Продолжительность: 36 лекций по 30 минут
Описание: Лекции по математическому анализу.
1. A Preview of Calculus
2. Review—Graphs, Models, and Functions
3. Review—Functions and Trigonometry
4. Finding Limits
5. An Introduction to Continuity
6. Infinite Limits and Limits at Infinity
7. The Derivative and the Tangent Line Problem
8. Basic Differentiation Rules
9. Product and Quotient Rules
10. The Chain Rule
11. Implicit Differentiation and Related Rates
12. Extrema on an Interval
13. Increasing and Decreasing Functions
14. Concavity and Points of Inflection
15. Curve Sketching and Linear Approximations
16. Applications—Optimization Problems, Part 1
17. Applications—Optimization Problems, Part 2
18. Antiderivatives and Basic Integration Rules
19. The Area Problem and the Definite Integral
20. The Fundamental Theorem of Calculus, Part 1
21. The Fundamental Theorem of Calculus, Part 2
22. Integration by Substitution
23. Numerical Integration
24. Natural Logarithmic Function—Differentiation
25. Natural Logarithmic Function—Integration
26. Exponential Function
27. Bases other than e
28. Inverse Trigonometric Functions
29. Area of a Region between 2 Curves
30. Volume—The Disk Method
31. Volume—The Shell Method
32. Applications—Arc Length and Surface Area
33. Basic Integration Rules
34. Other Techniques of Integration
35. Differential Equations and Slope Fields
36. Applications of Differential Equations
Calculus is the greatest mathematical breakthrough since the pioneering discoveries of the ancient Greeks. Without it, we wouldn't have spaceflight, skyscrapers, jet planes, economic modeling, accurate weather forecasting, modern medical technologies, or any of the countless other achievements we take for granted in today's world.
Indeed, calculus is so versatile and its techniques so diverse that it trains you to view problems, no matter how difficult, as solvable until proved otherwise. And the habit of turning a problem over in your mind, choosing an approach, and then working through a solution teaches you to think clearly—which is why the study of calculus is so crucial for improving your cognitive skills and why it is a prerequisite for admission to most top universities.
Understanding Calculus: Problems, Solutions, and Tips immerses you in the unrivaled learning adventure of this mathematical field in 36 half-hour lectures that cover all the major topics of a full-year calculus course in high school at the College Board Advanced Placement AB level or a first-semester course in college. With crystal-clear explanations of the beautiful ideas of calculus, frequent study tips, pitfalls to avoid, and—best of all—hundreds of examples and practice problems that are specifically designed to explain and reinforce major concepts, this course will be your sure and steady guide to conquering calculus.
Your teacher for this intensively illustrated DVD set is Professor Bruce H. Edwards, an award-winning instructor at the University of Florida and the coauthor of a best-selling series of calculus textbooks.
Accomplish Mathematical Wonders
Calculus is one of the most powerful and astonishing tools ever invented, yet it is a skill that can be learned by anyone with an understanding of high school mathematics.
Among its many uses, calculus teaches you to
* analyze a multitude of situations involving change, whether it's an accelerating rocket, the growth of a bacterial colony, or fluctuating stock prices;
* calculate optimum values, such as the greatest volume for a box with a given surface area or the highest feasible profit from the sales of an item;
* measure complex shapes—for example, the volume of a doughnut-shaped object called a torus or the area of a plot of land bounded by a river.
Learn about Precalculus and Limits . . .
Solving many types of calculus problems usually requires employing precalculus—algebra and trigonometry—to work out a solution. For this reason, Professor Edwards devotes the first few lectures to reviewing key topics in precalculus, then he covers some basic concepts such as limits and continuity before moving on to the two simple, yet brilliant ideas behind calculus—the derivative and the integral.
Despite the apparent differences between the derivative and integral, you discover that they are inextricably linked by the surprising fundamental theorem of calculus. Throughout the course, you will discover that simplicity is one of the hallmarks of the essential ideas of calculus.
. . . the Power of the Derivative . . .
The derivative is the foundation of differential calculus, which you study through Lecture 17, exploring its many applications in science, engineering, business, and other fields.
You start with a classic problem that illustrates one of the core ideas of calculus: Can you find the tangent line to a curve at a given point? This is the same as asking if the rate of change of the curve can be measured at that point—with a host of potential applications in situations where a quantity is changing, such as the speed of an accelerating vehicle. The answer is: Yes, and with amazing simplicity! After learning the steps involved, you have solved your first calculus problem.
* study a variety of ways to find derivatives, including the power rule, the constant multiple rule, the quotient rule, the chain rule, and implicit differentiation;
* learn how to find extrema—the absolute maximum and minimum values of functions, using derivatives; and
* apply derivatives to solve a variety of real-world problems.
. . . and the Importance of the Integral
Next, you are introduced to the integral, using a classic problem in which you are asked to find the area of a plot of land bounded by curves. To solve this problem, calculus provides us with the integral—a powerful tool that allows us to calculate areas, volumes, and other characteristics of complex shapes. The balance of the course is devoted to integral calculus and its applications. You study
* arc length and surface area—two applications of calculus that are at the heart of engineering;
* integration by substitution—a method that enables you to convert a difficult problem into one that's easier to solve; and
* the formulas for continuous compound interest, radioactive decay, and a host of other real-world applications.
A Calculus Course for All
Understanding Calculus is well suited for anyone who wants to take the leap into one of history's greatest intellectual achievements, whether for the first time or for review. Those who will benefit include these learners:
* Any student now studying calculus who would like personal coaching from a professor who has spent years honing his explanations for the areas that are most challenging to students. This course is specifically designed to cover all the major topics of a full-year calculus course in high school at the College Board Advanced Placement AB level or a first-semester course in college.
* Parents of students studying calculus, a subject with which they often give up trying to help their high-school-age children—at a critical turning point in their educational careers.
* Those who have already taken calculus and who need a thorough review.
* Anyone who didn't understand calculus on the first try and wants a lucid, in-depth presentation, with lots of interesting, well-explained practice problems.
The plentiful graphs, equations, and other visual aids in these lectures are clear and well-designed, allowing you to follow each step of Professor Edwards's presentation in detail. The accompanying workbook includes lecture summaries, sample problems and worked-out solutions, tips, and pitfalls; lists of formulas and theorems; a trigonometry review sheet; a glossary; and a removable study sheet to use for quick and easy reference during the lectures.
The Ideal Calculus Teacher
Professor Edwards is the ideal calculus teacher—friendly, animated, encouraging, and witty, but also focused on presenting the material in an organized and understandable way. For anyone who feels intimidated by calculus, there is a distinct joy in being able to calculate a derivative after just a few lessons. It's easier than one might have supposed, and it opens an amazing new world of insight.
As an educator who has been honored repeatedly, both for his teaching and for his textbooks, Professor Edwards is a fount of valuable advice. He offers frequent tips for success, including guidance for those preparing for the Advanced Placement Calculus AB exam, for which he has served as a grader and for which this course is excellent preparation. Among his suggestions are these:
* Graphing calculators: While some calculus teachers prefer that their students not use graphing calculators, the Advanced Placement exam requires them. Professor Edwards points out the strengths of graphing calculators as well as the weaknesses—for example, that in certain situations they can fool you.
* Memorization: Always memorize what your teacher assigns. However, no one can memorize all the formulas in calculus. A good approach is to commit to memory the idea behind a technique—for example, that the disk method of computing the volume of a solid involves slicing it into innumerable disks.
Ever since its inception in the 17th century, calculus has spawned a continuing flood of new ideas and techniques for solving problems. It's easy to be overwhelmed by the richness of this subject, which is why many beginning students find themselves struggling.
Through Professor Edwards's exceptional teaching in Understanding Calculus, you will come away with a deep appreciation for the extraordinary power of calculus, a grasp of which methods apply to different types of problems, and, with practice, a facility for unlocking the secrets of the ceaselessly changing world around us.
About Your Professor
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 Bogotá, 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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