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Understanding Trends Through Calculus I The Great Courses





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Uploaded on Jun 14, 2011

Welcome to calculus. It will be my great pleasure to guide you on a journey through a world of calculus during these 24 lectures. The tour begins at our doorsteps and takes us to the stars. Calculus is all around us every day of our lives. When we're driving down the road and we see where we are at every moment and we figure out how fast we're going, that's calculus. When we throw a baseball and see where it lands, that's calculus.

But, calculus is not restricted only to physical issues. When we lament the decline in the population of the spotted owl, that's calculus. When we analyze the stock market and we look at economic trends, that's calculus. Calculus comprises a collection of ideas that have had tremendous historical impact. And the reason is that calculus is enormously effective in allowing people to bend nature to human purpose. Much of the scientific description of our world is based on calculus; descriptions of motion, certainly; of electricity and magnetism; of sound waves or waterways—all of these involve calculus. But in addition to that, calculus is an essential tool for understanding social and biological sciences. It occurs every day when we describe economic trends, when we talk about population growth or decline, or medical treatments; all of the description are couched in terms of calculus. That is, its vocabulary, its notation, but most important, more important than any of those, are its ideas, its perspectives.

Well, in this course we're going to emphasize the ideas of calculus, the concepts of calculus, more than the mechanical side of calculus. I'd like to take the remaining part of this lecture to just tell you the structure of the upcoming lectures, the structure of the course.

So, we'll begin in Lectures Two, Three, and Four comprise a collection of lectures that really present the fundamental ideas of calculus. In Lecture Two we introduce the idea of the derivative and say what the definition is and what it means. Then, in Lecture Three, we do the same thing to the integral and say what the definition of the integral is, how it comes about. And, then, in Lecture Four, we introduce the fundamental theorem of calculus, which connects the two.

Well, after we get through Lecture Four, then we proceed to investigate each of these fundamental ideas in more detail. we're going to take the derivative and look at it from the point of view of its algebraic manifestations. One of the properties that makes the derivative so potent, and the integral, is that you can do it algebraically, and that's this mechanical side that students view as—the most common part that they deal with most is learning how to manipulate the algebra. So, we'll see the derivative then physically, that is, with the car moving; and then graphically; and then algebraically; and, then, we'll see it applied to different application areas such as volume, formulas for the volumes of objects—all of these things have manifestations about the derivative.

After that, we turn to the integral and we have a similar sequence of lectures that present the integral in these terms. That is to say, graphically, algebraically, and in terms of their applications to many different areas. So by taking these fundamental ideas and viewing them in different ways, that will show the richness of these themes. Then, the last half of this course will demonstrate the richness of these two ideas by showing lots of examples of their extensions, their variations, and their applications.

Well the purpose of these lectures is to explain clearly the concepts of calculus and to convince you that calculus can be understood from simple scenarios. Calculus is so effective because it deals with change and motion, and it allows us to view our world as a dynamic rather than just a static place. Calculus provides a tool for measuring change, whether it's a change in position, change in temperature, change in demand, or change in population. But, in addition to that, I like to think that calculus is intrinsically intriguing and beautiful, as well as just being merely important. So, calculus is a crowning intellectual achievement of humanity that all intelligent people can appreciate, enjoy, and understand. I look forward to exploring calculus with you during the next 23 lectures. Bye for now.

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