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Unformatted text preview: 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 AttributionNoncommercialShare 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 MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript Lecture 2 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. Okay so I'd like to begin the second lecture by reminding you what we did last time. So last time, we defined the derivative as the slope of a tangent line. So that was our geometric point of view and we also did a couple of computations. We worked out that the derivative of 1 / x was 1 / x^2. And we also computed the derivative of x ^ nth power for n = 1, 2, etc., and that turned out to be x, I'm sorry, nx^(n1). So that's what we did last time, and today I want to finish up with other points of view on what a derivative is. So this is extremely important, it's almost the most important thing I'll be saying in the class. But you'll have to think about it again when you start over and start using calculus in the real world. So again we're talking about what is a derivative and this is just a continuation of last time. So, as I said last time, we talked about geometric interpretations, and today what we're gonna talk about is rate of change as an interpretation of the derivative. So remember we drew graphs of functions, y = f(x) and we kept track of the change in x and here the change in y, let's say. And then from this new point of view a rate of change, keeping track of the rate of change of x and the rate of change of y, it's the relative rate of change we're interested in, and that's delta y / delta x and that has another interpretation. This is the average change. Usually we would think of that, if x were measuring time and so the average and that's when this becomes a rate, and the average is over the time interval delta x. And then the limiting value is denoted dy/dx and so this one is the average rate of change and this one is the instantaneous rate. Okay, so that's the point of view that I'd like to discuss now and give you just a couple of examples. So, let's see. Well, first of all, maybe some examples from physics here. So q is usually the name for a charge, and then dq/dt is what's known as current. So that's one physical example. A second example, which is probably the most tangible one, is we could denote know the letter s by distance and then the...
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 Fall '08
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 Calculus

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