Modeling the Dynamics of Life: Calculus and Probability for Life Scientists
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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Exponential Growth Model
The relation that represents an exponential function is:
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Chapter 2.1, Problem 24E

a

To determine

To calculate: To find the average rate of change between times 0 and 1.0 from the population size.

a

Expert Solution
Check Mark

Answer to Problem 24E

The average rate of change between times 0 and 1 is =1

Explanation of Solution

Given information: b(t)=2.0t

Calculation:

Consider the following equation for a population size.

  b(t)=2.0t

The population at time t=0 is

  b(0)=20           =1The population at time t = 1 is shown below:b(1)=21          =2

The population at time t= 0.1 is as follows:

  b(0.1)=20.1               =1.071773

The population at time t=0.01 is

  b(0.01)=20.01                 =1.006955

The population at time t=0.001 is shown below:

  b(0.001)=20.001                    =1.000693

For a given formula, the average rate of change between times t1 and t2 is

  average rate of change =𑨀b𑨀t                                                =b(t1)−b(t2)t1−t2

The average rate of change between times 0 and 1 is

  average rate of change =b(t1)−b(t2)t1−t2                                                  =b(0)−b(1)0−1                                                  =1−2−1                                                   =1                                                  

b

To determine

To calculate: To find the average rate of change between times 0 and 0.1 from the population size.

b

Expert Solution
Check Mark

Answer to Problem 24E

The average rate of change between times 0 and 0.1 is 0.71773

Explanation of Solution

Given information: b(t)=2.0t

Calculation:

The average rate of change between times 0 and 0.1 is as follows:

  average rate of change =b(t1)−b(t2)t1−t2                                                  =b(0)−b(0.1)0−0.1                                                  =1−1.071773−0.1                                                   =0.71773

c

To determine

To calculate: To find the average rate of change between times 0 and 0.01 from the population size.

c

Expert Solution
Check Mark

Answer to Problem 24E

The average rate of change between times 0 and 0.01 is 0.6955

Explanation of Solution

Given information: b(t)=2.0t

Calculation:

The average rate of change between times 0 and 0.01 is

  average rate of change =b(t1)−b(t2)t1−t2                                                  =b(0)−b(0.01)0−0.01                                                  =1−1.006955−0.01                                                   =0.6955

d

To determine

To calculate: To find the average rate of change between times 0 and 0.001 from the population size.

d

Expert Solution
Check Mark

Answer to Problem 24E

The average rate of change between times 0 and 0.001 is 0.693

Explanation of Solution

Given information: b(t)=2.0t

Calculation:

The average rate of change between times 0 and 0.001 is shown below:

  average rate of change =b(t1)−b(t2)t1−t2                                                  =b(0)−b(0.001)0−0.001                                                  =1−1.000693−0.001                                                   =0.693

e

To determine

To calculate: To find the limit using the obtained values

e

Expert Solution
Check Mark

Answer to Problem 24E

The limit is 0.693.

Explanation of Solution

Given information: b(t)=2.0t

Calculation:

The limit is 0.693.

f

To determine

To calculate: To graph the tangent line using the obtained values

f

Expert Solution
Check Mark

Answer to Problem 24E

Hence the graph is drawn and the equation of the tangent line is

  b^(t)=b(0)+m(t−0)          =1+0.693t

Explanation of Solution

Given information: b(t)=2.0t

Calculation:

Since the limit is 0.693, then the slope of the tangent is 0.693

The equation of the tangent line is

  b^(t)=b(0)+m(tβˆ’0)          =1+0.693t

The graph of the tangent line is shown below:

  Modeling the Dynamics of Life: Calculus and Probability for Life Scientists, Chapter 2.1, Problem 24E

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Chapter 2 Solutions

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

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