Overview From discussing the idea of limits in class ...
MATH 160 Calculus for Physical Scientists I Name:
Fall Semester, 2008
Calculator Laboratory Section:
Calculator:
Date due:
Numerical Investigation of Limits
The investigations in this lab require a calculator that can produce tables of function values and traceable graphs. While many makes and models of calculators have these capabilities, the author used Texas Instrument calculators as he wrote this lab. The lab does not include instructions for using a calculator. Use the manual for your calculator to learn how to perform the tasks in this lab efficiently and accurately. Manuals for Texas Instrument calculators can be read from the Texas Instrument web site. Go to . You can find instructions for many different makes of calculators at . You might also search for manuals for other calculators on the manufacturers’ web sites.
The calculator skills you develop doing this lab will serve you well throughout this and other courses. If you encounter difficulties, take your calculator and manual to your instructor and discuss the problem with him/her. Classmates may be able to help out, too.
The following factors will be considered in scoring your lab report:
• Completeness. Each investigation must be completed entirely, recorded fully, and explained or interpreted clearly.
• Mathematical and computational accuracy.
• Clarity and readability. Explanations must be written in complete sentences with correct spelling, capitalization, and punctuation and with reasonable margins and spacing. Handwriting must be legible. Tables and graphs must accurate and presented in a clear, readable format.
Space for writing your report is provided within the lab. However, if you wish to word process your lab report, your instructor will e-mail you a copy of this lab as an attached MS Word document. Submit your final lab report as a printed document.
PLEASE KEEP A COPY OF YOUR COMPLETED LAB REPORT.
You may need to refer to the work you did on this lab before it is graded and returned.
Overview
The problem of evaluating limx→cf(x) involves two questions.
Question 1: Is there a number L such that limx→cf(x) = L?
Question 2: If there is such a number L, what is it?
Scientific/graphical calculators can sometimes help you make informed guesses about the answers to these two questions. However, these tools have limitations that can lead you to wrong conclusions. Some of these limitations are inherent in all calculating devices. Others result from using the calculator uncritically. This lab is intended to acquaint you with some limitations and pitfalls of evaluating limits numerically.
The first investigations in this lab will acquaint you with inherent limitations of computing devices that can lead you to incorrect conclusions about the limit of a function. The second investigations show how calculations done carelessly or uncritically can mislead you.
From class discussion you understand that when we write limx→cf(x) = L we mean the function values f(x) can be made to approximate the number L as accurately as anyone wants (but not necessarily perfectly) by choosing the inputs x close enough to c (but ≠ c). It is easy to overlook the detail that every number x close enough to c
(but ≠ c) must produce a function value f(x) that approximates L to within the required error. This oversight, in combination with uncritical calculations, can lead to serious misteaks.
Investigation I We know that limx→2 [pic] = [pic]. For the moment, pretend we know that [pic] has a limit L as x approaches 2, but we don’t know what the number L is. Our problem is to find the number L.
Because we know what it means for [pic] to have limit L as x approaches 2, we know we can approximate L to any number of decimal places by evaluating f(x) = [pic] using x-values close enough to c = 2, but ( 2. Unfortunately, we don’t know how close to 2 these x-values must be to produce good approximations for L. Nevertheless, we can get some idea whether we have a good approximation for L by evaluating the function at several x-values that are very close to c = 2. The table feature of your calculator is a convenient tool for evaluating a function at several points. Refer to the manual for your calculator to learn how to create tables.
I.1. Use the table feature of your calculator to make a table of values of the function f(x) = [pic] for several values of x near c = 2 but not equal to 2. Include several x-values smaller than 2 and several larger than 2. The values displayed in the TI® calculator’s table are rounded off. The calculator produces more decimal places than shown. When a table entry is highlighted (by scrolling over it in the column of function values) the entry is displayed more accurately at the bottom of the table. In your table list function values to at least eight (8) decimal places.
[pic]
[pic]
I.2. None of the function values in the table you created are exactly equal to [pic], the value we know is the limit.
(a) Troy claims that limx→2[pic] is exactly 1.259921049894934159? Do the numbers from your table refute Troy’s claim? Explain why or why not.
(b) How accurately (i.e. to how many decimal places) do you believe the limit L can be estimated from the numerical evidence in the table you created? Explain why.
Investigation II Every calculating device has inherent limitations. If you aren’t aware of these limitations, you can be seriously misled by the numbers your calculator produces.
II.1. The function g(x) = [pic] has a limit as x approaches 0.
(a) Make a table of function values and use it to guess the number L so that limx→0[pic] = L.
Record your guess here: L = ______________ (Play fair! Don’t change your guess later!)
[pic]
[pic]
(b) Explain in terms of what it means to say that limx→0[pic] = L how the table you created
in (a) supports, but does not prove, the conclusion that the value of L is the number you wrote above.
II. 2. Graph the function g(x) = [pic] in several different windows. Include a window with a very narrow interval of x-values. In the space below, sketch one graph that tells you about limx→0[pic] .
Describe what you see in this graph that tell you about the value of limx→0[pic] and explain how what you see leads you to your conclusion. (On a TI-84® or TI-86®, try the Decimal window and also the window Xmin = –.02, Xmax = .02, Ymin = –.1, Ymax = .3. You may need a narrower window to see the expected phenomenon on the TI-89®.)
Investigation II (continued)
II.3. Evaluate limx→0[pic] algebraically by rationalizing the numerator. Show details.
II.4. Numerical, graphical, and algebraic methods should give you two different values for limx→0[pic].
(a) The limit of a function is unique – a limit can have at most one value.
Does the function even have a limit as x approaches 0? Why do you think it does or doesn’t?
If the function has a limit, what is the correct value of the limit? Explain why you think so.
(b) What do you think caused one (or more) of the methods above to produce an incorrect result?
(c) How can one recognize situations where the phenomenon you saw here might occur again?
(In other words, what should you watch out for?)
Investigation III. Our natural preferences for patterns can also lead us to be misled by our calculations. Again, being aware of this pitfall can help you avoid it.
III.1. Be sure your calculator is in radian mode. Then complete the table of values for H(x) = sin([pic]) below:
[pic]
Based solely on information from this table, what would you predict as the value of limx→0 sin([pic]) ? __________
Explain why the table suggests this value.
III.2. Complete the table of values for H(x) = sin([pic]) below:
[pic]
Based solely on information from this table, what would you predict as the value of limx→0 sin([pic]) ? _________
Explain why the table suggests this value.
III.3. From the above results, what do you conclude about limx→0sin([pic]) ? Explain in terms of what it means for a function to have a limit (or not to have a limit) how the tables in III.1 and III.2 support your conclusion.
Investigation III (continued)
III.4. The function H(x) = sin([pic]) oscillates rapidly between the values –1 and 1 near the point c = 0. The
x-values in the tables in III.1 and III.2 are points where the function always has the same value. You can be misled by a numerical investigation of the limit of any function with oscillatory behavior if you happen to follow an unlucky pattern in choosing the points where you evaluate the function.
Your calculator probably has a command that generates random numbers between 0 and 1. (On the TI-83® and TI-84® calculators, press the [math] key, choose the prb menu and look for rand. On the TI-89® you must enter a right parenthesis to get a random number between 0 and 1.) Devise a scheme that uses this random number generator to reduce the chances of choosing an unlucky pattern of evaluation points in the numerical investigation of a limit. Your scheme should meet the following requirements:
• It produces a list of evaluation points that “over all” are progressively closer to 0. By that we mean, while it is not necessary for each number in the list to be closer to zero than the preceding ones, as you progress down the list the numbers listed get arbitrarily close to 0.
• At each step the number the calculator produces is the next evaluation point in your list. (Generating random numbers until you finally get one smaller than all those previously generated doesn’t meet this requirement. If you are clever, you can arrange to get successive evaluation points by pressing the [enter] key repeatedly.)
Explain clearly what your scheme is and why you expect it to avoid an unlucky pattern of evaluation points.
III.5. Use the scheme you described in III.4 to investigate limx→0 sin([pic]) numerically.
(Make a table, draw a conclusion about the limit, and explain how the table supports your conclusion.)
[pic]
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