398 CHAPTER 6 Algebra, Graphs, and Functions All exponential functions of the form = y ax or = < < f x a a ( ) , 0 1, x will have the general shape of the graph illustrated in Fig. 6.50. Example 12 United States Population Growth The United States population in 2021 was about 331.9 million people. Assume that the U.S. population continues to grow exponentially at the growth rate of about 0.1% per year. The expected U.S. population, P t( ), in millions of people, t years after 2021, can be estimated by the formula = P t( ) 331.9(1.001) . t Use this formula to estimate the U.S. population in the year 2031. Source: Census.gov Solution Since 2031 is 10 years after 2021, we substitute = t 10 into the formula. = = ≈ ≈ P t P ( ) 331.9(1.001) (10) 331.9(1.001) 331.9(1.01004512) 335.2339754 t 10 Thus, in 2031, the U.S. population is expected to be about 335.2 million people. 7 Now try Exercise 77 Linear, Quadratic and Exponential Growth Natural Exponential Functions When the a in the exponential function = f x a ( ) x is replaced with a very special letter e, we get the natural exponential function = f x e ( ) . x The exponential growth or decay formula is = P t P a ( ) . kt 0 Replacing the letter a with the letter e in the exponential growth or decay formula leads us to the natural exponential growth or decay formula. Natural Exponential Growth or Decay Formula = P t P e ( ) kt 0 The letter e represents an irrational number called the natural base whose value is approximately 2.7183. The number e plays an important role in mathematics and is used in determining the solution to many application problems. To evaluate e0.009(10) on a calculator, as will be needed in Example 13, press × = 2 ln (0.009 10 ) 1.094174284 nd Notice that there is not a box around the left-hand parentheses. After the ln key is pressed the calculator automatically generates the left-hand parentheses. After the = key is pressed, the calculator displays the answer 1.094174284. To evaluate e 7.89 0.009(10) on a calculator, press × × = 7.89 2 ln (0.009 10 ) 8.633035098 nd In this calculation, after the = key is pressed, the calculator displays the answer 8.633035098. This calculation will be used in Example 13.
RkJQdWJsaXNoZXIy NjM5ODQ=