The population of the world for selected years from 1750 to 2009 is given in the

Question

The population of the world for selected years from 1750 to 2009 is given in the following table Year 1750 1800 1850 1900 1950 1990 2000 2009 Population791980 1,2601,6502,520 5,270 6,060 6,800 (millions) (a) Determine the exponential function that best fits the data. Use the func- tion to estimate the population in 1980. Make a plot of the points and the function (b) Curve-fit the data with a third-order polynomial. Use the polynomial to estimate the population in 1980. Make a plot of the points and the polyno- mial. (c) Fit the data with linear and spline interpolations. Estimate the population in 1975 with linear and spline interpolations. Make a plot of the data points and curves made of the interpolated points. In each part make a plot of the data points (circle markers) and the fit curve or the interpolation curves. Note that part (c) has two interpolation curves The actual population of the world in 1980 was 4453.8 million The following points are given 1.2 2.5 4 5.07 34 |-2.0 | -0.8 | 0 8.5 4.44.5 4 3.6 3.9 3.8 3.5 2.5 1.2 0.5 0.2 (a) Fit the data with a first-order polynomial. Make a plot of the points and (b) Fit the data with a second-order polynomial. Make a plot of the points and (c) Fit the data with a fourth-order polynomial. Make a plot of the points and (d) Fit the data with an eight-order polynomial. Make a plot of the points and the polvnomial. the polynomial the polynomial the polynomial

Explanation / Answer

1) a) let the function is f(x) = a.rx.....(i)

put (1750,791) in (i)

791 = a.r1750

so, a = 791/r1750........(ii)

now put (1800,980) in (i) and substitute value of a from (ii)

980 = 791.r1800/r1750

r50 = 980/791

r50 = 1.238

r = 1.2381/50 = 1.004

a = 791/r1750 = 0.73

so equation will be 0.73. 1.004x

now f91980) = 0.73. 1.0041980 = 1977

b) let f(x) = ax3+bx2+cx+d .....(i)

now put (1750,791),(1800,980),(1850,1260) and (1900,1650) in (i) we will have four equations and by solving them we will get the value of a,b,c,d and finally we will get the required polynomial.

c) let f(x) = a.x +b ....(i)

now putting (1750,791) and (1800,980) in (i) we get..

791 = 1750x + b and 980 = 1800x + b

by substracting both we will get 50x = 189

so, a = 3.78

and b = -5824

so linear equation is f(x) = 3.78x - 5824

now f(1975) = 7465.5 - 5824 = 1641

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