Explanation Given data: y The population of bacteria grow according to the function P ( t ) = 10 e t and individual mass function is m ( t ) = 1 − t 2 . Method/Formula used: If m ( t ) is the dynamic mass of individual bacteria and P ( t ) the bacteria growth function then at any time total mass is M ( t ) = P ( t ) × m ( t ) Calculation: From the given information, any time‘ t’ the total bacterial mass is M ( t ) = P ( t ) × m ( t ) = 10 e t × ( 1 − t 2 ) Or M ( t ) = 10 ( 1 − t 2 ) e t … (1) At t = 0 , the total mass is M ( 0 ) = 10 ( 1 − 0 2 ) e 0 = 10 We observe that the total mass function is zero when ( 1 − t 2 ) = 0 ⇒ t 2 =1 ⇒ t =± 1 ⇒ t =±1 The first derivative of M ( t ) is M ′ ( t ) = 10 [ ( 1 − t 2 ) ⋅ d ( e t ) d t + d ( 1 − t 2 ) d t ⋅ e t ] = 10 [ ( 1 − t 2 ) ⋅ e t − 2 t ⋅ e t ] = 10 [ ( 1 − t 2 ) ⋅ e t − 2 t ⋅ e t ] = 10 e t [ 1 − t 2 − 2 t ] = − 10 e t ( t 2 + 2 t − 1 ) ⇒ M ′ ( t ) = − 10 e t ( t 2 + 2 t − 1 ) … (2) From result (2) we observe that (i) The point t =-1+ 1 =0 .414 is the critical point of the function as M ′ ( t ) = 0 ⇒ − 10 e t ( t 2 + 2 t − 1 ) = 0 ⇒ t = − 2 ± 4 + 4 2 = − 1 + 2 , − 1 − 1

Modeling the Dynamics of Life: Calculus and Probability for Life Scientists

3rd Edition

ISBN: 9780840064189

Author: Frederick R. Adler

Publisher: Cengage Learning

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Chapter 2.8, Problem 37E

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To graph: The total mass function of bacteria by finding the first and second derivative of the mass function when population growth function and individual mass function are given. Also to tell, is the total mass ever greater than it is at t=0 and when does the total mass reach zero.

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Chapter 2.8, Problem 36E

Chapter 2.8, Problem 38E

Chapter 2 Solutions

Modeling the Dynamics of Life: Calculus and Probability for Life Scientists

Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

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To graph: The total mass function of bacteria by finding the first and second derivative of the mass function when population growth function and individual mass function are given. Also to tell, is the total mass ever greater than it is at t = 0 and when does the total mass reach zero.

Solution Summary: The author analyzes the total mass function of bacteria by finding the first and second derivatives of the mass functions.

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To graph: The total mass function of bacteria by finding the first and second derivative of the mass function when population growth function and individual mass function are given. Also to tell, is the total mass ever greater than it is at t=0 and when does the total mass reach zero.

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Chapter 2 Solutions