Exercises

The data in each of the following tables (Exercises 1-4) approximately fit either an exponential function or a power function. In each case, decide which, and find a formula that fits the data. Each formula should have one of the forms \(y=c b^t\), \(y=c e^{kt}\), or \(y=c t^k\), with explicit values for the constants. You may use the Plot Data tool to decide which formula to try, or you can do these exercises entirely with the numeric calculator. The Plot Data tool has the data for Exercise 1. You can edit the data entry for Exercises 2, 3, 4.

5. The data in Table E1 are graphed in Figures E1, E2, and E3, in Cartesian, semilog, and log-log plots, not necessarily in that order.
   

   1. Decide whether the data come from a linear function, a power function, an exponential function, or none of these.

   2. Estimate \(f(2)\).

   3. Estimate \(f\,' (2)\).

   
   

Table E1  Values of a function \(t\) \(f(t)\) 0.27 1.618 0.80 2.786 1.33 3.592 1.86 4.248 2.39 4.815 2.92 5.322 3.45 5.785 3.98 6.213 4.51 6.614 5.04 6.992 5.57 7.350	Figure E1  A plot of data in Table E1
Figure E2  A plot of data in Table E1	Figure E3  A plot of data in Table E1

6. The data in Table E2 are graphed in Figures E4, E5, and E6, in Cartesian, semilog, and log-log plots, not necessarily in that order.
   
   
   1. Decide whether the data come from a linear function, a power function, an exponential function, or none of these.
   2. Find a formula for \(f\) as a function of \(t\) .
      
      

Table E2  Values of a function \(t\) \(f(t)\) 0.27 1.618 0.80 2.240 1.33 2.664 1.86 3.008 2.39 3.512 2.92 3.936 3.45 4.360 3.98 4.784 4.51 5.208 5.04 5.632 5.57 6.056	Figure E4  A plot of data in Table E2
Figure E5  A plot of data in Table E2	Figure E6  A plot of data in Table E2

 

7. 

   Table E3   Population
   of Mexico

   Year	Population (millions)
   1980	67.38
   1981	69.13
   1982	70.93
   1983	72.77
   1984	74.66
   1985	76.60
   1986	78.59

   Table E3 shows the population of Mexico for each of the years 1980 through 1986.
   
   

   1. Find an exponential function that approximately fits these data.
      
      
   2. By what percentage did the population grow each year?
      
      
   3. What does your formula predict for the population in 2010? (The actual population in 2010 was about 111 million.)
      
      

    

    
8. 

   Table E4   Houston
   area population

   Year	Population
   1850	18,632
   1860	35,441
   1870	48,986
   1880	71,316
   1890	86,224
   1900	134,600
   1910	185,654
   1920	272,475
   1930	455,570
   1940	646,869
   1950	947,500
   1960	1,430,394
   1970	1,999,316
   1980	2,905,344
   1990	3,731,014
   2000	4,671,230
   2010	5,946,800

   Table E4 shows the U.S. Census data for the Houston, Texas, primary statistical metropolitan area over a period of 130 years.
   
   

   1. Show that the historic population data for Houston are approximated by an exponential function, and find a formula for the function.
      
      
   2. According to your model formula, how long does it take for Houston's population to double?
      
      
   3. If the growth pattern continues into the future, what will the Houston area population be in 2050?