14-8 Lagrange Multipliers.sagews

var('y')
implicit_plot(x^3 + y^3 == 16, (x, -4, 4), (y, -4, 4))
A = plot3d(exp(x*y), (x, -2.1, 2.1), (y, -2.1, 2.1), aspect_ratio = [20, 20, 1])
B = parametric_plot3d([t, (16 - t^3)^(1/3), exp(t*(16 - t^3)^(1/3))], (t, -2, 2.5), color = 'black')
show(A + B)
C = contour_plot(exp(x*y), (x, -4, 4), (y, -4, 4), contours = [20, 40, 60, 80, 100, 120], fill = False)
D = implicit_plot(x^3 + y^3 == 16, (x, -4, 4), (y, -4, 4), color = 'green')
show(C + D)

y

3D rendering not yet implemented

var('y', 't', 'z')
A = plot3d(x^2 - 4*x + 2*y^2, (x, -3.5, 3.5), (y, -3.5, 3.5))
B = parametric_plot3d([3*cos(t), 3*sin(t), z], (t, 0, 2*pi), (z, -5, 30), color = 'green')
C = parametric_plot3d([3*cos(t), 3*sin(t), 9 - 12*cos(t) + 9*(sin(t))^2], (t, 0, 2*pi), color = 'black')
show(A + B + C, aspect_ratio = [5, 5, 1])

(y, t, z)

3D rendering not yet implemented

var('y')
A = contour_plot(x^2 - 4*x + 2*y^2, (x, -4, 4), (y, -4, 4), labels = True, fill = False, contours = 12)
B = implicit_plot(x^2 + y^2 == 9, (x, -4, 4), (y, -4, 4), color = 'green')
show(A + B)

y

var('y', 'z', 't')
S1 = parametric_plot3d([2*cos(t), 2*sin(t), z], (t, 0, 2*pi), (z, -5, 5), color = 'green')
S2 = plot3d((6 - 3*x - 4*y)/5, (x, -5, 5), (y, -5, 5), color = 'cyan')
C = parametric_plot3d([2*cos(t), 2*sin(t), (6 - 6*cos(t) - 8*sin(t))/5], (t, 0, 2*pi), color = 'black')
show(S1 + S2 + C)

(y, z, t)

3D rendering not yet implemented