2 (MATH 4300-01) Due date: Friday, Sept. 25, 2015 1. Name (Print): The following data describe the changes of a certain population P in time t (in days). Develop a model for the data based on the logistic model P = a / [1 + b ect)]. t 1 2 3 4 5 P 43 58 76 90 97 6 7 106 112 a) Rewrite this function as a linear relation between redefined variables. b) Graph the data such that the linearity assumption can be tested. Compare the data in this plot with a linear function to find the model parameters. c) Present the model obtained and graph both the model and the original data given in the table. Graph the relative error of the model. d) Use the model to predict the time at which the initial population at t = 0 is increased by a factor of 2. 2. Consider the development of the world population in time from 1804-2050 according to the Decennial Censuses, U.S. Census Bureau, U.S. Dept. of Commerce (World Almanac 2010). The population P is measured in 109 and t refers to the year. The last two population values are projections. t P 1804 1927 1960 1974 1987 1999 2009 2025 2050 1.0 2.0 3.0 4.0 5.0 6.0 6.77 7.95 9.32 Assume that the population P can be described by the function a P d, 1 b ect where a, b, c, and d are any constants. For a certain time period before 1804, the population can be approximated by a constant value P = 1. Assume that the population density levels off finally at a value of P = 11. a) Rewrite the model for P as a linear relation between redefined variables. b) Graph the data such that the linearity assumption can be tested. Compare the data in this plot with a linear function to find the model parameters. c) Present the model obtained and graph both the model and the original data given in the table. Graph the relative error of the model. d) Find the time at which the population change dP / dt has a maximum. The thought process is below 1. t 1804 1927 1960 1974 1987 1999 2009 2025 2050 x 0 123 156 170 183 195 205 221 246 Y For the above data, the points for t are given starting from 1804. Assuming it to be t=0 and counting each year as 1 unit on the x-axis calculate the other points. Therefore, point 1999 is calculated as 1999-1804=195, 20091804=205 etc. 2. The logistics model is The addition of d to the logistics model shifts the curve upwards by d units. Since the minimum value of the curve is 1 unit implies the logistics curve will be shifted by 1 unit. Hence, the choice of d=1 and a=11 for our model Once, the values of a and d are approximated, the model is linearized by taking logs and Linear regression is used to calculate best-fit line. From the best fit line, the slope and intercept of the best fit line is calculated and from it the parameters b and c of the model are determined. The thought process is below 1. t 1804 1927 1960 1974 1987 1999 2009 2025 2050 x 0 123 156 170 183 195 205 221 246 Y For the above data, the points for t are given starting from 1804. Assuming it to be t=0 and counting each year as 1 unit on the x-axis calculate the other points. Therefore, point 1999 is calculated as 1999-1804=195, 20091804=205 etc. 2. The logistics model is The addition of d to the logistics model shifts the curve upwards by d units. Since the minimum value of the curve is 1 unit implies the logistics curve will be shifted by 1 unit. Hence, the choice of d=1 and a=11 for our model Once, the values of a and d are approximated, the model is linearized by taking logs and Linear regression is used to calculate best-fit line. From the best fit line, the slope and intercept of the best fit line is calculated and from it the parameters b and c of the model are determined