U.T. ARLINGTON CALCULUS I       LAB I.1           (Last revision March 2001.)
Objective
To understand the behavior of a function as x approaches some number, more specifically, to see if the y-coordinates of the function approach some number as x approaches a number.
Procedure
Answer the questions below on a sheet of paper unless otherwise instructed. Write your answers as completely and neatly as possible. Put your name at the top right of the paper and put the title ``Calculus I   Lab I.1'' at the top center.
Due-date
Your instructor should tell you if any of this lab is to be turned in for grading and what the due-date is.  If not, then ask.

PART A

Suppose f(x)=(x-2)/|x-2|  +  x - 2.   Our goal is to guess the limit of f(x) as x approaches 2.
You are going to use Netscape to access a certain site given below. For this to work properly, you should have "java" allowed in the Netscape preferences under the "Edit" menu above. You will alternate between that site and this one, so you might want to call up a new Netscape window, so that you can jump between that site and this one and look at them both at once. I recommend that you copy and paste the site's address into the new Netscape window rather than clicking on it.

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 A2.   Does f(1) 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.
 

PART B

In this part of the lab, we will focus on zooming in on the above function from Part A.
Question B1.
Construct a zoom-box around the point (2, 1) such that your zoom-box has height 5 units (i.e., 2.5 units above (2,1) and 2.5 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 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 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?
 
 
 

PART C

Suppose g(x) = 1 - |x - 2|. Notice that g(2) = 1, so the point (2,1) lies on the graph of g. Our goal is to guess the limit of g(x) as x approaches 2.
Repeat the work in Part A for this new function g, but first delete your previous graph in the applet.  You should enter   1 - abs(x-2)   for the new function.  This time the computer should have no trouble drawing the graph. Why not?
Question C1.   Does g(0) exist? If so, what is it, if not, why  not?
Question C2.   Does g(1) exist? If so, what is it, if not, why  not?

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.
 
 

PART D

In this part of the lab, we will focus on zooming in on the above function from Part C.  Recall that g(2) = 1.
Question D1.
Construct a zoom-box around the point (2, 1) such that your zoom-box has height 5 units (i.e., 2.5 units above (2,1) and 2.5 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 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?
 
 

PART E

In this part of the lab, we will relate the above work to the epsilon  (epsilon), delta (delta) discussion from lectures.
Question E1.
In B1, what is the value of epsilon?  In B2, what is the value of epsilon?   In B3, did you find a value for epsilon?
In D1, what is the value of epsilon?  In D2, what is the value of epsilon?   In D3, did you find a value for epsilon?

Question E2.
In B1, what are the possible values of delta?  In B2, what are the possible values of delta?
In B3, what are the possible values of delta?

In D1, what are the possible values of delta?  In D2, what are the possible values of delta?
In D3, what are the possible values of delta?

Question E3.
Suppose you zoom in without stopping for a long time on both the functions considered above.  This means that you are making epsilon be very very small.  For each function discuss in one or two sentences whether or not it is possible to always find a delta that corresponds to such an epsilon.

Question E4.
For the function in Parts A and B, given an arbitrary epsilon >0, are you able to find a delta>0  such that
0 < | x - 2 |  < delta  implies that   | f(x) - L | < epsilon   for some limit value L?
(Hint: consider your answer to E3.)

For the function in Parts C and D, given an arbitrary epsilon >0, are you able to find a delta>0  such that
0 < | x - 2 |  < delta  implies that   | g(x) - L | < epsilon   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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