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Math Notes - Matt Huston MA161-Exponential Functions-An...

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MA161: January 16, 2008 -Exponential Functions -An exponential function is of the form f(x)=a^x -If x = -n then we set a^-n = 1/(a^n) -If x= p/q, where p and q are integers and q > 0, we set a^x = a ^ (p/q) -How do we define a^x when x is irrational? -For example: 1.4 < sqrt(2) < 1.5 so… 3^(1.4) < 3^sqrt(2) < 3^(1.5) we set 3^(sqrt(2)) = y y = 4.72 -As a gets larger in the graph of f(x) = a^x, a^x grows more rapidly. -3^x is steeper than 2^x which is steeper than ½^x -If 0 < a < 1, a^x gets smaller as x gets larger -LAWS OF EXPONENTS -a^(x+y) = a^x * a^y -a^(x-y) = a^x / a^y -(a^x)^y = a^x*y -(a*b)^x = a^x * b^x -Exponential functions always grow faster than any polynomial over a short amount of time. -Malthus: population growth is exponential -Population Growth -For example: Suppose a population of bacteria double every hours and that: p(0)=100 =100 p(1)=2p(0) =200 p(2)=2p(1) = 400 p(3)=2p(2) = 800 so… p(n) = 2^n * 100 -There is a unique number e between 2 and 3 so the slope of the tangent line of y = e^x at (0,1) is =1. -e = 2.71828
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This essay was uploaded on 04/07/2008 for the course PHI 114 taught by Professor Faris during the Spring '08 term at Purdue.

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Math Notes - Matt Huston MA161-Exponential Functions-An...

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