1a)
total = vector([20414, 21071, 21586, 21510, 21356, 22454, 22044, 22345, 22702, 22361, 22922, 23365, 23912, 23637, 23865, 23873, 23674, 24812])
year = vector(range(1979, 1997))
plotPoints = year.transpose().augment(total.transpose())
myGraph = scatter_plot(plotPoints.rows(), facecolor='white', edgecolor='purple', markersize=20)
myGraph
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1b)
var('a, b, x')
f(x)= a*x + b
fit = find_fit(plotPoints.rows(), f, parameters=[a, b], variables=[x], solution_dict=true)
g(x) = fit[a]*x + fit[b]
myGraph = myGraph + plot(g, 1979, 1996, color='orange', plot_points=2000, legend_label='linear fit')
myGraph
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1c)
total2 = vector([20414, 21071, 21586, 21510, 21356, 22454, 22044, 22345, 22702, 22361, 22922, 23365, 23912, 23637, 23865, 23873])
year2 = vector(range(1979, 1995))
plotPoints2 = year2.transpose().augment(total2.transpose())
var('a, b, x')
f(x)= a*x + b
fit = find_fit(plotPoints.rows(), f, parameters=[a, b], variables=[x], solution_dict=true)
h(x) = fit[a]*x + fit[b]
myGraph2 = scatter_plot(plotPoints2.rows(), facecolor='white', edgecolor='purple', markersize=20) + plot(h, 1979, 1994, color='orange', plot_points=2000, legend_label='linear fit')
myGraph2
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The linear fit predicts the enrollment of 1995 and 1996 fairly well. The error was no more than ±3%
print h(x)
print "1995\t-->\t", int(h(1995))
print "1996\t-->\t", int(h(1996))
214.83695672116494*x - 404327.1737055374 1995 --> 24272 1996 --> 24487 214.83695672116494*x - 404327.1737055374 1995 --> 24272 1996 --> 24487 |
1d)
Calculate the enrollments for 1995 and 1996
print 'Year when enrollment should reach 30,000: ', int(solve([g(x)== 30000], x, solution_dict=true)[0][x])
print 'Year when enrollment should reach 35,000: ', int(solve([g(x)== 35000], x, solution_dict=true)[0][x])
Year when enrollment should reach 30,000: 2021 Year when enrollment should reach 35,000: 2044 Year when enrollment should reach 30,000: 2021 Year when enrollment should reach 35,000: 2044 |
5)
Rational function where m=2 and n=2
A = matrix(RR,[[1, i*pi/20, (i*pi/20)^2, -(i*pi/20)*tan(i*pi/20), -(i*pi/20)^2*tan(i*pi/20)] for i in range(5)])
y = matrix(RR,[[tan(i*pi/20)] for i in range(5)])
c = A\y
g(x) = (c[0][0] + c[1][0]*x + c[2][0]*x^2) / (1 + c[3][0]*x + c[4][0]*x^2)
p = plot(tan(x), color='orange', zorder=6, legend_label='tan(x)') + plot(g, color='blue', plot_points=50, marker='.', legend_label='rational function')
p.xmin(0)
p.xmax(pi/2)
p.ymin(0)
p.ymax(1)
p
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Rational function where m=3 and n=4
A = matrix(RR,[[1, i*pi/20, (i*pi/20)^2, (i*pi/20)^3, -(i*pi/20)*tan(i*pi/20), -(i*pi/20)^2*tan(i*pi/20), -(i*pi/20)^3*tan(i*pi/20), -(i*pi/20)^4*tan(i*pi/20)] for i in range(3)])
y = matrix(RR,[[tan(i*pi/20)] for i in range(3)])
c = A\y
g(x) = (c[0][0] + c[1][0]*x + c[2][0]*x^2) / (1 + c[3][0]*x + c[4][0]*x^2)
p = plot(tan(x), color='orange', zorder=6, legend_label='tan(x)') + plot(g, color='blue', plot_points=50, marker='.', legend_label='rational function')
p.xmin(0)
p.xmax(pi/2)
p.ymin(0)
p.ymax(1)
p
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Rational function where m=2 and n=4
A = matrix(RR,[[1, i*pi/20, (i*pi/20)^2, -(i*pi/20)*tan(i*pi/20), -(i*pi/20)^2*tan(i*pi/20), -(i*pi/20)^3*tan(i*pi/20), -(i*pi/20)^4*tan(i*pi/20)] for i in range(2)])
y = matrix(RR,[[tan(i*pi/20)] for i in range(2)])
c = A\y
g(x) = (c[0][0] + c[1][0]*x + c[2][0]*x^2) / (1 + c[3][0]*x + c[4][0]*x^2)
p = plot(tan(x), color='orange', zorder=6, legend_label='tan(x)') + plot(g, color='blue', plot_points=50, marker='.', legend_label='rational function')
p.xmin(0)
p.xmax(pi/2)
p.ymin(0)
p.ymax(1)
p
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