18-01F07-L15 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Please use the following citation format: David Jerison, 18.01 Single Variable Calculus, Fall 2007 . (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript – Lecture 15 The following content is provided under a Creative Commons license. your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Today we're moving on from theoretical things from the mean value theorem to the introduction to what's going to occupy us for the whole rest of the course, which is integration. So, in order to introduce that subject, I need to introduce for you a new notation, which is called differentials. I'm going to tell you what a differential is, and we'll get used to using it over time. If you have a function which is y = f ( x), then the differential of y is going to be denoted dy, and it's by definition f' ( x ) dx. So here's the notation. And because y is really equal to f, sometimes we also call it the differential of f. It's also called the differential of f. That's the notation, and it's the same thing as what happens if you formally just take this dx, act like it's a number and divide it into dy. So it means the same thing as this statement here. And this is more or less the Leibniz, not Leibniz, interpretation of derivatives. Of a derivative as a ratio of these so called differentials. It's a ratio of what are known as infinitesimals. Now, this is kind of a vague notion, this little bit here being an infinitesimal. It's sort of like an infinitely small quantity. And Leibniz perfected the idea of dealing with these intuitively. And subsequently, mathematicians use them all the time. They're way more effective than the notation that Newton used. You might think that notations are a small matter, but they allow you to think much faster, sometimes. When you have the right names and the right symbols for everything. And in this case it made it very big difference. Leibniz's notation was adopted on the Continent and Newton dominated in Britain and, as a result, the British fell behind by one or two hundred years in the development of calculus. It was really a serious matter. So it's really well worth your while to get used to this idea of ratios. And it comes up all over the place, both in this class and also in multivariable calculus. It's used in many contexts. So first of all, just to go a little bit easy. We'll illustrate it by its use in linear
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This note was uploaded on 01/18/2012 for the course MATH 18.01 taught by Professor Brubaker during the Fall '08 term at MIT.

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18-01F07-L15 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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