U.T. ARLINGTON | CALCULUS I LAB I.1 | (Last revision March 2001.) |
You should eventually (!) get a page with an xy-coordinate system and instructions. Once you get to that site, read the instructions there, click on the "new function" window and enter the function ((x-2)/(abs(x-2)))+x-2 by copying this function using the mouse and pasting it into the "new function" window at the other site.
Go to http://www.uta.edu/math/Calculus/grapher.html.
You should notice that the computer is having trouble drawing the graph properly at x = 2. Why?
Question A1. Does f(0) exist? If so, what is it, if not, why not?
Question A3. Does f(4) exist? If so, what is it, if not, why not?
Question A4. Does f(3) exist? If so, what is it, if not, why not?
The number 2 does not belong to the domain of f, which is why the graph looks weird at x = 2. This means that there is no obvious definition (at least, not with the current information) of what f(2) should be equal to.
Question A5.
Consider x approaching the number 2
from the left-hand side. What are the values of f(x) as x approaches
2 from the left? Produce a table like the following to help you record
their values.
Table: Values of f(x) = ((x -2) /|x - 2|) + x - 2 as x approaches 2 from the left.
x | -1 | 0 | 1 | 1.5 | 1.75 | 1.9 |
f(x) |
Question A6.
Consider x approaching the number 2
from
the right-hand side. What are the values of f(x) as x approaches
2 from the right? Produce a table like the following to help you
record their values.
Table: Values of f(x) = ((x -2) /|x - 2|) + x - 2 as x approaches 2 from the right.
x | 5 | 4 | 3 | 2.5 | 2.25 | 2.1 |
f(x) |
Question A7.
Using your data from question A5, what
do you guess is the limiting value of the y-coordinates as x
approaches
the number 2 from the left?
Question A8.
Using your data from question A6, what
do you guess is the limiting value of the y-coordinates as x
approaches
the number 2 from the right?
Question A9.
Explain by using your answers to questions
A7 and A8, whether or not the limit of f(x) as x approaches
the number 2 exists, and if you think it exists, give its value.
Question B2.
Construct a zoom-box around the point (2, 1)
such that your zoom-box has height 3 units (i.e., 1.5 units above (2,1)
and 1.5 units below (2,1)). For now, consider the width of the zoom-box
to be infinite. For what part of the domain of f is the graph of
f
lying in your zoom-box?
Question B3.
Does there exist a zoom-box that you can put
around the point (2,1) such that the part of the graph of f that
lies inside your zoom-box corresponds to those x in the domain of
f
such that either a < x < 2 or 2 <
x
< b (for some a and b of your choice) and such that both types of x
arise?
If so, then what are the dimensions of your zoom-box and where is it
centered? What are the dimensions of the shortest (in height) such
zoom-box that you can construct?
Question C3. Does g(4) exist? If so, what is it, if not, why not?
Question C4. Does g(3) exist? If so, what is it, if not, why not?
Question C5.
Consider x approaching the number 2
from the left-hand side. What are the values of g(x) as x approaches
2 from the left? Produce a table like the following to help you record
their values.
Table: Values of g(x) = 1 - |x - 2| as x approaches 2 from the left.
x | -1 | 0 | 1 | 1.5 | 1.75 | 1.9 |
g(x) |
Question C6.
Consider x approaching the number 2 from the
right-hand side. What are the values of g(x) as x approaches
2 from the right? Produce a table like the following to help you
record their values.
Table: Values of g(x) = 1 - |x - 2| as x approaches 2 from the right.
x | 5 | 4 | 3 | 2.5 | 2.25 | 2.1 |
g (x) |
Question C7.
Using your data from question C5, what
do you guess is the limiting value of the y-coordinates as x
approaches
the number 2 from the left?
Question C8.
Using your data from question C6, what
do you guess is the limiting value of the y-coordinates as x
approaches
the number 2 from the right?
Question C9.
Explain by using your answers to questions
C7 and C8, whether or not the limit of g(x) as x approaches
the number 2 exists, and if you think it exists, give its value.
Question D2.
Construct a zoom-box around the point (2, 1)
such that your zoom-box has height 3 units (i.e., 1.5 units above (2,1) and
1.5 units below (2,1)). For now, consider the width of the zoom-box to be
infinite. For what part of the domain of g is the graph of
g lying in your zoom-box?
Question D3.
Does there exist a zoom-box that you can put
around the point (2,1) such that the part of the graph of g that
lies inside your zoom-box corresponds to those x in the domain of
g
such that either a < x < 2 or 2 <
x
< b (for some a and b of your choice) and such that both types of x
arise?
If so, then what are the dimensions of your zoom-box and where is it
centered? What are the dimensions of the shortest (in height) such
zoom-box that you can construct?
Question E2.
In B1, what are the possible values of ?
In B2, what are the possible values of ?
In B3, what are the possible values of ?
In D1, what are the possible values of ?
In D2, what are the possible values of ?
In D3, what are the possible values of ?
Question E3.
Suppose you zoom in without stopping for a
long time on both the functions considered above. This means that
you are making be
very very small. For each function discuss in one or two sentences
whether or not it is possible to always find a
that corresponds to such an .
Question E4.
For the function in Parts A and B, given an
arbitrary >0, are
you able to find a >0
such that
0 < | x - 2 | <
implies that | f(x) - L | <
for some limit value L?
(Hint: consider your answer to E3.)
For the function in Parts C and D, given an
arbitrary >0, are
you able to find a >0
such that
0 < | x - 2 | <
implies that | g(x) - L | <
for some limit value L?
(Hint: consider your answer to E3.)
Explain using two or three sentences your reasoning
used in answering this question, E4.
END LAB I.1
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