(Solution) - Suppose yn represents the population of the world the nth -(2025 Original AI-Free Solution)

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Suppose yn represents the population of the world the nth year after 1 800. That is. y0 is the population in 1800, y1 is the population in 1801 and so on. From the definition of the derivative as the limit of a difference quotient, we can write

So Malthus's differential equation model dy/dt = 0.03y can be approximated by y(t + 1) - y(t) = 0.03y. that is. y(t + 1) = y(t) + 0.03y(t ). Thus writing yn for y(n) we obtain
yn + 1 = l .03yn·
(a) Use this discrete model (5 ) to estimate the population in the years 1801, 1802. 1803, . . . 1810.
(b) Estimate the world's population in the year 1 900using this discrete model. You might do this with a spreadsheet, using equation (5), or you might develop an algebraic formula by observing a pattern.
(c) Comment on the difference in results comparing this discrete process with Malthus's continuous model in Table 1.1.1 Explain.