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4.1 Exponential Growth

4.1 Exponential Growth
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  • 21. Logistic Curve

    In general, the size of a population (people, bacteria, cancer cells, etc.) follows a logistic growth pattern. The graph illustrates the growth patterns of a logistic curve. Compare an exponential curve to a logistic curve.

    • Worked-Out Solution

      Logistic growth starts off like exponential growth. At a certain point, however, the rate of growth begins to decrease. Eventually the population has reached a maximum size and is barely growing.

      Exponential growth does not taper off. It continues to grow more and more rapidly as time increases.

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     __   __    ______    _____      ___      _____   
     \ \\/ //  /_   _//  |__  //    / _ \\   /  ___|| 
      \   //    -| ||-     / //    / //\ \\ | // __   
      / . \\    _| ||_    / //__  |  ___  ||| \\_\ || 
     /_//\_\\  /_____//  /_____|| |_||  |_|| \____//  
     `-`  --`  `-----`   `-----`  `-`   `-`   `---`   
                                                      
    
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  • 22. Comparing Logistic and Exponential Growth

    In general, the size of a population (people, bacteria, cancer cells, etc.) follows a logistic growth pattern. The graph illustrates the growth patterns of a logistic curve. Why is the size of a population over time better represented by a logistic growth model than an exponential growth model?

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      ______   _    _     ____      _____     ______  
     /_   _// | || | ||  |  _ \\   |  ___||  /_   _// 
       | ||   | || | ||  | |_| ||  | ||__    `-| |,-  
      _| ||   | \\_/ ||  | .  //   | ||__      | ||   
     /__//     \____//   |_|\_\\   |_____||    |_||   
     `--`       `---`    `-` --`   `-----`     `-`'   
                                                      
    
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  • 23. World Population Growth

    In general, the size of a population (people, bacteria, cancer cells, etc.) follows a logistic growth pattern. The graph illustrates the growth patterns of a logistic curve. Based on the information in the table, what part of the graph represents the world population? Explain.

    • Worked-Out Solution

      The annual growth rate is decreasing. If this 30-year period is representative of the future growth of world population, then world population growth has moved into the blue part of the logistic curve.

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      ______    ______    ____     __   __   __   __  
     /_____//  /_   _//  |  _ \\   \ \\/ //  \ \\/ // 
     `____ `    -| ||-   | |_| ||   \ ` //    \   //  
     /___//     _| ||_   | .  //     | ||     / . \\  
     `__ `     /_____//  |_|\_\\     |_||    /_//\_\\ 
     /_//      `-----`   `-` --`     `-`'    `-`  --` 
     `-`                                              
    
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  • 24. Example 6

    To find the annual growth rate represented by Moore's Law, use the formula for exponential growth, A = P(1 + r)n . Because the number of transistors doubles every 2 years, you can substitute 2P for A and 2 for n in the formula. Then solve for r as shown. Suppose the number of transistors triples every two years. What is the annual growth rate?

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      _____     _____   __    __    _____    __   _   
     / ____||  |  ___|| \ \\ / //  |  ___|| | || | || 
    / //---`'  | ||__    \ \/ //   | ||__   | '--' || 
    \ \\___    | ||__     \  //    | ||__   | .--. || 
     \_____||  |_____||    \//     |_____|| |_|| |_|| 
      `----`   `-----`      `      `-----`  `-`  `-`  
                                                      
    
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  • 25. Tuition

    What is the annual growth rate of tuition?

    • Worked-Out Solution

      To estimate the annual rate of growth, you need to find a value of r such that

      After some trial and error, you can see that the value of r is about 0.154. This means that the annual rate of growth was about 15.4%.

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      ______    ______   _____       ___      ______  
     /_____//  /_   _// |  __ \\    / _ \\   /_   _// 
     `____ `    -| ||-  | |  \ ||  | / \ ||    | ||   
     /___//     _| ||_  | |__/ ||  | \_/ ||   _| ||   
     `__ `     /_____// |_____//    \___//   /__//    
     /_//      `-----`   -----`     `---`    `--`     
     `-`                                              
    
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  • 26. Tuition

    Using the rate from Exercise 25, what will tuition be in three more years?

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      _____    _    _                ___     ______   
     /  ___|| | || | ||   ____      / _ \\  |      \\ 
    | // __   | || | ||  |    \\   / //\ \\ |  --  // 
    | \\_\ || | \\_/ ||  | [] ||  |  ___  |||  --  \\ 
     \____//   \____//   |  __//  |_||  |_|||______// 
      `---`     `---`    |_|`-`   `-`   `-` `------`  
                         `-`                          
    
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