Vincent Selhorst-Jones

Application of Exponential and Logarithmic 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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### Application of Exponential and Logarithmic Functions

• Exponential and logarithmic functions have a huge array of applications. They are used in science, business, medicine, and even more fields. There are far too many applications to discuss them all in this lesson, so instead this lesson focuses on working a variety of examples.
• Exponential functions allow us to describe the growth (or decay) of a quantity whose rate of change is related to its current value. Here are some examples of applications:
• Compound interest,
• Depreciation (loss in value),
• Population growth,
• Half-life (decay of radioactive isotopes),
• Any many others!
• Many exponential functions have their own specific formulas, however, if you forget any of those formulas, it is sometimes possible to use the natural exponential growth model:
 Pert.
It will not use the same rate r as if you had used the specialized formula, but if you can figure out what the r needs to be for Pert from the problem, you can normally get away with using it instead.
• Logarithms have the ability to capture the information of a wide variety of inputs in a relatively small range of outputs. This behavior makes logarithms a great way to measure quantities that can be vastly different, but need an easy way to be compared and talked about:
• Earthquake magnitude (Richter scale),
• Sound intensity (decibels),
• Acid/base concentration (pH scale),
• And many others!
Logarithms also show up in formulas that analyze how exponential growth behaves.
• Carbon-14 is a radioactive isotope with a half-life of 5 730 years. It is created in the upper atmosphere by cosmic rays, and is then absorbed by living organisms (plants and animals). Carbon-14 makes up a small, but consistent, amount of the carbon in any live organism. However, once the organism dies, C-14 stops being absorbed. Without the isotope being replenished, the quantity of C-14 in a dead organism begins to decline. This allows for carbon-14 dating (AKA radiocarbon dating, carbon dating). By knowing how much C-14 would be in a live organism, then measuring the amount in a dead organism, we can do archeological dating based on how much C-14 remains.
• We can figure out how quickly something will heat up or cool down with Newton's Law of Cooling. The rate at which heat is transferred is proportional to the difference in temperature between the object and the surrounding environment. This results in the equation
 T = Ts + (Ti − Ts) ekt,
where T is the object's temperature, Ts is the surrounding environment, Ti is the object's initial temperature, k is a proportionality constant, and t is the time.

### Application of Exponential and Logarithmic Functions

A bank account that compounds continuously is opened with an original principal of $15 000 placed in it. After 10 years from the account's opening, it has a balance of$25 484. What will its balance be after 20 years from the account's opening?
• Note: All of the solution steps for the questions in this section assume a high degree of familiarity with solving exponential and logarithmic equations. While the set up of problems will be explained, there will not usually be a detailed explanation of the steps involved in solving the equations once set up. If you are not currently comfortable with solving such equations, make sure to watch the previous lesson, Solving Exponential & Logarithmic Equations.
• The amount A in a bank account with continuous compounding is modeled with the same equation as anything with continuous growth (or decay):
 A = Pert,
where P is the starting quantity, r is the rate coefficient, and t is the time elapsed.
• Notice that we do not know the rate r, so we cannot immediately plug in t=20 to find the balance after 20 years. However, we can plug in the information we know at t=10, then solve for r:
 25484 = 15000er·10
• Solve for r:
e10r = 25484

15000
⇒     10r = ln
25484

15000

⇒     r = ln ⎛⎝ 25484 15000 ⎞⎠

10
Plugging in to a calculator, we find that r = 0.053.
• Now that we know r, we have enough information to see what the balance is at t=20:
 A = 15000 e0.053 ·20     =     43 295.56
$43 295.56 A bank account compounds every quarter, and is initially opened with a principal of$22 000. After seven years, it has a balance of $28 273. What is the balance of the account 15 years after the account is opened? (Round to the nearest whole dollar.) • Note: All of the solution steps for the questions in this section assume a high degree of familiarity with solving exponential and logarithmic equations. While the set up of problems will be explained, there will not usually be a detailed explanation of the steps involved in solving the equations once set up. If you are not currently comfortable with solving such equations, make sure to watch the previous lesson, Solving Exponential & Logarithmic Equations. • From previous lessons, we know that for bank accounts that do not compound continuously, we can use the equation  A = P ⎛⎝ 1 + r n ⎞⎠ nt . Because the account compounds quarterly, we have n=4. To find the value at t=15, we first need to find the rate r by using what we know at t=7. • Setting it up at t=7, we have:  28273 = 22000 ⎛⎝ 1 + r 4 ⎞⎠ 4·7 Now we work towards solving for r (Notice how we take a root, not a logarithm. This is because we're interested in the base, not solving for the exponent.): 1 + r 4 28 = 28273 22000 ⇒ 1 + r 4 = 28  28273 22000 ⇒ r = 28  28273 22000 −1 ·4 Plugging in to a calculator, we find r = 0.036 • Now that we know r, we can find the balance at t=15:  A = 22000 ⎛⎝ 1 + 0.036 4 ⎞⎠ 4·15 ≈ 37 661 • Alternatively, we could solve the problem in another way. In the lesson, it is mentioned that we can often use the exponential growth model to replace many other formulas:  A = Pert We can do this here as well (see the lesson for an explanation of why we are able to do this). • Setting up with this formula, we can solve for r using information about the balance at t=7: 28273 = 22000 er·7 ⇒ r·7 = ln 28273 22000 ⇒ r = ln ⎛⎝ 28273 22000 ⎞⎠ 7 This gives us r=0.035 838. [Notice how this is a different value for r then the one we found when using the non-continuous compounding formula. Anytime you choose to use Pert instead of the standard formula, you will find a different value for your r rate.] • Now we can find the balance at t=15:  A = 22000 e0.035838 ·15 ≈ 37 661 This goes to show that in many situations where exponential growth is involved, we can often just default to using Pert if we so desire.$37 661
A bank account that compounds continuously is opened with an original principal of $7 000 placed in it. After 20 years from the account's opening, it has a balance of$19 400. What is the percentage rate of the account given to three decimal places?
• Note: All of the solution steps for the questions in this section assume a high degree of familiarity with solving exponential and logarithmic equations. While the set up of problems will be explained, there will not usually be a detailed explanation of the steps involved in solving the equations once set up. If you are not currently comfortable with solving such equations, make sure to watch the previous lesson, Solving Exponential & Logarithmic Equations.
• The amount A in a bank account with continuous compounding is modeled with the same equation as anything with continuous growth (or decay):
 A = Pert,
where P is the starting quantity, r is the rate coefficient, and t is the time elapsed.
• For this problem, we are looking to solve for r. Set up the equation:
 19400 = 7000er·20
• Solve for r:
e20r = 19400

7000
⇒     20 r = ln
19400

7000

⇒     r = ln ⎛⎝ 19400 7000 ⎞⎠

20
Plugging into a calculator, we find r = 0.050 968.
• We aren't quite done, though. The problem asked for r as a percentage rate, not a decimal coefficient. We convert it to a percentage by shifting the decimal place over two positions:
 r = 0.050 968     ⇒     5.0968%
Finally, the problem asked for the percentage to three decimal places, so we must round to three decimal places: 5.097%.
5.097%
A bank account that compounds continuously is opened with some original principal placed in it. After 10 years, the account has $1858 in it. After a further 10 years, it has$3456 in it. What was the original principal in the account, rounded to the nearest whole dollar?
• Note: All of the solution steps for the questions in this section assume a high degree of familiarity with solving exponential and logarithmic equations. While the set up of problems will be explained, there will not usually be a detailed explanation of the steps involved in solving the equations once set up. If you are not currently comfortable with solving such equations, make sure to watch the previous lesson, Solving Exponential & Logarithmic Equations.
• The amount A in a bank account with continuous compounding is modeled with the same equation as anything with continuous growth (or decay):
 A = Pert,
where P is the starting quantity, r is the rate coefficient, and t is the time elapsed.
• Notice that we do not know P or r currently. However, we know the account balance at t=10 and t=20, allowing us to set up the equations below:
 1858 = P er·10               3456 = Per·20
From here, we can work to solve through substitution.
• Working with the first of the above equations, we have
 P = 1858 e10r .
We can now plug this in to the other equation:
 ⎛⎝ 1858 e10r ⎞⎠ e20r = 3456
• From here, we work to solve for r:

1858

e10r

e20r = 3456     ⇒     1858 e10r = 3456     ⇒     10r = ln
3456

1858

⇒     r = ln ⎛⎝ 3456 1858 ⎞⎠

10
Plugging into a calculator, we find r=0.062 061.
• Now that we know r, we can plug it into our previous equation to find the value of P:
 P = 1858 e10·0.062061 ≈     999
$999 At a lab, a population of bacteria is growing exponentially. A scientist checks the bacteria culture at 8 and finds that there are 5 000 bactera in the culture. At 1, he checks again and finds that there are 417 000 bacteria. How many bacteria will be in the culture when the scientist leaves the lab at 5? (Round to the nearest thousand.) • Note: All of the solution steps for the questions in this section assume a high degree of familiarity with solving exponential and logarithmic equations. While the set up of problems will be explained, there will not usually be a detailed explanation of the steps involved in solving the equations once set up. If you are not currently comfortable with solving such equations, make sure to watch the previous lesson, Solving Exponential & Logarithmic Equations. • Because the population of bacteria grows exponentially, we can model it with the equation  A = Pert. • We can treat the first time the scientist checks the culture as t=0. Thus, the population at t=0 tells us our initial population: P=5 000. Next, we need to find the rate r of growth from the next piece of information. • Since the scientist checks the culture again five hours later, we can consider that t=5. This allows us to set up the below equation:  417000 = 5000 er ·5 • Solve for r: e5r = 417000 5000 ⇒ 5r = ln 417 5 ⇒ r = ln ⎛⎝ 417 5 ⎞⎠ 5 Using a calculator, we get r=0.884 730. • Finally, we want to know the population at 5, which is nine hours after beginning, so t=9:  A = 5000 e0.884730 ·9 = 14 357 215 We were told to round to the nearest thousand, so we get a population of 14 357 000. 14 357 000 bacteria Plutonium-238 is a radioactive isotope with a half-life of 87.7 years. If you start with 1kg of Pu-238, how long will it take to decay down to 1g? (Round to the nearest year.) • Note: All of the solution steps for the questions in this section assume a high degree of familiarity with solving exponential and logarithmic equations. While the set up of problems will be explained, there will not usually be a detailed explanation of the steps involved in solving the equations once set up. If you are not currently comfortable with solving such equations, make sure to watch the previous lesson, Solving Exponential & Logarithmic Equations. • First, notice that in the problem uses two different units for mass: kilograms (kg) and grams (g). Thus, before we get started, we want to convert to using just a single unit. Let's convert the kilograms: we start with 1000g of Pu-238, and we end up with 1g. • We can model the amount of undecayed Pu-238 as  A = 1000 ⎛⎝ 1 2 ⎞⎠ [t/87.7] , where t is the number of years since we started with the mass of Pu-238. [We could also model this with Pert ⇒ 1000 ert, but the downside is that we would have to begin by figuring out what r for the natural base (e) is by using the half-life value of 87.7 years: 500 = er·87.7 If instead we do it with the first equation given above, we don't have to figure out a rate coefficient, so it makes the problem slightly easier. But the method using Pert will work perfectly fine, it just requires an extra step. (Notice that the r coefficient is very different when we work with e as the base).] • We want to know what t is when the amount is A=1, so we plug in and solve:  1000 ⎛⎝ 1 2 ⎞⎠ [t/87.7] = 1 ⇒ ⎛⎝ 1 2 ⎞⎠ [t/87.7] = 1 1000 ⇒ t 87.7 log 1 2 = log 1 1000 [Remember, we can do the above because a logarithm of any base can be used to bring down an exponent. Thus base 10 or base e are good choices because they are easily accessible on most calculators.] t 87.7 = log 1 1000  log 1 2 ⇒ t =  log 1 1000  log 1 2 ·87.7 Using a calculator, we find t = 874. 874 years Silicon-32 is a radioactive isotope that decays over time. A scientist begins with 1.000 000 g of Si-32. A year later, she measures the amount of Si-32 to find there is only 0.995 931 g of the isotope remaining. What is the half-life of Si-32? (Round to the nearest year.) • Note: All of the solution steps for the questions in this section assume a high degree of familiarity with solving exponential and logarithmic equations. While the set up of problems will be explained, there will not usually be a detailed explanation of the steps involved in solving the equations once set up. If you are not currently comfortable with solving such equations, make sure to watch the previous lesson, Solving Exponential & Logarithmic Equations. • We can model the amount of undecayed Si-32 as  A = 1.000 000 · ⎛⎝ 1 2 ⎞⎠ [t/n] = ⎛⎝ 1 2 ⎞⎠ [t/n] , where t is the number of years since we started with the original mass of Si-32 and n is the half-life of the isotope. • Thus, we can solve for n by plugging in the information for a year later: 0.995 931 = 1 2 [1/n] ⇒ log(0.995931) = 1 n log 1 2 ⇒ n = log 1 2 log0.995931 Plugging into a calculator, we find n=170. • Alternatively, we could approach this problem using the Pert formula. First, we need to find the value of r:  0.995931 = 1 ·er·1 ⇒ 0.995931 = er ⇒ ln0.995931 = r Using a calculator, we find r = −0.004 077. Next, we need to figure the length of time for one half-life. • This requires a little bit of clever thought. We must think about how much isotope would be left over at the time of one half-life. By definition, a half-life is the amount of time it takes for half of the material to decay, so we must have half as much material at the end of one half-life cycle. For this problem, since we started with 1g of material, at the end of one half-life we would have 0.5g of material. Let t represent the time of one half-life, and we will have the equation  0.5 = 1·e−0.004077 ·t. We can now solve this equation for t to find the half-life:  e−0.004077 t = 0.5 ⇒ −0.004077 t = ln0.5 ⇒ t = ln0.5 −0.004077 Plugging into a calculator, we get our half-life time of t=170. The half-life is 170 years. Carbon-14 has a half-life of 5730 years and is present in living organisms at trace amounts and keeps a constant ratio with non-radioactive carbon while an organism is alive. A wooden spear is discovered in an archeological dig, and the wood has a C-14 ratio that is 0.34% of the ratio for living wood. Estimate the age of the spear to the nearest thousand years. • Note: This problem is based on the idea of carbon-14 dating. If you're curious to understand how the mechanism of carbon-14 dating (AKA radiocarbon dating, carbon dating) works, check out the video lesson for more information. The idea is introduced as a prelude to Example 2. • We can model the amount of undecayed C-14 as we have in the previous couple problems about isotopes, but we need to state how much C-14 we begin with when the tree dies and the spear is made. Since the spear is made just after the tree's death (or at least relatively soon, compared to the half-life of C-14), we can say that the spear had 100% of the ratio for living wood (because the C-14 has not had time to start decaying significantly). Thus, let's measure the amounts of C-14 with an arbitrary "points" system based on the percentages. When the spear is first made, it has 100 points. When the archeologists find it, it has 0.34 points. • Thus, we have the following equation, where t is the age of the spear when the archeologists find it:  0.34 = 100 · ⎛⎝ 1 2 ⎞⎠ [t/5730] • Solve for t: 1 2 [t/5730] = 0.34 100 ⇒ t 5730 log 1 2 = log 0.34 100 ⇒ t = log 0.34 100  log 1 2 ·5730 Plugging into our calculator, we find t ≈ 47 000. • As a side note, it would be possible to do this problem using Pert, but it would take a little more effort. We would have to begin by figuring out what r is for the half-life of 5730, which would then allow us to solve for the age of the spear. Nonetheless, we will get the same answer either way, so feel free to use the method you are more comfortable with. The spear is 47 000 years old. A pot of soup is on the stove and has a temperature of 85°C when you turn off the stove. You immediately move the pot of soup to the refrigerator, which has a constant interior temperature of 3°C. After five minutes of it being in the refrigerator, you check the soup and see that its temperature is now 72°C. If you can only eat the soup once it reaches 55°C, how much longer must you let the soup cool in the refrigerator after checking it? (Round to the nearest minute.) • Note: This problem is based on Newton's Law of Cooling, which allows us to model the temperature of an object based on its initial temperature and the temperature of its surroundings. The equation and the idea are introduced as a prelude to Examples 4 and 5 in the video lesson. The equation is  T = Ts + (Ti − Ts) ekt, where T is the object's temperature, Ts is the surrounding environment, Ti is the object's initial temperature, k is a proportionality constant we figure out for each problem, and t is the time. • The initial temperature of the soup is Ti = 85 and the surround temperature in the refrigerator is Ts=3. At t=5, we have the equation  72 = 3 + (85−3) ek·5, which we can now solve for the coefficient k. • Solving for k, we get 3 + 82 e5k = 72 ⇒ e5k = 69 82 ⇒ 5k = ln 69 82 ⇒ k = ln 69 82 5 Using a calculator, we find k = −0.034 523. • Now that we know the value for k, we can set up the equation for an unknown t when the temperature of the soup is at 55°C:  55 = 3 + (85−3) e−0.034523·t • Now we solve for t: 3 + 82e−0.034523t = 55 ⇒ e−0.034523t = 52 82 ⇒ t = ln 52 82 −0.034523 Plugging into a calculator, we find that the soup is cool enough at t ≈ 13. • However, the question asked for how much longer the soup had to cool. This was compared to when the temperature was checked at t=5. Thus, the amount of time longer is 13−5 = 8 minutes. You must let the soup cool for another 8 minutes. [Notice that it is 8 minutes after checking the temperature in the refrigerator, which comes to a total of 13 minutes after being initially taken off the stove.] During a cold winter night, the police get a call about a disturbance in an alleyway. They arrive on the scene a little while later, and find a body. They take the temperature of the body at 10:51, and it is 31.5°C. They proceed to secure the area and call a detective to the scene. The detective arrives at 11:10 and she immediately takes the temperature of the body, which is now at 27.3°C. She also takes the temperature of the alleyway, which has been holding at a constant −15°C for the past couple hours. Assuming that the victim had a standard human body temperature of 37°C, give the victim's time of death (to the nearest minute). • Note: This problem is based on Newton's Law of Cooling, which allows us to model the temperature of an object based on its initial temperature and the temperature of its surroundings. The equation and the idea are introduced as a prelude to Examples 4 and 5 in the video lesson. The equation is  T = Ts + (Ti − Ts) ekt, where T is the object's temperature, Ts is the surrounding environment, Ti is the object's initial temperature, k is a proportionality constant we figure out for each problem, and t is the time. • (It should be noted that this is a particularly challenging problem. If you find it difficult to follow the steps below, consult the video lesson for Example 5, which works out a very similar problem.) Let's begin by establishing how we look at the problem. Let t=0 be the time of the victim's death and let the unit of time be minutes. Notice that we do not know the time that the police first measure the temperature of the body. Since this is an unknown, let us call it t0. Thus, the police measure the body's temperature at t = t0, and therefore the detective measures the body's temperature 19 minutes later at t = t0 + 19. • Immediately at death, the body has an initial temperature of Ti = 37 and the surrounding environment is Ts = −15. Using Newton's Law of Cooling, the measuring done by the police and the measuring done by the detective each give us an equation to work with:  31.5 = −15 + (37 − (−15)) ek t0 27.3 = −15 + (37 − (−15)) ek (t0+19) Moving things around a bit in the equations, we get  46.5 = 52ek t0 42.3 = 52 ekt0 + 19k • At this point, we have two unknowns: t0 and k. Since we have two equations, we can use substitution to solve. Working towards this, notice that we can rewrite the second equation as  42.3 = 52 ekt0 + 19k ⇒ 42.3 = 52 ( ekt0 ·e19k ) ⇒ 42.3 = 52 ek t0 ·e19k At this point, we can substitute in our first equation, 46.5 = 52ek t0:  42.3 = 52 ek t0 ·e19k ⇒ 42.3 = (46.5) ·e19k • From here, we can now solve for k: 46.5 e19k = 42.3 ⇒ e19k = 42.3 46.5 ⇒ 19k = ln 42.3 46.5 ⇒ k = ln 42.3 46.5 19 Using a calculator, we find k = −0.004 982. • We can now plug this in to the first of our two temperature equations to solve for t0: 46.5 = 52e−0.004982 t0 ⇒ −0.004982 t0 = ln 46.5 52 ⇒ t0 = ln 46.5 52 −0.004982 Using a calculator, we find t0 = 22.439. • The problem asked for the time of death to the nearest minute, so we round to t0≈ 22. Remember, t0 represents the number of minutes after the victim's death. We know that t0 occurs at 10:51. Since that is 22 minutes after the victim's death, we see that the victim's death (t=0) occured at 10:29. This completes the problem. • The above is a very good, clear way to do the problem, and it is very similar to the method used in the video lesson. There is an interesting alternative way to do the problem, though. Instead of setting the victim's time of death as t=0, set the initial time of t=0 at the instant that the police first measure the body's temperature. This means we link 10:51 with t=0. Furthermore, because we've linked that moment with the initial time, it also means we link the body's temperature at that moment to the initial temperature: in other words, we set Ti = 31.5 (what the police measure at t=0↔ 10:51). From here, we can solve for the value of k using the detective's temperature measurement 19 minutes later at t=19:  27.3 = −15 + (31.5 − (−15) ) ek·19 • Solve for k: 46.5 e19k = 42.3 ⇒ e19k = 42.3 46.5 ⇒ 19k = ln 42.3 46.5 ⇒ k = ln 42.3 46.5 19 Plugging into a calculator, we get k = −0.004 982. (Notice that is the exact same value as we found the first time. This makes sense, because the coefficient k is just a mathematical expression of how quickly heat energy transfers between the object and its surrounding environment. It is based on how the object and the environment interact, not the specific temperatures involved.) • Now that we know k, we can work out what time t the body would have had a normal (living) body temperature of 37°C:  37 = −15 + (31.5 −(−15)) e−0.004982 t Solving for t, we find 46.5 e−0.004982 t = 52 ⇒ −0.004982 t = ln 52 46.5 ⇒ t = ln 52 46.5 −0.004982 Plugging into a calculator, we find that the time of death t=−22.439. Round to the nearest minute of t ≈ −22, and realize that this means the death occurred in "negative" time, that is prior to t=0. This makes perfect sense, since it's obvious that death must have occurred prior to the time the police measured the body's temperature. Working back 22 minutes from 10:51, we get a time of death at 10:29. Both methods of solving the problem are perfectly valid. The important part is just keeping track of what moment in time you decide to link with the "initial" moment of t=0. We can say the initial moment is the moment of death, or the initial moment is when the police first check the body. The important thing isn't which one we choose, just that we keep it straight throughout working on the problem. The victim died at 10:29. [If you're interested, there are two different methods to set up and solve the problem given in the steps for this question. The first part of the steps uses a method similar to the one in the video lesson, but after that is an alternative method where we look at the problem in a slightly different way.] *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 ### Application of Exponential and Logarithmic 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 0:06 • Applications of Exponential Functions 1:07 • A Secret! 2:17 • Natural Exponential Growth Model • Figure out r • A Secret!--Why Does It Work? 4:44 • e to the r Morphs • Example • Applications of Logarithmic Functions 8:32 • Examples • What Logarithms are Useful For • Example 1 11:29 • Example 2 15:30 • Example 3 26:22 • Example 4 32:05 • Example 5 39:19 ### Transcription: Application of Exponential and Logarithmic Functions Hi--welcome back to Educator.com.0000 Today we are going to talk about applications of exponential and logarithmic functions.0002 At this point, we have a good understanding of exponentiation and logarithms.0007 In this lesson, we will see some of the many applications that they have.0010 Exponential and logarithmic functions have a huge array of applications.0013 They are used in science, in business, in medicine, and even more fields.0017 They are used in all sorts of places.0021 There are far too many applications to discuss them all in this lesson, so instead we will focus on working a variety of examples.0024 We will begin with a brief overview of some other uses--some of the uses that we can see for exponential functions and logarithmic functions.0029 Then, we will look at many specific examples, so we can really get our hands dirty and see how word problems in this form work.0035 Now, before you watch this lesson, make sure you have an understanding0040 of exponents, logarithms, and how to solve equations involving both, before watching this.0043 We won't really be exploring why the actual nuts and bolts of this solving works--how these things work.0048 We are just going to be launching headfirst into some pretty complicated problems.0053 So, you really want to have an understanding of what is going on, because we are going to hit the ground running when we actually get to these examples.0057 Previous lessons will be really, really helpful here if you are not already used to this stuff.0062 OK, let's go: applications of exponential functions: exponential functions allow us to describe the growth or decay0066 of a quantity whose rate of change is related to its current value.0073 So, how fast it is changing is connected to what it is currently at.0078 So, some examples of applications: we can also see how their rate of change is related to its current value:0083 Compound interest--the amount of interest that an account earns is connected to how much money is already in the account.0088 If you have ten thousand dollars in an account, it will earn more than if it has one thousand dollars in the account,0096 or than if it has one hundred dollars in the account.0101 So, this is an example of seeing how the rate of change is related to the current value of the object.0103 Other things that we might see: depreciation--loss in value; compound interest and loss in value0109 are both used a lot in banking and business--anything that is fiscally oriented.0114 Population growth is used a lot in biology; half-life--the decay of radioactive isotopes--shows up a lot when we are talking about physics.0118 If you are studying anything in radiation, understanding half-life is very useful.0127 And many others--there is a whole bunch of stuff where exponential functions are going to show up.0131 It is really, really useful stuff.0135 Now, I have a secret for you: don't let anybody else know about this.0138 Many exponential functions have their own formula--things like compound interest, 1 plus the number of times that it compounds in a year, divided...0142 Oh, this should actually be the other way around; it should not be n/r; it should r/n.0152 The rate of it, divided by the number of times it compounds, to the number of times it compounds, times the time--0160 if you don't remember that one, remember our very first lesson on exponential functions that described why that is the case.0167 Population doubling is P, some original starting principal amount, times 2 to the rate that they double at, times t.0173 Half-life is some principal starting amount times 1/2 to the rate times time.0181 However, if you forget all of these formulas--there are a bunch of different formulas;0187 there are even more than just these; but it is sometimes possible to use the natural exponential growth model.0191 You can sometimes swap out any one of these more difficult-to-remember ones for simple "Pert"--0197 P, the original starting amount, times e, the natural base, to the r times t,0205 where r is the rate of the specific thing that we are modeling, times time.0210 r will change, depending on what different thing you are doing.0214 So, even if you are modeling isotopes--half-life in plutonium and half-life in uranium--you will get very different r's,0217 because the plutonium and uranium will have different rates of decay.0222 So, you are not going to use the same rate r.0226 Once again, if you are talking about half-life, the r here would be totally different than the r in our Pert if you had used the specialized formula.0229 So, the r for half-life of uranium using this formula would be totally different than the r for half-life using the Pert formula.0236 But if you can figure out what that r has to be for Pert form from the problem, you can get away with using it instead.0246 There are many situations where you might not remember any one of these specialized formulas.0255 But it can be OK if you have enough information from the problem to be able to figure out what r has to be.0260 There are lots of cases where that will end up being the case.0266 We will talk about a specific one on Example 2; we will see something where we could get away with not knowing0269 the specific formula, and still be able to figure things out by using this Pert formula.0275 We will talk about it in Example 2, if you want to see a specific example of being able to use this secret trick.0279 Why is this the case?--you probably wonder.0285 Why can you do this--how can you get away with using Pert when we have all of these special formulas?0287 How can you swap out some exponential formula for just natural exponential growth, the Pert form?0291 The reason why is because of this er part; er can morph into other forms.0297 For example, let's look specifically at a possible half-life formula.0303 We might have P times 1/2 to the t/5; we can see this as P times 1/2 to the rate of 1/5 times t.0307 That is what we have there: some principal starting amount, times 1/2 to the 1/5 times t.0318 So, for every, say, 5 years, we have half of the amount there that we originally had.0323 So, how can we get Pert to connect to this?0328 Well, if we use this very specific r, r = -0.1386, it also turns out that that is the same thing as -0.6931 times 1/5.0331 So, we can have our Pert form right here; we know what r is, so we swap that in for our r.0343 And we get P times e to the -0.1386 times time.0350 But we also know that -0.1386 is the same thing as -0.6931 times 1/5.0358 So, if we want, we can break this apart into a -0.6931 part and a 1/5 times t part that we might as well put outside.0366 We have e to the -0.6931, to the 1/5 times t, because by our rules from exponential properties, that is the same thing0374 as just having the 1/5 and the -0.6931 together, which is the r that we originally started with.0381 Now, it turns out that e-0.6931 comes out to be 1/2.0387 By this careful choice of r, we are able to get er to morph into something else.0396 We can get it to morph into this original 1/2; and now, we have this 1/5 here,0402 so it becomes just P times 1/2 to the t/5, which is what we originally started with as the half-life formula.0406 So, by this careful choice of r--and notice, the r here is equal to 1/5; the r here is equal to the very different 0.1386;0414 we get totally different r's here; but by choosing r carefully, if we have enough information from the problem0426 (sometimes you will; sometimes you won't; you will have to know that special formula)--sometimes,0433 you will be able to get enough information from the formula, and you will be able to figure out what r is.0438 So, you can have forgotten the special formula--you can forget the special formula occasionally, when you are lucky.0441 And you would be able to just use Pert instead.0446 By choosing the right r for Pert, we morph it into something that works the same as the other formula.0448 Now, of course, you do have to figure out the appropriate r from the problem.0454 You are just saying it--you have to be able to get what that r is.0457 And remember: it is the r for Pert, which may be (and probably is) going to be totally different0460 than the r for the special formula that we would use for that kind of problem.0465 But if you can figure out what the r is from the problem, you can end up using Pert instead.0469 Once again, we will talk about a specific use of this in Example 2, where we will show how you can actually use this if you end up forgetting the formula.0473 Now, I want you to know that the above isn't precisely true.0480 e-0.6931 isn't precisely 1/2; it is actually .500023, which is really, really, really close to 1/2; but it is not exactly 1/2.0484 But it is a really close approximation, and it is normally going to do fine for most problems.0496 It is such a close approximation that it will normally end up working.0500 And if you need even more accuracy, you could have ended up figuring out what r is, just to more decimal places.0504 And you could have used this more accurate value for r.0509 Applications of logarithmic functions: logarithms have the ability to capture the information of a wide variety of inputs in a relatively small range of outputs.0513 Consider the common logarithm, base 10: if we have log(x) equaling y, log(x) going to y--0522 over here we have our input, which is the x, and our output, which is the value y--that is what is coming out of log(x).0528 x can vary anywhere from 1 to 10 billion; and our output will only vary between 0 and 10.0537 That is really, really tiny variance in our output, but massive variance in our input.0545 Why is this happening? Because 1 is the same thing as 100,0550 and 10 billion is the same thing as 1010,because we have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 zeroes,0555 so that is the same thing as 100 to 1010.0568 And because it is log base 10--it is the common logarithm--when log base 10 operates on 100, we will get 0.0571 When it operates on 1010, we will get 10; and as it operates on everything in between, we will get everything in between, as well.0577 So, there is massive variance in our inputs and massive different possible inputs that we can put in.0585 There is a very, very small range of outputs that we will end up getting out of it.0589 This behavior makes logarithms a great way to measure quantities that can be vastly different--0594 things that can have really huge variance in what you are measuring.0599 But we want an easy way to compare or talk about them; we have to be able to talk about these things.0602 They come up regularly, and we don't want to have to say numbers like 10 billion or 9 billion 572 million.0607 We want some number that is fairly compact--that doesn't require all of this talking.0614 So, we use logarithms to turn it into this much smaller, more manageable number that makes sense, and we can understand, relative to these other things.0618 Earthquake magnitude is one of the things that is measured on it.0625 It is measured on the Richter scale, which is a logarithmic scale.0629 Sound intensity is another one; it is measured in decibels, which is another logarithmic scale.0632 Acid or base concentration is measured on the pH scale; we will actually have examples about that in Example 3.0637 And that one is measured, once again, on a logarithmic scale.0643 And many others--there are many other logarithmic scales,0646 when you have a really, really large pool of information that can be going in as an input,0649 but you want to be able to narrow that to a fairly small, manageable, sensible range of values.0653 0 to 10 is going to have lots of decimals, when you evaluate log of 8 billion and 72 million.0658 It is going to have lots of possible decimals to it, but it is going to be a fairly small, manageable number for thinking about.0664 Logarithms will also show up in formulas that are analyzing exponential growth,0670 because if we are building a formula that is going to be connected to exponential growth,0675 if we are trying to break down and figure out what its power is raised,0678 we are going to end up having logarithms show up when we are solving it;0682 so they will end up showing up in the formula, as well.0684 Logarithms show up in formulas for analyzing exponential growth.0686 All right, let's get to some examples: A principal investment of$4700 is made in an account that compounds quarterly.0690

If no further money is deposited, and the account is worth $5457.57 in 5 years, what will it be worth after a total of 10 years?0698 The first thing: what kind of formula are we working with here?0707 Well, we have compounding, but not continuous compounding; so we go and look that up.0710 It is principal, times 1 + the rate, divided by the number of times that compounding occurs,0714 raised to the number of times that the compounding occurs, times the amount of time elapsed in years.0721 So, our principal investment here is P = 4700; and we know that, at time 5, at t = 5, we have 5457 dollars and 57 cents.0727 So,$5457.57 is equal to...what was our principal amount? 4700 dollars, times 1 plus...what is our rate?0743

That is one of the things we don't know yet--we don't know what our rate is.0754

And that is why we are setting up at t = 5, instead of just hopping immediately to the 10 years question:0758

we need to figure out what our rate is first, so that is what we are figuring out now.0763

1 + r/n; our n is quarterly, so that is an n of 4, because it happens four times in a year, in each of the four quarters of the year.0767

So, 1 + r/4, raised to the 4 times t--do we know what t is in this case?0778

We do know what t is; otherwise we would have two unknowns to solve in one equation.0785

It is 5, because we are looking at the 5-year mark.0789

So, we work this out; we get 5457 dollars and 57 cents, over 4700 dollars, equals (1 + r/4)20.0793

Now, at this point, we would probably be tempted--how can we use logs?--how are logs connected to this?0806

We could take the natural log of both sides and bring down the 20; but then we would end up having ln(1 + r/4).0810

We are actually losing sight of a much simpler way.0816

When we raise to a power, how do we get rid of powers?0819

If we are trying to figure out what is in the base, and the power is a known value, we just take a root.0822

We take a root, depending on what the power is.0827

If it is squared, we take the square root; in this case, it is to the 20th, so we are going to raise both sides to the 1/20.0829

We are taking the twentieth root of this, so it is raised to the 1/20.0836

That cancels out here, and we are left with (1 + r/4); we take the 1/20 power, which is also the twentieth root,0842

of 5457 dollars and 57 cents, divided by 4700; and that comes out to be 1.0075.0855

We subtract 1 on both sides; we get 0.0075 = r/4.0864

We multiply by 4 on both sides, and we finally get 0.03 equals our rate.0869

So, our rate, in this case, is a modest 3% return on investment.0875

Now, we can use this information to look at the ten years, at t = 10.0881

At this point, we have everything that we need to know, except for the amount.0886

The amount is the unknown--we don't know what it is going to be worth.0889

But we do know what the initial principal was, \$4700; we do know that it is 1 + a rate of 0.03/4, quarterly,0892

to the 4 times...our number of years is 10; so our amount is 4700(1 + 0.03/4)40.0902

We work that all out with a calculator, and it ends up coming out to 6337 dollars and 24 cents.0914

And that is the final amount in the account at the 10-year mark.0925

The second example: carbon-14 dating: first, we are going to talk about the idea, and then we will actually work on a specific problem.0931

Carbon-14 is a radioactive isotope; a radioactive isotope is an element that is an isotope,0937

one version of an element, that breaks down over time; it decays into something else.0944

It is something for a while, and then something happens; it undergoes fission; it breaks apart; and it turns into a different element.0949

It is a radioactive isotope with a half-life of 5,730 years.0964

It is created in the upper atmosphere by cosmic rays.0968

Cosmic rays from the sun or other stars hit the upper atmosphere of the earth, and they create carbon-14 isotopes.0971

Then, these carbon-14 isotopes are absorbed by living organisms--that is, plants and animals.0982

They get absorbed by trees through the carbon cycle, and then the carbon ends up getting into the animals that eat those trees,0990

and the animals that eat those animals that eat those trees.0998

Any plant that is absorbing carbon will end up absorbing this, so it ends up getting into the cycle of biology.1001

Now, carbon-14--there is a very small amount of this isotope, so it is going to make up a small, but consistent, amount of the carbon of any live organism.1010

So, it is a very, very tiny amount, but it is going to be a regular amount.1018

As long as an organism is alive, it is going to continue consuming things.1022

It is going to continue bringing carbon-14 into its body; so that amount will stay basically consistent throughout its life.1027

However, once the organism dies, carbon-14 stops being absorbed.1033

There will be no more carbon-14 pulled in, because now, since the creature is dead,1039

since the organism is dead, it is not pulling any more carbon-14 into its body.1042

It is not pulling anything else into its body.1046

Without the isotope being replenished...remember, carbon-14 is radioactive; it begins to break down over time.1048

So, the C-14 (another name for carbon-14) in our dead organism will begin to decline,1054

because the creature is no longer pulling in carbon-14 to keep its levels consistent.1060

So, the carbon-14 begins to slowly, slowly drip away; it begins to slowly dissipate, break down, and decay into other things.1064

This fact allows for carbon-14 dating, also known as radiocarbon dating (because it is based on radioactive carbon), or simply carbon dating.1072

By knowing how much C-14 would be in a live organism (we know how much that is in a live organism,1082

because we can just measure it on real, live organisms--we can figure out that this creature is alive;1088

it has this much C-14 in it--a live creature, something that was very recently alive1093

or is currently alive--has this consistent amount of C-14), then once the creature dies, it begins to steadily decline.1099

So, we can measure the amount in a dead organism.1106

And since it is steadily declining--it is declining by rules that we know about--it is declining by half-life rules--1109

we can figure out, based on the amount of carbon-14 left in the dead organism at this point, when the creature died--1115

when it went from being alive and keeping a steady state of carbon-14 to dead,1124

where it is starting to let its carbon-14 just sort of disappear.1128

So, by knowing how much carbon-14 remains, we can get an estimate of when the creature was originally alive.1132

All right, now we will look at a specific example.1139

A skeleton of an ancient human is discovered during an archaeological dig.1141

If the level of carbon-14 in the bones is 7% of the carbon-14 ratio present in living organisms, how long ago did the human die?1145

And we want to round our answer to the nearest 1000 years.1152

And we are also reminded that the carbon-14 half-life is 5,730 years.1154

It is the amount that we originally start with, times 1/2 to the rate times time.1162

Now, in this case, the rate is going to be based on this 5730 years.1168

So, our amount is going to be equal to P--whatever our original amount is that we have of carbon-14--1172

to the 1/2, and we want a 1/2 to occur every 5730 years, so it is t/5730 years.1179

That way, when t = 5731, we will have an exponent of 1, and it will cause the 1/2 to go once.1191

If it is 5730 years twice, we will have an exponent of 2, and we will get 1/2 times 1/2: 1/4.1196

Great; so it makes sense what is going on there.1203

Now, what is our P going to be? We probably want to figure out what the P that we are going to use is.1205

Well, notice: it said the level of carbon-14 in the bones is 7% of the carbon-14 ratio.1210

So, we were never given absolute values; we weren't given quantities of carbon-14; we were told ratios.1215

If that is the case, well, what would we want the amount to be when it is 0--when we start at 0--when the creature was very first alive?1220

a = P(1/2)0/5730; well, that is going to be a number raised to the 0, which just becomes 1.1229

So, we have a = P; the amount that we start with is equal to the principal amount (that is exactly what it is there for).1237

If the ratio slowly declines, let's just say that it started at a ratio of 1.1246

We will say P = 1; this is an idea that we are figuring out.1251

We are saying, "Well, what would the ratio in a living organism be to a living organism? That would be 1, right?"1256

So, just before the creature dies at 0 time, it is going to have a ratio of living organism to living organism,1261

so it will be a ratio of 1; the ratio will not have changed.1268

So, initially, this P thing is just going to be 1, because the ratio at the instant of death is just 1,1272

because it is the normal thing with other living organisms.1278

So, our amount is 1/2 raised to the t/5730.1280

Now, at this point, we want to figure out what is the time that it took to get to where we are now.1286

So, at t = ? (we don't know what the time is), when we have it at 7%, we know that the amount is going to be 0.07.1293

The ratio if it is 7% is .07 with what it originally was: (1/2)t/5730.1305

Now, at this point, we could take log base 1/2, but I just feel like taking the natural log of both sides,1315

because we are going to have to do a change of base later anyway.1320

So, we have the natural log of 0.07 equals the natural log of 1/2 to the t/5730.1323

So, we can bring this down out front; we have t/5730.1333

ln(0.07) will divide by ln(1/2) on both sides; that equals t/5730, which finally brings us to t.1340

We multiply both sides by 5730; that equals 5730 times ln(0.07)/ln(1/2).1352

We punch that into a calculator, and we are going to end up getting about 21,983 as our time, which we round to 22,000 years.1363

This stuff isn't perfectly accurate, because...1374

So, the creature died 22,000 years ago; great.1378

This stuff can't be perfectly accurate, because there is always going to be a little bit of error in measurement.1382

And since we only had 7%, that is not as accurate as 21,983; so it makes sense that we can only get to a certain amount.1386

And that is why we are being told to round to the nearest 1000 years, because we only have so much specific data here.1393

Alternatively, if we had forgotten this a = P(1/2)rt, we could have started at the Pert form.1399

So, the amount equals Pert; once again, it is the same idea, this P = 1 business.1408

Let's start at a ratio of 1; we can plug that in, so we have a = ert.1414

Now, we need to figure out what r is; r is not going to be connected just directly to this 5730 years.1418

Instead, what we know is that, originally, at time = 0, we have 1 as the ratio.1425

At time = 5730, we are going to have 1/2 as the ratio.1436

And that is our trick here: 1/2 = er(5730).1441

We can take the natural log of both sides, so we have ln(1/2) = r(5730).1448

We divide both sides by 5730; we have ln(1/2)/5730 = r, which ends up getting us a rate of -0.00012097.1455

Now, that is long; it is a little complex; but it does mean that, even in the bad situation1475

where we have forgotten the formula for half-life--how half-life works--we can still get something that is going to work.1479

So now, we have this new thing we can work with, a = e-0.00012097t.1484

So, with that in mind, we know that we can look at 0.07 (put a little marker/divider down here);1497

0.07 = e to the same exponent, 0.00012097t.1505

We take the natural log of both sides; we have the natural log of 0.07 equals -0.00012097t,1513

because the exponent just comes down; we take the natural log.1524

So, now we have to divide this natural log of 0.07 divided by this huge...well, actually very tiny, but very long number, 0.00012097, equals t.1527

We punch that into a calculator, and we end up getting that t is approximately equal to 21,982,1540

which is very, very, very close to 21,983; so it ends up making sense that the reason why it is not quite the same1550

is because we ended up having to use a decimal approximation.1559

r wasn't exactly equal to -0.00012097; it was really close--it was approximately equal to that value.1562

So, we got something that was approximately the same thing as our answer.1571

And in this case, we are still going to end up rounding to 22,000 years as our final answer; cool.1574

The third example: first, let's talk about the pH scale.1583

The acid or base level of a solution (acid and base are two opposite ideas) is the concentration of positive hydrogen ions that are in solution.1585

This concentration is measured in moles per liter (moles is just a way of measuring how many of an atomic object you have), which is called molarity.1597

And it can vary hugely from one solution to another, so it makes sense that we want to use a logarithmic scale, since it can vary hugely.1606

To describe these massive possible differences, we measure acidity or alkalinity,1613

(acidity being the name for an acid measurement--how acid something is;1617

and base, basic, alkalinity, alkaline--these are all the same thing, for how basic something is--how "base" is a solution?)1622

with a logarithmic scale, the pH scale, which comes from powers of hydrogen,1630

which is connected to the fact that we are looking at a logarithm of our amount of hydrogen ions.1635

If we denote the molarity of hydrogen ions with the symbol H+, the pH is pH = negative log (common log,1640

so it is a base 10 log) of the molarity of hydrogen ions.1648

OK, with this idea in mind, we can now look at some interesting things.1653

The pH of cow's milk is 6.6, while tomatoes have the pH of 4.3.1657

How many times more hydrogen ions are in tomatoes than milk?1663

First, we will look at milk; and for milk, we will describe its hydrogen ion concentration with the letter M.1666

So, its pH was 6.6, and that is equal to the negative log of M.1674

We move the negative over; we get -6.6 = log(M); we can raise both sides to the 10; this cancels out,1680

and we end up getting that 10-6.6 is equal to the molarity concentration of milk; cool.1689

Next, let's do tomatoes: we will use a T to denote their hydrogen ion concentration molarity.1696

So, we were told 4.3 is equal to -log(T); -4.3 = log(T); once again, we raise them both up with a 10 underneath.1706

So, we get 10-4.3 (let me keep the same color there, so we see what is going on) is equal to...1719

and remember: since 10 and common log (log base 10) cancel out, we are left with just t.1728

So, 10-4.3 is the molarity concentration of ions in tomatoes; 10-6.6 is the molarity concentration of milk.1735

If we want to look at the ratio of ions in tomatoes to milk, the ratio of T to milk,1743

well, then we just plug in the numbers we have; we have 10-4.3 divided by 10-6.6,1754

which comes out to be 102.3, which is about 200 times.1764

So, the ratio of hydrogen ions in tomatoes to the hydrogen ions in milk is 200 times more hydrogen ions in tomatoes than there are in milk.1774

Now, let's look at lemon juice: if we have lemons--and we will use an L to denote it--1787

then we were told that 2.2 = -log(L); let me use a curly L, so we don't get confused with l in our log.1797

So, negative on both sides: -2.2 = log of our lemons.1806

We raise both sides to the 10; that cancels out with the common log,1814

so we have that 10-2.2 is equal to the molarity concentration of hydrogen ions in lemons.1819

At this point, we can now compare lemon juice to milk and tomatoes; let's compare it to tomatoes first.1827

Ratio of lemons to tomatoes is going to be 10-2.2 over tomatoes at 10-4.3,1833

which is equal to 102.1, which comes out to be around 125 times--really close; 126, actually, is closer.1844

But it is 125 times, so there are 125 times more hydrogen ions in lemons than in tomatoes.1858

Finally, let's look at the huge value that it is for milk versus lemons.1865

The ratio of hydrogen ions in lemons compared to milk is equal to 10-2.2 divided by 10-6.6,1870

which comes out to be 104.4, which is approximately 25 thousand times.1883

So, there are 25 thousand times more hydrogen ions in lemon juice than there are in milk--25 thousand!1893

This is what we are seeing with the pH scale--it is a logarithmic scale.1903

It is not just showing that there is a difference of 4.4 between milk and lemon juice.1906

We are showing this massive difference, because we are being able to encapsulate1913

this information of a huge difference with a fairly small number being able to show that.1916

We are using a logarithmic scale, because there can be these massive differences in pH reading.1921

All right, the final set of examples: Newton's Law of Cooling:1925

If you take a warm mug of tea outside on a cold winter day, the tea will cool down over time.1928

It is in this cold environment, so it is going to cool down, because it is warm, and the outside is colder.1934

So, it is going to give off its heat energy to the surrounding environment.1940

The surrounding environment is colder than it, so it will end up sucking out that energy from our mug of tea.1943

Conversely, if you put a warm mug of tea into a hot oven--we have this hot oven,1948

and we put the mug of tea in--it is going to heat up, because the hot oven is a hot environment.1953

It is hotter than the mug of tea is, so it is going to put heat energy into the mug of tea.1959

Our mug of tea will absorb heat energy from its surrounding environment.1965

If we have a hot object in a cold environment, the hot object will radiate out energy, and it will go to the cold environment.1970

If we have a cold object in a hot environment, it will pick up that energy and become hot, just like the environment it is in.1977

The rate at which heat is transferred is proportional to the difference in temperature between the object and the surrounding environment.1983

That makes a lot of sense: if you put your hand outside of a car window, and it is, say,1989

a warm day outside--a little cool--it is going to cool off your hand, slowly but surely.1998

But if you put your hand out of a car window on a freezing day, where it is colder than freezing,2003

it is going to actually cause your hand to drop in temperature really, really, really quickly.2009

So, this makes sense: if something is put into a really hot environment, it is going to heat up faster than if it is only put into a pretty hot environment.2014

The rate at which heat is transferred is proportional to the difference in temperature between the object and the surrounding environment.2020

Since the amount of heat transfer is connected to a proportion of what this difference is, it ends up drawing on exponential equations.2025

This idea is called Newton's Law of Cooling, and it is modeled by the temperature of the object2033

is equal to the temperature of the surrounding environment, added to the quantity (the object's initial temperature,2040

minus the temperature of the surrounding environment) times ekt.2046

This is the idea...the ekt is always going to end up coming out to be...k will end up being a negative number,2051

always, in these cases, when we are working these problems, because the environment2058

is going to cause this part here to eventually go and drop down to 0,2061

until eventually the object becomes the surrounding environment.2066

It will make more sense as we see some examples; so let's get started.2070

Example 4: A pot of soup is on a stove at a temperature of 85 degrees centigrade when you turn off the stove.2076

The kitchen is at 20 degrees, and after 10 minutes, the pot has cooled to 70 degrees.2082

If the hottest temperature you can eat the soup at is 55 degrees, how much longer do you need to wait?2086

All right, let's figure out our things here: what is the initial temperature of our soup?2091

The initial temperature of the soup is 85 degrees; we are told that it started at 85 degrees.2096

We are told that the kitchen is at 20 degrees; so the surrounding environment temperature is 20.2102

Now, we don't know what the ratio is; we don't know what the proportionality constant is yet.2109

And we don't know what the time is going to be in all of these cases.2113

However, we do have one specific example: at t = 10, we know that it cools to 70.2116

So, let's look at that: at time = 10 minutes, we know that the temperature of our object is 70.2122

And then, we can set up everything else: the surrounding environment is 20,2130

plus the initial temperature of 85, minus the surrounding at 20, e to the k,2133

and we know that it is 10 minutes: t = 10 in this case; so times 10--great.2139

We start working this out; we start working towards the ek(10).2145

We subtract 20 on both sides; 85 - 20 in there becomes 65, times e to the k times 10.2150

Divide by 65 on both sides; we have 50/65 = e10k.2157

Take the natural log of both sides; we have ln(50/65) = 10k, because the e disappears when I take the natural log of both sides.2162

We divide by 10 on both sides; we have ln(50/65), divided by 10, equals k.2171

And that ends up coming out to be that k is approximately equal to -0.02624.2178

So, because of the fact that this proportionality constant came out to be negative,2190

and it always will come out to be negative whenever we are looking at a Newton's Law of Cooling problem,2193

because it came out to be negative, we are going to end up seeing that this part here...2197

as t becomes larger and larger, e to the negative larger and larger thing will become smaller and smaller.2203

And this Ti - Ts portion will get crushed down to 0 as the e to the negative thing becomes smaller and smaller.2209

It will crush that down, and we will end up being left at just the temperature of our surrounding environment.2215

All right, let's figure out the other half here.2220

We want to figure out at t = [what?] we end up having a temperature of 55 degrees, because at that point we will be able to eat our soup.2222

So, at 55 temperature, we know we will have surrounding at 20, plus initial temperature was 85, minus 20, times e to our constant, -0.02624.2232

I will just leave it as k right now, and we can swap it out later when we need to figure out actual numbers.2250

That is one trick to avoid having to do a lot of writing; so ekt.2255

Subtract 20 on both sides; we get 35 = 65ekt.2259

Don't worry: we know what k is; we will just plug it in when we get to needing it.2264

35/65...we can take the natural log of both sides in just a moment to get rid of that e.2268

We have ln(35/65) = kt; divide by k on both sides; we can now plug that in, and we have ln(35/65)/-0.02624;2274

that is equal to our time, and that ends up coming out to be approximately 23.6.2293

So, 23.6 minutes from the moment that we turn off the stove (because when we turned off the stove, that was our time of T = 0)2301

is going to be when the soup is at 55 degrees Celsius.2322

However, we were asked how much longer we have to wait.2325

So, we have already waited 10 minutes, because at 10 minutes, we were told 70 degrees.2329

So, how much longer do we need to wait?2334

We need to wait 23.6 (what we got here) minus the 10 minutes that we have already waited, so 13.6 minutes, until the soup is cool enough to eat it.2336

All right, the final example--this one is going to be a little bit complicated.2359

But this is about as hard as we will end up seeing any of this sort of thing get.2364

All right, police discover a body in an air-conditioned room with a temperature of 22 degrees centigrade (Celsius).2368

When the medical examiner inspects the body at 10 PM, it has a temperature of 33 degrees.2374

At 11 PM, it is at 31 degrees; when did the person die?2379

We are also told that human body temperature is 37 degrees.2384

And finally, we are also told to give our answer to the nearest tenth of an hour.2387

All right, what is the idea going on here?2391

The police find this body, and so they can take the temperature of the body at one moment in time.2393

And then, they wait a little bit longer, and they can figure out that the body has cooled.2398

From that, they can figure out how fast the body cools.2402

And because they know what it was at one time, they can go backwards.2405

They can use the math to go backwards and figure out when the body was at a normal human body temperature--when the body was still alive.2408

And because, when it is 37 degrees, that will be the moment of death, that will tell us when the person died.2416

So, this forensic science; this is the sort of information, the sort of math,2421

that crime scene investigators can use to figure out when a person died, and be able to create a good murder case based on that.2424

All right, let's get our pieces of data from this.2432

We are told that the room has a temperature of 22 degrees.2434

So, the surrounding temperature of our room is 22; the initial temperature of our body is 37 degrees.2437

Human bodies are 37 degrees when they are alive.2446

When the medical examiner inspects the body at 10 PM, it has a temperature of 33 degrees.2449

So now, we have this confusing thing of time.2453

In the last example, we knew what time 0 meant; we knew where it was, and everything was told relative to time 0.2456

Time 0 was when you turned off the stove, and then time t = 10 was 10 minutes from turning off the stove.2462

But at this point, the t = 0 is time of death; but we don't know what 10 PM is to that yet.2468

But let's start by saying t = 0 is time of death; the moment the person dies is t = 0.2474

And from there, we are just counting there; so t = 0 is time of death; t = 1 is one hour after time of death,2486

because currently we are dealing with 10 PM, and our answer was told to be given in hours; so let's work with hours as our form here.2492

t = 0 is time of death, so t = 1 is one hour after death; t = 2 is two hours after death; t = 3 is three hours after death, etc., etc.2500

Now, that still doesn't quite tell us what 10 PM is, so we are going to have to name symbols for this.2510

Let's say that t10 is equal to the hours to get from the moment of death to 10 PM.2514

And similarly, t11 will be the hours to 11 PM.2525

Great; with all of these ideas in mind, we are ready to start working things out.2531

So, at time of 10 PM (which...we don't know how many hours it is after death, but we can still talk about it as t10),2535

we know that the body is at 33 degrees at 10 PM; we have that 33 degrees at 10 PM is equal to2544

surrounding temperature, 22, plus initial temperature, 37,2551

minus surrounding of 22ek(t10), because that is our time, t10.2556

OK, subtract 22 on both sides: we get 11 = 37 - 22 is 15ekt10.2563

So, we divide by that; we have 11/15 = ekt10.2571

Finally, we take the natural log of both sides, and we have ln(11/15) = kt10.2577

Now, at this point, we say, "Oh, no, there are two unknowns, and this is only one equation."2584

Well, we are going to have to bring a little bit more information to the table.2589

Let's look at the 11 PM hour: at t11 (that is not t = 11; that is t11,2592

which is just the name that we gave to the number of hours after the time of death when it gets to 11 PM),2600

how many hours after death is 11 PM?--at t11, we have a temperature of 31 degrees, at 11 PM.2608

And then, the surrounding environment is still 22, plus initial temperature was still 37, minus 22e to the k...2620

and now we are using a time of 11, t11.2628

Once again, I want to just point out that it is not time equals 11; it is definitely not that.2633

It is just that we named this thing, hours to 10 PM, as t10, and hours to 11 PM as t11.2638

They could be 1 and 2; they could be 5 and 6; they could be 80 and 81; we don't know yet.2644

We just came up with names for these things so that we could start working things out.2651

We can subtract 22 on both sides; we will get 9 = 37 - 22 is 15, ekt11.2654

We can divide by the 15; we get 9/15 = ekt11.2662

Take the natural log of both sides: ln(9/15) = kt11.2668

Oh, no; we have, once again, two unknowns and one equation.2674

And between all of them, we have k, t10, t11; that is 3 unknowns and 2 equations.2678

We need some other piece of information that will connect these things together.2685

How is k connected to t10? How is t10 connected to t11?2688

Of course--how is t10 connected to t11? Well, 11 PM is one hour after 10 PM!2692

To get from 10 PM to 11 PM, you go up one hour; so however many hours after death t10 is,2700

we know that, if we add one to that, we will end up being at the 11 PM mark.2706

t10 + 1 = t11; this key realization will allow us to solve things.2710

We can plug this in over here; so we have ln(9/15) = kt10 + 1.2717

kt10 + k...over here, we know what kt10 is--it is ln(11/15).2727

Let's bring this back down; ln(9/15) is equal to kt10; we swap that out for ln(11/15) + k.2735

We subtract ln(9/15) - ln(11/15) = k.2745

Now, at this point, we could just punch that into a calculator, or we could remember the properties that we know,2752

if we want to do a little bit less on our calculator.2756

That is ln(9/15), divided (because subtraction outside of logarithms becomes division inside) by 11/15.2758

So, since we are dividing by 15 on the top and the bottom, they cancel out, and we are left with ln(9/11) is equal to k.2767

We take ln(9/11) in our calculators, and we end up getting -0.20067 as our constant for k.2773

Our proportionality constant of k is -0.20067; great.2787

That means we can now take this piece of information right here.2793

We know what k is; we put that in over here; we have ln(11/15) is equal to...sorry, I accidentally wrote an extra 0...-0.20067t10.2796

We divide by that; we have natural log of 11/15, divided by -0.20067, equals t10.2814

So, t10 comes out to be exactly...well, not exactly; we will truncate it; we will cut it down,2823

and cut off some of those decimal places; we will round it to 1.5456.2835

It comes out to be approximately 1.5456.2840

However, they asked for the nearest tenth of an hour, so that ends up being about 1.5 hours.2844

Now, we want to know what time the person died at--when did the person die?2851

Well, if we are at t10, that is the hours to get to 10 PM from time of death.2856

So, if it is 1.5 hours to get to t10, then that means that death occurs 1.5 hours before t10.2862

So, we know that 10 PM is 1.5 hours after death.2870

If we go back 1.5 hours from 10 PM, we get 8:30 PM as the time of death.2882

Great; and this idea right here is actually a basis for something that real detectives2895

and real medical examiners can use--this sort of idea of how these things work out.2902

Math has a lot of really powerful applications, and this is just one example that is something that is almost out of a detective story.2906

All right, cool; we will see you at Educator.com later.2914

This is the end of logarithms and exponential things, so we are going to move into a new topic.2917

I hope everything here made sense, because it is time to get started on something else.2921

All right, goodbye!2925