MATH 160 Calculus for Physical Scientists I Name:Spring Semester, 2008Calculator Laboratory Section:Calculator:Date due:Numerical Investigation of LimitsMATH 160 Calculus for Physical Scientists I Name: ____________________________________Spring Semester, 2008Calculator Laboratory Section: _____________________________Calculator:___________________________Date due: ____________________________Numerical Investigation of LimitsThe investigations in this lab require a calculator that can produce scatter plots of data 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 efficientlyand accurately. Manuals for Texas Instrument calculators can be read from the Texas Instrument website. Go to http://education.ti.com/us/global/guides.html. You can find instructions for many different makes of calculators at http://www.prenhall.com/divisions/esm/app/calc_v2/frameset_83.html. 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, tooThe 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 yourlab 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. © 2006 Kenneth F. Klopfenstein, Fort Collins, CONumerical Investigation of LimitsOverviewThe 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 youto 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 theinputs 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.© 2006 Kenneth F. Klopfenstein,Fort Collins, CO page 1 of 6Numerical Investigation of LimitsInvestigation I We know that limx→2 3x = 32. For the moment, pretend we know that 3x has a limit Las 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 3x to have limit L as x approaches 2, we know we can approximate L to any number of decimal places by evaluating f(x) = 3x 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 evaluatinga function at several points.I.1. Use the TABLE feature of your calculator to make a table of values of the function f(x) = 3x 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.x 1.85 3xx 2.15 3xI.2. None of the function values in the table you created are exactly equal to 32, the value we know is the limit. (a) Troy claims that limx→23x is ex a c t l y 1.259921049894934159? Do the numbers from your table refute this 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.Page total _________© 2006 Kenneth F. Klopfenstein,Fort Collins, CO page 2 of 6Numerical Investigation of LimitsInvestigation 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) = 4424xx  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→04424xx  = L.Record your guess here: L = ______________ (Play fair! Don’t change your guess later!) x 0.01g (x )x -0.01g (x ) (b) Explain in terms of what it means to say that limx→04424xx