18-01F07-L03 - 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 3 The following 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: In today's lecture I want to develop several more formulas that will allow us to reach our goal of differentiating everything. So these are derivative formulas, and they come in two flavors. The first kind is specific, so some specific function we're giving the derivative of. And that would be, for example, x^n or (1/x) . Those are the ones that we did a couple of lectures ago. And then there are general formulas, and the general ones don't actually give you a formula for a specific function but tell you something like, if you take two functions and add them together, their derivative is the sum of the derivatives. Or if you multiply by a constant, for example, so (cu), the derivative of that is (cu)' where c is constant. All right, so these kinds of formulas are very useful, both the specific and the general kind. For example, we need both kinds for polynomials. And more generally, pretty much any set of forumulas that we give you, will give you a few functions to start out with and then you'll be able to generate lots more by these general formulas. So today, we wanna concentrate on the trig functions, and so we'll start out with some specific formulas. And they're going to be the formulas for the derivative of the sine function and the cosine function. So that's what we'll spend the first part of the lecture on, and at the same time I hope to get you very used to dealing with trig functions, although that's something that you should think of as a gradual process. Alright, so in order to calculate these, I'm gonna start over here and just start the calculation. So here we go. Let's check what happens with the sine function. So, I take sin (x delta x), I subtract sin x and I divide by delta x. Right, so this is the difference quotient and eventually I'm gonna have to take the limit as delta x goes to 0. And there's really only one thing we can do with this to simplify or change it, and that is to use the sum formula for the sine function. So, that's this. That's sin x co delta x plus Oh, that's not what it is? OK, so what is it? Sin x sin delta x. OK, good. Plus cosine.
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18-01F07-L03 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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