Explanation Given information: The per capita production is 2.0 b t ( 1 − b t 1000 ) and values of b t are assumed to be less than 1000. Formula used: The new population is the product of old population and per capita production in a model of growing population. The power rule of differentiation , d d x x n = n x n − 1 . If the second derivative of the function is negative then the function is concave down. If the second derivative of the function is positive then the function is concave up. Calculation: It is provided that the per capita production is 2.0 b t ( 1 − b t 1000 ) and values of b t are assumed to be less than 1000. Recall that the new population is the product of old population and per capita production in a model of growing population. Denote the old population by b , so the new population f ( b ) is, f ( b ) = 2.0 b ( 1 − b 1000 ) ⋅ b = 2.0 b 2 ( 1 − b 1000 ) = 2.0 b 2 − 2.0 b 3 1000 Differentiate the above expression with respect to b . Recall the power rule of differentiation, d d x x n = n x n − 1 . Apply it, d d b f ( b ) = d d b [ 2.0 b 2 − 2.0 b 3 1000 ] = 4

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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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 2.7, Problem 42E

To determine

To calculate: The second derivative of the new population as a function of old population where per capita production is 2.0bt(1−bt1000) . Sketch the graph of the function obtained.

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Chapter 2.7, Problem 41E

Chapter 2.7, Problem 43E

Chapter 2 Solutions

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

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The second derivative of the new population as a function of old population where per capita production is 2.0 b t ( 1 − b t 1000 ) . Sketch the graph of the function obtained.

Solution Summary: The author calculates the second derivative of the new population as a function of old population.

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To calculate: The second derivative of the new population as a function of old population where per capita production is 2.0bt(1−bt1000) . Sketch the graph of the function obtained.

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Chapter 2 Solutions