# Each cell is evaluated individually, by either clicking the 'evaluate' link
# below the cell, or by pressing 'shift + enter'
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2+7
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integrate(x^3,x)
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show(integrate(x^3,x))
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diff(1/(1+x^2),x)
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# Now let's look at some 3-dimensional graphs.
# We first have to specify which variables we'll be using.
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x,y,z=var('x,y,z')
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# In the plane, we're all familiar with the graph of y=x^2
plot(x^2,(x,-5,5))
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# When another dimension is introduced, the same equation produces a
# PARABOLIC CYLINDER:
plot3d(x^2, (x,-5,5), (y,-5,5))
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# If we introduce the third variable into the equation,
# we'll get the three-dimensional version of a parabola.
# A PARABOLOID is the graph of f(x,y) = ax^2 + by^2
plot3d(x^2 + y^2, (x,-8,8), (y,-5,5),aspect_ratio=(10,10,1))
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# We can even graph equations that can't be written as functions
implicit_plot(x^2 + y^2 == 9, (x,-5,5), (y,-5,5))
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# The CIRCULAR CYLINDER
implicit_plot3d(x^2 + y^2 == 9, (x,-5,5), (y, -5,5), (z, -5,5))
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# The SPHERE
implicit_plot3d(x^2 + y^2 + z^2 == 9, (x,-5,5), (y, -5,5), (z, -5,5))
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# An ELLIPSE in the plane
implicit_plot(x^2/7 + y^2/2 == 1, (x,-3,3), (y,-3,3), aspect_ratio=1)
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# An ELLIPTICAL CYLINDER
implicit_plot3d(x^2/7 + y^2/2 == 1, (x,-3,3), (y,-3,3), (z,-3,3), aspect_ratio=1)
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# An ELLIPSOID
implicit_plot3d(x^2/7 + y^2/2 + z^2 == 1, (x,-3,3), (y,-3,3), (z,-3,3), aspect_ratio=1)
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# A HYPERBOLA in the plane
implicit_plot(x^2 - y^2 == 1, (x,-3,3), (y,-3,3), aspect_ratio=1)
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# A HYPERBOLIC CYLINDER
implicit_plot3d(x^2 - y^2 == 1, (x,-3,3), (y,-3,3), (z, -3,3), aspect_ratio=1)
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# A HYPERBOLOID (of ONE sheet)
implicit_plot3d(x^2 + z^2 - y^2 == 1, (x,-3,3), (y,-3,3), (z,-3,3), aspect_ratio=1)
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# A HYPERBOLOID (of TWO sheets)
implicit_plot3d(x^2 - y^2 - z^2 == 1, (x,-3,3), (y,-3,3), (z, -3,3), aspect_ratio=1)
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A=implicit_plot3d(x^2 - y^2 - z^2 == 1, (x,-3,3), (y,-3,3), (z, -3,3), aspect_ratio=1, color="red")
B=implicit_plot3d(-x^2 + y^2 + z^2 == 1, (x,-3,3), (y,-3,3), (z, -3,3), aspect_ratio=1, color="green")
A+B
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# Remember the paraboloid
plot3d(x^2 + y^2, (x,-8,8), (y,-5,5))
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# Changing one of the signs produces the HYPERBOLIC PARABOLOID
plot3d(x^2 - y^2, (x,-8,8), (y,-5,5))
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#AN ELLIPTICAL CONE
implicit_plot3d(2*y^2 + 3*x^2 - z^2 == 0,(x,-5,5),(y,-5,5),(z,-5,5))
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