course Mth 272
Š¦óêäǪ︜šŸ¿ªS‹G⾉…†YÁ»Æ÷’assignment #024
024.
Applied Calculus II
11-09-2008
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18:45:50
What is the name of this quadric surface, and why?
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RESPONSE -->
Elliptic cone
z^2= x^2 + y^2/2
x^2 + y^2/2 - z^2 = 0
this equation corresponds to the elliptic cone formula:
x^2/a^2 + y^2/b^2 - z^2/c^2 = 0
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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18:46:23
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 -->
y= 0
x^2 - z^2= 0
hyperbolic
confidence assessment: 2
** 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. **
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18:53:48
Describe in detail the z = 2 trace of this surface.
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RESPONSE -->
z=2
4= x^2 + y^2/2
x^2 + y^2/2 - 4 = 0
I am unsure of how to graph this by hand. Am I supposed to use the calculator? If so I am unsure how to plug this into the TI-83 Plus to show the graph. I am in need of help on this one.
confidence assessment: 0
** 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. **
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&#Let me know if you have questions. &#
For some background on circles, parabolas, ellipses and hyperbolas you can look up Conic Sections on the Web.
The link http://www.vhcc.edu/pc2fall9/frames%20pages/class_notes.htm takes you to my Class Notes for Precalculus II. You might find those pages helpful.
A quick summary:
The equation y = a x^2 is the basic parabola opening vertically, while x = a y^2 is a parabola opening horizontally. In each case the vertex is the origin.
If the vertex is the point (h, k) then we replace x and y respectively by x - h and y - k, which shifts the graph in the horizontal and vertical directions so that the origin moves to the point (h, k).
The equation x^2 / a^2 + y^2 / b^2 = 1 is an ellipse which will fit into the rectangular box defined by the lines x = a, x = -a, y = b and y = -b. This ellipse is centered at the origin.
If the center of the ellipse is the point (h, k) then we replace x and y respectively by x - h and y - k, which shifts the graph in the horizontal and vertical directions so that the center moves to the point (h, k).
The equations x^2 / a^2 - y^2 / b^2 = 1 and the equation - x^2 / a^2 + y^2 / b^2 = 1 are hyperbolas which can be sketched using the rectangular box defined by the lines x = a, x = -a, y = b and y = -b. In either case the hyperbola will be asymptotic to 'diagonal' lines of the box (imagine extending the diagonals of the recangle indefinitely in both directions). The equation x^2 / a^2 - y^2 / b^2 = 1 has vertices at the points (a, 0) and (-a, 0), the points where the x axis intersects the box; it opens to the right and to the left. The equation -x^2 / a^2 + y^2 / b^2 = 1 has vertices at the points (0, b) and (0, -b), the points where the y axis intersects the box; this hyperbola opens up and down.
The two asymptotes of either of these hyperbolas meet at the origin. If the asymptotes meet at the point (h, k) then we replace x and y respectively by x - h and y - k, which shifts the graph in the horizontal and vertical directions so that the intersection of the asymptotes moves to the point (h, k).