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Mathematics Describing the Real World: Precalculus and Trigonometry

Mathematics Describing the Real World: Precalculus and Trigonometry

Professor Bruce H. Edwards Ph.D.
University of Florida
Course No.  1005
Course No.  1005
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Course Overview

About This Course

36 lectures  |  31 minutes per lecture

What's the sure road to success in calculus? The answer is simple: Precalculus. Traditionally studied after Algebra II, this mathematical field covers advanced algebra, trigonometry, exponents, logarithms, and much more. These interrelated topics are essential for solving calculus problems, and by themselves are powerful methods for describing the real world, permeating all areas of science and every branch of mathematics. Little wonder, then, that precalculus is a core course in high schools throughout the country and an important review subject in college.

Unfortunately, many students struggle in precalculus because they fail to see the links between different topics—between one approach to finding an answer and a startlingly different, often miraculously simpler, technique. As a result, they lose out on the enjoyment and fascination of mastering an amazingly useful tool box of problem-solving strategies.

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What's the sure road to success in calculus? The answer is simple: Precalculus. Traditionally studied after Algebra II, this mathematical field covers advanced algebra, trigonometry, exponents, logarithms, and much more. These interrelated topics are essential for solving calculus problems, and by themselves are powerful methods for describing the real world, permeating all areas of science and every branch of mathematics. Little wonder, then, that precalculus is a core course in high schools throughout the country and an important review subject in college.

Unfortunately, many students struggle in precalculus because they fail to see the links between different topics—between one approach to finding an answer and a startlingly different, often miraculously simpler, technique. As a result, they lose out on the enjoyment and fascination of mastering an amazingly useful tool box of problem-solving strategies.

And even if you're not planning to take calculus, understanding the fundamentals of precalculus can give you a versatile set of skills that can be applied to a wide range of fields—from computer science and engineering to business and health care.

Mathematics Describing the Real World: Precalculus and Trigonometry

is your unrivaled introduction to this crucial subject, taught by award-winning Professor Bruce Edwards of the University of Florida. Professor Edwards is coauthor of one of the most widely used textbooks on precalculus and an expert in getting students over the trouble spots of this challenging phase of their mathematics education.

"Calculus is difficult because of the precalculus skills needed for success," Professor Edwards points out, adding, "In my many years of teaching, I have found that success in calculus is assured if students have a strong background in precalculus."

A Math Milestone Made Clear

In 36 intensively illustrated half-hour lectures, supplemented by a workbook with additional explanations and problems, Mathematics Describing the Real World takes you through all the major topics of a typical precalculus course taught in high school or college. Those who will especially benefit from Professor Edwards's lucid and engaging approach include

  • high school and college students currently enrolled in precalculus who feel overwhelmed and want coaching from an inspiring teacher who knows where students stumble;
  • parents of students, who may feel out of their depth with the advanced concepts taught in precalculus;
  • those who have finished Algebra II and are eager to get a head start on the next milestone on the road to calculus;
  • beginning calculus students who want to review and hone their skills in crucial precalculus topics;
  • anyone motivated to learn precalculus on his or her own, whether as a home-schooled pupil or as an adult preparing for a new career.

The Powerful Tools of Precalculus

With precalculus, you start to see all of mathematics as a unified whole—as a group of often radically different techniques for representing data, analyzing problems, and finding solutions. And you discover that these techniques are ultimately connected in a beautiful way. Perceiving these connections helps you choose the best tool for a given problem:

  • Algebraic functions: Including polynomial functions and rational functions, these equations relate the input value of a variable to a single output value, corresponding to countless everyday situations in which one event depends on another.
  • Trigonometry: Originally dealing with the measurement of triangles, this subject has been vastly enriched by the concept of the trigonometric function, which models many types of cyclical processes, such as waves, orbits, and vibrations.
  • Exponential and logarithmic functions: Often involving the natural base, e, these functions are built on terms with exponents and their inverse, logarithms, and describe phenomena such as population growth and the magnitude of an earthquake on the Richter scale.
  • Complex numbers: Seemingly logic-defying, complex numbers are based on the square root of –1, designated by the symbol i. They are essential for solving many technical problems and are the basis for the beautiful patterns in fractal geometry.
  • Vectors: Quantities like velocity have both magnitude and direction. Vectors allow the direction component to be specified in a form that allows addition, multiplication, and other operations that are crucial in fields such as physics.
  • Matrices: A matrix is a rectangular array of numbers with special rules that permit two matrices to be added or multiplied. Practically any situation where data are collected in columns and rows can be treated mathematically as a matrix.

In addition, Professor Edwards devotes two lectures to conic sections, slicing a cone mathematically into circles, parabolas, ellipses, and hyperbolas. You also learn when it's useful to switch from Cartesian to polar coordinates; how infinite sequences and series lead to the concept of the limit in calculus; and two approaches to counting questions: permutations and combinations. You close with an introduction to probability and a final lecture that features an actual calculus problem, which your experience in precalculus makes ... elementary!

Real-World Mathematics

Believing that students learn mathematics most effectively when they see it in the context of the world around them, Professor Edwards uses scores of interesting problems that are fun, engaging, and often relevant to real life. Among the many applications of precalculus that you'll encounter are these:

  • Public health: A student with a new strain of flu arrives at college. How long before every susceptible person is infected? An exponential function called the logistic growth model shows how quickly an epidemic spreads.
  • Surveying: Suppose you have to measure the diagonal width of a marsh without getting wet. It's a simple matter of walking two sides of a triangle on dry land and then using trigonometry to determine the length of the third side that spans the marsh.
  • Astronomy: One of the most famous cases involving the sine and cosine functions that model periodic phenomena occurred in 1967, when astronomer Jocelyn Bell detected a radio signal from space at 1.3373-second intervals. It proved to be the first pulsar ever observed.
  • Acoustics: The special properties of an ellipse explain why a person standing at a given spot in the U.S. Capitol's Statuary Hall can hear a whisper from someone standing 85 feet away.
  • Computer graphics: How do you make an object appear to rotate on a computer screen? Matrix algebra allows you to move each pixel in an image by a specified angle by multiplying two matrices together.
  • Probability: Have you ever forgotten your four-digit ATM PIN number? What is the probability that you can guess it? A simple calculation shows that you would have to punch numbers nonstop for many hours before being assured of success.
An Adventure in Mathematical Learning

A three-time Teacher of the Year in the College of Liberal Arts and Sciences at the University of Florida, Professor Edwards has a time-tested approach to making difficult material accessible. In Mathematics Describing the Real World, he enlivens his lectures with study tips and a feature he calls "You Be the Teacher," in which he puts you in the professor's shoes by asking how you would design a particular test problem or answer one of the frequently asked questions he gets in the classroom. For example, are all exponential functions increasing? After you hear Professor Edwards's explanation, you'll know that when someone uses the term "exponentially," you should ask, "Do you mean exponential growth or decay?"—for it can go in either direction. He also gives valuable tips on using graphing calculators, pointing out their amazing capabilities—and pitfalls.

View Less
36 Lectures
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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? x
  • 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. x
  • 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. x
  • 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. x
  • 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! x
  • 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. x
  • 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. x
  • 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. x
  • 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. x

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Bruce H. Edwards
Ph.D. Bruce H. Edwards
University of Florida
Dr. Bruce H. Edwards is Professor of Mathematics at the University of Florida. Professor Edwards received his B.S. in Mathematics from Stanford University and his Ph.D. in Mathematics from Dartmouth College. After his years at Stanford, he taught mathematics at a university near Bogot·, Colombia, as a Peace Corps volunteer. Professor Edwards has won many teaching awards at the University of Florida, including Teacher of the Year in the College of Liberal Arts and Sciences, Liberal Arts and Sciences Student Council Teacher of the Year, and the University of Florida Honors Program Teacher of the Year. He was selected by the Office of Alumni Affairs to be the Distinguished Alumni Professor for 1991-1993. Professor Edwards has taught a variety of mathematics courses at the University of Florida, from first-year calculus to graduate-level classes in algebra and numerical analysis. He has been a frequent speaker at research conferences and meetings of the National Council of Teachers of Mathematics. He has also coauthored a wide range of mathematics textbooks with Professor Ron Larson. Their textbooks have been honored with various awards from the Text and Academic Authors Association.
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Reviews

Rated 4.4 out of 5 by 16 reviewers.
Rated 5 out of 5 by PreCalculus can be made clear and logical Let me begin by stating I am a retired mathematics instructor who had a very successful and rewarding career of 39 years and that I actually used the PreCalculus and Calculus textbooks, which I thought were the best available, written by Dr. Edwards and his coauthors. Professor Edwards’ presentation in this excellent Teaching Company course is logical and clear. His carefully chosen examples are especially insightful and illuminating. WARNING: This is not a superficial survey course but instead uses an in-depth approach, like his Calculus course, that will leave you feeling knowledgeable, satisfied, and confident upon its completion. It successfully bridges the gap between TTC’s “Algebra II” and “Understanding Calculus: Problems, Solutions, and Tips.” I hope TTC will have Dr. Edwards add Calculus II and Calculus III classes to complete its basic mathematics curriculum series. June 19, 2011
Rated 2 out of 5 by Not For Me I bought the course hoping to review junior and senior high school math I had forgotten decades ago. The content is good but the teacher's presentation is usually "this next concept is scary" or "don't freak out". I loved math and it was easy for me, so this attitude is patronizing. When I found myself yelling at my computer, I stopped watching. Will switch to a textbook. July 8, 2014
Rated 5 out of 5 by I first bought and dug into Professor Edwards' calculus course. It was lucid and I was learning, but Professor E. repeatedly emphasized that success in calculus depends on strong precalculus skills. I was convinced. As Sal Khan of the Khan Academy has put it, I had a lot of "holes in my cheese". So I tried to order the precalculus course at a discounted rate (having bought the calculus course only a month earlier) but learned it wasn't on sale. So, I checked out this course from our city library and was halfway through it when the announcement came that I could buy it for the reduced price. No question. I want to own it as a resource, and I want the workbook for the problems in it. I looked at the negative criticisms of the precalculus course and found them largely groundless. Prof. Edwards has his quirky mannerisms, and his lectures are formulaic (e.g. he always ends with an application). I find that comforting and helpful. It's a joy to pause the video when he poses a question and try to figure it out on my own before going on, and when I miss something, it's great to be able to go back and listen to his explanations again. Trust me. Unless you are a math whiz, you'll need to. Prof. Edwards was educated in the pre-calculator era as was I. The handheld graphing calculator has revolutionized math education. Trig tables and log tables are no longer needed. Prof. Edwards acknowledges that and uses calculators to advantage in his course. I am also a Khan Academy aficionado. The two approaches are complementary and synergistic. Khan's on-line problems are the best, bar none, and his warm explanations are first-rate. Khan's YouTube clips and Edwards' DVD explanations are different but each is valuable. We are on the threshold of a new era in mathematics literacy, thanks to these novel approaches. Highly recommended. May 25, 2014
Rated 5 out of 5 by Excellent Teaching Honestly, I have purchased some courses from this company which, quite frankly, just stunk. This was not one of them. Professor Edwards is by far one of the best teachers I have ever seen, and he made precalculus easy to understand. I would highly recommend this course to anyone interested in learning or reviewing precalculus. December 26, 2013
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