Ron Larson
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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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