13-1-36-stewart.nb

Problem 13-1-36 in Stewart, 5'th ed.

They want us to find teh space curve that is the intersection of the elliptical paraboloid with the parabolic cylinder . We wrote this in class, by direct substitution: .

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$R[t_] = {t, t^2, 4t^2 + t^4} ;$

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[Graphics:HTMLFiles/index_7.gif]

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Here's the parabolic cylinder.

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$RowBox[{parcyl, =, RowBox[{ParametricPlot3D, [, RowBox[{{x, x^2, z}, ,, RowBox[{{, RowBox[{x, ,, RowBox[{-, 1.5}], ,, 1.5}], }}], ,, {z, 0, 5}, ,, ViewPointvp}], ]}]}]$

[Graphics:HTMLFiles/index_10.gif]

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The parabolic ellipsoid. Consider a cross-section at and write the result as an ellipse in standard form. You should get that the semi-major and semi-minor axes have lengths and . Then recall the usual parameterization of an ellipse.

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$parellipsoid = ParametricPlot3D[{s^(1/2)/2 Cos[t], s^(1/2) Sin[t], s}, {t, 0, 2Pi}, {s, 0, 5}, ViewPointvp]$

[Graphics:HTMLFiles/index_16.gif]

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Together.

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[Graphics:HTMLFiles/index_19.gif]

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<<Graphics`Shapes`

Wait, pearls are whiter...

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pearls = Show[Graphics3D[{SurfaceColor[RGBColor[1, 1, 1], RGBColor[1, 1, 1]], Table[TranslateShape[Sphere[.08, 5, 5], R[t]], {t, -1, 1, .05}]}], ViewPointvp]