time,2537

is equal to...how much did we start with? We started with 1 kilogram; times...what happens every cycle?2545

1/2...we halve it every time we put it through a cycle; so how fast is a cycle?2551

The number of days--we will make t into the number of days, because we see that we are dealing with days, based on this here.2559

So, t divided by 6.75--let's do a really quick check.2567

We check, because we know that, after 6.75 days, we should have 1/2 of a kilogram.2572

So, let's check that by plugging it in: a(6.75) is going to be 1 times 1/2 raised to the 6.75 over 6.75,2580

which is the same thing as just 1/2 to the 1, which equals 1/2.2592

So sure enough, it checks out--it seems like the way we have set this up passes muster,2596

because it is going to divide by half every time the 6.75 days pass.2601

So, if we plugged in double 6.75, it would divide by half twice, because it would be 1/2 squared.2604

It seems to make sense; we have set it up well; and we can see that this also can be just written as 1/2 times...2610

let's just leave it as it is; it gives us a better idea of how this works in general, for half-life breakdowns.2616

So, now we are going to ask ourselves how long--what is the time that we are dealing with?2621

In our case, t is one year; what is one year in days (because we set up our units as days,2625

because that is what our half-life was given to us in)?2632

One year is 365 days; so at the end of that, we plug in 365 = 1 (the amount that we started with),2634

times...the half-life will occur every 6.75 days (and we are still having 365 days go in).2643

We plug that all into a calculator, and we get the amazingly tiny number of 5.273x10-17 kilograms--a really, really, really small number.2652

To appreciate how small that is, let's try to expand it a bit more.2669

1 kilogram is 1000 grams; so that means that a kilogram is 103 grams.2672

We could also write this as 5.273x10...if it is 1000 grams for a kilogram, then that means we are going to increase by 3 in our scientific exponent.2681

So, in the scientific notation, we are now at 5.237x10-14 grams of our material.2695

which, if we wanted to write this whole thing out...we would be able to write it as 0.00000 (five so far) 00000 (10 so far) 000 (13)...2701

and let's see why that is the case--we can stop writing there--because if we were to bring that 10-14 here,2720

(and remember, it is in grams, because we had grams here), that would count as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,2728

because we can move the decimal places 14 times to the right by having 10-14.2740

And that is how that scientific notation there is working.2745

Or alternatively, we could also write this with kilograms as the incredibly tiny 0.00000000000000005273 kilograms.2749

And if we counted that one out as well, we would have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17...2767

so we have that 5.273x10-17 kilograms there, as well.2779

So, it is much easier to write it with scientific notation; that is also probably what a calculator would put out,2783

because it is hard to write a number like this, this long, on a calculator.2788

So, we are much more likely to see it in scientific notation, 5.273x10-17 kilograms,2792

which is an absolutely miniscule amount of radioactive material left, considering that we started at 1 kilogram.2799

That shows us how decay works.2805

All right, cool: we have a pretty good base in exponential functions.2807

Next, we will see logarithms, and see how logarithms allow us to flip this idea of exponentiation.2810

And then, in a little while, we will see how logarithms and exponential functions...how we can oppose the two against each other.2814

It is pretty cool--we can find out a lot of stuff with this.2819

All right, we will see you at Educator.com later--goodbye!2821