#$&*
course Mth163
july 18,2013 2:10pm
I'm going to ask you to submit your answers to the following questions, which should clarify vertical asymptotes and add to your understanding of the meaning of division:
If you understand this it won't take you long. If you don't, it's worth the time you spend on it:
Sketch a line segment at least a couple of inches long.
Sketch a line segment 1/3 the length of the original.
How many of the shorter segment are required to equal the length of the longer?
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2 more 1/3 segments or 2/3 including the first 1/3 segment
In all you need 3 segments
&&&
****
#$&*
what is 1 divided by 1/3 and how do your line segments reveal the meaning of this division?
****
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(3/3)/(1/3)=3
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****
#$&*
Now sketch a line segment 1/10 the length of the original.
How many of the shorter segment are required to equal the length of the longer?
****
#$&*
&&&
10 shorter segments because it is a 10th shorter than your original, so on the last one 3/3 would be your 100% of length in this case now 10/10 is your 100%
&&&
what is 1 divided by 1/10 and how do your line segments reveal the meaning of this division?
****
#$&*
&&&&
(10/10)/(1/10)
One divided by ten = ten
Ten divided by ten =one
&&&
How can you make sense of the division of 1 by 1/n using this geometrical reasoning?
****
#$&*
&&&
whatever your denominator is will be equal to the total of your length of your string(or side of object) because your using a number to describe a part of the total and the total is the denominator of the string (or side of object)
&&&
#$&*
On a y vs. x graph, scale the x axis so that the interval from 0 to 1 on that axis matches the length of the original line segment you
sketched above.
Mark the points 1/3 and 1/10 on the x axis.
What is the value of y = 1 / x for x = 1, 1/3 and 1/10, and how is this related to the sketches you made above?
****
#$&*
&&&
They are the same lengths
@&
You haven't given the values of 1 / x and you haven't related the values to the sketches.
*@
&&&&
****
#$&*
Scale your y axis so it accommodates these y values.
Plot the points on a graph of y = 1 / x corresponding to x values 1, 1/3 and 1/10. Describe your graph.
****
#$&*
&&&
My graph is linear
@&
A graph drawn according to these instructions will not be linear.
What are the coordinates of the three points you plotted?
If you sketch the curve suggested by these points, is it increasing or decreasing, and is it doing so at an increasing or a decreasing rate?
*@
&&&&
#$&*
What are the values of x = 1 / n for n = 1, 2, 3, 5 and 10?
****
#$&*
&&&
X=
1
1 /2
1 /3
1 /5
1 /10
&&&
Which of these x values were used in sketching your graph?
****
#$&*
&&&
1/10
@&
You would have used a point with this x value, but this is not the only x value you would have used. You need to indicate them all.
*@
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As n gets larger, what happens to the point you would mark on the x axis to represent x = 1 / n?
****
#$&*
&&&
As n gets larger x gets smaller the bigger the number divided by one the smaller the fraction
@&
Your phrase "the bigger the number divided by one the smaller the fraction" doesn't seem to make sense.
If you divide a number by 1 you get the same number, so if you divide a bigger number by 1 you get a bigger result.
Nothing in the question involves dividing a value of x by 1.
*@
&&&
****
#$&*
Mark a new point on your x axis which is as close as possible to the origin, but still distinct from the origin.
How many times closer to the origin is this point than the point you marked for x = 1/10?
****
#$&*
&&&
.1+0=.1 times closer
@&
Something that is twice as close is closer. Something that is 50 times as close is closer.
But something that is .1 times as close is 10 times further. It's not closer.
*@
.1*0=0 times closer and that’s not true because it should be closer to origin right??? That’s why I added in the first place
@&
It's not clear why the answer to this question would involve multiplication by 0.
*@
&&&
****
#$&*
If this point represents x = 1 / n, what do you think is the value of n?
****
&&&
Zero
@&
At this point, x has a value.
If x = 1 / 0, though, x does not have a value, since 1 / 0 is undefined (for reasons these questions are designed to illuminate).
*@
&&&
#$&*
If you were to sketch a very short line segment whose length is equal to the distance from the origin to your new point, how many such
segments would be required to equal the length of your very first line segment, the one you drew for the very first question?
****
#$&*
&&&
I drew 48 very small segments the distance of my new point from zero across the length of my large segment
&&&
#$&*
If x is the value represented by your new point, what then is the value of 1 / x?
****
#$&*
&&&
1/.1 or 1/(1/1000)
=1000
@&
.1 is not 1 / 1000, so 1 / .1 is not 1 / (1 / 1000). However it's difficult to see what any of these number has to do with the small segment that appears to go into length 1 a total of 48 times.
*@
(I could have drawn my line closer then this would have made sense instead of 48)
@&
If you drew 48 segments to span a distance of 1, then the length of the segment divides in to length 1 a total of 48 times. This is what division means.
If x represents the length of one of the 48 segments, what then is the value of x?
*@
&&&
Where would the corresponding point on the graph of y = 1 / x be? Would you be able to draw it on your graph?
****
#$&*
&&&
Yes literally 1/1000 th away from the origin
&&&&
#$&*
Imagine you had a powerful magnifying glass that could magnify the small interval between the origin and your new point until it looked as
long as your very first line segment. Imagine you have precise instruments that could mark a microscopic point on this interval as close to
the origin as possible, but still distinct from the origin, as well as the microscopic line segment from the origin to this point.
How many times closer to the origin would this point be than the new point you marked previously?
****
#$&*
&&&
The difference would be 1/1000 - 1/1000000
=999/1000000 or .000999
@&
If the first point was at x = 1 / 1000 then if a second point was drawn this much closer to the origin, it would be at 1 / 1000 * 1 / 1000 = 1 / 1 000 000.
However the difference of the two would not answer the question of 'how many times closer'.
How many times closer to the origin is x = 5 than x = 10? Answer: It's twice as close.
How many times closer to the origin is x = 5 than x = 100?
How do you determine how many times closer to the origin one value of x is than another?
*@
@&
You drew a point which we will assume is 48 times closer to the origin than the x = 1 point.
If you magnify the interval between the origin and this new point, then mark another point 48 times closer to the origin, how many times would the corresponding segment go into 1?
*@
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How many times further from the x axis would the corresponding graph point be, compared to the preceding graph point?
****
#$&*
&&&
1/1000000
I got this by doubling my zeros because your twice as zoomed in as you were before
&&&
How far from the x axis would be the corresponding point on the graph of y = 1 / x?
****
#$&*
&&&
1/1000000
@&
Assuming your first point was 48 times closer to the origin than the x = 1 point, and that the new point is 48 times closer to the origin that that point:
How many times closer to the origin is the new point than the one before?
How therefore would the new value of y = 1/x correspond to the preceding value of y = 1/x?
How did the preceding value of y = 1/x correspond to the values of y = 1 / x when x had value 1?
*@
@&
You didn't make your first point 1000 times closer to the origin than x = 1. But suppose you had, and suppose that your new point was again that many times closer to the origin.
For your first point how would the new value of y = 1/x correspond to the value of y = 1/x when x = 1?
How therefore would the new value of y = 1/x correspond to the preceding value of y = 1/x, and how would that value correspond to the value of y when x = 1?
*@
@&
What are the implications of all this for the graph of y = 1 / x, as x approaches 0?
*@
&&&
If we continue to repeat this process, every time magnifying the interval to mark an additional points closer to the origin and plotting the
corresponding point on the graph of y = 1 / x, what happens to the graph?
****
#$&*
&&&
It will eventually reach some sort of an asymptote
@&
It's not clear just what the asymptote would be or why, in terms of your preceding answers, it would occur.
*@
&&&
#$&*
What does all this have to do with vertical asymptotes?
****
#$&*
&&&
Your graph is moving by a factor of 1/x every time???
@&
x is a variable, so 1/x doesn't refer to any specific factor.
Something appears to have changed by a factor of 48, and then by another factor of 48.
What is it that changed by factor 48, and what does this have to do with the values of y = 1 / x?
*@
&&&
#$&*
How does this help you understand the graph of y = x^(-p)?
*@
&&&
Y=x^(-p)
Is the same thing as
Y= p/x
&&&
@&
y = x^(-p) is not the same thing as y = p / x.
*@
@&
I promise I'm not trying to drive you nuts with these questions, but it's very important that you understand this.
`gr99
*@
#$&*
course Mth163
july 18,2013 2:10pm
I'm going to ask you to submit your answers to the following questions, which should clarify vertical asymptotes and add to your understanding of the meaning of division:
If you understand this it won't take you long. If you don't, it's worth the time you spend on it:
Sketch a line segment at least a couple of inches long.
Sketch a line segment 1/3 the length of the original.
How many of the shorter segment are required to equal the length of the longer?
&&&&
2 more 1/3 segments or 2/3 including the first 1/3 segment
In all you need 3 segments
&&&
****
#$&*
what is 1 divided by 1/3 and how do your line segments reveal the meaning of this division?
****
&&&
(3/3)/(1/3)=3
&&&
****
#$&*
Now sketch a line segment 1/10 the length of the original.
How many of the shorter segment are required to equal the length of the longer?
****
#$&*
&&&
10 shorter segments because it is a 10th shorter than your original, so on the last one 3/3 would be your 100% of length in this case now 10/10 is your 100%
&&&
what is 1 divided by 1/10 and how do your line segments reveal the meaning of this division?
****
#$&*
&&&&
(10/10)/(1/10)
One divided by ten = ten
Ten divided by ten =one
&&&
How can you make sense of the division of 1 by 1/n using this geometrical reasoning?
****
#$&*
&&&
whatever your denominator is will be equal to the total of your length of your string(or side of object) because your using a number to describe a part of the total and the total is the denominator of the string (or side of object)
&&&
#$&*
On a y vs. x graph, scale the x axis so that the interval from 0 to 1 on that axis matches the length of the original line segment you
sketched above.
Mark the points 1/3 and 1/10 on the x axis.
What is the value of y = 1 / x for x = 1, 1/3 and 1/10, and how is this related to the sketches you made above?
****
#$&*
&&&
They are the same lengths
@&
You haven't given the values of 1 / x and you haven't related the values to the sketches.
*@
&&&&
****
#$&*
Scale your y axis so it accommodates these y values.
Plot the points on a graph of y = 1 / x corresponding to x values 1, 1/3 and 1/10. Describe your graph.
****
#$&*
&&&
My graph is linear
@&
A graph drawn according to these instructions will not be linear.
What are the coordinates of the three points you plotted?
If you sketch the curve suggested by these points, is it increasing or decreasing, and is it doing so at an increasing or a decreasing rate?
*@
&&&&
#$&*
What are the values of x = 1 / n for n = 1, 2, 3, 5 and 10?
****
#$&*
&&&
X=
1
1 /2
1 /3
1 /5
1 /10
&&&
Which of these x values were used in sketching your graph?
****
#$&*
&&&
1/10
@&
You would have used a point with this x value, but this is not the only x value you would have used. You need to indicate them all.
*@
&&&
As n gets larger, what happens to the point you would mark on the x axis to represent x = 1 / n?
****
#$&*
&&&
As n gets larger x gets smaller the bigger the number divided by one the smaller the fraction
@&
Your phrase "the bigger the number divided by one the smaller the fraction" doesn't seem to make sense.
If you divide a number by 1 you get the same number, so if you divide a bigger number by 1 you get a bigger result.
Nothing in the question involves dividing a value of x by 1.
*@
&&&
****
#$&*
Mark a new point on your x axis which is as close as possible to the origin, but still distinct from the origin.
How many times closer to the origin is this point than the point you marked for x = 1/10?
****
#$&*
&&&
.1+0=.1 times closer
@&
Something that is twice as close is closer. Something that is 50 times as close is closer.
But something that is .1 times as close is 10 times further. It's not closer.
*@
.1*0=0 times closer and that’s not true because it should be closer to origin right??? That’s why I added in the first place
@&
It's not clear why the answer to this question would involve multiplication by 0.
*@
&&&
****
#$&*
If this point represents x = 1 / n, what do you think is the value of n?
****
&&&
Zero
@&
At this point, x has a value.
If x = 1 / 0, though, x does not have a value, since 1 / 0 is undefined (for reasons these questions are designed to illuminate).
*@
&&&
#$&*
If you were to sketch a very short line segment whose length is equal to the distance from the origin to your new point, how many such
segments would be required to equal the length of your very first line segment, the one you drew for the very first question?
****
#$&*
&&&
I drew 48 very small segments the distance of my new point from zero across the length of my large segment
&&&
#$&*
If x is the value represented by your new point, what then is the value of 1 / x?
****
#$&*
&&&
1/.1 or 1/(1/1000)
=1000
@&
.1 is not 1 / 1000, so 1 / .1 is not 1 / (1 / 1000). However it's difficult to see what any of these number has to do with the small segment that appears to go into length 1 a total of 48 times.
*@
(I could have drawn my line closer then this would have made sense instead of 48)
@&
If you drew 48 segments to span a distance of 1, then the length of the segment divides in to length 1 a total of 48 times. This is what division means.
If x represents the length of one of the 48 segments, what then is the value of x?
*@
&&&
Where would the corresponding point on the graph of y = 1 / x be? Would you be able to draw it on your graph?
****
#$&*
&&&
Yes literally 1/1000 th away from the origin
&&&&
#$&*
Imagine you had a powerful magnifying glass that could magnify the small interval between the origin and your new point until it looked as
long as your very first line segment. Imagine you have precise instruments that could mark a microscopic point on this interval as close to
the origin as possible, but still distinct from the origin, as well as the microscopic line segment from the origin to this point.
How many times closer to the origin would this point be than the new point you marked previously?
****
#$&*
&&&
The difference would be 1/1000 - 1/1000000
=999/1000000 or .000999
@&
If the first point was at x = 1 / 1000 then if a second point was drawn this much closer to the origin, it would be at 1 / 1000 * 1 / 1000 = 1 / 1 000 000.
However the difference of the two would not answer the question of 'how many times closer'.
How many times closer to the origin is x = 5 than x = 10? Answer: It's twice as close.
How many times closer to the origin is x = 5 than x = 100?
How do you determine how many times closer to the origin one value of x is than another?
*@
@&
You drew a point which we will assume is 48 times closer to the origin than the x = 1 point.
If you magnify the interval between the origin and this new point, then mark another point 48 times closer to the origin, how many times would the corresponding segment go into 1?
*@
&&&
How many times further from the x axis would the corresponding graph point be, compared to the preceding graph point?
****
#$&*
&&&
1/1000000
I got this by doubling my zeros because your twice as zoomed in as you were before
&&&
How far from the x axis would be the corresponding point on the graph of y = 1 / x?
****
#$&*
&&&
1/1000000
@&
Assuming your first point was 48 times closer to the origin than the x = 1 point, and that the new point is 48 times closer to the origin that that point:
How many times closer to the origin is the new point than the one before?
How therefore would the new value of y = 1/x correspond to the preceding value of y = 1/x?
How did the preceding value of y = 1/x correspond to the values of y = 1 / x when x had value 1?
*@
@&
You didn't make your first point 1000 times closer to the origin than x = 1. But suppose you had, and suppose that your new point was again that many times closer to the origin.
For your first point how would the new value of y = 1/x correspond to the value of y = 1/x when x = 1?
How therefore would the new value of y = 1/x correspond to the preceding value of y = 1/x, and how would that value correspond to the value of y when x = 1?
*@
@&
What are the implications of all this for the graph of y = 1 / x, as x approaches 0?
*@
&&&
If we continue to repeat this process, every time magnifying the interval to mark an additional points closer to the origin and plotting the
corresponding point on the graph of y = 1 / x, what happens to the graph?
****
#$&*
&&&
It will eventually reach some sort of an asymptote
@&
It's not clear just what the asymptote would be or why, in terms of your preceding answers, it would occur.
*@
&&&
#$&*
What does all this have to do with vertical asymptotes?
****
#$&*
&&&
Your graph is moving by a factor of 1/x every time???
@&
x is a variable, so 1/x doesn't refer to any specific factor.
Something appears to have changed by a factor of 48, and then by another factor of 48.
What is it that changed by factor 48, and what does this have to do with the values of y = 1 / x?
*@
&&&
#$&*
How does this help you understand the graph of y = x^(-p)?
*@
&&&
Y=x^(-p)
Is the same thing as
Y= p/x
&&&
@&
y = x^(-p) is not the same thing as y = p / x.
*@
@&
I promise I'm not trying to drive you nuts with these questions, but it's very important that you understand this.
&#Please see my notes and submit a copy of this document with revisions, comments and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end).
Be sure to include the entire document, including my notes. &#
*@