Calculus II Notes 7.2 - Calculus II-Stewart Dr. Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 7.2 7.2 Calculus with Exponential Functions The Exponential Function Bacteria double their population at regular intervals through cell division. We speak of the half-life of radioactive material. Exponential functions arise naturally in the real world. Definition The exponential function with base b , with b > 0 and b 1 , has the form f ( x ) = b x . Given a positive integer n , it is easy to define b n as the product of n copies of b . Then we can define b n = 1 b n . For a rational number x = p / q with q > 0 , we define b p / q = q ( ) p = p q . Irrational exponents are a little harder. If x is irrational we define b x = lim r x b r for r rational. Example A We present the graphs of f ( x ) = 1/2 ( ) x and g ( x ) = 2 x . f ( x ) = ( ) x g ( x ) = 2 x
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Calculus II- Stewart Dr. Berg Spring 2010 Page 2 7.2 Theorem The exponential function f ( x ) = b x is a continuous function with domain ( −∞ , ) and range (0, ) . It is an increasing function if b > 1 and decreasing if 0 < b < 1 . Additional properties are: 1) a x a y = a x + y 2) a x y = a x b y 3) a x ( ) y = a xy 4) ab ( ) x = a x b x Note: The graph has a horizontal asymptote coincident with the x -axis.
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Calculus II Notes 7.2 - Calculus II-Stewart Dr. Berg Spring...

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