course Mth 272 ?x??z????????assignment #024024.
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16:06:38 Query problem 7.2.52 (was 7.2.48) identify quadric surface z^2 = x^2 + y^2/2.
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RESPONSE --> ok confidence assessment:
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16:08:32 What is the name of this quadric surface, and why?
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RESPONSE --> it is an elliptic paraboloid. becuase the ellipse is parallel to the xy plane, and parabola's to the xz and yz planes confidence assessment: 3 ** If z = c, a constant, then x^2 + y^2/2 = c^2, or x^2 / c^2 + y^2 / (`sqrt(2) * c)^2 = 1. This gives you ellipse with major axis c and minor axis `sqrt(2) * c. Thus for any plane parallel to the x-y plane and lying at distance c from the x-y plane, the trace of the surface is an ellipse. In the x-z plane the trace is x^2 - z^2 = 0, or x^2 = z^2, or x = +- z. Thus the trace in the x-z plane is two straight lines. In the y-z plane the trace is y^2 - z^2/2 = 0, or y^2 = z^2/2, or y = +- z * `sqrt(2) / 2. Thus the trace in the y-z plane is two straight lines. The x-z and y-z traces show you that the ellipses in the 'horizontal' planes change linearly with their distance from the x-y plane. This is the way cones grow, with straight lines running up and down from the apex. Thus the surface is an elliptical cone. **
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16:34:28 Give the equation of the xz trace of this surface and describe its shape, including a justification for your answer.
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RESPONSE --> x^2= z^2 for the xz trace. it looks like a parabola. confidence assessment: 3
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16:36:21 Describe in detail the z = 2 trace of this surface.
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RESPONSE --> it looks like an ellipse confidence assessment: 3
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