Vincent Selhorst-Jones

Word Problems

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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• ## Related Books

### Word Problems

• Word problems form the bedrock foundation of why we should care about math. While arithmetic and algebra are useful tools that we must learn to do well in math, we connect math to the real world through word problems. It is through word problems that we find value in math.
• Learning how to solve word problems teaches us crucial skills such as logic, thoughtfulness, and problem-solving. Even if you have no interest in doing math later in life, these skills are important for virtually anything you might be interested in doing later on. Studying word problems isn't just about doing well in math: it's about preparing yourself for dealing with puzzling situations for the rest of your life.
• There is no one way to solve all word problems because they can take so many shapes. However, there are some general guidelines that will help us work on them. We will use a four step process for approaching word problems. By following this method, you'll have clear, concrete tasks to accomplish at every step. While creativity and thought are still necessary, these guidelines can be used on virtually any problem.
#1 Understand the Problem. Figure out what the problem is asking about. You don't have to solve anything right now, or even figure out exactly what you're looking for: you just want to have some idea of what's going on. This might seem obvious, but remember, you can't solve something before you know what's going on.
#2 What Are You Looking For? Once you understand what's going on, you need to figure out what you are trying to find. In math (especially for the next few years), this will often take the form of setting up variables. Define any variable the problem asks for, along with any others you will need to solve it. Make sure to write down a reminder about what each variable means so you don't forget later.
#3 Set Up Relationships. Use what you know to set up relationships between your unknowns and whatever information the problem gives you. In math (especially for the next few years), this will usually take the the form of making equations. You will set up equations involving your unknown variable(s) and whatever else you know. Sometimes you'll realize you have more unknowns than you originally thought. That's okay: just make some new variables, then figure out equations that will involve these new variables as well.
#4 Solve it! Once you've done the above three steps, you're ready to solve it! Often this is the easiest part. After all, it's like doing any other exercise now: you have some equations to solve. Just roll up your sleeves and work on it like a normal (non-word) problem. [As a rule of thumb, to solve a problem you need as many relationships as you have unknowns. For example, if you need to figure out three variables, you must have three equations relating them to each other.]
• This general method of problem-solving is great: it will work in pretty much any situation, no matter what you're working on. But let's simplify it a little and consider the specific approach you'll need in math for the next couple years:
1. Understand what the problem is talking about,
2. Set up and name any variables you need to know,
3. Set up equations connecting those variables to the information in the problem statement,
4. Use the equations to solve for the answer.
While a few word problems ( ∼ 5-10%: mostly concept questions and proofs) won't use that exact formula, you can always fall back on the more general method.
• Here are some extra tips for working on word problems:
• Tip-Draw Pictures: While it may not be possible for every problem, it can help massively when you can. Basically, if a problem talks about geometry, shapes, or something that is physically happening, you want a picture to look at. If the problem doesn't give you one, draw it yourself! When you're not sure how something works or what to do, sometimes a quick sketch can clear things up.
• Tip-Sum of Parts = Whole;  Whole = Sum of Parts: You will often have problems where you can't directly solve for something, but the thing is part of a larger whole or built out of smaller pieces that you can solve for. In that case, figure out how it relates to those other things and solve for those instead, then use that information to get what you want.
• Tip-Try Out Hypothetical Numbers: Sometimes it can be hard to figure out what's going on because we aren't working with numbers. Try plugging in hypothetical numbers to help you understand what is going on. This is a great way to test the equations you set up. It's easy to make a mistake while setting up equations, so check them afterwards with values you understand.
• Tip-Student Logic: The hardest part is often figuring out the relationships the problem gives you. Luckily, you've got a secret weapon: you're a student. This means you can use student logic. You can pretty much be certain the problem is based on whatever you're currently learning. For example, if you're studying logarithms, you know the problem can almost certainly be solved with logarithms.
• Tip-Jump in! Working on word problems can sometimes be paralyzing. You're not sure where to start, you don't know exactly what you're looking for, you don't see how to solve it. It's okay! That happens to everyone sometimes. The important thing is to not freeze up. Instead, just try something. Even if you can't set up the equation right the first time, or you pick the wrong variable, you'll wind up learning from your mistakes. As long as you pay attention to what you're doing and think about what makes sense, you will see where you went wrong. By seeing where you went wrong, you can realize what you have to do for the problem. The fastest way to learn can be by making a mistake!

### Word Problems

While there are many ways to solve word problems, this lesson went over a general method for solving most math word problems. What were the four steps?
• First, you need to know what's going on. If the problem doesn't make sense, how can you work it?
• Once you understand the problem, you need a way to talk about what you want to find out. You can represent a number you don't know yet with a variable.
• Now that you have your variable(s) in hand, you need to connect them to the rest of the information in the problem. You can do this by carefully reading the problem and creating equations.
• After you have all your equations in place, solve them for the answer!
1) Understand: Make sense of the problem-what is it asking?
2) Variable(s): Set up variables so you can talk about the values you need to know for solving the problem. Always make sure to write a note to yourself about what each variable represents.
3) Equation(s): Use the information in the problem to set up equations using the variables you just created.
4) Solve: Now that you have all your equations, use them to find the answer. This will normally be a matter of solving for what one of the variables is.
Ray and Lola are picking oranges. Together, they have picked 25 oranges in total. If Lola has picked 17, how many has Ray picked?
• Begin by making sense of the problem. They're picking oranges, and they have each picked some portion of the total share.
• Set the thing we want to know as a variable:
 x=number of oranges Ray picked.
[We could use any symbol, but x is fine and pretty common.]
• Using the variable, set up an equation. If we add x to the number Lola picked, we get the total:
 x+17 = 25.
• Solve the equation for x.
Ray picked 8 oranges.
In the kingdom of Dunsinane, there are three libraries, with a total of 892 books amongst them all. The three libraries are named Porter, Hecate, and Siward. If Porter has 127 books and Siward has 390, then how many books does Hecate Library have?
• Begin by making sense of the problem. There are three libraries with some number of books amongst them. Each of the libraries contributes to that total number.
• Set the thing we want to know as the variable:
 x = number of books in Hecate Library.
[We could use any symbol, but x is fine and pretty common.]
• Using the variable, set up an equation. If we add x to the number of books in each of the other two libraries, we will have the total number of books:
 127 + x + 390 = 892.
• Solve the equation for x.
There are 375 books in Hecate Library.
The area of a rectangle is 160 cm2. If the length of the rectangle is 20 cm, what is its width?
• Begin by making sense of the problem. Because the problem has to do with a geometric shape, it would help to draw a picture.
• We want to know the width, so we set it as a variable:
 w = width of the rectangle.
[We already wound up doing this over the course of drawing the picture, so it would probably be fine to skip this step if we wanted.]
• The area of a rectangle is equal to the product of the length and width. [If we had forgotten this fact, we know that we need to somehow connect the width and length to the area, so we could start looking up formulas for the area of a rectangle.]
 160 = 20 ·w.
• Solve the equation for w. Don't forget that it needs to have a unit on it because it's a measurement.
The width of the rectangle is 8 cm.
What two consecutive integers add up to 109?
• Begin by making sense of the problem. To understand this, we need to know what `consecutive integers' are. If we don't know it, we can look it up. Consecutive integers are two numbers that are "next to each other": 1 and 2, 7 and 8, 512 and 513 are all examples.
• Set up the variables. We want to know both, so we could set up variables for both:
 x = the first integer               y = the second integer
[We could actually write this with just one variable as well, as we will see in the next step.]
• We know that the two numbers add up together to 109:
 x+y = 109.
However, that's not enough information to solve it. So we have to go back to the question ands see what other equations we can create. We know that the numbers are consecutive, so y must be 1 greater than x:
 x+1 = y.
[Alternatively, we could have realized this from the beginning, and never written out y when we were setting up the variables. We could have just set the two "numbers" as x and x+1 since we knew they were consecutive from the start.]
• Using this information, we have
 x + (x+1) = 109.
• Solve the equation for x. Don't forget that the question asks for both numbers, not just x. This means we have to give the value of x and x+1.
54 and 55
This time, Ray and Lola are picking strawberries. So far, they have picked a total of 210 strawberries. If Lola has picked 120 fewer strawberries than five times the amount that Ray has picked, how many have each of them picked?
• Begin by making sense of the problem. Two people are picking strawberries. Each of them has picked some number, and together they have picked a total of 210.
• Set up the variables. We want to know how many each of them have picked:
 R = Ray′s number of strawberries        L = Lola′s number
• From the problem, we know they have picked 210 total:
 R + L = 210.
However, that's not enough information to solve it. So we go back to the problem and see what other equations we can make. We have a relationship between how many Ray has picked and how many Lola has picked-"Lola has picked 120 fewer strawberries than five times the amount that Ray has picked":
 L = 5R − 120.
[Note: Turning a sentence like that into math can be tough. If you try to make an equation, but you're not sure you set it up right, test it out with some hypothetical numbers. See if it does the same thing as the sentence said.]
• Substituting one equation into the other, we get
 R + (5R − 120) = 210.
• Solve the equation for R, then use that to find L as well.
Ray picked 55 strawberries and Lola picked 155 strawberries.
Birnam Publishing House has come to Dunsinane, bringing many new books! Now amongst the three libraries of Dunsinane (Porter, Hecate, and Siward), there are a total of 2397 books.
Hecate Library has three times as many books as Porter Library. Siward Library has 120 more books than half the number of books in Hecate. How many books are in each of the three libraries?
• Begin by making sense of the problem. There are three libraries with some number of books amongst them. Each of the libraries contributes to that total number.
• We don't know the number of books in any of the libraries, so let's make a variable for each one:
 P = number of books in Porter        H = Hecate        S = Siward
• From the problem, we know that there are 2397 books in total:
 P + H + S = 2397.
However, that's not enough information to solve it. So we go back to the problem and see what other equations we can make. We see we are given two other relationships amongst the libraries. "Hecate Library has three times as many books as Porter Library":
 H = 3 P.
"Siward Library has 120 more books than half the number of books in Hecate":
 S = 1 2 H + 120.
• We can put each of the libraries in terms of P:
 H = 3 P               S = 1 2 (3P) + 120
We can now plug these in to our equation involving all three libraries:
 P + 3P + 1 2 (3P) + 120 = 2397.
• Solve the equation for P, then use that information to find out H and S as well.
Porter has 414 books, Hecate has 1242 books, and Siward has 741 books.
A circle is inscribed (drawn inside) of a square so that the circle touches all four sides of the square. The radius of the circle is 4. What is the area of the shaded portion in the picture below?
• Begin by making sense of the problem. Since we have a picture already, we just need to understand the picture. We can think of it as if the square has had a circular portion removed from its area.
• In this case, we could get away without creating any variables because we won't need to do any complicated algebra with the unknown. Still, let's set one up anyway:
 A = area of shaded portion.
• To find out what the shaded area is, we need to break this into pieces. We can think of the square as being a combination of the shaded area and the circle.
 Asquare = A+ Acircle        ⇒        A = Asquare − Acircle
This means we need formulas for the area of a square and the area of a circle. If we don't know them already, we can look them up and find out:
 Asquare = s2        Acircle = πr2
• We already know that the radius of the circle is r=4 from the problem, but what is the length of the side of the square? We can look at our picture again to figure this out. Since the radius spans from the center to the edge of the circle, we can use two radii to create a side as in this picture:
• Putting this all together, we get
 A = 82 − π·42 .
64 − 16 π ≈ 13.735
Malcolm has precisely 14 coins in his pockets. He has only quarters ($0.25), dimes ($0.10), and pennies ($0.01). If he has a total of$2.09 in coins and three dimes, how many quarters does he have?
• Begin by making sense of the problem. He has various types of coins and they all add up to a certain amount of money.
• We want to know how many quarters he has so we will definitely need a variable for that. It seems likely that we will need to know the number of pennies to work out the problem, so we'll make a variable for that as well:
 q = number of quarters        p = number of pennies
• We know the total value of all his coins, so we can create an equation:
 q ·0.25 + 3 ·0.10 + p ·0.01 = 2.09.
We will need another equation to solve since we have two unknowns. Going back to the problem, we also know the total number of coins:
 q + 3 + p = 14.
• We can solve the second equation for p in terms of q:
 q + 3 + p = 14        ⇒        p = 11 − q
Then we can substitute that in to the first equation to find an equation that only uses q:
 q ·0.25 + 3 ·0.10 + (11−q) ·0.01 = 2.09.
• Solve the equation for q.
Malcolm has seven quarters.
A farmer wishes to make three identical pens by subdividing a single large rectangular pen into three smaller pens. He will do so by creating one large rectangular fence, then putting two fences up inside the large fence. If he wants the shared side(s) of each pen to be 3 m and the total area of all the pens to be 54 m2, what length of fencing material does he need?
• Begin by making sense of the problem. This is a difficult problem to understand, but we can make it a lot easier by representing it with a picture: Our goal is to find out what the total length of all the fence is (the black lines).
• To figure out the total amount of fence, we will need to know the width of the pens: w = width of each pen.
• We know the total area is 54 m2, so if we multiply the area of one pen (3·w) by three, that would be equal to the total area:
 3 ·(3w) = 54.
• We can solve this to find that the width of each pen is w = 6. Now we need to find what the total length of fencing is.
• The width (w=6) is used a total of six times because it makes the top and bottom of each pen. The length (3) is used only four times though because it is shared by some pens. This means the total amount of fencing is
 6 ·6 + 4 ·3.
The farmer needs 48 m of fencing material.

*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.

### Word Problems

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

• Intro 0:00
• Introduction 0:05
• What is a Word Problem? 0:45
• Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
• Requires Us to Think
• Why Are They So Hard? 2:11
• Reason 1: No Simple Formula to Solve Them
• Reason 2: Harder to Teach Word Problems
• You Can Learn How to Do Them!
• 'But I'm Never Going to Use This In Real Life'
• Solving Word Problems 12:58
• First: Understand the Problem
• Second: What Are You Looking For?
• Third: Set Up Relationships
• Fourth: Solve It!
• 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
• Set Up and Name Any Variables You Need to Know
• Set Up Equations Connecting Those Variables to the Information in the Problem Statement
• Use the Equations to Solve for an Answer
• Tip 26:58
• Draw Pictures
• Breaking Into Pieces
• Try Out Hypothetical Numbers
• Student Logic
• Jump In!
• Example 1 34:03
• Example 2 39:15
• Example 3 44:22
• Example 4 50:24

### Transcription: Word Problems

Hi--welcome back to Educator.com.0000

Today, we are going to talk about word problems.0002

This lesson is going to tackle solving those dreaded word problems.0005

Hopefully, when this lesson is over, you are going to dread them a little less.0008

We are going to start by talking about why we should care about them, and essentially why we should care about learning in general.0011

Then, we are going to go over a general structure for approaching and solving word problems, and really any form of problem.0017

We will see how it applies to what we are working on now.0022

And then, once we understand that method, we will see a variety of different tips and strategies to help us get the most out of it.0025

So, we will go over some specific tips that apply to that general strategy that we talked about.0031

And then finally, we will work a bunch of examples where we will actually see directly how we are using this step-by-step strategy,0035

so we can see how the method gets applied on real word problems.0041

All right, let's go: first, what is a word problem?0044

The term word problem gets used a lot; it is pretty much just a catch-all.0048

It is for any problem that is primarily being described with words, as opposed to math symbols.0052

That doesn't mean very much; you can use words to describe a lot of things.0058

So, it means that there is no one way to approach word problems, because you could be talking about so many different kinds of problems.0061

Since there are so many different possible problems, we have to be ready to adapt when we are working on a word problem.0067

Trying to answer the question, trying to figure out the one way to solve word problems, would be like saying, "How do you play sports?"0073

There are a lot of different sports; there are a lot of different ways to play within each of those sports.0079

It depends on the specific situation; that is going to tell us what we have to do.0083

The best method depends on our specific situation, so there is no one way to do all word problems.0087

But we do know this one fact: pretty much all word problems are going to require us to think.0093

We have to pay attention and be creative, because we are being asked to do more than just follow a formula.0098

As opposed to a lot of problems that we will get in math, where it is just "here is the method that we do it;0103

let's apply that method one hundred times in a row."0108

It is going to be "understand this idea, and then do some critical thinking about it."0111

So, we have to actually be thinking and paying attention when we are working on word problems.0115

We should always be thinking, and we should always be paying attention, when we are doing everything in life.0118

But for word problems, it is going to be especially important that we are really thinking about what we are doing, because we are trying to understand it.0123

Otherwise, it is just not going to make sense.0128

Why are they so hard--why are people constantly thinking that word problems are the hardest kind out there?0132

Well, first, there is no simple formula for them: there is no one way that you do it--just plug things in it-- because each word problem can be different.0137

So, when we are working non-word problems, we normally have examples that we can follow step-by-step.0145

There is some formulaic method that we can just rely on.0149

But with a word problem, there is no such formulaic method.0152

It is up to us to be paying attention, to be ready to be creative and thoughtful.0155

The fact that it actually requires thought--it requires some creativity--that is one of the reasons why word problems are generally harder than non-word problems.0159

Second, it is a lot harder to teach word problems.0168

It is easy to teach simple, repeatable instructions--perhaps not easy all the time, but it is easier to teach simple, repeatable instructions,0171

things like formulas or step-by-step guides--anything like that where the idea is "do what I did; do what I did; do what I did."0179

That sort of thing is pretty easy, because you can just follow the "monkey see, monkey do" saying--you do what they led, and it works out.0186

But with word problems, you have to teach creativity; you have to teach an ability to understand what is going on.0194

You have to really teach analyzing a whole bunch of things at once, and understanding things.0198

As opposed to just teaching to follow a method, you have to teach how to understand and how to think.0202

That is a much bigger task than just teaching a few quick instructions.0206

So often, sadly, they are overlooked, because it is easier to teach that, and a lot of education is based0211

on taking standardized tests, so we end up seeing us teaching to standardized tests, as opposed to teaching0217

to a larger scale of thinking and understanding, which is the sort of thing0223

that is absolutely necessary if we are going to do well on word problems.0226

Don't let this make you be worried: don't despair--you can learn how to do them.0231

Just because they are a little bit harder, and you don't get much learning about them (usually),0237

doesn't mean you can't get great at them--you can get great at doing word problems.0241

It is just going to take some though, some imagination and patience.0246

Like everything else, it is going to be a skill that you can practice--you just need to practice the skill.0248

In this case, if, for example, your teacher doesn't assign you many word problems,0255

it might be a good idea to work an extra word problem out of your book.0259

Make sure it is one of the problems...most math textbooks have answers to the odd exercises, or sometimes the even exercises, in the back.0262

So, choose the one that you have the answer to, but that wasn't assigned to you,0272

so that you get the chance to have some extra experience with word problems.0275

So, if you know that you have difficulty with word problems, you have to focus on it; you have to work on it a little bit extra.0278

I know, I am asking for you to do more; but there is pretty much no other way to get better at something.0282

You get better at things because you practiced, not because you just want to be better at them.0286

So, word problems are a skill you can practice, and a skill you can definitely improve.0291

But you do have to practice it, if you want to improve.0295

But if you practice, you will definitely get better; so consider that.0297

If you have a lot of difficulty with word problems, just start tackling some easier word problems.0299

Work your way up to medium ones; work your way up to hard ones.0303

Just sort of do that in the background, as we are working on precalculus.0305

It will be a really useful skill that will really help you in a lot of things later--not just math, even--all sorts of things.0308

Why should we care about word problems?0315

While word problems may be difficult, they are also incredibly important.0317

In some way, they are the point of math: word problems turn math into something more than a meaningless series of exercises.0321

As opposed to just solving one equation after another, where it is just meaningless symbols, word problems "ground" what we are doing.0327

They give it context; they make the math mean something, which can make it, sometimes, beautiful, and usually important to what we are doing.0334

Well, when we use math in science, we can better understand and appreciate our universe.0342

It is through math that we can have physics be able to turn into equations that we can describe our world with.0350

It is through math that we can figure out specific things in chemistry and talk about quantities.0356

It is through math that we can do statistics in biology and sociology.0360

It is through math that allows us to understand the world around us.0364

When we use it in engineering, that allows us to build amazing feats of human ingenuity.0368

It allows us to build huge bridges; it allows us to build giant dams, skyscrapers...0373

It lets us build devices like phones that fit in the size in your hand...well, yes..."computers" is actually what I meant to say,0379

because most smartphones these days are becoming even more powerful and more powerful.0386

The incredible miniaturization of technology--what we have right now--the fact that I can be teaching you0389

when I am in a totally different place than you, and you are watching it right now...0394

Engineering: math has been put in all the engineering, and the computer things when we are building things--we need math0398

for us to be able to make all these things: math runs all of the technology that we have.0407

At heart, technology is based on what we have been able to figure out in math.0411

In many ways, it is applied math through other things.0415

And if we use math just on its own, we can find deep truths about the nature of logic and knowledge.0418

Geometry: when you were studying geometry, it was all just mathematical ideas.0423

They have applicability in the real world, but they are also just kind of beautiful, interesting ideas.0427

I have studied a lot of math, and I think there are some really, really cool things that you can see in math, that are just purely ideas in math.0431

But to describe them, we need words; just a bunch of symbols isn't going to do it.0438

And without language to connect math to these really deep, interesting ideas, we just have symbols.0442

So, we need language to be able to give us a deeper context, to ground what we are doing with our math.0448

It is through word problems that we find value in math.0455

Word problems are our connection between wanting to do things and learn things, and this interesting symbolic language that lets us solve things.0458

Word problems are the point of connection between wanting to know things and being able to solve things.0466

They are really, really important for that reason; in many ways, word problems are what gives us knowledge.0471

Now, not everyone is going to be convinced by my impassioned appeal to the inherent value of beauty and knowledge.0478

I think that wanting to learn, and the fact that learning is an amazing, really cool thing--that is a great reason to learn, in and of itself.0484

But if you would like a more material reason--grades.0492

You can get an A, and you can possibly even get a B, but you are never going to be able to achieve the highest marks0496

in a math class if you don't understand word problems.0500

To do your best in class, you need to be able to solve word problems.0504

Every math is going to include, or at least really should include, some word problems.0507

So, it is important that you know how to approach them.0512

With time and practice, you can understand them better, and you can improve your grades.0515

Remember what I was saying before: if you practice this, you will be able to improve at it.0518

And it is not just math class: any standardized test (like if you are going to take the SAT,0522

because you are interested in applying to college; or if you are currently in college,0526

and one day you want to take the GRE so that you can apply to graduate school)--0529

they are going to use word problems in there, as well.0533

Pretty much any test that you will take anywhere will have some word problem ideas going in it.0535

So, you want the ability to solve problems like that--being able to solve word problems is crucial in a lot of situations.0539

Also, if you want to do anything like engineer and build things, or do science,0547

or do anything that has really hard scientific connections, or is really analyzing0551

and measuring the world around us, you are going to need to understand math.0558

So, there are lots of really good reasons to understand math, in just a "material-benefit-to-us" way.0560

Finally, there is always that gem of an excuse, "I am never going to use this in real life!"0567

"I know that you might say it is useful in science, but when am I ever going to use partial fraction decomposition0572

(which you will learn about later)--I am never going to use this in real life, later on, so why should I learn it now?0577

Is that true--"I am never going to use this in real life"?--yes, honestly, you are probably right.0585

There is a very good chance that the things you will learn in this course, or any math course,0592

will never be the sort of thing that you get paid for doing later in life.0597

You will not be having to analyze functions, and that is going to be what gets you your paycheck every day.0601

That is not the point: that is not the point of learning these things.0607

You are not in school to learn the things that you are going to use later in life.0611

You are not in school just to learn those things that are going to get you a paycheck later in life.0615

You are there in school to learn how to learn.0619

Think about this: musicians do not play scales at concerts; boxers to fight punching bags in the ring; and surgeons do not operate on cadavers.0623

But the practice that they get by doing each of those things is absolutely necessary for them being able to perform well later in life.0632

So, you might not end up using this immediately in real life, or you might never use it in "real life" (whatever that means).0640

But what you are getting now is practice: whatever you do later in life, you are going to need to learn new things over and over.0648

So much of any interesting job, so much of whatever you do later in life, is going to be learning new things and getting good at those new skills.0657

School is your chance to practice this process, this skill of being able to learn something, get good at it, and use it.0665

This skill is what you are learning right now, and it is absolutely necessary if you want a job that pays well, is interesting, or is enjoyable.0673

And if you want one that is well-paying, interesting, and enjoyable, you are definitely going to have to be able0681

to learn lots of things quickly, in your situation, and do whatever you need to.0687

You need the ability to adapt, the ability to learn many things and operate in many conditions.0690

The way that you do that is: you practice it now.0695

You do not get better by just wishing you were going to be better; you get better by practicing it.0697

School, learning, is your chance to practice the learning that you will need for the rest of your life.0703

If you do not practice it now, because you are not going to use those things, yes, you are technically right.0709

But that is like saying, "I am not going to drive this car later in life; I am going to drive some other car; so I won't practice driving this first car."0713

To be able to drive that second car, you are going to have to have learned how to drive a car somewhere.0722

So, you can't skip learning how to drive the first car.0727

Even though there are going to be some differences between the two cars (they might be completely different cars),0729

there are a lot of parallels here--it is going to be very similar driving one car or the other.0734

It may be hard to see right now, but please trust me on this.0739

I am being as truthful as...this is just from the bottom of my heart: put effort into learning now.0742

The process of learning is going to give you skills that you will use for the rest of your life.0750

It can be difficult to see how valuable those skills are to you now.0755

But trust me, in 5 or 10 years, when you look back later, you are going to be so thankful0758

that you took the time and effort to really understand what you were doing--to go through0764

all of that practice of learning--because those skills of being able to learn quickly0767

and do well and understand things--they are going to make the rest of your life so much better and so much easier; it is really important.0771

All right, moving on to actually solving word problems: they are so important, so we really want to be able to solve them.0779

Now, as we discussed, there is this problem that we can't just solve all word problems with one method.0789

But luckily, there are some general guidelines that will help us work on them.0795

We are now going to see a four-step process for approaching word problems.0799

If you follow this method, you will have clear, concrete tasks that you can accomplish at every step.0802

Now, creativity and thought are still going to be absolutely necessary.0807

But these guidelines can be used on virtually any problem that you encounter, so it is a really useful thing here.0811

Let's look at it: the very first step is that you need to understand the problem--begin by figuring out what the problem is asking about.0816

You don't have to solve anything right now; you don't even have to figure out exactly what you are looking for.0823

But you need to figure out what the problem is asking about.0827

You just want to have some idea of what is going on--what is the general thing that is happening here?0834

Many word problems are going to unfold like a story, in some way; so you just want to understand--"What is this story telling me?"0839

Now, this step may seem obvious, like everybody is going to think, "Well, of course I have to understand the problem!"0845

But so many people gloss over it, and they just try to hop right into solving the problem.0850

But you need to understand it first; how can you solve something if you don't even know what is going on?0855

You have to know what is happening in the problem, what the problem is about, what the ideas going on are,0861

before you are going to be able to have any chance of actually solving it.0867

So first, just get a sense of what is going on.0869

Once you have that, second, what are you looking for?0873

Set up your variables: what are you looking for?0876

Once you know what is happening in the problem, you want to figure out what you are trying to find.0878

What are the ideas that are central here, that you need to be able to work out this problem?0884

Now, almost always, especially for the next few years, before you get into really advanced math classes,0889

if you continue to study math in college--for precalculus, calculus, and even the next couple of years--0895

this will take the form of setting up variables: you will set up variables and define any variable that the problem asks for.0899

You might also need to define other variables, such as values that are talked about, but never explicitly given.0906

Now, I want to make absolutely sure that you write down what each variable means.0911

I really want to make sure that you do this; I am not kidding.0916

Until you are extraordinarily comfortable with word problems, you should be literally writing out a little reminder of what each variable means.0920

For example, if you had a problem where you had to talk about some number of tables, then you might say, before you started, "t = # of tables."0927

And then, you also might have to talk about chairs in the problem, so "c = # of chairs."0939

And that is enough right there; but very often, students will be working on it, and they will forget what this letter meant or what was really there.0945

This is a way of anchoring your work, so you know what you are searching for, what these ideas are about.0952

And by writing it down, it will make it that much easier to work on the rest of the problem.0957

Really, really, really: write down what the variable means--it should be completely obvious to you,0960

either because you will have a picture (which we will talk about later), or because you have literally written out what the variable means.0964

That way, you can't forget it while you are in the middle of working on the problem, and get confused.0970

So, you set up what you are looking for at this point: what you are looking for, what is unknown, what is going to help you get to figuring out the answer.0973

Third, you set up relationships: you want to figure out what the problem is telling you about what you are looking for.0982

What is the problem telling you about what you are looking for?0988

You are trying to solve something at this point, which you have figured out from #1.0993

And you know the sort of ideas that you are looking for; that was #2.0996

So now, you need to figure out what the problem is telling you about those things that you are looking for (those variables, if we are doing a math problem).1000

Use what you know; use what the problem gave you to set relationships between your unknowns and whatever information the problem gives you.1007

Now, in math class, especially for the next few years (once again), this will usually take the form of making equations.1015

This is what it is normally going to come down to: you will be making equations.1022

You will set up equations involving your unknown variable or variables, and whatever else you know from how the problem is set up.1025

The problem will tell you some information, and you will use that information to build equations.1031

And if it is something that is not specifically a math problem, or a more general form of math problem,1035

you might be just using it to set up things of what you know and what you can rely on as you start to work through it.1040

Sometimes, when you are working on these equations, you will realize that you have more unknowns than you originally thought.1045

That is perfectly fine; just make up new variables at this point.1049

You realize, "Oh, to use this idea, I need to also have this new variable introduced"--introduce a new variable!1052

Go back to step #2; make a new variable; write down what it means; and then plug it into the equation where it is going to go.1058

Figure out the equations that will involve that new variable that you realize you need to use.1063

Finally, once you understand what the problem is about, you know what you are looking for,1068

and you have all of your relationships (equations) set up, you are ready to solve it.1072

Normally, in many ways, this is going to actually be the easiest part, because it is just like doing any other exercise now.1076

You have equations to solve: you are looking to solve for something, and you have equations.1082

It is just going to be a matter of doing the math that you have been doing in every other section,1086

in every other part of the section that you are working on.1089

It will normally have many exercises, and it will only finally culminate in some word problems.1092

So, it is just like doing all of the exercises you have been doing before;1096

it is just that it had this framing of how we get to those equations--how we get to doing the exercise.1098

So, you have some equations to solve or something like that; roll up your sleeves and work on it like a normal non-word problem.1104

And as a general rule, to solve a problem, you need as many relationships as you have unknowns.1109

For example, if you need to find out three variables, then you will have to have three equations (or more) that relate the three variables together.1116

So, you will need the same number of relationships, the same number of equations,1125

as the pieces of information you are trying to find out.1129

So, it is like each equation is a key that you can fit into an unknown lock.1131

Each equation is a piece of knowledge that you can use to shine the light on some unknown thing.1138

In summary, complex problems can be approached by first understanding what you are working with;1145

then figuring out what you are looking to find (figure out what you want or need to know);1150

third, you work out relationships--how do the things that you know already from the problem,1157

and the things that you are looking to find--how are they connected to each other?1163

And then finally, you put it all together, and you get the answer.1167

So, that is our basic summary of method.1171

Now, it may seem surprising, but this method can actually be applied to pretty much any problem--any issue you have, not just math.1174

This is such a general tool for approaching complex ideas that you can use it1181

for pretty much anything where you are trying to figure out a way to solve an issue.1185

You are probably used to doing this in all sorts of situations already; it is just a good way to approach problem-solving in general.1189

So, let's actually see two examples that are not connected to math, just so we can get an idea of how we solve problems on a general scale.1194

So, consider these two situations: in the first situation, your car is making weird noises when you drive.1201

And in the second situation, your friend suddenly stops talking to you one day.1207

They are very different things, but both issues that you would want to deal with.1211

So, to begin with, our first thing is that we understand what is going on.1214

Your car is making weird noises when you drive: well, if my car is making weird noises, there is probably some sort of issue going on.1217

And so, now you also have...in a word problem in math, we are going to have a more specific being-told-what-to-do.1222

In real life, our issues are normally more amorphous, more uncertain in what we need to do to get to the right thing.1228

But with this case, let's say we have figured out that our car has a problem.1234

We need to figure out what that problem is, and then we need to fix it.1238

Now, let's say, in this world, we don't want to spend the money on taking it to a mechanic.1241

So, that means we have to be the one to figure it out, and we have to be the one to fix the problem.1245

So, your car is making weird noises when you drive; that means that what we need to do here--1250

our understanding of it is that we need to figure out what the problem is, and then we need to fix it.1253

That is what our objective is: "Fix the problem."1258

In the version where we are talking about our friend who stops talking to us, what we want to do now is:1260

We are going to assume that, yes, they are actually your friend; you want them to be friendly again.1269

But this is one of our things where we are trying to understand the problem.1273

We actually have to understand and think about what it is--how does it connect to the rest of what is going on?1275

All right, if your friend suddenly stops talking to you one day, but you actually just realized that you don't actually like them at all,1280

then they are no longer your friend; just make them no longer your friend.1284

But in this case, we are saying that, yes, our objective is to get our friend to be friendly again.1287

We want them to actually be our friend, not just be this person who won't talk to us.1292

All right, now we have our objectives in mind; now we can figure out what I need to know.1296

What are the things that I don't know that will help me get to that objective?1300

The second step: in the car example again, if our car is making weird noises, and it has some sort of problem (is our assumption),1305

we want to figure out what is making that noise; we also want to figure out1311

if the thing that is causing the noise is the result of some other issue.1317

And if there is some other root issue, has it damaged anything in the car?1320

And then finally, how can we fix any and all damage?1325

Now really, this is the only thing that matters: we don't really care...in truth, if some genie appeared out of nowhere,1327

and said, "I will fix your car, just because I am a nice guy," we would say, "Yes, genie, fix that car--yay!"1335

And they would fix our car; so really, if we could somehow get a genie to show up here at the end,1342

we wouldn't need to know the answer to these first three things.1346

But it is real life; it is likely going to be the case that these first three issues, these first three unknowns, would really help us to fix the damage.1348

If we can figure out what is causing the noise, if we can figure out if there is an issue that is making that noise,1357

and if we can figure out if that issue has damaged anything, those are all things that are going to help us1362

to fix the car and make it stop making that noise.1366

So, while our final unknown is the "how do we make it better?" we really want to know other things on the way,1369

if we are ever going to be able to figure out how to make it better.1375

In the version where we are looking at our friend, we might want to know why our friend is not talking to us, right?1378

And then, what has happened recently would probably be a good way to get a sense of why they are not talking to you.1386

And finally, what is the best method to make them friendly?1390

Once again, we might actually not care about all of the other things--although, with the first two things,1392

we would probably be curious about why your friend isn't talking to you.1399

But once again, if that genie appeared and said, "I will make your friend be your friend again--no problem!1401

I am a nice guy!" you would say, "Thanks, genie--you are the best!"1405

This genie would be pretty awesome, and they would make our friendship better.1409

But in real life, we have to deal with our problems; we have to make things better.1412

We have to figure out how to solve the problem; we don't have some genie who is just going to do it for us.1416

So, we have to figure out how to solve it.1421

So, while the only thing we really want to know is this final idea, to be able to get to the point1422

where we can figure out that final idea, it would probably be very useful to know1427

why our friend isn't talking to us, and what has happened recently that frames what is going on right now.1431

So, this is an idea of how we set up what are the unknowns that would help me answer that original question.1436

For car troubles, if we want to answer what those unknowns are--get a sense of those things--there are a bunch of things we could do.1442

We could look under the hood; we could get a service manual for the car.1448

A service manual tells us how a car is put together--how that car is put together.1451

We would talk to friends and family who know cars well; we might research the problem online,1455

or anything else we can think of that would tell us more about what is going on with the car.1459

These would help us answer the unknown questions of what is wrong.1462

What is the issue? Has there been any damage? And how do we fix whatever damage it is, and fix the underlying issue?1465

With the friend, we want to think about what happened to your friend recently.1471

We want to ask our other friends if they know what is wrong--perhaps they have some information that we are unaware of right now.1477

We want to pay attention to how our friend is acting when we are not around, or when we are not interacting with them directly.1482

That will give us some sense of what is going on, and be able to help us understand the situation.1488

And we can use all of those pieces of information to help us figure out how to make them friendly again, and make things better.1492

With all of this information, we now just figure out what is wrong with the car and what we need to do to fix the issue; and then, we do it.1498

In the friend version, we use all of this information to figure out1504

what would put our friend in a good mood to be friendly towards us again; and we do it.1507

We use the information to give us many things that we can now to reach our objective.1511

That is what is going on in complex problem-solving, just generally all the time.1515

We figure out what the problem is; we figure out what we need to know to deal with that problem.1519

We figure out how we can answer those things that we need to know.1523

And then, we use all of that information to get what we want.1526

Math-specific method...that is really great for any complex problem, and it can be used on anything at all that you need to solve.1530

But what about what we are going to see exactly in math?1537

This general method is great, but let's see what we are going to use in the next couple of years.1540

Let's look at the specific approach that we will be using.1545

Once again, we understand what the problem is about: we want to just get a sense of what is going on.1547

Then, we set up and name variables: variables represent our unknowns, the things that we are going to need1552

and have to have a handle on to be able to answer the problem.1558

Then, we set up equations; we use the information that is given to us in the problem statement1562

to get us the equations that we need to be able to answer the question.1570

And then finally, we have equations; we solve for that answer.1574

We have equations; we know what we are looking for because we understood the problem in the beginning.1578

We just work it all out, and we get an answer.1582

There are going to be a few word problems, probably around 5 to 10 percent (it depends).1585

But these are going to be mostly concept questions and proofs.1590

So, there are a few word problems that won't use this exact formula of "Understand, Variables, Equations, Solve."1592

All right, that is our formula in general when we are approaching word problems.1599

But sometimes we are going to have to prove things; or sometimes it is asking a general concept question1602

of "Is this true? Is this false? Why does this happen?" and in those situations, we won't be able to use this exact method.1606

But we can fall back on that more general method, where we just think in large-scale, complex terms.1611

Tips: let's start talking about tips that help us improve the efficacy of this method.1618

We understand how the method works now; it is "Go through and understand;1623

set up the things that you are looking for (your variables); set up what you know (your equations);1626

and then finally put it all together and solve it."1630

Here are some tips that will help us to use this method.1633

First, draw pictures: this is probably the single most useful tip in all of this.1636

Draw pictures: it may not be possible for every problem--some problems are going to be completely abstract, and you won't be able to do this.1642

But when you can, it is going to help massively.1647

Basically, if a problem is asking about geometry, shapes, or something that is physically happening--1649

if something is a real-world thing that we can imagine--it is really useful to draw a picture.1655

If the problem doesn't give you a picture right away, draw it yourself--just draw a picture.1660

And don't worry about making the picture look nice--you just want something that you can sketch in a few seconds, and that makes sense to you.1664

Cars can become boxes; people can become little stick figures; you can change things to dots and just straight lines.1670

You don't have to draw an entire road, if you are talking about a road; it is enough to just have a street line.1675

You just want something that you can use as a reference point, so that you can see it in your mind's eye.1680

It is not absolutely necessary to do this; but a visual representation of a problem can help so much.1684

When you are not sure how something works or what to do, sometimes a quick sketch1689

(so we can see what is going on) can help clear things up so much--it can be really, really useful.1693

Drawing pictures is a great way to approach word problems.1697

It really helps us understand it, helps us figure out what we are looking for, helps us set up our equations, and it can help us solve it.1700

It is really, really useful stuff.1706

The next tip is breaking things into pieces: Imagine you have a plate in your hands.1709

You drop it, and it breaks into many pieces.1714

If you collect all of the pieces and you put them back together, you will have a plate again, right?1716

And if you pull apart the rebuilt plate, you will have all of the pieces again.1720

You can take something into the pieces that make it up, and you can put it back together to make the whole.1723

The whole is made out of its parts, and the parts make up the whole.1727

The sum of the parts equals the whole; the whole equals the sum of the parts.1730

That is what this idea is expressed by; it is a simple idea, but it is important, and it comes up in math a lot.1734

Consider this problem right here: if we want to figure out what the area of the shaded area is--1740

we want to figure out how much shaded area there is--by breaking this figure into its pieces, we can figure out how to do this.1745

We see that that shaded part and that semicircle--they come together to form a square.1752

Well, I know what the square is; I can figure out what the semicircle is.1757

And so, I can use that information to figure out the shaded area.1760

You can break things into their pieces; and this will normally be applied to geometrical things, where we are talking about shapes and objects.1763

But once in a while, it will actually be applied to mathematical things, where it is just in terms of letters and things.1769

And we will actually break things into their pieces then, and use each of the pieces on its own.1773

We will see those things later on, potentially; and we will talk about it then, if it happens.1778

But normally, you are going to end up seeing it in pictures, where we are dealing with shapes, geometry, and things like that.1782

But it is a really useful idea and comes up in a lot of different places.1789

Next, try out hypothetical numbers: sometimes, it can be hard to find out what is going on in a problem,1792

because we are not working with numbers--we are really good at working with numbers,1797

because we have been doing it for a long time at this point.1800

So, you can try plugging in hypothetical numbers to help you understand what is going on.1803

Now, if you try a hypothetical number, make sure you try out numbers that follow the rules set in your problem.1808

If your problem says that the number is even, you want to make sure whatever hypothetical number you use1813

is something like 2, 4, 6, 8, so that you are following the rules that they tell you, so that you can go along with that path.1818

This is a great way to test the equations you set up.1825

We can set up an equation and say, "Well, this should tell me how much money the company made."1827

And what we do is plug in some numbers and say, "Well, what if they made 10 widgets?"1832

Well, we could probably figure out what it should be for 10 widgets, and so we could make sure that our equation1838

is giving out the same thing that we know 10 widgets should make them, in terms of money.1842

And so, we can make sure that, yes, our equation seems to be checking out.1846

It is working when I use it on simple numbers, that I can easily understand.1849

So, now it is up and running, and we can use it on variables and trust it.1852

So, if you have difficulty setting up equations, try plugging in hypothetical numbers to understand what is going on in there.1857

It is easy to make mistakes when you are setting up equations, so check them afterwards by plugging in hypothetical numbers, if you are not sure.1862

You can plug in a number where you understand what should happen there.1868

And then you will see that that makes sense; and we will see some examples when we work on the examples part of this lesson.1871

If things don't work out with your hypothetical number, you know you made an error in your equation, or possibly an error in your hypothetical number.1877

And there is something you need to go back and work on; make sure you fix it before you move on and try to solve it.1882

Student logic: this is an interesting idea.1888

The hardest part often can be figuring out what relationships are going on.1891

But you have a secret weapon--you are a student.1896

What I mean by this is that you can use student logic: you can be pretty much certain that whatever your problem is going to be about,1899

it is going to use what you are currently learning.1906

If you are a student, they are not going to have you do something that you haven't learned.1909

And they are probably mainly going to be focusing on the things that you have been working on.1914

So, whatever you are currently learning is exactly what you will probably use to set up the relationships.1917

For example, if you are currently studying parabolas, you know that the problem1922

can almost certainly be solved by using something that you learned about parabolas.1925

So, you will set up your word problem, and then you will say, "Oh, right, what do I know about parabolas?"1929

"Oh, the top of a parabola happens at this equation," and so you will see if that would be useful here; and it probably would be.1932

Try to figure out how the problem is related to what you have recently been studying.1938

How is this connected to what you have been working on in this section, in this lesson, in this chapter?1941

And then, use those ideas to help you set up the equation.1946

You don't have to normally worry about things that are completely unrelated to what you have just been learning,1949

because your teacher is normally trying to teach you the things that you have been working on.1953

So, they are going to be using those ideas in the word problems, as well.1957

Jump in: this is a good suggestion--you don't want to just get paralyzed.1961

Working on word problems can sometimes freeze you, and you are thinking,1967

"I don't know what to do! I don't know what to do!" so just calm down and try something.1969

You are not sure where to start; you don't know exactly what you are looking for; you don't know how to solve it, so you get scared.1974

But instead--that is OK; that happens to everyone; instead, throw in something; try to do something.1979

If you can't set up the equation right the first time, that is OK.1985

If you pick the wrong variable, that is OK; if you plug in the wrong hypothetical number, that is OK.1987

You are probably going to learn from your mistake.1992

Normally, by making a mistake, we think, "Oh, well, that didn't work..."1995

Oh, but because that didn't work, this other thing would work...and then you are right on the way to doing it.1999

As long as you pay attention to what you are doing, and you think about what makes sense and what doesn't make sense,2003

and you are double-checking your work and thinking "Is this sane? Is this a reasonable thing to be doing?"--2007

you are normally going to see where you went wrong.2012

And by seeing where you went wrong, you can realize what you need to do in the problem.2015

The fastest way you can learn is often by just making a mistake.2018

So, reach in; get your hands dirty; it is OK if things go wrong at first, because eventually they will work out.2022

If you just stand back and keep thinking, "I don't know how to do this! I don't know how to do this!"2027

yes, you are right--you are not going to know how to do it, because you are just saying one thing; you are pulled back.2030

You need to try something; trying something is almost always going to work better than doing nothing.2035

So, just jump in and get started, if you get stuck.2040

All right, let's look at some examples: a four-part method: first, understand; 2) Set up variables; 3) Set up equations; 4) Solve the thing.2044

So, first, we need to understand this; #1--we read through it.2053

"Sally has a job selling cars. Her monthly base pay is $2,000, along with a 1.7% commission on all the cars she sells."2057 "If she earned$5,315 in March, what was the total cost of the cars she sold in March?"2065

So, the first thing we have to do is understand what is happening here.2070

Sally has a job selling cars--that makes sense.2073

She gets paid some base pay; she gets paid some amount, and then..."along with"...she gets paid $2000, plus some other thing, this commission.2076 Now, we might say, "I don't know what a commission means."2085 So, what do we do? We look it up! You put the word "commission" into an Internet search, or you look up "commission" in a dictionary.2088 It would probably be a good first step.2095 If you don't know what the word "commission" is, you look it up.2097 We look up "commission," and we find out that it is normally a percentage fee that you get for selling something.2099 So, if somebody has a 10% commission, if they sell$100 of things, they personally get $10 back.2105 So, a commission is a way of making a profit off of what you sell to other people.2112 A salesman gets a commission on their sales, generally, to encourage them to sell more.2116 So, she gets a 1.7% commission: 1.7% of whatever she sells, she gets back; OK.2120 So, she gets$2000, plus 1.7% of the amount of the cars that she sold in the month.2128

If she earns $5,315 in March, what was the total cost of the cars she sold in March?2134 Oh, so we have a piece of information about March; we are looking to know about the cost of that.2137 So, we are looking for a connection between how we know her money breaks down and what the cost of those cars must have been.2142 All right, so number 1 is done: we understand what it is about.2147 #2: What are the things that we need to know?2151 Well, we want to be able to talk about her pay--how much does she get paid?2153 So, we want to have some way of being able to talk about how that varied based on her commission.2158 How about we make c equal, not commission, but the cost of cars sold.2163 This is the amount of money she makes off of the cars she sells.2170 So now, let's figure out a way of being able to make this$2000, along with 1.7% commission, into something.2174

She gets $2000; so #2 is done--we figured out the things that we need to talk about, her pay and the amount of the cars that are sold.2180 Those are the two variables that we are really looking at.2188 #3: We set up equations--we want to have some way of being able to talk about her pay.2190$2000 + 1.7% of the cost of those cars that are sold...how does that work out?2195

You might say, "Well, I don't really remember how percent works," so let's test some things.2203

We know that 10% of 100 should end up being 10; we know that 5% of 100 should end up being 5.2207

We know that 1% of 100 should end up being 1; so, we might say, "Oh, right, you move the decimal 2 over, and then you multiply."2216

0.017; let's check and make sure that that makes sense.2224

If she sold $100 car (a cheap car, but...) if she were to sell$100, 1.7% of 100 would be one dollar and 70 cents.2228

So, she should make $1.70: .017...let's check that out: 0.017 times 100 would end up being...2240 we move the decimal 2 over...we get 1.7, or$1.70, which would be a dollar and 70 cents.2249

That makes sense, so .017 times the cost of the cars sold is equal to the amount that she gets paid in any given month.2258

Now, what month are we looking at specifically? March.2267

We know that p, in this case, is equal to $5,315.2269 So, we take this information; we plug it in here; we have 2000 + 0.017 times the cost of the cars = 5315.2275 From here, it is just step #4: we set up our equations; we know how the things interact.2286 We know all this; now we just have an equation to solve.2294 We want to know c; c is what we are looking for; it asks what was the total cost.2297 We are looking for c = ?, so we just solve for what c has to be.2302 Subtract 2000 from both sides; we get 0.017c = 3315.2308 And then, we divide both sides by .017; so we get c = 3315/0.017.2318 We plug that into a calculator to make it easy for us, and we get 195,000 dollars.2334 So, she makes$195,000 in terms of what she sells, which brings her a commission of $3315, so a total pay of$5315.2340

The total cost of all the cars she sold was $195,000; great.2350 Example 2: the first thing we want to do is understand what is going on.2355 The very first thing...we have a semicircle of radius 8 inscribed inside of a rectangle.2360 So, we might say, "What is a semicircle?" Well, we look at the picture--a semicircle is half of a circle.2364 Semi- means half; half of a circle makes sense.2370 So, it is radius 8; from center to edge is 8; great--it is inside of a rectangle, so it is inscribed.2373 We see exactly what it looks like; what is the area of the shaded portion?2380 We want to figure out how much is the shade in here.2384 We understand what we are looking for: we have half a circle held inside of a rectangle, so it barely touches the edges;2387 and we are looking to figure out what is the part that isn't the circle, but is still contained in the rectangle--what is that shaded portion.2393 So, #1 is done; #2--what would allow us to know this?2399 Well, if we knew what the area of the rectangle was (area of rectangle); if we know what the s for the shaded area;2403 and if we knew the circle's area; we would be in pretty good shape to be able to figure this out.2421 So, in #2, those are the three things we are looking for.2429 R equals the area of the rectangle; S equals the shaded area; C equals the circle's area.2432 #3: we start looking for some equations--can we figure out what the area of a rectangle is?2439 We say, "Oh, yes, it is length times width"; so R = length times width.2445 So now, let's go back, and let's look at our picture.2451 Well, if this is 8 from center to edge, look, over here is the edge of the circle; so this is 8;2453 over here is the edge of the circle, so this is 8; so the entire length is 16.2460 Vertically, you have...we go from here, and we go directly up; then this is 8 here, so it must be 8 on this side, as well.2469 So, we have length times width; we know that it is 16 times 8, which comes out to be 128.2478 So now, we know what the area of the rectangle is; great.2487 What about the area of the shaded area--do we know what that is?2490 No, that is what we are looking for--it is what the problem asked, so that is our question mark--it is our unknown.2494 C: can we figure out what is the semicircle's (let me write this out...it is the semicircle, not the circle,2498 because it is half of a circle)...C equals the semicircle's area.2504 Now, we say, "Well, what is the area for a circle?"2508 If I knew the area for a circle or a semicircle, I would be good.2512 Well, we probably remember that the area for a circle is πr2.2515 And even if we don't remember what the area for a circle is, we say, "I have learned this before; let's type it into an Internet search!"2518 You type in "area of circle"; you have it right there.2526 Or if you have a math book, you can possibly look in the cover, and it will already have that formula right there.2530 The area of a circle is πr2; so the circle equals πr2; the area for a circle is equal to πr2.2535 Now, notice: this is r, not this capital R; the area of our rectangle is very different than r.2547 What is little r here? We look once again, and try to remember--what is it?2553 Oh, right, it is the radius of the circle, from the center of the circle out.2556 The radius of our circle is 8, from the center of the circle out.2560 Now, there is one difference between the area of a circle and the area of our semicircle.2565 What is a semicircle? It is half of a circle, so it is 1/2 times πr2, so 1/2 times π times 82.2569 That equals 1/2 times π times 82...82 is 64, so that gets us 32π.2582 Now, that is 2 pieces of information: we know that r equals 128, and we know c equals 32π.2589 But we are still looking for this other thing: we have three unknowns to start with, R, S, and C.2598 And we have two pieces of information; so we need some way to connect S to R and C.2602 Well, we look at this, and we think, "Oh, the area of the rectangle, the shaded area, and the semicircle--they are all connected!"2606 The area of the rectangle contains both of the other ones, put together, so it is S + C.2612 So, at this point, we are ready to go on to step 4: we have all of our equations set up.2619 Step 4: R = S + C--we have numbers, so let's get S on its own, because that is what we are looking to solve.2623 So, R - C = S; the area of the shaded area is equal to the rectangle's area, minus the area of the semicircle.2629 So, we plug in the area for the rectangle; that is 128, minus the area for the semicircle--that is 32π; and that is equal our shaded area.2639 And that is our answer, because there is no way to combine 32π and 128--they speak different languages.2649 If we want, we could plug it into a calculator and get 32 times 3.14, and then get something.2654 But 128 - 32π...that is a great answer; there we are.2659 The next example: Tobias has precisely 17 coins in his pocket; so the first thing we are doing is understanding what is going on.2662 #1: Tobias has some coins in his pocket--that makes sense; 17--cool.2669 Coins come in three types, so he has three different coin types: quarters, nickels, and pennies.2674 Now, if we didn't remember, we would say, "Oh, what is a quarter?" and we would look it up; a quarter is 25 cents.2678 What is a nickel? 5 cents. What is a penny? 1 cent.2683 He has a total of 2 dollars and 17 cents in coins; if he has 2 pennies, how many quarters does he have?2687 Let's see...everything here makes sense; the kid has some change in his pocket.2694 They come in three different types of coins that make up that change.2699 And he has a total amount of money, and now we want to find out what the specific number of quarters is.2703 So, what might be useful to know? First, we are certainly going to need to know what is the number of quarters.2708 That is our very first thing; so here in step 2, the first unknown that we definitely have to figure out is number of quarters.2714 Well, we will probably also want to know how many nickels and how many pennies he has,2720 because they are connected to the other pieces of information right here.2723 n = number of nickels; and finally, p = number of pennies.2726 Those little reminders serve us to understand what is going on as we work through.2737 Now, #3: How can we connect these ideas together?2740 Now, we notice...well, how can I talk about the number of coins in his pocket?2744 Well, I know what the number of quarters is; I know what the number of nickels is; I know what the number of pennies is;2749 do I literally know what they are?--no, but I have names for them.2752 So, we can talk about...well, how can we say how many coins he has, in terms of these variables?2756 Well, he is going to have his number of quarters, plus his number of nickels, plus his numbers of pennies.2760 That should describe the amount of coins--the number of coins in his pocket.2766 Let's check and make sure: let's do a really quick check.2770 What if he had 2 quarters, one nickel, and one penny?2772 Then that is 2 quarters for q, 1 nickel, and 1 penny (p); so 2 + 1 + 1 is 4 coins, and he would have 4 coins.2776 Great; that makes sense; in this case, we are told that he had 17 coins, so q + n + p = 17.2784 What else were we told? We were also told that he had$2.17.2792

Oh, so how can we figure out how much money he has?2794

Well, if he had two quarters, that would be 50 cents.2797

Let's make things easy, and let's, in fact, instead of talking in terms of dollars...we will talk about 217 cents.2800

We will think just in terms of cents: if he has just one quarter, how many cents does he have? 25.2808

If he has two quarters, he has 50 cents; it is 25 times the number of quarters, so 25 times q is how much money, how many cents, he has from quarters.2812

What about nickels? Well, 5 times the number of nickels--let's check and make sure.2822

3 nickels would be 15 cents; 5 times 3 would be 15; it works out.2826

So, 5 times nickels, plus...what if we had pennies? Well, what number would we multiply by pennies?2830

Oh, pennies would just be by 1; so it is just the number of pennies.2837

That is going to be the amount of money in his pocket; in this case, we know how many cents he has--he has 217 cents, \$2.17.2841

2.17 dollars is the same thing as 217 cents; we are thinking about this in terms of cents.2850

That way, we just don't have decimal numbers.2855

Finally, do we have any other information?--because currently, we have three unknowns and three relationships.2857

Oh, yes, he has two pennies--so we know immediately that p = 2.2861

Three pieces of information; three unknowns; we are good to go.2868

So, #4: we start working things out--well, we are going to have to somehow use p = 2 in one of these two.2873

So, let's plug it into this one first: so q + n + 2 = 17; that means q + n = 17, or...2879

let's go with n; we will solve for n, because we want to work out q over here.2889

So, we will plug it in here, and we can keep working up through substitution.2892

n = 17 - q; great; we will take that information, and we plug it in over here.2896

So, 25q + 5...remember, we have to substitute with quantities...(17 - q) + p = 217.2903

Do we know what p is? Yes, we certainly know what p is; p is just 2.2917

So, 25q...we will distribute that 5...+ 5(17)...5 times 17 is...oops, I made a mistake here...2922

sorry about that; hopefully you caught that and were thinking, "What are you doing?"...2935

q + n...we subtracted 2 here; this is a good example of why we have to make sure we do the exact same thing on both sides.2940

15; 15; so this shouldn't be 17 - q over here; it should be 15 - q over here.2947

Great; so 25q + 5(15 - q)...we get 75 - 5q; and then, we will also subtract 2 right now, just to get rid of it.2954

So, we will get...equals 215; great, 25q + 75 - 5q.2964

Let's consolidate that: we will get 20q, and then let's also move the 75 over to the other side; -75, -75...2971

So, we get 20q =...215 - 75 is 140, so q is equal to 7.2979

The number of quarters he has in his pocket must be 7 quarters, because we were able to work things out from those original equations.2987

And if we wanted to figure out what the number of nickels he had was, we could just plug it right here, and we would get,2994

"Oh, he must also have 8 nickels"; so if n = 8 and p = 2, q = 7, we could do a really easy check.2998

Is 7(25) + 5(8) + 2 equal to 217; yes, it ends up being that that does check out.3007

If we multiply the number that the quarters should be worth and that and 7 + 8 + 2,3014

that equals 17, so everything checks out; our answer makes sense; q = 7; great.3019

Final example: we have a tank for holding water.3024

The first step we need to do: we need to understand what is going on.3027

The tank for holding water is shaped like a circular cone with its point towards the ground.3030

What does that look like? Oh, yes, OK...you draw something circular, and then it comes to a point, like a cone.3035

That makes sense; the tank is 10 feet tall, and has a diameter of 6 feet at the top.3045

Let's put that into this: it is 10 feet tall, and has a diameter that is 6 feet across.3049

What is the volume when the tank is full?3057

And then, we have a second part, which is "What is the volume when the water is only 5 feet deep?"3060

So, let's just start by breaking it here, and we will answer, "What is the volume in the tank when it is full?"3063

Our first thing makes sense: we have this cone full of water; we fill it up to the top; how much water is going to be in it?3068

The second part, second idea: we want to get what we need to know here.3076

It seems like it would be useful to talk about the height; well, we actually know what the height is.3081

Height h will equal the height of the cone; and let's say d equals...we were told the diameter, so let's say the diameter,3085

even though these are really just going to be values; we can talk about them as if they are like that.3094

And that seems like everything we need to know right now, so let's go and let's see...3099

do we have a good way to relate these two things together?3103

How can we relate the height and diameter of a cone to its volume?3106

When I think, "Well, how do I get the volume of a cone? I have learned this before--they told me in geometry..."3110

Right, it is this: we remember the formula--or maybe we don't remember the formula, like,3115

"Well, I know I have been told it..." if you have been told it, it is out there on the Internet, right?3120

Or it is in a math book; either crack open a math book, or do a quick web search, and you will be able to find it really quickly.3125

And you find out that the volume of a cone is equal to 1/3 times what the volume of its cylinder would have been.3132

The volume of its cylinder would have been the area of the top, πr2, the circle, times the height of the cylinder.3140

OK, now at this point, we think, "Wait a second; we are talking about volume; we want to know what volume is."3149

So, volume = ?; that is what we are really searching for.3155

But did we have r show up before? No, we didn't have r show up before, so let's make a new one: r = radius.3157

We were told some things here: we were told the diameter is 6 feet, so how does a radius connect to that?3165

Well, we say, "Oh, right, the diameter is just double the radius; the radius is half the diameter; so r equals 3."3171

Halfway would just be...if the whole thing is 6, then halfway is going to be 3; so r = 3.3179

Our h equals...our height was 10 feet tall; so h = 10; now we have only one unknown left in this equation here.3188

V = 1/3 times πr2 times height: we know what the radius is; we know what the height is;3198

π is just some number; 1/3 is just some number; the only thing we are looking for is the volume--really easy.3203

We just plug our numbers in at this point; so, the fourth step is just solving it.3207

Our volume is equal to...plugging in the values...1/3 times π times 32 times 10.3212

That is going to end up being...the 1/3 here will cancel the squaring that we have here.3220

So, we have π times 3 times 10, or 30π; and we have to talk about it in terms of units.3226

So, if it is volume, and we have been doing feet before, it must be cubic feet.3232

And there is our answer; all right.3237

Now, what if we wanted to this other portion of it?3240

We will change over to a new version; now, it is nice, because it is going to follow a lot of parallel ideas.3244

So, we can just use what we have already figured out.3251

Instead of being 10 feet tall, it is only 5 feet tall; so, the water really only comes up to here.3254

Now, if that is the case, if the diameter is 6 up here, is the diameter going to be the same down here?3261

No, that makes no sense: the diameter can't be the same, because it is a cone.3269

It shrinks down, the farther down we get.3274

So, if we were all the way at the bottom, the diameter would be 0; if we go all the way up to the top, it would be 10.3277

It makes sense that the diameter is going to be half of what it would have been before, because we are now at half the height.3280

So, the diameter in the middle is 3; so that means our radius is equal to 1.5; our height is 5; and that same formula from before,3286

our volume formula for a cylindrical cone, still works; so volume equals 1/3 times πr2 times height.3296

We plug in 1/3 times π times (1.5)2 times 5.3306

We work that out: 1/3 times π times 11.25 simplifies to 3.75π; and it is in cubic feet.3316

And there is our answer: we are basically following the same outline we did before.3335

So, we don't have to worry about doing it step-by-step, because we can just work from our previous idea.3341

We figured out how it was done in the complex way; now it is just a matter of using new values3346

and making sure that the values we are using are right.3350

r changes, because we are at a different place; height changes because we are at a different place.3353

But all of the relationships are still the same, which makes sense--we want to be thinking about it,3357

but it makes sense that all of the relationships are the same, because we are still just looking3360

to figure out what is the volume inside of this cone.3364

It is just now a cone inside of a cone.3367

All right, I hope that made sense; I hope that gives you a slightly better understanding of how to approach word problems.3371

Don't be that scared by them; it is just a matter of breaking it down and understanding what is going on.3376

setting up what you are looking for, figuring out the relationships that connect3380

what you are looking for to what you know, and then finally just solving it like a normal math problem.3384

All right, we will see you at Educator.com later--goodbye!3389