M CC 160 Calculus for Physical Scientists I



MATH 160 Calculus for Physical Scientists I Name:

Fall, 2008

Calculator Laboratory Section:

Date due:

Calculator:

Making the Idea of Limit Precise

What does it mean for a function to have number L as its limit?

The investigations in this lab require a calculator that can produce 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 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 be 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.

Introduction

In numerical terms, 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 could want (but not necessarily perfectly) by choosing the inputs x close enough to c but not equal to c. The meaning of the limit relationship can be stated more accurately by using notation for how accurately the function values f(x) are required to approximate L and for how close x must be to c to achieve this accuracy. More precisely, then, when we write limx→c f(x) = L we mean that for every ε > 0 there exists δ > 0 so that for every x in the domain of f with 0 < |x – c| < δ we have |f(x) – L| < ε (see Weir, Hass & Giordano, page 92). The number ε (epsilon) specifies how accurately the function values f(x) are required to approximate L. Epsilon is greater than zero because it’s an error tolerance and we must tolerate some error. The number δ (delta) is the “control”. If x is controlled to be within distance δ of c (but ≠  c), the function value f(x) is guaranteed to approximate L to within the error tolerance ε.

It is important to realize that only values of x that satisfy the inequality 0 < |x – c| < δ are required to produce function values within the prescribed error tolerance ε of L. In other words, the limit of a function as x approaches a number c depends only on the values of the function for x’s near the number c and not on the value of the function at the point x = c itself. The value of the function at the number c is completely irrelevant.

Graphically function values are y-values and are identified with points on the vertical axis. Input values are x-values and are identified with points on the horizontal axis. Stated graphically, then, limx→cf(x) = L means we can make the function values f(x) lie as close as anyone could want to the number L (but not necessarily coincide with L) on the vertical axis by choosing the inputs x close enough to c (but distinct from c) on the horizontal axis.

In this lab we will use the calculator to investigate the precise definition of limit numerically and graphically. We will not be concerned with how to find the limit L of a function, but rather, how the values of the independent variable x, the number c, function values f(x), and the number L are related when limx→cf(x) = L .

Investigation I Exploring the Idea of Limit Numerically

It’s intuitively clear that limx→2.25 [pic] = 1.5. When we write limx→2.25 [pic] = 1.5 we are claiming that if we wish to approximate L = 1.5 to within any specified error tolerance (that’s ε > 0) by values of the square root function f(x) = [pic], we can do so by inputting x-values chosen from within some distance (that’s δ > 0) of c = 2.25 but different from 2.25. Of course, evaluating the square root function with

x = 2.25 gives the value L = 1.5 exactly, but, as explained in the Introduction, this fact is not relevant to the limit L being 1.5.

I.1 Suppose we wish to approximate L = 1.5 to within an error tolerance of ε ’ 0.2 by values of the square root function f(x) = [pic] calculated using x-values within some distance δ > 0 of c = 2.25 but different from 2.25. The problem is to figure out a suitable “control distance”, δ.

(a) We’ll first make a table that gives experimental evidence that δ = 0.56 is a suitable “control”. That is to say, the table will give evidence that numbers x within distance δ = 0.56 of c = 2.25 but different from 2.25 produce function values [pic] within ε = 0.2 of L = 1.5.

First, find the interval of numbers x within δ = 0.56 of c = 2.25: __________ < x < __________

Next, choose several x-values (all different from 2.25) within this interval and write them in the first row of the table below. Choose x-values that aren’t uniformly spaced. Choose some x-values quite close to 2.25. One x-values has been chosen to illustrate.

Finally, calculate f(x) = [pic] and |f(x) – L| from these x-values. Round to five decimal places and record the results.

[pic]

Calculator note: Here is an efficient way to calculate f(x) = [pic] and |f(x) – L| on a TI-84® calculator. Enter [pic] and |[pic] – 1.5 | as y1 and y2 on the Y= screen. Press [2nd] tblset (row 1, col 2) to access the Table Setup menu. In this menu, scroll down to Indpnt, then scroll right to highlight Ask. Press [enter]. Press [2nd] table (row 1, col 5) to access the screen for creating a table. Enter the x-values you chose in the first column. The calculator will calculate f(x) = [pic] and |f(x) – L| = |[pic] – 1.5 | automatically and display these values in the next two columns. Move the cursor over a table entry to see more decimal places displayed at the bottom of the screen. Read the manual to learn more about table.

(b) What is the largest error when the limit L = 1.5 is approximated by calculating values of the function f(x) = [pic] using x-values listed in the table you created in part (a)? ______________

(c) Explain why the table you created above suggests, but does not show conclusively, that all function values calculated from x-values in this interval are within ε = 0.2 of the limit L = 1.5.

I.2 Suppose we wish to approximate L = 1.5 to within an error tolerance of ε ’ 0.1 by values of the square root function calculated using x-values (x ≠  2.25) within some distance δ > 0 of c = 2.25.

(a) Experiment numerically (use trial and error) to find a value for δ that meets this requirement.

δ = ________________

Explain what you did to find the value you chose for δ.

(b) Make a table analogous to the one you made in question I.1(a) that gives experimental evidence that numbers x different from 2.25 within the distance δ you chose in (a) of c = 2.25, produce function values [pic] within ε = 0.1 of the limit L = 1.5. Round entries to five decimal places.

[pic]

(c) Explain why the table you created above suggests, but does not show conclusively, that all function values calculated from x-values within the distance δ you chose of c = 2.25 are within ε = 0.1 of the limit L = 1.5.

(d) The δ you found in (a) is not the only acceptable control distance that will keep the function values within error tolerance ε = 0.1 of the limit L = 1.5. For the particular error tolerance ε = 0.1, is there a smallest acceptable control distance δ? If so, what is it? If not, why not? Explain.

(e) For the particular error tolerance ε = 0.1, is there a largest acceptable control distance δ? If so, what is it? If not, why not? Explain.

I.3 Give two reasons why your work in I.1 and I.2 does not, in itself, show that the conditions the definition of requires in order for limx→2.25 [pic] to be equal to 1.5 are all met. (The second reason is different than the limitations of the tables you made above in I.1(a) and I.2(b).)

Reason 1:

Reason 2:

Investigation II Exploring the Idea of Limit Graphically

The assertion limx→2.25 [pic] = 1.5 means the value of the function f(x) = [pic] can be made to approximate the number L = 1.5 as accurately as anyone could want (but not necessarily perfectly) by choosing the inputs x close enough to c = 2.25 but not equal to 2.25.

Graphically, the value of the square root function at a specific point x (on the horizontal axis) approximates the number L = 1.5 accurately when the vertical distance between the points where the vertical line at x intersects the graph of the square root function and the horizontal line y = 1.5 (recall that L = 1.5) is small. As illustrated in Figure 1, this vertical distance is equal to the absolute value of the numerical difference between the value of the square root function at the point x and the limit L =1.5.

[pic]Figure 1

Generate the graph of the function f(x) = [pic] and the horizontal line y = 1.5 as shown in Figure 1 on your calculator. Use the window suggested by the figure (Xmin = 0, Xmax = 4, Ymin = 0, Ymax = 2).

II.1 Suppose we wish to approximate L = 1.5 to within an error tolerance of ε = 0.25 by evaluating f(x) = [pic] using x-values different from 2.25 but within some distance δ > 0 of 2.25. Then the problem is to find a number δ so that choosing x (x ≠ 2.25) between 2.25 – δ and 2.25 + δ forces the values of y = [pic] to be between

L – ε = 1.5 – .25 = 1.25 and L + ε = 1.5 + .25 = 1.75.

Points that have y – values between 1.25 and 1.75 lie between the horizontal lines y = 1.25 and

y = 1.75. Draw these two horizontal lines in Figure 1 and add them to the graph you created on your calculator. We will use these two lines and the graph of y = [pic] to find a suitable value for δ.

All the points that are on the graph of y = [pic] and between the horizontal lines y = 1.25 and y = 1.75 have y-values between 1.25 and 1.75. Therefore, the points on the graph of y = [pic] within the horizontal strip between these two lines have y-coordinates that differ from L = 1.5 by no more than ε = 0.25. We need to find an interval around the point c = 2.25 on the x-axis so that the part of the graph of y = [pic] above this interval lines in the strip between the horizontal lines y = 1.25 and y = 1.75.

II.2 What is the exact value of the x-coordinate of the point where

the line y = 1.25 intersects the graph of y = [pic] . x = _______________________

What is the exact value of the x-coordinates of the point where

the line y = 1.75 intersects the graph of y = [pic] . x = _______________________

The two x-values you just found represent points on the x-axis.

Find the distance between each of these points and the point

c = 2.25 on the x-axis. Set δ equal to the smaller of these

two distances. δ = _____________________

II.3 Explain how to see from the graph that all numbers x within this distance δ of c = 2.25 (except x = 2.25; we don’t care about it), produce function values [pic] within distance ε = 0.25 of L = 1.5.

II.4 Choose a (small) positive fraction as an error tolerance ε. (May we suggest ε =[pic] or ε =[pic]?)

Use the method from above to find a corresponding δ.

ε = _______________________________

δ = _______________________________

II.5 (a) Explain why the work above (II.1 – II.4) doesn’t by itself establish conclusively that limx→2.25[pic] = 1.5.

(b) Explain how to see from the graph that all the requirements (involving ε and δ) to conclusively establish that limx→2.25 [pic] = 1.5 are almost certainly met.

Investigation III

One can find the limit of the function f(x) = [pic] as x approaches c = 1 by doing some not-too-fancy algebra. The algebra involved isn’t our main concern, so we won’t worry about it now (but you may want to try it). Instead, we’ll guess the limit L from the graph of the function and then look at the connection between the f(x) = [pic] and its limit L graphically.

Generate the graph of f(x) = [pic] on your calculator. Use a window that includes an interval of x-values that extends some distance on both sides of x = 1. If you are using a TI–84®, try the window

–0.4 < x < 4.3, –0.4 < y < 2.7. Notice that f(1) is not defined, so there is a hole in the graph at the point above x = 1. If necessary, adjust the viewing window so you can see this hole in your graph.

III.1 Use your graph to guess the number L so that limx→1 [pic] = L.

(L is a simple fraction you can guess exactly.) L = _____________________

Explain what you see in the graph that leads you to guess this number as the limit.

III.2. Use your number L and take ε = 0.05. Use the graphical techniques from this lab to find a corresponding δ. δ = _____________________

III.3. What, if anything, do you see in the graph that persuades you that all the requirements (involving ε and δ) to establish conclusively that limx→1 [pic] is equal to your number L are met.

(Best answers will say something about ε and something about δ.)

IV. Summary Question

The quality of a product is usually determined by the quality of the inputs used to produce it. For example, think of the café mochas served in Sweet Sinsations, the coffee shop in Lory Student Center. The quality of the drink (the product) is determined by the quality of the inputs – the quality of the chocolate; the variety, quality, grind, and freshness of the coffee; the freshness of the milk; the quality and condition of the espresso machine; and the skill of the barista who makes the drink. The problem of controlling the quality of a product (like a coffee drink) by controlling the quality of the materials and labor that go into producing the product is called a Quality Control Problem.

Explain how the connection between a function y = f(x) and its limit L as x approaches c can be interpreted as a quality control problem. Explain how ε and δ are to be interpreted in terms of quality control.

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