Using the Spreadsheet to Develop an Intuitive Understanding of the Limit Concept

Phyllis Brudney, Marilyn Keir and Mary Viruleg

Abstract

Students who have successfully completed algebra and have studied some geometry should have success with this module. The purpose is to lead the student to an intuitive understanding of the limit concept. It includes three activities which may be used independently or sequentially. The activities take the student from the discreet to the continuous beginning with a geometric model which is easily visualized and culminating with the more theoretical case of the limit of a function.

Technology

This module requires the use of a spreadsheet program such as Excel or Microsoft WORKS. Students should be familiar with basic spreadsheet capabilities, including the use of formulas and replication.

Teacher Notes

The three activities which accompany this module may be used independently. Each involves the use of the spreadsheet to observe and draw conclusions based on patterns in a sequence. Activity 0 is a geometry activity. Activity 1 should be introduced once students have studied and discussed different kinds of sequences and their behavior. Activity 2 gives students who understand sequences an opportunity to explore infinite geometric series and to discover the conditions under which they have a sum. Activity 3 allows students to examine the classic - definition of the limit of a function and to work toward an intuitive understanding of this fundamental idea.

Activity 0 : Using Spreadsheets to Study Circle Measurements

Use your spreadsheet to find a relationship between regular polygons and circles. If we consider regular polygons with radius, r, and n sides, beginning with n = 3 and watch what happens to the perimeter of the polygon as the number of sides increases, we will discover a relationship between the sequence of perimeters and the circumference of the circle with the same radius. A similar relationship exists between polygonal areas and the area of the circle.

To set up your worksheet:

Col A number of sides, n

Col B perimeter of polygon with n sides

Col C area of polygon with n sides.

Set aside cells for the value of r, and for the circumference and area of the circle with radius, r.

To calculate the perimeter of an n-sides polygon with radius, r:, (Column C)

= = x =

s = 2 r sin (/n)

perimeter = n s

= 2 n r sin(/n)

To calculate the area of each regular polygon: (Column D)

Area = .5 a p (perimeter is in Col C)

= .5 (r cos (/n) p

Calculate the circumference of the circle with radius, r. Compare your result with the sequence of perimeters. What do you find?

Calculate the area of the circle with radius, r. Compare it with the sequence of areas. What do you find?

Activity 1: Infinite Geometric Sequence and Iteration

INSTRUCTIONS:

1. Set up spreadsheet to find N(R) = _.

2. Label columns N, N(R), and N(N(R)).

3. Off to the side, label entry R and put a constant value in for R.

4. Start N at 0 counting by increases of 1 for the first experiment.

5. Increment N by more than 1 for following trials in each experiment.

EXPERIMENT #1:

1. Set an arbitrary R in a cell off to the right.

2. Fill in down columns labeled N, R, N(R), and N(N(R)).

3. Record observations about N(R) & N(N(R)).

4. Reset increments of N. Record observations.

5. Reset values for R 20 times. Record observations each time.

6. Answer questions: For which values of R do N(R) and N(N(R)) have limits?

Does incrementing N differently make any difference?

Are there different limits for N(R) and N(N(R))?

EXPERIMENT #2:

1. Now try N(R) = K *.

2. Set up a cell off to the right for K and enter a value.

3. Repeat process in experiment #1 for different values of R and K.

4. Record observations.

5. Answer the same questions as #6 in experiment #1.

EXPERIMENT #3:

1. Now try N(R) = K*+C where C is a constant.

2. Set up a cell off to the right for C and enter a value.

3. Repeat process in experiments #1 and #2 for different values of R, K, and C.

4. Record observations.

5. Answer the same set of questions as before.

CONCLUSION:

1. Do you see any overall patterns? Explain.

2. Do any specific R's produce different results? Explain.

3. Do any specific K's produce different results? Explain.

4. Do any specific C's produce different results? Explain.

5. Do any of your observations suggest that N(R) approaches a limit?

6. Do any of your observations suggest that N(N(R)) approaches a limit?

7. Is this a discrete or a continuous model?

8. What can you say overall for N(R) = K* + C?

Investigating the Sum of an Infinite Geometric Series

Technology: This activity requires the use of a spreadsheet program such as Excel or Microsoft WORKS. Students should be familiar with basic spreadsheet capabilities, including the use of formulas and replication.

Teacher notes: Introduce this activity once students have studied the behavior of different kinds of sequences and have been introduced to series.

Series: If you add the terms of a sequence, the sum is called a series. For example, the sequence: 3, 7, 11, 15,... yields the series: 3 + 7 + 11 + 15 + ...

The n-th partial sum of a series is the sum of the first n terms of that series and is represented by S_n . For the example above,

S₁ = 3

S₂ = 3 + 7 = 10

S₃ = 3 + 7 + 11 = 21

S₄ = 3 + 7 + 11 + 15 = 36

Notice that the partial sums also form a sequence: 3, 10, 21, 36, ... A partial sum is often represented using "sigma" notation. If t _{k represents the k-th term of the sequence, the n-th partial sum can be represented:}

S_n =

If the series is infinite, it may or may not be useful to study the behavior of the sequence of its partial sums. For example, the series: 1 + 6 + 11 + 16 + 21 + ... has partial sums that just keep getting bigger as n increases. Examining the partial sums of infinite geometric series leads to more interesting conclusions.

Consider the infinite geometric series: 5 + + + + ...

The first few partial sums are: S₁ = 5

S_{2 = 5 +} = 7

S₃ = _{5 +} + = 8

S₄ = _{5 +} + + = 9

S₅ = _{5 +} + + + = 9

The partial sums are increasing, but they seem to be getting closer (converging) to 10. We say that series converges to a number, S, if its sequence of partial sums, S_n , converges to that number, S.

Exploring Infinite Series on the Spreadsheet

Use your spreadsheet to construct the following:

Col A The value of n (the number of the term).

Col B The terms of the sequence from 1 to n.

Col C The sequence of partial sums.

For each sequence, list an approximation for the i-th term requested and for the i-th partial sum. Also, give the limit of the sequence, if it exists, and the limit of the sequence of partial sums. Generate at least twenty terms of each sequence and associated series.

1. The geometric sequence with first term: 4 and r =

t₁₈ = __________________ S₁₀ = __________________________

limit t_n = ______________ limit S_n = ______________________

2. The geometric sequence with first term: .4 and r = -1.2

t₁₄ = __________________ S₁₄ = _________________________

limit t_n = ______________ limit S_n = _____________________

3. The geometric sequence with first term: 1.75 and r =

t₁₉ = __________________ S₁₃= _________________________

limit t_n = ______________ limit S_n = _____________________

4. The geometric sequence with first term: 4 and r =

t₉ = __________________ S₁₄ = _________________________

limit t_n = ______________ limit S_n = _____________________

5. The geometric sequence with first term: 100 and r = .8

t₁₂= __________________ S₂₀ = _________________________

limit t_n = ______________ limit S_n = _____________________

Conjecture: An infinite geometric series converges to a number, S, when the value

of r is: _____________________________________________________________

Bouncing Ball Problem. Suppose that you drop a ball from a window 18 meters above the ground. The ball bounces up to 80% of its previous height with each bounce. How far does the ball travel between the first and second bounce? Between the second and third bounce? Between the third and fourth bounce?

If the ball continues to bounce this way until coming to rest, how far has it traveled from the time it was dropped from the window?

Nested Squares Problem. A set of nested squares is drawn inside a square of edge 1 unit. The corners of the next square are the midpoints of the sides of the preceding square.

Set up four columns on your spreadsheet:

Col A The level of your drawing. Let the original square be level 1.

Col B The length of a side of each new square in the figure.

Col C The area of each new square formed.

Col D The cumulative sum of all the squares starting with square 1.

Use your worksheet to answer the following:

1. What is the length of the side of the 11th square formed by this process? _______

2. How is the length of the sides of the squares changing? ____________________

3. What is the area of the fourth square? _____ ...of the eighth square? _________

4. How is the area of the squares changing? ________________________________

5. What is the sum of the areas of the first 6 squares? _____ ...of the first 7? _____

6. What finite number does this area sum seem to be approaching?______________

7. Does the sum of the perimeters seem to be approaching a finite number? _______

Activity 3: Informal Investigation of the Limit of a Function

Technology: This activity requires the use of a spreadsheet program such as Excel. The students should have enough familiarity with spreadsheet use to put the appropriate formulas in the spreadsheet.

Teacher notes: The teacher should set up the spreadsheet in advance. The formulas used are listed here: (Thanks to David Bannard for the spreadsheet layout.)

C D E F G H

Setting up the graph chart takes a bit of doing. To get both left and right limits on the same graph, you must begin with the , , [add series] commands after you have the initial graph. To set it up for the right, chart C to D for the function, C to G and C to H for the epsilon line. On the left for the same lines, chart E to F, E to G and E to H.

The student screen should be so that the chart shows as well as the spreadsheet. The students should be encouraged to play around with it awhile and get comfortable with what each of the columns is showing, and how to tell what the limit is and when you have selected a good epsilon for a given delta.

Student Activity Sheet on the Limit of a Function

Idea of a limit: The mathematical statement of a limit is written : f(x) = L

The intuitive idea of a function approaching a limit says that as x gets close to some value, a the value of the function f(x) gets close to some limit, L. You can generally get a good idea of what limit a function is approaching by using your graphing calculator.

A more formal definition of a limit:

f(x) = L means that for any small epsilon (e) you select, you can find a delta ( ) such that whenever 0<|x-a| < then 0<|f(x)-L|< e.

Instructions: Enter each function on the spreadsheet in both column D and F using column C and column E as the x value, respectively. Be certain to copy the formula down in the column. For each, pick an x start value near the a value and try to determine the limit of the function. When you have determined the limit, find a delta which will give you an epsilon < .001. When you have succeeded, your graph on the spread sheet should `fill' the area between the two lines of the epsilon. Like this:

If the function crosses the epsilon line, then your delta is too large. If it doesn't `fill' the area, it is too small.

1. (3x-2) = 2. () =

3. () = 4. () =

5. (xcos x) = 6. ( ) =

What you will see on the spreadsheet is shown on the following page.

Sample Spreadsheet

A B C D E F G H I

Woodrow Wilson Leadership Program in Mathematics lpt@www.woodrow.org
The Woodrow Wilson National Fellowship Foundation webmaster@woodrow.org
CN 5281, Princeton NJ 08543-5281 Tel:(609)452-7007 Fax:(609)452-0066