Stewart6e1_5

# Stewart6e1_5 - MATH 191, Sections 1 and 3 Calculus I Fall...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 1.5: Fun with Exponential Functions In order to understand the function f ( x ) = a x (here, and always, a > 0), we can play around with what a x must be for various values of x : 1. Positive integers. Of course, a 1 = a , a 2 = a · a , and so forth. That is, a n is a multiplied by itself n times, whenever n is any positive integer. This gives our ﬁrst rule for exponents, as follows: since a m · a n = ( a · a · ··· · a )( a · a · ··· · a ) where there are a s in the ﬁrst term, and a s in the second, we have a m · a n = for any positive integers m and n . 2. Other integers. For the above rule to hold for all integers, not just positive ones, we must be able to let n = 0: a m · a 0 = a m +0 = a m . For this to be so, we must have a 0 = for any a . Also, letting m = - n , we get a - n · a n = a - n + n = a 0 = 1 . From this it follows that a - n = must be true, ﬁnishing our compu- tation of a n for any integer. Note this also gives another, related, rule for

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## This note was uploaded on 04/08/2008 for the course MATH 191 taught by Professor Bahls during the Fall '07 term at UNC Asheville.

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Stewart6e1_5 - MATH 191, Sections 1 and 3 Calculus I Fall...

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