Assignment 24

course MTH 272

......!!!!!!!!...................................Applied Calculus II

Your work has been received. Please scroll through the document to see any inserted notes (inserted at the appropriate place in the document, in boldface) and a note at the end. The note at the end of the file will confirm that the file has been reviewed; be sure to read that note. If there is no note at the end, notify the instructor through the Submit Work form, and include the date of the posting to your access page.

Asst # 24

07-06-2006

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16:26:55

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**** Query problem 7.2.52 (was 7.2.48) identify quadric surface z^2 = x^2 + y^2/4.

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16:30:32

z^2 = x^2 + y^2/4

sqrt (z^2) = sqrt (x^2) + sqrt (y^2 / 4)

z = x + y/2

This fits the equation for an elliptic paraboloid: z= x^2 / a^2 + y^2 / b^2

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16:30:33

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**** What is the name of this quadric surface, and why?

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16:31:35

The name of this quadric surface is an elliptic paraboloid because the xy-trace is an ellipse, while the xz and yz-planes are parabolas.

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16:31:36

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**** Give the equation of the xz trace of this surface and describe its shape, including a justification for your answer.

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16:35:30

Set y=0

z^2 = x^2 + (y^2)/4

z^2 = x^2 + 0/4

z^2 = x^2 +0

z^2 = x^2

This is a parabola because it contains a squared term.

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16:35:31

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**** Describe in detail the z = 2 trace of this surface.

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16:39:34

2^2 = x^2 + (y^2)/4

4 = x^2 + (y^2)/4

I'm not really sure what to do from here. Maybe you could take the square root of both sides, leaving you with:

sqrt(4) = sqrt(x^2) + sqrt(y^2 /4)

but then that leaves you with all of the terms being either positive or negative

+,- 2 = +,- x + +,- (y/2)

I don't think this is the right answer, so maybe you don't take the square root, and it leaves you with:

2^2 = (x-0)^2 + (y/2 - 0)^2, which would be an ellipse with a center of (0,0) and a radius of 2.

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16:39:37

Your response has been entered.

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"

Your work has not been reviewed.

Please notify your instructor of the error, using the Submit Work form, and be sure to include the date 07-07-2006.

This looks good. Let me know if you have questions.

Assignment 24

course MTH 272

......!!!!!!!!...................................Applied Calculus II

Your work has been received. Please scroll through the document to see any inserted notes (inserted at the appropriate place in the document, in boldface) and a note at the end. The note at the end of the file will confirm that the file has been reviewed; be sure to read that note. If there is no note at the end, notify the instructor through the Submit Work form, and include the date of the posting to your access page.

Asst # 24

07-06-2006

......!!!!!!!!...................................

16:26:55

......!!!!!!!!...................................

**** Query problem 7.2.52 (was 7.2.48) identify quadric surface z^2 = x^2 + y^2/4.

......!!!!!!!!...................................

16:30:32

z^2 = x^2 + y^2/4

sqrt (z^2) = sqrt (x^2) + sqrt (y^2 / 4)

z = x + y/2

This fits the equation for an elliptic paraboloid: z= x^2 / a^2 + y^2 / b^2

** 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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16:30:33

......!!!!!!!!...................................

**** What is the name of this quadric surface, and why?

......!!!!!!!!...................................

16:31:35

The name of this quadric surface is an elliptic paraboloid because the xy-trace is an ellipse, while the xz and yz-planes are parabolas.

.........................................

16:31:36

......!!!!!!!!...................................

**** Give the equation of the xz trace of this surface and describe its shape, including a justification for your answer.

......!!!!!!!!...................................

16:35:30

Set y=0

z^2 = x^2 + (y^2)/4

z^2 = x^2 + 0/4

z^2 = x^2 +0

z^2 = x^2

This is a parabola because it contains a squared term.

A parabola is square in one variable, linear in the other.

z^2 = x^2 is equivalent to the two linear equations

z = x and

z = -x,

which give you two straight lines.

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16:35:31

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**** Describe in detail the z = 2 trace of this surface.

......!!!!!!!!...................................

16:39:34

2^2 = x^2 + (y^2)/4

4 = x^2 + (y^2)/4

I'm not really sure what to do from here. Maybe you could take the square root of both sides, leaving you with:

sqrt(4) = sqrt(x^2) + sqrt(y^2 /4)

Careful. You can take the square root of both sides, but you can't take the square root of every term. sqrt(a^b + b^2) is not a + b; the simplest way to see this is to see that the square of a + b is not a^2 + b^2 but a^2 + 2 a b + b^2.

but then that leaves you with all of the terms being either positive or negative

+,- 2 = +,- x + +,- (y/2)

I don't think this is the right answer, so maybe you don't take the square root, and it leaves you with:

2^2 = (x-0)^2 + (y/2 - 0)^2, which would be an ellipse with a center of (0,0) and a radius of 2.

The equation (which should read 2^2 = x^2 + (y^2)/2) is an ellipse

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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16:39:37

Your response has been entered.

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"

Again you have the right ideas; you just need a little practice distinguishing and identifying circles, hyperbolas, parabolas and ellipses.