plot3d xy ’ h3x^2 2y^2l 8x, 8y, ,axeslabel fi x,y,z h...
TRANSCRIPT
Plot3D@x y � H3 x^2 + 2 y^2L, 8x, -1, 1<, 8y, -1, 1<, AxesLabel ® 8"x", "y", "z"<DH* Limit yx�H3x^2+2y^2L as Hx,yL®H0,0L DNE. If you approach the origin along y=kx,
z®k�Hk^2+3L. f has a non removable discontinuity at the origin. *L
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ContourPlot@x y � H3 x^2 + 2 y^2L, 8x, -1, 1<, 8y, -1, 1<, PlotPoints ® 200D
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Continuity.nb 1
Plot3D@x^3 y � H2 x^6 + y^2L, 8x, -1, 1<, 8y, -1, 1<, AxesLabel ® 8"x", "y", "z"<,ViewPoint -> 80.210, -2.165, 0.862<, BoxRatios ® 81, 1, 1<D
H* Limit yx^3�Hy^2+2y^6L as Hx,yL®H0,0L DNE. If you approach the origin along y=kx^3,
z®k�Hk^2+2L. f has a non removable discontinuity at the origin. *L
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ContourPlot@x^3 y � H2 x^6 + y^2L, 8x, -1, 1<,8y, -1, 1<, PlotPoints ® 200, Contours ® 80, .1, -.1<D
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Continuity.nb 2
Plot3D@x y^3 � Hx^2 + y^6L, 8x, -8, 8<, 8y, -2, 2<, AxesLabel ® 8"x", "y", "z"<,BoxRatios ® 81, 1, 1<, ViewPoint -> 82.241, -3.053, 3.446<D
H* Limit xy^3�Hx^2+y^6L as Hx,yL®H0,0L DNE. If you approach the origin along x=ky^3,
z®k�Hk^2+1L. f has a non removable discontinuity at the origin. *L
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ContourPlot@x y^3 � Hx^2 + y^6L, 8x, -10, 10<,8y, -10, 10<, ContourShading ® False, Contours ® 10D
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Continuity.nb 3
Plot3D@x y^2 � Hx^2 + y^4L, 8x, -9, 9<, 8y, -3, 3<, AxesLabel ® 8"x", "y", "z"<,BoxRatios ® 82, 2, 4<, ViewPoint -> 81.626, -7.854, 4.083<D
H* Limit xy^2�Hx^2+y^4L as Hx,yL®H0,0L DNE. If you approach the origin along x=ky^2,
z®k�Hk^2+1L. f has a non removable discontinuity at the origin. *L
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ContourPlot@x y^2 � Hx^2 + y^4L, 8x, -9, 9<, 8y, -3, 3<D
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Continuity.nb 4
Plot3D@x y � Hx^2 + y^2L, 8x, -2, 2<, 8y, -2, 2<, AxesLabel ® 8"x,", "y,", "z,"<,BoxRatios ® 81, 1, 1<, ViewPoint -> 80.958, -1.875, 3.151<D
H* Limit xy�Hx^2+y^2L as Hx,yL®H0,0L DNE. If you approach the origin along x=ky,
or y=kx z®k�Hk^2+1L.f has a non removable discontinuity at the origin. *L-2
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ContourPlot@x y � Hx^2 + y^2L, 8x, -2, 2<, 8y, -2, 2<D
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Continuity.nb 5
Plot3D@Hx + yL � Hx + y^2L, 8x, -1, 1<, 8y, -.9, .9<,AxesLabel ® 8"x", "y", "z"<, BoxRatios ® 81, 1, 1<D
H* f is ripped along the surface of the cylinder x=-y2. As you approach x=-y2,
z®±¥ depending on which side of the cylinder you are on and on the value of y. *L-1.0
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ContourPlot@Hx + yL � Hx + y^2L, 8x, -1, 1<, 8y, -.9, .9<, PlotPoints ® 200D
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Continuity.nb 6
Plot3D@x y^2 � Hx^2 + y^2L, 8x, -2, 2<, 8y, -2, 2<, AxesLabel ® 8"x,", "y,", "z,"<,BoxRatios ® 81, 1, 1<, ViewPoint -> 80.958, -1.875, 3.151<D
H* Limit xy^2�Hx^2+y^2L as Hx,yL->H0,0L=0 by Squeeze theorem,
0< xy^2�Hx^2+y^2L < x . f can be made continuous by defining fH0,0L =0*L-2
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ContourPlot@x y^2 � Hx^2 + y^2L, 8x, -2, 2<, 8y, -2, 2<, Contours ® 50D
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Continuity.nb 7