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India is the second most populous country in the world with a population of about $\text{\hspace{0.17em}}1.25\text{\hspace{0.17em}}$ billion people in 2013. The population is growing at a rate of about $\text{\hspace{0.17em}}1.2\%\text{\hspace{0.17em}}$ each year http://www.worldometers.info/world-population/. Accessed February 24, 2014. . If this rate continues, the population of India will exceed China’s population by the year $\text{\hspace{0.17em}}2031.$ When populations grow rapidly, we often say that the growth is “exponential,” meaning that something is growing very rapidly. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential functions , which model this kind of rapid growth.
When exploring linear growth, we observed a constant rate of change—a constant number by which the output increased for each unit increase in input. For example, in the equation $\text{\hspace{0.17em}}f(x)=3x+4,$ the slope tells us the output increases by 3 each time the input increases by 1. The scenario in the India population example is different because we have a percent change per unit time (rather than a constant change) in the number of people.
A study found that the percent of the population who are vegans in the United States doubled from 2009 to 2011. In 2011, 2.5% of the population was vegan, adhering to a diet that does not include any animal products—no meat, poultry, fish, dairy, or eggs. If this rate continues, vegans will make up 10% of the U.S. population in 2015, 40% in 2019, and 80% in 2050.
What exactly does it mean to grow exponentially ? What does the word double have in common with percent increase ? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.
For us to gain a clear understanding of exponential growth , let us contrast exponential growth with linear growth . We will construct two functions. The first function is exponential. We will start with an input of 0, and increase each input by 1. We will double the corresponding consecutive outputs. The second function is linear. We will start with an input of 0, and increase each input by 1. We will add 2 to the corresponding consecutive outputs. See [link] .
$x$ | $f(x)={2}^{x}$ | $g(x)=2x$ |
---|---|---|
0 | 1 | 0 |
1 | 2 | 2 |
2 | 4 | 4 |
3 | 8 | 6 |
4 | 16 | 8 |
5 | 32 | 10 |
6 | 64 | 12 |
From [link] we can infer that for these two functions, exponential growth dwarfs linear growth.
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