Expression 29: "x" squared plus "y" squared equals 4 left brace, "z" equals 9 , right bracex2+y2=4z=9
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Expression 30: "x" squared plus "y" squared equals 4 left brace, "z" equals negative 3 , right bracex2+y2=4z=−3
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Expression 31: "x" squared plus "y" squared equals 4 left brace, "z" equals negative 6 , right bracex2+y2=4z=−6
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Expression 32: "x" squared plus "y" squared equals 4 left brace, "z" equals negative 9 , right bracex2+y2=4z=−9
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So how will the set of all points (x,y.z) satisfying x^2 + y^2=4 look?
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Expression 34: "x" squared plus "y" Superscript, 2 , Baseline equals 4x2+y2=4
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So the set of all points (x,y,z) satisfying x^2+y^2 = 4 is the white circular cylinder.
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Clean-up: close and unclick above folder. Also unclick all surfaces/items showing.
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Step 3: Graph in set of points (x,y,z) satisfying z=my.
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We have seen that z=3y in 3D is a plane generated by the 2D (in the yz plane) line slope 3. So to sketch z=my, let's just vary the m (say between -6 and 6)
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Expression 39: "z" equals "m" "y"z=my
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Expression 40: "m" equals 3m=3
negative 6−6
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Let's stop/set the slider (at m=3). In 3d, the graph of z=my
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Put back in the cylinder x^2>y^2 = 4.
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Expression 43: "x" squared plus "y" squared equals 4x2+y2=4
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Points (x,y,z) on the intersection of the white cylinder and gray curve have the form (think back to parameriztion curves in 2D) note need x^2+y^2 = 4 AND z=my
44
Expression 45: left parenthesis, 2cos "t" , 2sin "t" , "m" left parenthesis, 2sin "t" , right parenthesis , right parenthesis2cost,2sint,m2sint
00
domain t Minimum:
less than or equal to "t" less than or equal to≤t≤
domain t Maximum: 2 pi2π
45
So the intersection of the white cylinder and gray plane is the green curve.
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Next let m vary again. Go up to where it says what m is (just under z=my) and click on "m=" to let m vary.
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What happens as m varies.
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What is the set of ALL points (x,y,z) satisfying x^2+y^2 = 4 AND z=my. I.e., what is the intersection of white cylinder x^2+y^2 = 4 and plane z=my?
