If your solution to a stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
Your solution, attempt at solution: If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
023.
Question: `qQuery problem 7.2.6 intercepts and sketch graph of 2x - y + z = 4.
Your solution:
Confidence Assessment:
Given Solution:
`a The x-intercept occurs when y and z are 0, giving us 2x = 4 so x = 2.
The y-intercept occurs when x and z are 0, giving us -y = 4 so y = -4.
The z-intercept occurs when x and y are 0, giving us z = 4.
The intercepts are therefore (2, 0, 0), (0, -4, 0) and (0, 0, 4).
These three points form a triangle and this triangle defines the plane 2x - y + z = 4. This plane contains the triangle but extends beyond the triangle, extending infinitely far in all directions.
Self-critique (if necessary):
Self-critique Rating:
Question: `qIf 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?
Your solution:
Confidence Assessment:
Given Solution:
`a 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.
Self-critique (if necessary):
Self-critique Rating:
Question: `qIf 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?
Your solution:
Confidence Assessment:
Given Solution:
`a The line from (0,0,4) to (2,0,0), in the x-y plane, has slope 2 and is therefore steeper than the line from (0, 0, 4) to (0, -4, 0) in the yz plane, which has slope of magnitude 1.
Self-critique (if necessary):
Self-critique Rating:
Question: `qQuery problem 7.2.34 (was 7.2.30) match y^2 = 4x^2 + 9z^2 with graph
Which graph matches the equation?
Your solution:
Confidence Assessment:
Question: `qThe graph couldn't be (e). Explain why not.
Your solution:
Confidence Assessment:
Given Solution: `a The
equation for e) is set equal to 1 and the needed equation is set equal to 0. So
one has a constant term while the other does not.
Self-critique (if necessary):
Self-critique Rating:
Question: `qThe 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.
Your solution:
Confidence Assessment:
Given Solution:
`a 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.
Self-critique (if necessary):
Self-critique Rating:
Question: `qThe graph couldn't be (c). Explain why not.
Your solution:
Confidence Assessment:
Given Solution:
`a 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.
Self-critique (if necessary):
Self-critique Rating:
Question: `qThe 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.
Your solution:
Confidence Assessment:
Given Solution:
`a If y^2 = 4x^2 + 9z^2 then the xy trace, which occurs when z = 0, is y^2 = 4 x^2. This is equivalent to the two equations y = 2x and y = -2x, two straight lines.
Self-critique (if necessary):
Self-critique Rating:
Question: `qWhat is the shape of the trace of the graph in the plane y = 1?
Your solution:
Confidence Assessment:
Given Solution:
`a 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.
Self-critique (if necessary):
Self-critique Rating:
Question: `qWhat is the shape of the trace of the graph in the plane x = 1?
Your solution:
Confidence Assessment:
Given Solution:
`a 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.
STUDENT QUESTION
#### From our form of equation, or after solving for x = 1, how is the hyperbola exactly found? I see where this is in the text, but just not getting this exactly clear, at least at this very point.
INSTRUCTOR RESPONSE
The ellipse and hyperbolas corresponding to the equation
+- x^2 / a^2 +- y^2 / b^2 = 1
are all constructed based on the rectangular 'box' bounded by the lines x = +- a and y = +- b.
The x and y axes. and these two lines, are plotted in the figure below.
The same figure with the ellipes x^2 / a^2 + y^2 / b^2 = 1:
Note that the intercepts (a, 0), (-a, 0), (0, b), (0, -b) lie on the graph of the ellipse. If you plug the coordinates of any of these points into the equation you get 1 = 1.
The original 'box' with the lines y = b/a * x and y = -b/a * x.
The same, with the hyperbola x^2 / a^2 - y^2 / b^2 = 1
Note that of the intercepts (a, 0), (-a, 0), (0, b), (0, -b) of the box, only (a, 0) and (-a, 0) lie on the graph of the hyperbola. If you plug in the coordinates of (0, +- b) you get -1 = 1.
Furthermore if | y | < b/a * | x | the left-hand side is negative, so the graph is excluded completely from the corresponding region of the plane.
For large x and y, the 1 on the right-hand side becomes insignificant and the graph approaches one of the lines y = +- b/a * x.
Note that of the intercepts (a, 0), (-a, 0), (0, b), (0, -b) of the box, only (0, b) and (0, -b) lie on the graph of the hyperbola. If you plug in the coordinates of (+- a, 0) you get -1 = 1.
Furthermore if | y | > b/a * | x | the left-hand side is negative, so the graph is excluded completely from the corresponding region of the plane.
For large x and y, the 1 on the right-hand side becomes insignificant and the graph approaches one of the lines y = +- b/a * x.
Self-critique (if necessary):
Self-critique Rating:
Question: `qWhat is the shape of the trace of the graph in the plane z = 1?
Your solution:
Confidence Assessment:
Given Solution:
`a In the plane z = 1 the trace of y^2 = 4x^2 + 9z^2 is
y^2 - 4 x^2 = 9,
a hyperbola with vertices at x = 0 and y = +- 3 (i.e., at points (0, +- 3, 1) ) and asymptotes y = 2x and y = -2x in the plane z = 1.
Self-critique (if necessary):
Self-critique Rating:
Question: `qQuery Add comments on any surprises or insights you experienced as a result of this assignment.