lec21 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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Lecture 21 18.01 Fall 2006 Lecture 21: Applications to Logarithms and Geometry Application of FTC 2 to Logarithms The integral defnition ±unctions like C ( x ), S ( x ) Fresnel makes them nearly as easy to use as elementary ±unctions. It is possible to draw their graphs and tabulate values. You are asked to carry out an example or two this on your problem set. To get used to using defnite integrals and FTC2, we will discuss in detail the simplest integral that gives rise to a relatively new ±unction, namely the logarithm. Recall that n +1 x x n dx = + c n + 1 except when n = 1. It ±ollows that the antiderivative 1 /x is not a power, but something else. So let us defne a ±unction L ( x ) by x dt L ( x ) = t 1 (This ±unction turns out to be the logarithm. But recall that our approach to the logarithm was ±airly involved. We frst analyzed a x , and then defned the number e , and fnally defned the logarithm as x the inverse ±unction to e . The direct approach using this integral ±ormula will be easier.) All the basic properties L ( x ) ±ollow directly ±rom its defnition. Note that L ( x ) is defned ±or 0 < x < . (We will not extend the defnition past x = 0 because 1 /t is infnite at t = 0.) Next, the ±undamental theorem calculus (FTC2) implies 1 L ( x ) = x Also, because we have started the integration with lower limit 1, we see that 1 dt L (1) = = 0 t 1 Thus L is increasing and crosses the x -axis at x = 1: L ( x ) < 0 ±or 0 < x < 1 and L ( x ) > 0 ±or x > 1. Differentiating a second time, L ( x ) = 1 /x 2 It ±ollows that L is
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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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lec21 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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