Question

A country's census lists the population of the country as 242 million in 1990, 286 million in 2000, and 322 million in 2010. Fit a second- degree polynomial passing through these three points. (Let the year 2000 be x 0 and let p(x) represent the population in millions.) p(x) million Use this polynomial to predict the populations in 2020 and in 2030. 2020 2030 million million

Answer 1

Let by given condition, the three data points for p (x) be x_0 = -10 x_1 = 0 x_2 = 10 and p (x_0) = 242 p (x_1) = 286 p (x_2) = 322 Find the Lagranges polynomial, which is the unique second degree polynomial satisfy given data for p (x) then, p (x) = (x - x_0) (x - x_1)/(x_2 - x_0) (x_2 - x_1) middot p (x_2) + (x - x_0) (x - x_2)/(x_1 - x_0) (x_1 - x_2) middot p (x_1) + (x - x_1) (x - x_2)/(x_0 - x_1) (x_0 - x_2) middot p (x_0) = x (x + 10)/200 middot 322 + (x + 10) (x - 10)/(- 100) middot 286 + x (x - 10)/200 middot 242 = 1/100 (161. (x^2 + 10 x) - (x^2 - 100) middot 286 + 121 (x^2 - 10 x) = 1/100 (- 4 x^2 + 400 x + 28600) = -(0.04) x^2 + (4 x)

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