#$&* course 272 11/30 7:45p If your solution to a stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: ? &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qWhat is the name of this quadric surface, and why? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This surface is an elliptic cone xz plane: x^2-z^2 = 0 or x^2=z^2 or x = +/- z showing trace is 2 straight lines yz plane: y^2 - z^2/2 = 0 = y^2 = z^2/2 = y = z* (sqrt2)/2 showing trace is 2 straight lines xz and yz traces show elipses in horizontal planes are changing linearly showing this quadratic surface is elliptical cone confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a f 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qGive the equation of the xz trace of this surface and describe its shape, including a justification for your answer. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: xz trace: z^2 = x^2+y^2/2 y = 0 so z^2 = x^2+0^2/2 = z^2 = x^2 or so graph consists of two lines in yz plane when z = +/- x confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a The xz trace consists of the y = 0 points, which for z^2 = x^2 + y^2/2 is z^2 = x^2 + 0^2/2 or just z^2 = x^2. The graph of z^2 = x^2 consists of the two lines z = x and z = -x in the yz plane. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `qDescribe in detail the z = 2 trace of this surface. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: when z = 2: z^2= x^2+y^2/2 2^2 = x^2 + y^2/2 4 = x^2 + y^2/2 1 = (x^2)/4 + (y^2)/8 or (x^2)/2^2 + (y^2)/(2sqrt2)^2 = 1 showing ellipse with major axis 2 in the x direction and 2 sqrt 2 in the y direction confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a If z = 2 then z^2 = x^2 + y^2/2 becomes 2^2 = x^2 + y^2 / 2, or x^2 + y^2 / 2 = 4. This is an ellipse. If we divide both sides by 4 we can get the standard form: x^2 / 4 + y^2 / 8 = 1, or x^2 / 2^2 + y^2 / (2 `sqrt(2))^2 = 1. This is an ellipse with major axis 2 `sqrt(2) in the y direction and 2 in the x direction. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qDescribe in detail the z = 2 trace of this surface. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: when z = 2: z^2= x^2+y^2/2 2^2 = x^2 + y^2/2 4 = x^2 + y^2/2 1 = (x^2)/4 + (y^2)/8 or (x^2)/2^2 + (y^2)/(2sqrt2)^2 = 1 showing ellipse with major axis 2 in the x direction and 2 sqrt 2 in the y direction confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a If z = 2 then z^2 = x^2 + y^2/2 becomes 2^2 = x^2 + y^2 / 2, or x^2 + y^2 / 2 = 4. This is an ellipse. If we divide both sides by 4 we can get the standard form: x^2 / 4 + y^2 / 8 = 1, or x^2 / 2^2 + y^2 / (2 `sqrt(2))^2 = 1. This is an ellipse with major axis 2 `sqrt(2) in the y direction and 2 in the x direction. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!