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36. GPS Devices and Looking Forward to Calculus
June 17, 2011
In a final application, locate a position on the surface of the earth with a twodimensional 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.
35. Elementary Probability
June 17, 2011
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.
34. Counting Principles
June 17, 2011
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 series to distinguish between permutations and combinations and provide precise counts for each.
33. Sequences and Series
June 17, 2011
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.
32. Polar Coordinates
June 17, 2011
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!
30. Ellipses and Hyperbolas
June 17, 2011
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 US Capitol's "whispering gallery," an ellipseshaped room with fascinating acoustical properties.
29. Circles and Parabolas
June 17, 2011
In the first of two episodes on conic sections, examine the properties of circles and parabolas. Learn the formal definition and standard equation for each, and solve a reallife problem involving the reflector found in a typical car headlight.
28. Applications of Linear Systems and Matrices
June 17, 2011
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?
27. Inverses and Determinants of Matrices
June 17, 2011
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.
26. Operations with Matrices
June 17, 2011
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.
25. Systems of Linear Equations and Matrices
June 17, 2011
Embark on the first of four episodes 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.
24. Trigonometric Form of a Complex Number
June 17, 2011
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.
23. Introduction to Vectors
June 17, 2011
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.
21. Law of Sines
June 17, 2011
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 sinebased formula for the area of a triangle.
20. Sum and Difference Formulas
June 17, 2011
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.
19. Trigonometric Equations
June 17, 2011
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.
18. Trigonometric Identities
June 17, 2011
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.
17. Inverse Trigonometric Functions
June 17, 2011
For a given trigonometric function, only a small part of its graph qualifies as an inverse function. 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.
16. Graphs of Other Trigonometric Functions
June 17, 2011
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.
15. Graphs of Sine and Cosine Functions
June 17, 2011
The graphs of sine and cosine functions form a distinctive wavelike 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.
13. Trigonometric Functions  Right Triangle Definition
June 17, 2011
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.
12. Introduction to Trigonometry and Angles
June 17, 2011
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.
10. Exponential and Logarithmic Equations
June 17, 2011
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.
9. Properties of Logarithms
June 17, 2011
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 and are widely used in calculus. Close by focusing on applications, including the pH system in chemistry and the Richter scale in geology.
8. Logarithmic Functions
June 17, 2011
A logarithmic function is the inverse of the exponential function, with all the characteristics of inverse functions covered earlier. 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.
7. Exponential Functions
June 17, 2011
Explore exponential functions, which 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.
6. Solving Inequalities
June 17, 2011
You've 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.
5. Inverse Functions
June 17, 2011
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.
4. Rational Functions
June 17, 2011
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.
3. Complex Numbers
June 17, 2011
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.
2. Polynomial Functions and Zeros
June 17, 2011
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 realnumber roots for polynomial functions.
1. An Introduction to Precalculus  Functions
June 17, 2011
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 series. #Science & Mathematics
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Where to Watch Mathematics Describing the Real World: Precalculus and Trigonometry
Mathematics Describing the Real World: Precalculus and Trigonometry is available for streaming on the The Great Courses Signature Collection website, both individual episodes and full seasons. You can also watch Mathematics Describing the Real World: Precalculus and Trigonometry on demand at Amazon Prime and Amazon.
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Premiere DateJune 17, 2011