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lec6 - 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 6 18.01 Fall 2006 Lecture 6: Exponential and Log, Logarithmic Differentiation, Hyperbolic Functions Taking the derivatives of exponentials and logarithms Background We always assume the base, a , is greater than 1. a 0 = 1; a 1 = a ; a 2 = a a ; . . . · a x 1 + x 2 = a x 1 a x 2 ( a x 1 ) x 2 = a x 1 x 2 p q q a = a p (where p and q are integers) r To define a for real numbers r , fill in by continuity. d Today’s main task: find a x dx We can write d a x x x x a = lim a dx Δ x 0 Δ x We can factor out the a x : x x x Δ x Δ x lim a a = lim a x a 1 = a x lim a 1 Δ x 0 Δ x Δ x 0 Δ x Δ x 0 Δ x Let’s call M ( a ) lim a Δ x 1 Δ x 0 Δ x We don’t yet know what M ( a ) is, but we can say d a x = M ( a ) a x dx Here are two ways to describe M ( a ): d 1. Analytically M ( a ) = a x at x = 0. dx Indeed, M ( a ) = lim a 0+Δ x a 0 = d a x Δ x 0 Δ x dx x =0 1
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Lecture 6 18.01 Fall 2006 M(a) (slope of a x at x=0) a x Figure 1: Geometric definition of M ( a ) x 2. Geometrically, M ( a ) is the slope of the graph y = a at x = 0. The trick to figuring out what M ( a ) is is to beg the question and define e as the number such that M ( e ) = 1. Now can we be sure there is such a number e ? First notice that as the base a x increases, the graph a gets steeper. Next, we will estimate the slope M ( a ) for a = 2 and a = 4 geometrically. Look
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