plots (or scatter diagrams) and least squares linear fit.

Chapter 3, Problem 23E

(choose chapter or problem)

Regression Line Read the documentation for your CAS on scatter plots (or scatter diagrams) and least squares linear fit The straight line that best fits a set of data points is called a regression line or a least squares line. Your task is to construct a logistic model for the population of the United States, defining f (P) in (2) as an equation of a regression line based on the population data in the table in Problem 4. One way of doing this is to approximate the left-hand side \(\frac{1}{P} \frac{d P}{d t}\) of the first equation in (2), using the forward difference

quotient in place of dP/dt:

\(Q(t)=\frac{1}{P(t)} \frac{P(t+h)-P(t)}{h} .\)

(a) Make a table of the values t, P(t), and Q(t) using t = 0, 10, 20, . . . , 160 and h = 10. For example, the first line of the table should contain t = 0, P(0), and Q(0). With P(0) = 3.929 and P(10) = 5.308,

\(Q(0)=\frac{1}{P(0)} \frac{P(10)-P(0)}{10}=0.035\)

Note that Q(160) depends on the 1960 census population P(170). Look up this value.

(b) Use a CAS to obtain a scatter plot of the data (P(t), Q(t)) computed in part (a). Also use a CAS to find an equation of the regression line and to superimpose its graph on the scatter plot.

(c) Construct a logistic model dP/dt  Pf(P), where f(P) is the equation of the regression line found in part (b).

(d) Solve the model in part (c) using the initial condition P(0) = 3.929.

(e) Use a CAS to obtain another scatter plot, this time of the ordered pairs (t, P(t)) from your table in part (a). Use your CAS to superimpose the graph of the solution in part (d) on the scatter plot.

(f) Look up the U.S. census data for 1970, 1980, and 1990. What population does the logistic model in part (c) predict for these years? What does the model predict for the U.S. population P(t) as \(t \longrightarrow \infty\)?

Text Transcription:

frac{1}{P} \frac{d P}{d t}

Q(t)=\frac{1}{P(t)} \frac{P(t+h)-P(t)}{h}

t \longrightarrow \infty