36
Lectures
30
minutes/lecture
1.
An Introduction to the Course
Professor Sellers introduces the general topics and themes for the course, describing his approach and recommending a strategy for making the best use of the lessons and supplementary workbook. Warm up with some simple problems that demonstrate signed numbers and operations.
1.
An Introduction to the Course
|
19.
Factoring Trinomials
Begin to find solutions for quadratic equations, starting with the FOIL technique in reverse to find the binomial factors of a quadratic trinomial (a binomial expression consists of two terms, a trinomial of three). Professor Sellers explains the tricks of factoring such expressions, which is a process almost like solving a mystery.
19.
Factoring Trinomials
|
2.
Order of Operations
The order in which you do simple operations of arithmetic can make a big difference. Learn how to solve problems that combine adding, subtracting, multiplying, and dividing, as well as raising numbers to various powers. These same concepts also apply when you need to simplify algebraic expressions, making it critical to master them now.
2.
Order of Operations
|
20.
Quadratic Equations—Factoring
In some circumstances, quadratic expressions are given in a special form that allows them to be factored quickly. Focus on two such forms: perfect square trinomials and differences of two squares. Learning to recognize these cases makes factoring easy.
20.
Quadratic Equations—Factoring
|
3.
Percents, Decimals, and Fractions
Continue your study of math fundamentals by exploring various procedures for converting between percents, decimals, and fractions. Professor Sellers notes that it helps to see these procedures as ways of presenting the same information in different forms.
3.
Percents, Decimals, and Fractions
|
21.
Quadratic Equations—The Quadratic Formula
For those cases that defy simple factoring, the quadratic formula provides a powerful technique for solving quadratic equations. Discover that this formidable-looking expression is not as difficult as it appears and is well worth committing to memory. Also learn how to determine if a quadratic equation has no solutions.
21.
Quadratic Equations—The Quadratic Formula
|
4.
Variables and Algebraic Expressions
Advance to the next level of problem solving by using variables as the building blocks to create algebraic expressions, which are combinations of mathematical symbols that might include numbers, variables, and operation symbols. Also learn some tricks for translating the language of problems (phrases in English) into the language of math (algebraic expressions).
4.
Variables and Algebraic Expressions
|
22.
Quadratic Equations—Completing the Square
After learning the definition of a function, investigate an additional approach to solving quadratic equations: completing the square. This technique is very useful when rewriting the equation of a quadratic function in such a way that the graph of the function is easily sketched.
22.
Quadratic Equations—Completing the Square
|
5.
Operations and Expressions
Discover that by following basic rules on how to treat coefficients and exponents, you can reduce very complicated algebraic expressions to much simpler ones. You start by using the commutative property of multiplication to rearrange the terms of an expression, making combining them relatively easy.
5.
Operations and Expressions
|
23.
Representations of Quadratic Functions
Drawing on your experience solving quadratic functions, analyze the parabolic shapes produced by such functions when represented on a graph. Use your algebraic skills to determine the parabola's vertex, its x and y intercepts, and whether it opens in an upward "cup" or downward in a "cap."
23.
Representations of Quadratic Functions
|
6.
Principles of Graphing in 2 Dimensions
Using graph paper and pencil, begin your exploration of the coordinate plane, also known as the Cartesian plane. Learn how to plot points in the four quadrants of the plane, how to choose a scale for labeling the x and y axes, and how to graph a linear equation.
6.
Principles of Graphing in 2 Dimensions
|
24.
Quadratic Equations in the Real World
Quadratic functions often arise in real-world settings. Explore a number of problems, including calculating the maximum height of a rocket and determining how long an object dropped from a tree takes to reach the ground. Learn that in finding a solution, graphing can often help.
24.
Quadratic Equations in the Real World
|
7.
Solving Linear Equations, Part 1
In this lesson, work through simple one- and two-step linear equations, learning how to isolate the variable by different operations. Professor Sellers also presents a word problem involving a two-step equation and gives tips for how to solve it.
7.
Solving Linear Equations, Part 1
|
25.
The Pythagorean Theorem
Because it involves terms raised to the second power, the famous Pythagorean theorem, a2 + b2 = c2, is actually a quadratic equation. Discover how techniques you have previously learned for analyzing quadratic functions can be used for solving problems involving right triangles.
25.
The Pythagorean Theorem
|
8.
Solving Linear Equations, Part 2
Investigating more complicated examples of linear equations, learn that linear equations fall into three categories. First, the equation might have exactly one solution. Second, it might have no solutions at all. Third, it might be an identity, which means every number is a solution.
8.
Solving Linear Equations, Part 2
|
26.
Polynomials of Higher Degree
Most of the expressions you've studied in the course so far have been polynomials. Learn what characterizes a polynomial and how to recognize polynomials in both algebraic functions and in graphical form. Professor Sellers defines several terms, including the degree of an equation, the leading coefficient, and the domain.
26.
Polynomials of Higher Degree
|
9.
Slope of a Line
Explore the concept of slope, which for a given straight line is its rate of change, defined as the rise over run. Learn the formula for calculating slope with coordinates only, and what it means to have a positive, negative, and undefined slope.
9.
Slope of a Line
|
27.
Operations and Polynomials
Much of what you've learned about linear and quadratic expressions applies to adding, subtracting, multiplying, and dividing polynomials. Discover how the FOIL operation can be extended to multiplying large polynomials, and a version of long division works for dividing one polynomial by another.
27.
Operations and Polynomials
|
10.
Graphing Linear Equations, Part 1
Use what you've learned about slope to graph linear equations in the slope-intercept form, y = mx + b, where m is the slope, and b is the y intercept. Experiment with examples in which you calculate the equation from a graph and from a table of pairs of points.
10.
Graphing Linear Equations, Part 1
|
28.
Rational Expressions, Part 1
When one polynomial is divided by another, the result is called a rational function because it is the ratio of two polynomials. These functions play an important role in algebra. Learn how to add and subtract rational functions by first finding their common divisor.
28.
Rational Expressions, Part 1
|
11.
Graphing Linear Equations, Part 2
A more versatile approach to writing the equation of a line is the point-slope form, in which only two points are required, and neither needs to intercept the y axis. Work through several examples and become comfortable determining the equation using the line and the line using the equation
11.
Graphing Linear Equations, Part 2
|
29.
Rational Expressions, Part 2
Continuing your exploration of rational expressions, try your hand at multiplying and dividing them. The key to solving these complicated-looking equations is to proceed one step at a time. Close the lesson with a problem that brings together all you've learned about rational functions.
29.
Rational Expressions, Part 2
|
12.
Parallel and Perpendicular Lines
Apply what you've discovered about equations of lines to two very special types of lines: parallel and perpendicular. Learn how to tell if lines are parallel or perpendicular from their equations alone, without having to see the lines themselves. Also try your hand at word problems that feature both types of lines.
12.
Parallel and Perpendicular Lines
|
30.
Graphing Rational Functions, Part 1
Examine the distinctive graphs formed by rational functions, which may form vertical or horizontal curves that aren't even connected on a graph. Learn to identify the intercepts and the vertical and horizontal asymptotes of these fascinating curves.
30.
Graphing Rational Functions, Part 1
|
13.
Solving Word Problems with Linear Equations
Linear equations reflect the behavior of real-life phenomena. Practice evaluating tables of numbers to determine if they can be represented as linear equations. Conclude with an example about the yearly growth of a tree. Does it increase in size at a linear rate?
13.
Solving Word Problems with Linear Equations
|
31.
Graphing Rational Functions, Part 2
Sketch the graphs of several rational functions by first calculating the vertical and horizontal asymptotes, the x and y intercepts, and then plotting several points in the function. In the final exercise, you must simplify the expression in order to extract the needed information.
31.
Graphing Rational Functions, Part 2
|
14.
Linear Equations for Real-World Data
Investigating more real-world applications of linear equations, derive the formula for converting degrees Celsius to Fahrenheit; determine the boiling point of water in Denver, Colorado; and calculate the speed of a rising balloon and the time for an elevator to descend to the ground floor.
14.
Linear Equations for Real-World Data
|
32.
Radical Expressions
Anytime you see a root symbol—for example, the symbol for a square root—then you're dealing with what mathematicians call a radical. Learn how to simplify radical expressions and perform operations on them, such as multiplication, division, addition, and subtraction, as well as combinations of these operations.
32.
Radical Expressions
|
15.
Systems of Linear Equations, Part 1
When two lines intersect, they form a system of linear equations. Discover two methods for finding a solution to such a system: by graphing and by substitution. Then try out a real-world example, involving a farmer who wants to plant different crops in different proportions.
15.
Systems of Linear Equations, Part 1
|
33.
Solving Radical Equations
Discover how to solve equations that contain radical expressions. A key step is isolating the radical term and then squaring both sides. As always, it's important to check the solution by plugging it into the equation to see if it makes sense. This is especially true with radical equations, which can sometimes yield extraneous, or invalid, solutions.
33.
Solving Radical Equations
|
16.
Systems of Linear Equations, Part 2
Expand your tools for solving systems of linear equations by exploring the method of solving by elimination. This technique allows you to eliminate one variable by performing addition, subtraction, or multiplication on both sides of an equation, allowing a straightforward solution for the remaining variable.
16.
Systems of Linear Equations, Part 2
|
34.
Graphing Radical Functions
In previous lessons, you moved from linear, quadratic, and rational functions to the graphs that display them. Now do the same with radical functions. For these, it's important to pay attention to the domain of the functions to ensure that negative values are not introduced beneath the root symbol.
34.
Graphing Radical Functions
|
17.
Linear Inequalities
Shift gears to consider linear inequalities, which are mathematical expressions featuring a less than sign or a greater than sign instead of an equal sign. Discover that these kinds of problems have some very interesting twists, and they come up frequently in business applications.
17.
Linear Inequalities
|
35.
Sequences and Pattern Recognition, Part 1
Pattern recognition is an important and fascinating mathematical skill. Investigate two types of number patterns: geometric sequences and arithmetic sequences. Learn how to analyze such patterns and work out a formula that predicts any term in the sequence
35.
Sequences and Pattern Recognition, Part 1
|
18.
An Introduction to Quadratic Polynomials
Transition to a more complex type of algebraic expression, which incorporates squared terms and is therefore known as quadratic. Learn how to use the FOIL method (first, outer, inner, last) to multiply linear terms to get a quadratic expression.
18.
An Introduction to Quadratic Polynomials
|
36.
Sequences and Pattern Recognition, Part 2
Conclude the course by examining more types of number sequences, discovering how rich and enjoyable the mathematics of pattern recognition can be. As in previous lessons, employ your reasoning skills and growing command of algebra to find order—and beauty—where once all was a confusion of numbers.
36.
Sequences and Pattern Recognition, Part 2
|
36
Lectures
30
minutes/lecture
1.
An Introduction to Algebra II
Professor Sellers explains the topics covered in the course, the importance of algebra, and how you can get the most out of these lessons. You then launch into the fundamentals of algebra by reviewing the order of operations and trying your hand at several problems.
1.
An Introduction to Algebra II
|
19.
An Introduction to Polynomials
Pause to examine the nature of polynomials—a class of algebraic expressions that you've been working with since the beginning of the course. Professor Sellers introduces several useful concepts, such as the standard form of polynomials and their degree, domain, range, and leading coefficients.
19.
An Introduction to Polynomials
|
2.
Solving Linear Equations
Explore linear equations, starting with one-step equations and then advancing to those requiring two or more steps to solve. Next, apply the distributive property to simplify certain problems, and then learn about the three categories of linear equations.
2.
Solving Linear Equations
|
20.
Graphing Polynomial Functions
Deepen your insight into polynomial functions by graphing them to see how they differ from non-polynomials. Then learn how the general shape of the graph can be predicted from the highest exponent of the polynomial, known as its degree. Finally, explore how other terms in the function also affect the graph.
20.
Graphing Polynomial Functions
|
3.
Solving Equations Involving Absolute Values
Taking your knowledge of linear equations a step further, look at examples involving absolute values, which can be thought of as a distance on a number line, always expressed as a positive value. Use your critical-thinking skills to recognize absolute value problems that have limited or no solutions.
3.
Solving Equations Involving Absolute Values
|
21.
Combining Polynomials
Switch from graphs to the algebraic side of polynomial functions, learning how to combine them in many different ways, including addition, subtraction, multiplication, and even long division, which is easier than it seems. Discover which of these operations produce new polynomials and which do not.
21.
Combining Polynomials
|
4.
Linear Equations and Functions
Moving into the visual realm, learn how linear equations are represented as straight lines on graphs using either the slope-intercept or point-slope forms of the function. Next, investigate parallel and perpendicular lines and how to identify them by the value of their slopes.
4.
Linear Equations and Functions
|
22.
Solving Special Polynomial Equations
Learn how to solve polynomial equations where the degree is greater than two by turning them into expressions you already know how to handle. Your "toolbox" includes techniques called the difference of two squares, the difference of two cubes, and the sum of two cubes.
22.
Solving Special Polynomial Equations
|
5.
Graphing Essentials
Reversing the procedure from the previous lesson, start with an equation and draw the line that corresponds to it. Then test your knowledge by matching four linear equations to their graphs. Finally, learn how to rewrite an equation to move its graph up, down, left, or right—or flip it entirely.
5.
Graphing Essentials
|
23.
Rational Roots of Polynomial Equations
Going beyond the approaches you've learned so far, discover how to solve polynomial equations by applying two powerful tools for finding rational roots: the rational roots theorem and the factor theorem. Both will prove very useful in succeeding lessons.
23.
Rational Roots of Polynomial Equations
|
6.
Functions—Introduction, Examples, Terminology
Functions are crucially important not only for algebra, but for precalculus, calculus, and higher mathematics. Learn the definition of a function, the notation, and associated concepts such as domain and range. Then try out the vertical line test for determining whether a given curve is a graph of a function.
6.
Functions—Introduction, Examples, Terminology
|
24.
The Fundamental Theorem of Algebra
Explore two additional tools for identifying the roots of polynomial equations: Descartes' rule of signs, which narrows down the number of possible positive and negative real roots; and the fundamental theorem of algebra, which gives the total of all roots for a given polynomial.
24.
The Fundamental Theorem of Algebra
|
7.
Systems of 2 Linear Equations, Part 1
Practice solving systems of two linear equations by graphing the corresponding lines and looking for the intersection point. Discover that there are three possible outcomes: no solution, infinitely many solutions, and exactly one solution.
7.
Systems of 2 Linear Equations, Part 1
|
25.
Roots and Radical Expressions
Shift gears away from polynomials to focus on expressions involving roots, including square roots, cube roots, and roots of higher degrees—all known as radical expressions. Practice multiplying, dividing, adding, and subtracting a wide variety of radical expressions.
25.
Roots and Radical Expressions
|
8.
Systems of 2 Linear Equations, Part 2
Explore two other techniques for solving systems of two linear equations. First, the method of substitution solves one of the equations and substitutes the result into the other. Second, the method of elimination adds or subtracts the equations to see if a variable can be eliminated.
8.
Systems of 2 Linear Equations, Part 2
|
26.
Solving Equations Involving Radicals
Drawing on your experience with roots and radicals from the previous lesson, try your hand at solving equations with these expressions. Begin by learning how to manipulate rational, or fractional, exponents. Then practice with simple equations, while being on the lookout for extraneous, or "imposter," solutions.
26.
Solving Equations Involving Radicals
|
9.
Systems of 3 Linear Equations
As the number of variables increases, it becomes unwieldy to solve systems of linear equations by graphing. Learn that these problems are not as hard as they look and that systems of three linear equations often yield to the strategy of successively eliminating variables.
9.
Systems of 3 Linear Equations
|
27.
Graphing Power, Radical, and Root Functions
Using graph paper, experiment with curves formed by simple radical functions. First, determine the domain of the function, which tells you the general location of the graph on the coordinate plane. Then, investigate how different terms in the function alter the graph in predictable ways.
27.
Graphing Power, Radical, and Root Functions
|
10.
Solving Systems of Linear Inequalities
Make the leap into systems of linear inequalities, where the solution is a set of values on one side or another of a graphed line. An inequality is an assertion such as "less than" or "greater than," which encompasses a range of values.
10.
Solving Systems of Linear Inequalities
|
28.
An Introduction to Rational Functions
Shift your focus to graphs of rational functions—functions that are the ratio of two polynomials. These graphs are more complicated than those from the previous lesson, but their general characteristics can be quickly determined by calculating the domain, the x- and y-intercepts, and the vertical and horizontal asymptotes.
28.
An Introduction to Rational Functions
|
11.
An Introduction to Quadratic Functions
Begin your investigation of quadratic functions by visualizing what these functions look like when graphed. They always form a U-shaped curve called a parabola, whose location on the coordinate plane can be predicted based on the individual terms of the equation.
11.
An Introduction to Quadratic Functions
|
29.
The Algebra of Rational Functions
Combine rational functions using addition, subtraction, multiplication, division, and composition. The trick is to start each problem by putting the expressions in factored form, which makes the calculations go more smoothly. Leaving the answer in factored form also allows other operations, such as graphing, to be easily performed.
29.
The Algebra of Rational Functions
|
12.
Quadratic Equations—Factoring
One of the most important skills related to quadratics is factoring. Review the basics of factoring, and learn to recognize a very useful special case known as the difference of two squares. Close by working on a word problem that translates into a quadratic equation.
12.
Quadratic Equations—Factoring
|
30.
Partial Fractions
Now that you know how to add rational expressions, try the opposite procedure of splitting a more complicated rational expression into its component parts. Called partial fraction decomposition, this approach is a topic in introductory calculus and is used for solving a wide range of more advanced math problems.
30.
Partial Fractions
|
13.
Quadratic Equations—Square Roots
The square root approach to solving quadratic equations works not just for perfect squares, such as 3 × 3 = 9, but also for values that don't seem to involve squares at all. Probe the idea behind this technique, and also venture into the strange world of complex numbers.
13.
Quadratic Equations—Square Roots
|
31.
An Introduction to Exponential Functions
Exponential functions are important in real-world applications involving growth and decay rates, such as compound interest and depreciation. Experiment with simple exponential functions, exploring such concepts as the base, growth factor, and decay factor, and how different values for these terms affect the graph of the function.
31.
An Introduction to Exponential Functions
|
14.
Completing the Square
Turn a quadratic equation into an easily solvable form that includes a perfect square—a technique called completing the square. An important benefit of this approach is that the rewritten form gives the coordinates for the vertex of the parabola represented by the equation.
14.
Completing the Square
|
32.
An Introduction to Logarithmic Functions
Plot a logarithmic function on the coordinate plane to see how it is the mirror image of a corresponding exponential function. Just like a mirror image, logarithms can be disorienting at first; but by studying their properties you will discover how they make certain calculations much simpler.
32.
An Introduction to Logarithmic Functions
|
15.
Using the Quadratic Formula
When other approaches fail, one tool can solve every quadratic equation: the quadratic formula. Practice this formula on a wide range of problems, learning how a special expression called the discriminant immediately tells how many real-number solutions the equation has.
15.
Using the Quadratic Formula
|
33.
Uses of Exponential and Logarithmic Functions
Delve deeper into exponential and logarithmic functions with the goal of solving a typical financial investment problem using the "Pert" formula. To prepare, study the change of base formula for logarithms and the special function of the base called e.
33.
Uses of Exponential and Logarithmic Functions
|
16.
Solving Quadratic Inequalities
Extending the exercises on inequalities from lecture 10, step into the realm of quadratic inequalities, where the boundary graph is not a straight line but a parabola. Use your skills analyzing quadratic expressions to sketch graphs quickly and solve systems of quadratic inequalities.
16.
Solving Quadratic Inequalities
|
34.
The Binomial Theorem
Pascal's triangle is a famous triangular array of numbers that corresponds to the coefficients of binomials of different powers. In a lesson connecting a branch of mathematics called combinatorics with algebra, investigate the formula for each value in Pascal's triangle, the factorial function, and the binomial theorem.
34.
The Binomial Theorem
|
17.
Conic Sections—Parabolas and Hyperbolas
Delve into the algebra of conic sections, which are the cross-sectional shapes produced by slicing a cone at different angles. In this lesson, study parabolas and hyperbolas, which differ in how many variable terms are squared in each. Also learn how to sketch a hyperbola from its equation.
17.
Conic Sections—Parabolas and Hyperbolas
|
35.
Permutations and Combinations
Continue your study of the link between combinatorics and algebra by using the factorial function to solve problems in permutations and combinations. For example, what are all the permutations of the letters a, b, c? And how many combinations of four books are possible when you have six to choose from?
35.
Permutations and Combinations
|
18.
Conic Sections—Circles and Ellipses
Investigate the algebraic properties of the other two conic sections: ellipses and circles. Ellipses resemble stretched circles and are defined by their major and minor axes, whose ratio determines the ellipse's eccentricity. Circles are ellipses whose eccentricity = 1, with the major and minor axes equal.
18.
Conic Sections—Circles and Ellipses
|
36.
Elementary Probability
After a short introduction to probability, celebrate your completion of the course with a deck of cards. Can you use the principles of probability, permutations, and combinations to calculate the probability of being dealt different hands? As with the rest of algebra, once you know the rules, it's simplicity itself!
36.
Elementary Probability
|
24
Lectures
30
minutes/lecture
1.
Addition and Subtraction
This introductory lecture starts with Professor Sellers’ overview of the general topics and themes you’ll encounter throughout the course. Then, plunge into an engaging review of the addition and subtraction of whole numbers, complete with several helpful tips designed to help you approach these types of problems with more confidence.
1.
Addition and Subtraction
|
13.
Exponents and Order of Operations
Explore a fifth fundamental mathematical operation: exponentiation. First, take a step-by-step look at the order of operations for handling longer calculations that involve multiple tasks—complete with invaluable tips to help you handle them with ease. Then, see where exponentiation fits in this larger process.
13.
Exponents and Order of Operations
|
2.
Multiplication
Continue your quick review of basic mathematical operations, this time with a focus on the multiplication of whole numbers. In addition to uncovering the relationship between addition and multiplication, you’ll get plenty of opportunities to strengthen your ability to multiply two 2-digit numbers, two 3-digit numbers, and more.
2.
Multiplication
|
14.
Negative and Positive Integers
Improve your confidence in dealing with negative numbers. You’ll learn to use the number line to help visualize these numbers; discover how to rewrite subtraction problems involving negative numbers as addition problems to make them easier; examine the rules involved in multiplying and dividing with them; and much more.
14.
Negative and Positive Integers
|
3.
Long Division
Turn now to the opposite of multiplication: division. Learn how to properly set up a long division problem, how to check your answers to make sure they’re correct, how to handle zeroes when they appear in a problem, and what to do when a long division problem ends with a remainder.
3.
Long Division
|
15.
Introduction to Square Roots
In this lecture, finally make sense of square roots. Professor Sellers offers examples to help you sidestep issues many students express frustration with, shows you how to simplify radical expressions involving addition and subtraction, and reveals how to find the approximate value of a square root without using a calculator.
15.
Introduction to Square Roots
|
4.
Introduction to Fractions
Mathematics is also filled with “parts” of whole numbers, or fractions. In the first of several lectures on fractions, define key terms and focus on powerful techniques for determining if fractions are equivalent, finding out which of two fractions is larger, and reducing fractions to their lowest terms.
4.
Introduction to Fractions
|
16.
Negative and Fractional Powers
What happens when you have to raise numbers to a fraction of a power? How about when you have to deal with negative exponents? Or negative fractional exponents? No need to worry —Professor Sellers guides you through this tricky mathematical territory, arming you with invaluable techniques for approaching these scenarios.
16.
Negative and Fractional Powers
|
5.
Adding and Subtracting Fractions
Fractions with the same denominator. Fractions with different denominators. Mixed numbers. Here, learn ways to add and subtract them all (and sometimes even in the same problem) and get tips for reducing your answers to their lowest terms. Math with fractions, you’ll discover, doesn’t have to be intimidating—it can even be fun!
5.
Adding and Subtracting Fractions
|
17.
Graphing in the Coordinate Plane
Grab some graph paper and learn how to graph objects in the coordinate (or xy) plane. You’ll find out how to plot points, how to determine which quadrant they go in, how to sketch the graph of a line, how to determine a line’s slope, and more.
17.
Graphing in the Coordinate Plane
|
6.
Multiplying Fractions
Continue having fun with fractions, this time by mastering how to multiply them and reduce your answer to its lowest term. Professor Sellers shows you how to approach and solve multiplication problems involving fractions (with both similar and different denominators), fractions and whole numbers, and fractions and mixed numbers.
6.
Multiplying Fractions
|
18.
Geometry—Triangles and Quadrilaterals
Continue exploring the visual side of mathematics with this look at the basics of two-dimensional geometry. Among the topics you’ll focus on here are the various types of triangles (including scalene and obtuse triangles) and quadrilaterals (such as rectangles and squares), as well as methods for measuring angles, area, and perimeter.
18.
Geometry—Triangles and Quadrilaterals
|
7.
Dividing Fractions
Professor Sellers walks you step-by-step through the process for speedily solving division problems involving fractions in this lecture filled with helpful practice problems. You’ll also learn how to better handle calculations involving different notations, fractions, and whole numbers, and even word problems involving the division of fractions.
7.
Dividing Fractions
|
19.
Geometry—Polygons and Circles
Gain a greater appreciation for the interaction between arithmetic and geometry. First, learn how to recognize and approach large polygons, including hexagons and decagons. Then, explore the various concepts behind circles (such as radius, diameter, and the always intriguing pi), as well as methods for calculating their circumference, area, and perimeter.
19.
Geometry—Polygons and Circles
|
8.
Adding and Subtracting Decimals
What’s 29.42 + 84.67? Or 643 + 82.987? What about 25.7 – 10.483? Problems like these are the focus of this helpful lecture on adding and subtracting decimals. One tip for making these sorts of calculations easier: making sure your decimal points are all lined up vertically.
8.
Adding and Subtracting Decimals
|
20.
Number Theory—Prime Numbers and Divisors
Shift gears and demystify number theory, which takes as its focus the study of the properties of whole numbers. Concepts that Professor Sellers discusses and teaches you how to engage with in this insightful lecture include divisors, prime numbers, prime factorizations, greatest common divisors, and factor trees.
20.
Number Theory—Prime Numbers and Divisors
|
9.
Multiplying and Dividing Decimals
Investigate the best ways to multiply and divide decimal numbers. You’ll get insights into when and when not to ignore the decimal point in your calculations, how to check your answer to ensure that your result has the correct number of decimal places, and how to express remainders in decimals.
9.
Multiplying and Dividing Decimals
|
21.
Number Theory—Divisibility Tricks
In this second lecture on the world of number theory, take a closer look at the relationships between even and odd numbers, as well as the rules of divisibility for particular numbers. By the end, you’ll be surprised that something as intimidating as number theory could be made so accessible.
21.
Number Theory—Divisibility Tricks
|
10.
Fractions, Decimals, and Percents
Take a closer look at converting between percents, decimals, and fractions—an area of basic mathematics that many people have a hard time with. After learning the techniques in this lecture and using them on numerous practice problems, you’ll be surprised at how easy this type of conversion is to master.
10.
Fractions, Decimals, and Percents
|
22.
Introduction to Statistics
Get a solid introduction to statistics, one of the most useful areas of mathematics. Here, you’ll focus on the four basic “measurements” statisticians use when gleaning meaning from data: mean, media, mode, and range. Also, see these concepts at work in everyday scenarios in which statistics plays a key role.
22.
Introduction to Statistics
|
11.
Percent Problems
Use the skills you developed in the last lecture to better approach and solve different kinds of percentage problems you’d most likely encounter in your everyday life. Among these everyday scenarios: calculating the tip at a restaurant and determining how much money you’re saving on a store’s discount.
11.
Percent Problems
|
23.
Introduction to Probability
Learn more about probability, a cousin of statistics and another mathematical field that helps us make sense of the seemingly unexplainable nature of the world. You’ll consider basic questions and concepts from probability, drawing on the knowledge and skills of the fundamentals of mathematics you acquired in earlier lectures.
23.
Introduction to Probability
|
12.
Ratios and Proportions
How do ratios and proportions work? How can you figure out if a particular problem is merely just a ratio or proportion problem in disguise? What are some pitfalls to watch out for? And how can a better understanding of these subjects help save you money? Find out here.
12.
Ratios and Proportions
|
24.
Introduction to Algebra
Professor Sellers reviews the importance of math in daily life and previews the next logical step in your studies: Algebra I (which involves variables). Whether you’re planning to take more Great Courses in mathematics or simply looking to sharpen your mind, you’ll be sent off with new levels of confidence.
24.
Introduction to Algebra
|
36
Lectures
30
minutes/lecture
1.
An Introduction to Precalculus—Functions
Precalculus is important preparation for calculus, but it’s also a useful set of skills in its own right, drawing on algebra, trigonometry, and other topics. As an introduction, review the essential concept of the function, try your hand at simple problems, and hear Professor Edwards’s recommendations for approaching the course.
1.
An Introduction to Precalculus—Functions
|
19.
Trigonometric Equations
In calculus, the difficult part is often not the steps of a problem that use calculus but the equation that’s left when you’re finished, which takes precalculus to solve. Hone your skills for this challenge by identifying all the values of the variable that satisfy a given trigonometric equation.
19.
Trigonometric Equations
|
2.
Polynomial Functions and Zeros
The most common type of algebraic function is a polynomial function. As examples, investigate linear and quadratic functions, probing different techniques for finding roots, or “zeros.” A valuable tool in this search is the intermediate value theorem, which identifies real-number roots for polynomial functions.
2.
Polynomial Functions and Zeros
|
20.
Sum and Difference Formulas
Study the important formulas for the sum and difference of sines, cosines, and tangents. Then use these tools to get a preview of calculus by finding the slope of a tangent line on the cosine graph. In the process, you discover the derivative of the cosine function.
20.
Sum and Difference Formulas
|
3.
Complex Numbers
Step into the strange and fascinating world of complex numbers, also known as imaginary numbers, where i is defined as the square root of -1. Learn how to calculate and find roots of polynomials using complex numbers, and how certain complex expressions produce beautiful fractal patterns when graphed.
3.
Complex Numbers
|
21.
Law of Sines
Return to the subject of triangles to investigate the law of sines, which allows the sides and angles of any triangle to be determined, given the value of two angles and one side, or two sides and one opposite angle. Also learn a sine-based formula for the area of a triangle.
21.
Law of Sines
|
4.
Rational Functions
Investigate rational functions, which are quotients of polynomials. First, find the domain of the function. Then, learn how to recognize the vertical and horizontal asymptotes, both by graphing and comparing the values of the numerator and denominator. Finally, look at some applications of rational functions.
4.
Rational Functions
|
22.
Law of Cosines
Given three sides of a triangle, can you find the three angles? Use a generalized form of the Pythagorean theorem called the law of cosines to succeed. This formula also allows the determination of all sides and angles of a triangle when you know any two sides and their included angle.
22.
Law of Cosines
|
5.
Inverse Functions
Discover how functions can be combined in various ways, including addition, multiplication, and composition. A special case of composition is the inverse function, which has important applications. One way to recognize inverse functions is on a graph, where the function and its inverse form mirror images across the line y = x.
5.
Inverse Functions
|
23.
Introduction to Vectors
Vectors symbolize quantities that have both magnitude and direction, such as force, velocity, and acceleration. They are depicted by a directed line segment on a graph. Experiment with finding equivalent vectors, adding vectors, and multiplying vectors by scalars.
23.
Introduction to Vectors
|
6.
Solving Inequalities
You have already used inequalities to express the set of values in the domain of a function. Now study the notation for inequalities, how to represent inequalities on graphs, and techniques for solving inequalities, including those involving absolute value, which occur frequently in calculus.
6.
Solving Inequalities
|
24.
Trigonometric Form of a Complex Number
Apply your trigonometric skills to the abstract realm of complex numbers, seeing how to represent complex numbers in a trigonometric form that allows easy multiplication and division. Also investigate De Moivre’s theorem, a shortcut for raising complex numbers to any power.
24.
Trigonometric Form of a Complex Number
|
7.
Exponential Functions
Explore exponential functions—functions that have a base greater than 1 and a variable as the exponent. Survey the properties of exponents, the graphs of exponential functions, and the unique properties of the natural base e. Then sample a typical problem in compound interest.
7.
Exponential Functions
|
25.
Systems of Linear Equations and Matrices
Embark on the first of four lectures on systems of linear equations and matrices. Begin by using the method of substitution to solve a simple system of two equations and two unknowns. Then practice the technique of Gaussian elimination, and get a taste of matrix representation of a linear system.
25.
Systems of Linear Equations and Matrices
|
8.
Logarithmic Functions
A logarithmic function is the inverse of the exponential function, with all the characteristics of inverse functions covered in Lecture 5. Examine common logarithms (those with base 10) and natural logarithms (those with base e), and study such applications as the “rule of 70” in banking.
8.
Logarithmic Functions
|
26.
Operations with Matrices
Deepen your understanding of matrices by learning how to do simple operations: addition, scalar multiplication, and matrix multiplication. After looking at several examples, apply matrix arithmetic to a commonly encountered problem by finding the parabola that passes through three given points.
26.
Operations with Matrices
|
9.
Properties of Logarithms
Learn the secret of converting logarithms to any base. Then review the three major properties of logarithms, which allow simplification or expansion of logarithmic expressions—methods widely used in calculus. Close by focusing on applications, including the pH system in chemistry and the Richter scale in geology.
9.
Properties of Logarithms
|
27.
Inverses and Determinants of Matrices
Get ready for applications involving matrices by exploring two additional concepts: the inverse of a matrix and the determinant. The algorithm for calculating the inverse of a matrix relies on Gaussian elimination, while the determinant is a scalar value associated with every square matrix.
27.
Inverses and Determinants of Matrices
|
10.
Exponential and Logarithmic Equations
Practice solving a range of equations involving logarithms and exponents, seeing how logarithms are used to bring exponents “down to earth” for easier calculation. Then try your hand at a problem that models the heights of males and females, analyzing how the models are put together.
10.
Exponential and Logarithmic Equations
|
28.
Applications of Linear Systems and Matrices
Use linear systems and matrices to analyze such questions as these: How can the stopping distance of a car be estimated based on three data points? How does computer graphics perform transformations and rotations? How can traffic flow along a network of roads be modeled?
28.
Applications of Linear Systems and Matrices
|
11.
Exponential and Logarithmic Models
Finish the algebra portion of the course by delving deeper into exponential and logarithmic equations, using them to model real-life phenomena, including population growth, radioactive decay, SAT math scores, the spread of a virus, and the cooling rate of a cup of coffee.
11.
Exponential and Logarithmic Models
|
29.
Circles and Parabolas
In the first of two lectures on conic sections, examine the properties of circles and parabolas. Learn the formal definition and standard equation for each, and solve a real-life problem involving the reflector found in a typical car headlight.
29.
Circles and Parabolas
|
12.
Introduction to Trigonometry and Angles
Trigonometry is a key topic in applied math and calculus with uses in a wide range of applications. Begin your investigation with the two techniques for measuring angles: degrees and radians. Typically used in calculus, the radian system makes calculations with angles easier.
12.
Introduction to Trigonometry and Angles
|
30.
Ellipses and Hyperbolas
Continue your survey of conic sections by looking at ellipses and hyperbolas, studying their standard equations and probing a few of their many applications. For example, calculate the dimensions of the U.S. Capitol’s “whispering gallery,” an ellipse-shaped room with fascinating acoustical properties.
30.
Ellipses and Hyperbolas
|
13.
Trigonometric Functions—Right Triangle Definition
The Pythagorean theorem, which deals with the relationship of the sides of a right triangle, is the starting point for the six trigonometric functions. Discover the close connection of sine, cosine, tangent, cosecant, secant, and cotangent, and focus on some simple formulas that are well worth memorizing.
13.
Trigonometric Functions—Right Triangle Definition
|
31.
Parametric Equations
How do you model a situation involving three variables, such as a motion problem that introduces time as a third variable in addition to position and velocity? Discover that parametric equations are an efficient technique for solving such problems. In one application, you calculate whether a baseball hit at a certain angle and speed will be a home run.
31.
Parametric Equations
|
14.
Trigonometric Functions—Arbitrary Angle Definition
Trigonometric functions need not be confined to acute angles in right triangles; they apply to virtually any angle. Using the coordinate plane, learn to calculate trigonometric values for arbitrary angles. Also see how a table of common angles and their trigonometric values has wide application.
14.
Trigonometric Functions—Arbitrary Angle Definition
|
32.
Polar Coordinates
Take a different mathematical approach to graphing: polar coordinates. With this system, a point’s location is specified by its distance from the origin and the angle it makes with the positive x axis. Polar coordinates are surprisingly useful for many applications, including writing the formula for a valentine heart!
32.
Polar Coordinates
|
15.
Graphs of Sine and Cosine Functions
The graphs of sine and cosine functions form a distinctive wave-like pattern. Experiment with functions that have additional terms, and see how these change the period, amplitude, and phase of the waves. Such behavior occurs throughout nature and led to the discovery of rapidly rotating stars called pulsars in 1967.
15.
Graphs of Sine and Cosine Functions
|
33.
Sequences and Series
Get a taste of calculus by probing infinite sequences and series—topics that lead to the concept of limits, the summation notation using the Greek letter sigma, and the solution to such problems as Zeno’s famous paradox. Also investigate Fibonacci numbers and an infinite series that produces the number e.
33.
Sequences and Series
|
16.
Graphs of Other Trigonometric Functions
Continue your study of the graphs of trigonometric functions by looking at the curves made by tangent, cosecant, secant, and cotangent expressions. Then bring several precalculus skills together by using a decaying exponential term in a sine function to model damped harmonic motion.
16.
Graphs of Other Trigonometric Functions
|
34.
Counting Principles
Counting problems occur frequently in real life, from the possible batting lineups on a baseball team to the different ways of organizing a committee. Use concepts you’ve learned in the course to distinguish between permutations and combinations and provide precise counts for each.
34.
Counting Principles
|
17.
Inverse Trigonometric Functions
For a given trigonometric function, only a small part of its graph qualifies as an inverse function as defined in Lecture 5. However, these inverse trigonometric functions are very important in calculus. Test your skill at identifying and working with them, and try a problem involving a rocket launch.
17.
Inverse Trigonometric Functions
|
35.
Elementary Probability
What are your chances of winning the lottery? Of rolling a seven with two dice? Of guessing your ATM PIN number when you’ve forgotten it? Delve into the rudiments of probability, learning basic vocabulary and formulas so that you know the odds.
35.
Elementary Probability
|
18.
Trigonometric Identities
An equation that is true for every possible value of a variable is called an identity. Review several trigonometric identities, seeing how they can be proved by choosing one side of the equation and then simplifying it until a true statement remains. Such identities are crucial for solving complicated trigonometric equations.
18.
Trigonometric Identities
|
36.
GPS Devices and Looking Forward to Calculus
In a final application, locate a position on the surface of the earth with a two-dimensional version of GPS technology. Then close by finding the tangent line to a parabola, thereby solving a problem in differential calculus and witnessing how precalculus paves the way for the next big mathematical adventure.
36.
GPS Devices and Looking Forward to Calculus
|
36
Lectures
30
minutes/lecture
1.
A Preview of Calculus
Calculus is the mathematics of change, a field with many important applications in science, engineering, medicine, business, and other disciplines. Begin by surveying the goals of the course. Then get your feet wet by investigating the classic tangent line problem, which illustrates the concept of limits.
1.
A Preview of Calculus
|
19.
The Area Problem and the Definite Integral
One of the classic problems of integral calculus is finding areas bounded by curves. This was solved for simple curves by the ancient Greeks. See how a more powerful method was later developed that produces a number called the definite integral, and learn the relevant notation.
19.
The Area Problem and the Definite Integral
|
2.
Review—Graphs, Models, and Functions
In the first of two review lectures on precalculus, examine graphs of equations and properties such as symmetry and intercepts. Also explore the use of equations to model real life and begin your study of functions, which Professor Edwards calls the most important concept in mathematics.
2.
Review—Graphs, Models, and Functions
|
20.
The Fundamental Theorem of Calculus, Part 1
The two essential ideas of this course—derivatives and integrals—are connected by the fundamental theorem of calculus, one of the most important theorems in mathematics. Get an intuitive grasp of this deep relationship by working several problems and surveying a proof.
20.
The Fundamental Theorem of Calculus, Part 1
|
3.
Review—Functions and Trigonometry
Continue your review of precalculus by looking at different types of functions and how they can be identified by their distinctive shapes when graphed. Then review trigonometric functions, using both the right triangle definition as well as the unit circle definition, which measures angles in radians rather than degrees.
3.
Review—Functions and Trigonometry
|
21.
The Fundamental Theorem of Calculus, Part 2
Try examples using the second fundamental theorem of calculus, which allows you to let the upper limit of integration be a variable. In the process, explore more relationships between differentiation and integration, and discover how they are almost inverses of each other.
21.
The Fundamental Theorem of Calculus, Part 2
|
4.
Finding Limits
Jump into real calculus by going deeper into the concept of limits introduced in Lecture 1. Learn the informal, working definition of limits and how to determine a limit in three different ways: numerically, graphically, and analytically. Also discover how to recognize when a given function does not have a limit.
4.
Finding Limits
|
22.
Integration by Substitution
Investigate a straightforward technique for finding antiderivatives, called integration by substitution. Based on the chain rule, it enables you to convert a difficult problem into one that's easier to solve by using the variable u to represent a more complicated expression.
22.
Integration by Substitution
|
5.
An Introduction to Continuity
Broadly speaking, a function is continuous if there is no interruption in the curve when its graph is drawn. Explore the three conditions that must be met for continuity—along with applications of associated ideas, such as the greatest integer function and the intermediate value theorem.
5.
An Introduction to Continuity
|
23.
Numerical Integration
When calculating a definite integral, the first step of finding the antiderivative can be difficult or even impossible. Learn the trapezoid rule, one of several techniques that yield a close approximation to the definite integral. Then do a problem involving a plot of land bounded by a river.
23.
Numerical Integration
|
6.
Infinite Limits and Limits at Infinity
Infinite limits describe the behavior of functions that increase or decrease without bound, in which the asymptote is the specific value that the function approaches without ever reaching it. Learn how to analyze these functions, and try some examples from relativity theory and biology.
6.
Infinite Limits and Limits at Infinity
|
24.
Natural Logarithmic Function—Differentiation
Review the properties of logarithms in base 10. Then see how the so-called natural base for logarithms, e, has important uses in calculus and is one of the most significant numbers in mathematics. Learn how such natural logarithms help to simplify derivative calculations.
24.
Natural Logarithmic Function—Differentiation
|
7.
The Derivative and the Tangent Line Problem
Building on what you have learned about limits and continuity, investigate derivatives, which are the foundation of differential calculus. Develop the formula for defining a derivative, and survey the history of the concept and its different forms of notation.
7.
The Derivative and the Tangent Line Problem
|
25.
Natural Logarithmic Function—Integration
Continue your investigation of logarithms by looking at some of the consequences of the integral formula developed in the previous lecture. Next, change gears and review inverse functions at the precalculus level, preparing the way for a deeper exploration of the subject in coming lectures.
25.
Natural Logarithmic Function—Integration
|
8.
Basic Differentiation Rules
Practice several techniques that make finding derivatives relatively easy: the power rule, the constant multiple rule, sum and difference rules, plus a shortcut to use when sine and cosine functions are involved. Then see how derivatives are the key to determining the rate of change in problems involving objects in motion.
8.
Basic Differentiation Rules
|
26.
Exponential Function
The inverse of the natural logarithmic function is the exponential function, perhaps the most important function in all of calculus. Discover that this function has an amazing property: It is its own derivative! Also see the connection between the exponential function and the bell-shaped curve in probability.
26.
Exponential Function
|
9.
Product and Quotient Rules
Learn the formulas for finding derivatives of products and quotients of functions. Then use the quotient rule to derive formulas for the trigonometric functions not covered in the previous lecture. Also investigate higher-order derivatives, differential equations, and horizontal tangents.
9.
Product and Quotient Rules
|
27.
Bases other than e
Extend the use of the logarithmic and exponential functions to bases other than e, exploiting this approach to solve a problem in radioactive decay. Also learn to find the derivatives of such functions, and see how e emerges in other mathematical contexts, including the formula for continuous compound interest.
27.
Bases other than e
|
10.
The Chain Rule
Discover one of the most useful of the differentiation rules, the chain rule, which allows you to find the derivative of a composite of two functions. Explore different examples of this technique, including a problem from physics that involves the motion of a pendulum.
10.
The Chain Rule
|
28.
Inverse Trigonometric Functions
Turn to the last set of functions you will need in your study of calculus, inverse trigonometric functions. Practice using some of the formulas for differentiating these functions. Then do an entertaining problem involving how fast the rotating light on a police car sweeps across a wall and whether you can evade it.
28.
Inverse Trigonometric Functions
|
11.
Implicit Differentiation and Related Rates
Conquer the final strategy for finding derivatives: implicit differentiation, used when it's difficult to solve a function for y. Apply this rule to problems in related rates—for example, the rate at which a camera must move to track the space shuttle at a specified time after launch.
11.
Implicit Differentiation and Related Rates
|
29.
Area of a Region between 2 Curves
Revisit the area problem and discover how to find the area of a region bounded by two curves. First imagine that the region is divided into representative rectangles. Then add up an infinite number of these rectangles, which corresponds to a definite integral.
29.
Area of a Region between 2 Curves
|
12.
Extrema on an Interval
Having covered the rules for finding derivatives, embark on the first of five lectures dealing with applications of these techniques. Derivatives can be used to find the absolute maximum and minimum values of functions, known as extrema, a vital tool for analyzing many real-life situations.
12.
Extrema on an Interval
|
30.
Volume—The Disk Method
Learn how to calculate the volume of a solid of revolution—an object that is symmetrical around its axis of rotation. As in the area problem in the previous lecture, you imagine adding up an infinite number of slices—in this case, of disks rather than rectangles—which yields a definite integral.
30.
Volume—The Disk Method
|
13.
Increasing and Decreasing Functions
Use the first derivative to determine where graphs are increasing or decreasing. Next, investigate Rolle's theorem and the mean value theorem, one of whose consequences is that during a car trip, your actual speed must correspond to your average speed during at least one point of your journey.
13.
Increasing and Decreasing Functions
|
31.
Volume—The Shell Method
Apply the shell method for measuring volumes, comparing it with the disk method on the same shape. Then find the volume of a doughnut-shaped object called a torus, along with the volume for a figure called Gabriel's Horn, which is infinitely long but has finite volume.
31.
Volume—The Shell Method
|
14.
Concavity and Points of Inflection
What does the second derivative reveal about a graph? It describes how the curve bends—whether it is concave upward or downward. You determine concavity much as you found the intervals where a graph was increasing or decreasing, except this time you use the second derivative.
14.
Concavity and Points of Inflection
|
32.
Applications—Arc Length and Surface Area
Investigate two applications of calculus that are at the heart of engineering: measuring arc length and surface area. One of your problems is to determine the length of a cable hung between two towers, a shape known as a catenary. Then examine a peculiar paradox of Gabriel's Horn.
32.
Applications—Arc Length and Surface Area
|
15.
Curve Sketching and Linear Approximations
By using calculus, you can be certain that you have discovered all the properties of the graph of a function. After learning how this is done, focus on the tangent line to a graph, which is a convenient approximation for values of the function that lie close to the point of tangency.
15.
Curve Sketching and Linear Approximations
|
33.
Basic Integration Rules
Review integration formulas studied so far, and see how to apply them in various examples. Then explore cases in which a calculator gives different answers from the ones obtained by hand calculation, learning why this occurs. Finally, Professor Edwards gives advice on how to succeed in introductory calculus.
33.
Basic Integration Rules
|
16.
Applications—Optimization Problems, Part 1
Attack real-life problems in optimization, which requires finding the relative extrema of different functions by differentiation. Calculate the optimum size for a box, and the largest area that can be enclosed by a circle and a square made from a given length of wire.
16.
Applications—Optimization Problems, Part 1
|
34.
Other Techniques of Integration
Closing your study of integration techniques, explore a powerful method for finding antiderivatives: integration by parts, which is based on the product rule for derivatives. Use this technique to calculate area and volume. Then focus on integrals involving products of trigonometric functions.
34.
Other Techniques of Integration
|
17.
Applications—Optimization Problems, Part 2
Conclude your investigation of differential calculus with additional problems in optimization. For success with such word problems, Professor Edwards stresses the importance of first framing the problem with precalculus, reducing the equation to one independent variable, and then using calculus to find and verify the answer.
17.
Applications—Optimization Problems, Part 2
|
35.
Differential Equations and Slope Fields
Explore slope fields as a method for getting a picture of possible solutions to a differential equation without having to solve it, examining several problems of the type that appear on the Advanced Placement exam. Also look at a solution technique for differential equations called separation of variables.
35.
Differential Equations and Slope Fields
|
18.
Antiderivatives and Basic Integration Rules
Up until now, you've calculated a derivative based on a given function. Discover how to reverse the procedure and determine the function based on the derivative. This approach is known as obtaining the antiderivative, or integration. Also learn the notation for integration.
18.
Antiderivatives and Basic Integration Rules
|
36.
Applications of Differential Equations
Use your calculus skills in three applications of differential equations: first, calculate the radioactive decay of a quantity of plutonium; second, determine the initial population of a colony of fruit flies; and third, solve one of Professor Edwards's favorite problems by using Newton's law of cooling to predict the cooling time for a cup of coffee.
36.
Applications of Differential Equations
|