Project 2.6 -- Families of Implicit Curves¶
In this notebook we complete the project for section 2.6 of Stewart's book.
This notebook was made using SageMath (http://sagemath.org) in a browser-based Jupyter notebook (http://jupyter.org).
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version()
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%display latex
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(x,y) = var('x y');
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left = y**2 - 2*x**2*(x+8);
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right = (y+1)**2*(y+9) - x**2;
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c =[None]*2;
c[0] = implicit_plot(left == 0,(x,-1,1),(y,-1.5,-0.5));
c[1] = implicit_plot(left == 2*right,(x,-1,1),(y,-1.5,-0.5),color='red');
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show(sum(c),title='Part 1.a')
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c.append(implicit_plot(left == 5*right,(x,-1,1),(y,-1.5,-0.5),color='green'));
c.append(implicit_plot(left == 10*right,(x,-1,1),(y,-1.5,-0.5),color='purple'));
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show(sum(c),title='Part 1.b')
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k = var('k');
f = x**2 + y**2 + k*x**2*y**2;
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cc = [None];
for j in [-6..6]:
if j == -6:
cc[0] = implicit_plot(f(k = j/3) == 1,(x,-2,2),(y,-2,2),color=hue((12+j)/18));
else:
cc.append(implicit_plot(f(k = j/3) == 1,(x,-2,2),(y,-2,2),color=hue((12+j)/18)));
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show(sum(cc),title='Part 2.a')
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implicit_plot(f(k = -1) == 1,(x,-2,2),(y,-2,2),title='Part 2.b')
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var('x k');
y = function('y')(x);
ff = x**2 + y**2 + k*x**2*y**2;
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impd = ff.diff(x);
z = solve(impd == 0,y.diff(x))
z[0]
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In [16]:
impd(k = -1)
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