terminology “per capita” appropriate? (By the way, you may be tempted to think of P′ as the derivative of P –and it is–but for the moment this is not as useful as thinking of it as the net growth rate. In any event since we do not have a formula for P we can’t “derive” P′ .)

2. In 1985 the per capita growth rate of Poland was 9 persons/year per 1000 persons. For Afghanistan it was 21.6 persons/year per 1000 persons. If P denotes the population of Poland and A the population of Afghanistan, write the rate equations for P′ and A′ that govern the growth of these two populations.

3. In 1985 the population of Poland was 37.5 million, and of Afghanistan 15 million. What were the net growth rates in 1985? Is it true that a population with a larger per capita growth rate also has a larger net growth rate? What is the correct conclusion that one can draw in comparing populations with different per capita growth rates?

4. If we use Δt to denote a time interval, and ΔP to denote the change in the population of Poland during this time interval, write an equation that shows how the quantities ΔP , P′ , and Δt are connected to one another (hint: if you consider the units for these quantities, that may help). Use this equation to compute the population of Poland in 1990, based on the available information.

5. Critique the solution you have just given. Can you see a way to to get a better estimate for the population in 1990? (Hint: what happens to P′ as soon as P changes?) Use your improved method, say with 10 steps, to track P from 1985 to 1990. Do the same for A. Sketch the graphs illustrating the population growth. The method is credited to Euler, and the graphs that you obtained are called piecewise linear.

The analysis that you made above can be put into a more familiar setting. Recall that P′ = lim_{Δt→0} \frac{ΔP}{Δt} . So if Δt is small, P′ is approximately equal to

\frac{ΔP}{Δt} or, equivalently, and much more usefully, ΔP ≈ P′Δt . What you have done above is apply this approximation over and over again: an initial value of P gives an initial value of P′ = rP , which gives ΔP , which enables us to update the value of P , which gives a new P′ , and on and on it goes.

6. Find the analytic solution (formula) for this differential equation P′ = rP , that is, find a formula for P as a function of t. (Hint: rewrite \frac{dP}{dt} = rP , separate variables, integrate both sides to “undo” the derivative, consolidate constants, do some algebra. The constant of integration can be expressed in terms of P_{0} , the value of P at t = 0 .)

7. Apply your formula to the cases of Poland and Afghanistan. What does the formula predict for the population of each in 1990? How long will it take for each population to double in size? What is plainly wrong with this model of population growth?