College Algebra (Math 110) Video Course

Listing of the Videos and Their Contents

College Algebra (Math 110) - Recorded in 1998
Video Professor:
Richard Delaware

• No reference on the videos is made to a particular text, although the course is based on the instructor's experience and the text: College Algebra Enhanced with Graphing Utilities, by Sullivan & Sullivan, 1st Ed., (1996), Prentice Hall.
• Times are approximate.
• There are many internal "Stop Tape" pauses for students to work problems or review.

Welcome to the Course
Unit 0 - BASICS: Remembrance of Things Past
Unit 1 - GRAPHS
Unit 2 - FUNCTIONS AND THEIR GRAPHS
Unit 3 - EQUATIONS & INEQUALITIES
Unit 4 - POLYNOMIAL & RATIONAL FUNCTIONS
Unit 5 - EXPONENTIAL & LOGARITHMIC FUNCTIONS
Unit 6 - SYSTEMS OF EQUATIONS
Unit 7 - SOME DISCRETE TOPICS
The X-Y Files: The Proof is in Here - Several Proofs and other topics keyed to the previous Units

Welcome to the Course (Accidentally omitted from the tapes)

Welcome to College Algebra.
G.H. Hardy, a famous 19th Century mathematician wrote:

"A mathematician, like a painter or poet, is a maker of patterns.
If his patterns are more permanent than theirs, it is because they are made with ideas..."

This will be a course with emphasis on IDEAS. An idea, like a picture, is worth 1000 words!
With the ideas clear, language & techniques become easier to remember;
they hang together in your mind.

This course will prepare you for further courses, such as
Analytic Geometry, Precalculus, and Calculus.

Please note: A graphing device is appropriate for this course, and will be discussed.
However, I expect YOU to master the basic features of your device yourself.

My recommendations to you:

Replay any tape segment whenever you feel it's necessary!
This is the great advantage of a course on video.

Pay attention to the language I use in speaking about Algebra.
Try to become fluent in this language yourself.

Adopt an organized style of work, as I do.
You need not mimic me exactly, but learn to leave a clear "paper trail",
especially when you use a graphing device.

Don't get behind! Work lots of problems.

Now it's time we roll up our sleeves and get down to the nuts-and-bolts of Algebra!

NUMBERS [1 hr. 14 min.]
• Sets of Objects [4 min.]
• Natural Numbers [7 min.]
• Integers [5 min.]
• Rational Numbers [13 min.]
• Irrational Numbers [2 min.]
• Real Numbers [1 min.]
• The Real Line: From Numbers to Points [4 min.]
• Real Numbers are Ordered [19 min,]
• The Real Line: Distance Between Points [8 min.]
• Real Numbers as Decimals [11 min.]
THE LANGUAGE OF MATHEMATICS [27 min.]
• Learning to Read Mathematics [7 min.]
• Some Symbols of Algebra [9 min.]
• Nouns, Pronouns, & The Main Verb of Algebra [4 min.]
• Theorems, Corollaries, Lemmas, and All That [7 min.]
THE POWERS THAT BE: EXPONENTS [1 hr. 29 min.]
• Integer Exponents [12 min.]
• Operations with Integer Exponents [31 min.]
• Square Roots: A Pair Of Equal Factors [13 min.]
• Nth Roots & Rational Exponents [12 min.]
• Operations with Rational Exponents [21 min.]
POLYNOMIAL EXPRESSIONS [3 hr. 2 min.]
• What is a Polynomial? [18 min.]
• Adding & Subtracting Polynomials [3 min.]
• Multiplying Polynomials [19 min.]
• A Common Error [5 min.]
• Handy Polynomial Products [15 min.]
• Un-Multiplying (Factoring) Polynomials [28 min.]
• Completing A Perfect Square [10 min.]
• Dividing Polynomials: Rational Expressions [24 min.]
• The Art of Simplification [14 min.]
• Solving Some Polynomial & Rational Equations [46 min.]
MORE NUMBERS [19 min.]
• Beyond Real Numbers: Complex Numbers [19 min.]
A LITTLE GEOMETRY [20 min.]
• Some Area Formulas [8 min.]
• The Pythagorean Theorem & A Visual Proof [12 min.]

UNIT 1 - GRAPHS [3 hr. 53 min.]

GRAPHS [3 hr. 53 min.]
• Rectangular Coordinates: Geometry Meets Algebra [11 min.]
• Distance Between Points [18 min.]
• Midpoint of a Line Segment [7 min.]
• Your Graphing Device [17 min.]
• Graphs of Equations [25 min.]
• Intercepts: Crossing the Axes [12 min.]
• Symmetry of Graphs [14 min.]
• Lines: Defining "Slope" [34 min.]
• Lines & Their Equations [24 min.]
• Parallel & Perpendicular Lines [26 min.]
• Circles & Their Equations [28 min.]
• Some Exercises Explained [16 min.]

UNIT 2 - FUNCTIONS AND THEIR GRAPHS [7 hr. 2 min.]

FUNCTIONS AND THEIR GRAPHS [7 hr. 2 min]
• Function: The Central Idea of Mathematics [26 min.]
• The Language & Notation of Functions [20 min.]
• More On Domains [19 min.]
• Function Notation Practice [16 min.]
• Visualizing Functions: Graphs of (x, f(x)) Pairs [26 min.]
• Increasing & Decreasing Functions [23 min.]
• Local Maximums & Local Minimums [18 min.]
• Even & Odd Functions [20 min.]
• A Library of Important Functions [20 min.]
• Piecewise Defined Functions [19 min.]
• Some Exercises Explained [11 min.]
• Graphing Techniques: Vertical & Horizontal Shifts [28 min.]
• Graphing Techniques: Compressions & Stretches [27 min.]
• Graphing Techniques: Reflections Across the Axes [17 min.]
• Putting It All Together: Moving The Graphs of Functions Around The Plane [36 min.]
• The Algebra of Functions [16 min.]
• A New Operation Unique To Functions: Composition [24 min.]
• Mathematical Models of Real World Problems: Constructing Functions [55 min.]

UNIT 3 - EQUATIONS & INEQUALITIES [7 hr. 16 min.]

EQUATIONS IN 1 VARIABLE [4 hr. 38 min.]
• Solving Equations (Approximately) with a Graphing Device & The Intermediate Value Theorem [44 min.]
• Solving Linear Equations: The "Linear Formula" or, Graphing [23 min.]
• Solving Non-Linear Equations That Lead To Linear Equations [13 min.]
• Solving Quadratic Equations: Factoring, or, Graphing [11 min.]
• A Complex Reminder & The Principal Square Root of a Negative Number [12 min.]
• Solving Quadratic Equations: The "Quadratic Formula" & The Discriminant or, Graphing [50 min.]
• Some Linear & Quadratic Equation Exercises Explained [24 min.]
• Setting Up Equations: More Mathematical Models [65 min.]
• Solving "Radical" Equations [14 min.]
• Solving Equations "Quadratic in Form" [15 min.]
• Solving Factorable Equations [7 min.]
INEQUALITIES IN 1 VARIABLE [1 hr. 54 min.]
• Properties of Inequalities [20 min.]
• Solving Inequalities In General [17 min.]
• Solving Linear Inequalities [19 min.]
• Solving Quadratic Inequalities [17 min.]
• Solving Higher-Degree Polynomial Inequalities [14 min.]
• Solving Rational Inequalities [27 min.]
EQUATIONS & INEQUALITIES IN 1 VARIABLE [44 min.]
• When Absolute Value Appears: Equations [12 min.]
• When Absolute Value Appears: Inequalities [17 min.]
• More Exercises Explained [15 min.]

UNIT 4 - POLYNOMIAL & RATIONAL FUNCTIONS [5 hr. 23 min.]

POLYNOMIAL FUNCTIONS [2 hr. 30 min.]
• Degree 2: Quadratic Functions [19 min.]
• Graphing Quadratic Functions [39 min.]
• Quadratic Functions As Mathematical Models [25 min.]
• Degree "n": General Polynomial Functions [8 min.]
• Special Case: Power Functions & Their Graphs [18 min.]
• Graphing General Polynomial Functions: Zeros, Multiplicity, Turning Points & End Behavior [40 min.]
LOCATING THE ZEROS OF A POLYNOMIAL FUNCTION [1 hr. 28 min.]
• How Many Zeros Are There? [12 min.]
• How Many Zeros Are Real? [13 min.]
• How Many Real Zeros Are Positive? Negative? [16 min.]
• Where (On What Interval) Are All Real Zeros? [22 min.]
• How Can You Guess The Location of Real Zeros? [4 min.]
• How Can You Reduce The Number of Real Zeros? [21 min.]
• Strategy & Tools: A Practical Checklist [14 min.]
• Some Polynomial Exercises Explained [22 min.]
RATIONAL FUNCTIONS [1 hr. 25 min.]
• General Rational Functions [14 min.]
• What Is An Asymptote? [17 min.]
• Finding Asymptotes of Rational Functions [35 min.]
• Graphing Rational Functions [19 min.]

UNIT 5 - EXPONENTIAL & LOGARITHMIC FUNCTIONS [3 hr. 42 min.]
EXPONENTIAL FUNCTIONS [55 min.]

• One-To-One Functions [14 min.]
• Exponential Functions & Their Graphs [26 min.]
• The Natural Exponential Function [15 min.]
LOGARITHMIC FUNCTIONS [1 hr. 59 min.]
• Inverse Functions [23 min.]
• Logarithmic Functions & Their Graphs [24 min.]
• The Natural Logarithmic Function [4 min.]
• Properties of Logarithms [17 min.]
• All Logarithms are Natural (or Common)! [15 min.]
• Solving Logarithmic Equations [20 min.]
• Logarithmic Models: Sound (Loudness) & Fury (Earthquakes) [16 min.]
EXPONENTIAL FUNCTIONS (con.) [47 min.]
• Solving Exponential Equations [15 min.]
• Exponential Models: Compounded Interest, and Growth & Decay [32 min.]

UNIT 6 - SYSTEMS OF EQUATIONS [2 hr. 44 min.]

SYSTEMS OF LINEAR EQUATIONS [2 hr. 29 min.]
• Systems of Linear Equations in General [19 min.]
• Solving A System of 2 (or 3) Linear Equations in 2 (or 3) Variables: Substitution [20 min.]
• Solving A System of 2 (or 3) Linear Equations in 2 (or 3) Variables: Elimination [36 min.]
• Some Exercises Explained [31 min.]
• An Application: Writing Proper Rational Functions As Sums of Simpler Proper Rational Functions (Partial Fractions) [43 min.]
SYSTEMS OF NON-LINEAR EQUATIONS [14 min]
• Solving (Mostly Graphically) a System of 2 Non-Linear Equations in 2 Variables [14 min.]

UNIT 7 - SOME DISCRETE TOPICS [3 hr. 40 min.]

SEQUENCES [1 hr. 41 min.]
• Infinite Sequences: Functions With Domain N [28 min.]
• The Factorial Symbol: ! [8 min.]
• Adding The First n Terms of a Sequence: nth Partial Sums & Summation Notation [17 min.]
• Geometric Sequences: Multiplying Your Way to Infinity [30 min.]
SERIES & INDUCTION [56 min.]
• Geometric Series & Their (Infinite) Sums [26 min.]
• The Principle of Mathematical Induction [30 min.]
THE BINOMIAL THEOREM [1 hr. 2 min.]
• The "Binomial Coefficient" Symbol [11 min.]
• Pascal's Triangle [19 min.]
• The Binomial Theorem: How to Expand (x+a)n [32 min.]

THE X-Y FILES: The Proof is in Here [1 hr. 27 min.]

FOR UNIT 0 [29 min.]
• A Proof That Ö2 Is Irrational [6 min.]
• A Proof That There Are The Same Number of Rational Numbers as Natural Numbers! [9 min.]
• A Proof That There Are More Real Numbers Than Natural Numbers! [14 min.]
FOR UNIT 2 [17 min.]
• Algebra For Science: Variation [17 min.]
FOR UNIT 4 [21 min.]
• How Can You Find Rational Zeros (if any) of a Polynomial Function? [21 min.]
FOR UNIT 7 [19 min.]
• A Proof That "e" Is Irrational [19 min.]