Assignment 23

course Mth 272

??????????????assignment #023

023.

Applied Calculus II

04-17-2009

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14:35:19

Query problem 7.2.6 intercepts and sketch graph of 2x - y + z = 4.

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RESPONSE -->

finding the intercepts

for x int. 2x-0+0 = 4

x= (2,0,0)

for y 2(0)- y +0 = 4

y= (0,-4,0)

for z 2(0) - 0 + z = 4

z= (0,0,4)

so the graph looks like a triangular region

confidence assessment: 3

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14:35:24

List the intercepts of the graph.

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RESPONSE -->

confidence assessment: 3

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14:35:31

Describe the graph of the plane.

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RESPONSE -->

answered

confidence assessment: 3

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14:37:07

If you released a marble on the plane at the point where it intercepts the z axis, it would roll down the incline. When the marble reached the xy plane would it be closer to the x axis or to the y axis?

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RESPONSE -->

it would be closer to the x axis, becuase if it rolled directly downward from the highest point on the z axis, the steeper of the lines would be closest to the z, which is the x axis.

confidence assessment: 3

The marble would travel the steepest possible path. The line from (0,0,4) to (2,0,0), in the xz plane, is steeper than the line from (0, 0, 4) to (0, -4, 0) in the yz plane. So the marble would reach the xy plane closer to the x axis than to the y axis.

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14:38:09

If you were climbing the plane straight from your starting point to the point for the plane intercepts the z axis, with your climb be steeper if you started from the x intercept or from the y intercept?

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RESPONSE -->

It would be steeper from the x axis, since it is much closer to the z axis. The y axis to the z axis would be a more gradual climb and it is out at 4 and not in at 2 like the x axis.

confidence assessment: 3

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15:04:21

Query problem 7.2.34 (was 7.2.30) match y^2 = 4x^2 + 9z^2 with graph

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RESPONSE -->

the graph that corresponds is G. The axis all meet at 0.

confidence assessment: 3

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15:04:30

Which graph matches the equation?

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RESPONSE -->

answered

confidence assessment: 3

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15:05:54

The graph couldn't be (e). Explain why not.

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RESPONSE -->

becuase the traces are not parallel to each other the way an elliptic cone is.

Also note that this form is

a x^2 - b y^2+ c z^2 = 0, whereas the form of (e) is
a x^2 - b y^2+ c z^2 = 1.

confidence assessment: 3

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15:15:22

The graph could not be (c) because the picture shows that the surface is not defined for | y | < 1, while 4x^2 + 9z^2 = .25, for example, is the trace for y = 1/2, and is a perfectly good ellipse. State this in your own words.

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RESPONSE -->

it cant be (c) becuase the other traces are not parabolas

xz plane for instance is x= 3/2z

the other traces must be accounted for

confidence assessment: 3

In the plane y = 1/2 the trace of y^2 = 4x^2 + 9z^2 is found by substituting y = 1/2 into this equation. We obtain (1/2)^2 = 4x^2 + 9z^2, or 1/4 = 4x^2 + 9z^2. Multiplying both sides by 4 we get the 16 x^2 + 36 z^2 = 1, which can be expressed as x^2 / [1/4^2] + y^2 / [ 1/6^2]. This is the standard form of an ellipse with major axis 1/4 in the x direction and minor axis 1/6 in the y direction.

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15:18:54

The graph couldn't be (c). Explain why not.

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RESPONSE -->

answered

confidence assessment: 3

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15:23:55

The trace of this graph exists in each of the coordinate planes, and is an ellipse in each. The graph of the given equation consists only of a single point in the xz plane, since there y = 0 and 4x^2 + 9z^2 = 0 only if x = z = 0. Explain why the xy trace is not an ellipse.

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RESPONSE -->

im not sure why it's not an ellipse. is it becuase the x and y are both squared, so that the lines are not parabolic, they are parallel and straight?

confidence assessment: 3

The xy trace occurs when z = 0. The equation is y^2 = 4x^2 + 9z^2, so the x-y trace is
y^2 = 4 x^2, which has solutions
y = +- 2 x.

The x-y trace therefore consists of two straight lines.

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15:50:04

What is the shape of the trace of the graph in the plane y = 1?

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RESPONSE -->

a circle

confidence assessment: 3

In the plane y = 1 the trace of y^2 = 4x^2 + 9z^2 becomes 4 x^2 + 9 z^2 = 1, which is an ellipse.

In standard form the ellipse is

x^2 / [ 1 / 2^2 ] + z^2 / [ 1 / 3^2 ] = 1,

so has major axis 1/2 in the x direction and minor axis 1/3 in the z direction.

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15:51:55

What is the shape of the trace of the graph in the plane z = 1?

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RESPONSE -->

two straight lines for both

confidence assessment: 3

In the plane x = 1 the trace of y^2 = 4x^2 + 9z^2 is

y^2 - 9 z^2 = 4,

which is a hyperbola with vertices at y = +- 2, z = 0 (i.e., at points (1, +-2, 0) since x = 1); the asymptotes are the lines y = 3z and y = -3z in the plane x = 1.

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15:52:47

Query Add comments on any surprises or insights you experienced as a result of this assignment.

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RESPONSE -->

The last part about the traces i wasn't too sure how to figure that out. If you could explain that to me i would appreciate it. Thanks.

confidence assessment: 3

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In any plane parallel to one of the coordinate planes one of the three variables is constant.

If you want the trace in such a plane, just substitute the constant value of this variable. You are left with an equation in the other two variables.

If the function is a quadric form, then all of these traces are conic sections (ellipses, hyperbolas, parabolas and straight lines).

Assignment 23

** The marble would travel the steepest possible path. The line from (0,0,4) to (2,0,0), in the xz plane, is steeper than the line from (0, 0, 4) to (0, -4, 0) in the yz plane. So the marble would reach the xy plane closer to the x axis than to the y axis. **

Also note that this form is a x^2 - b y^2+ c z^2 = 0, whereas the form of (e) is a x^2 - b y^2+ c z^2 = 1.

The xy trace occurs when z = 0. The equation is y^2 = 4x^2 + 9z^2, so the x-y trace is y^2 = 4 x^2, which has solutions y = +- 2 x. The x-y trace therefore consists of two straight lines.

** In the plane y = 1 the trace of y^2 = 4x^2 + 9z^2 becomes 4 x^2 + 9 z^2 = 1, which is an ellipse. In standard form the ellipse is x^2 / [ 1 / 2^2 ] + z^2 / [ 1 / 3^2 ] = 1, so has major axis 1/2 in the x direction and minor axis 1/3 in the z direction. **

** In the plane x = 1 the trace of y^2 = 4x^2 + 9z^2 is y^2 - 9 z^2 = 4, which is a hyperbola with vertices at y = +- 2, z = 0 (i.e., at points (1, +-2, 0) since x = 1); the asymptotes are the lines y = 3z and y = -3z in the plane x = 1. **

&#Good work. See my notes and let me know if you have questions. &#

The marble would travel the steepest possible path. The line from (0,0,4) to (2,0,0), in the xz plane, is steeper than the line from (0, 0, 4) to (0, -4, 0) in the yz plane. So the marble would reach the xy plane closer to the x axis than to the y axis.

Also note that this form is

a x^2 - b y^2+ c z^2 = 0, whereas the form of (e) is
a x^2 - b y^2+ c z^2 = 1.

The xy trace occurs when z = 0. The equation is y^2 = 4x^2 + 9z^2, so the x-y trace is
y^2 = 4 x^2, which has solutions
y = +- 2 x.

The x-y trace therefore consists of two straight lines.

In the plane y = 1 the trace of y^2 = 4x^2 + 9z^2 becomes 4 x^2 + 9 z^2 = 1, which is an ellipse.

In standard form the ellipse is

x^2 / [ 1 / 2^2 ] + z^2 / [ 1 / 3^2 ] = 1,

so has major axis 1/2 in the x direction and minor axis 1/3 in the z direction.

In the plane x = 1 the trace of y^2 = 4x^2 + 9z^2 is

y^2 - 9 z^2 = 4,

which is a hyperbola with vertices at y = +- 2, z = 0 (i.e., at points (1, +-2, 0) since x = 1); the asymptotes are the lines y = 3z and y = -3z in the plane x = 1.