PP Section 5.7 - ExponentialGrowthandDecay Definition A...

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Exponential Growth and Decay
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Definition A quantity is said to grow or decay exponentially if the size of the quantity at any time t is given by 1 ), 0 ( , ) ( 0 0 = = b Q Q b Q t Q kt If k > 0, there is exponential growth. If k < 0, there is exponential decay. The most common model is when b = e.
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Example 10 6 bugs are mating and multiplying according to the equation P(t) = 10 6 (2 0.03t ). Assume t is in hours. What is the population in 12 hours? This is exponential growth with b = 2. 898 . 425 , 283 , 1 ) 2 ( 10 ) 12 ( ) 2 ( 10 ) ( ) 12 ( 03 . 0 6 03 . 0 6 = = P t P t
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Since we are counting bugs, we can NOT round up the answer. Hence, after 12 hours, there are 1,283,425 bugs. Unless otherwise specified, we will assume that exponential growth and decay refers to b = e. That is, the model for the behavior will be ) 0 ( , ) ( 0 0 Q Q e Q t Q kt = =
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Example N N 0 100 1 175 = = ( ) N t N e kt ( ) = 0 N e k ( ) ( ) 1 175 100 1 = = 175 . = e k ln . 175 = k k 05596 . A culture of bacteria increases exponentially.
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PP Section 5.7 - ExponentialGrowthandDecay Definition A...

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