discovering calculus

Discovering CalculusAuthor(s): ROBERT DECKERSource: The Mathematics Teacher, Vol. 82, No. 7 (OCTOBER 1989), pp. 558-563Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27966392 .

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Discovering Calculus By ROBERT DECKER

What

can you do in a mathematics class when a computer is available that you

can't do otherwise? How do you actually in

corporate the computer into a mathematics

class without rewriting the textbooks? This

article addresses these questions by relating my experience using computers in a mathe

matics class. In a standard first-year calculus course I

assigned four projects, or laboratories, that

could be completed only with the use of a

calculus software package. The students worked in groups of two outside of class. I

gave them as much help as they needed in

completing the projects. After the projects were handed in, the students shared what

they had discovered. The software used was

written for an IBM PC. The disk was sup

plied by the author of the textbook that was

used for the course (Fraleigh 1985). Any calculus software could be used that has the

ability to?

1. graph functions and their derivatives; 2. graph secant lines and calculate their

slopes; 3. magnify portions of the graph; and

4. calculate and display areas.

I have two reasons for introducing the

computer into my calculus classes. Experi enced mathematicians often think geometri cally, yet getting students to do so is dif ficult. Making graphs by hand is tedious, and so students miss the interplay of the

geometric and algebraic viewpoints. With the computer, students can create good graphs of virtually any function quickly and

easily. In addition, computers allow me to introduce the discovery method into the

Bob Decker is an assistant professor of mathematics at

the University of Hartford, West Hartford, CT 06117. He

is interested in bringing the computer into the mathe

matics classroom at all levels, as well as in the role of the

computer in mathematical research.

study of calculus. Calculus is primarily con

cerned with very small changes, so compar

ing a function to its derivative or antideriv ative when these values are calculated

numerically is again very tedious. With the

computer it becomes possible to have stu dents actually discover new relationships. In

particular, my students were able to dis cover the fundamental theorem of calculus.

In laboratory 1, students experiment with the coefficients of a sixth-degree poly nomial to see how they affect the shape of the graph. (Refer to the student handout.) The students are told to come up with a

graph that might reasonably represent the

speed of a car s(t) during a trip from one

town to another with a third small town in between. Students are instructed to find a curve that has two humps in it and that

passes through the origin, that is, a kind of

very simple curve fitting. This approach gets the students acquainted with the software, helps develop algebra-geometry connec

tions, and sets the stage for the other

projects. This laboratory is given at the very

beginning of the course, before any calculus is covered. (Fig. 1 shows an example of what the students see on the computer screen for this project; a student group actually came

up with the function shown.)

Laboratory 2 asks the student to find the acceleration at the beginning and end of the

trip and to locate by inspection the points where the acceleration is zero. This labora

tory is given when the derivative is first introduced as a difference quotient but be fore any derivative formulas are developed. To find the accelerations, the software draws the graph of both the function and the secant line corresponding to a given point and

given increment. The software also prints the slope of the secant. Thus, the student sees the sequence of approximating secants as well as their slopes. To find the points of zero acceleration, the student is asked to

558 Mathematics Teacher



LABORATORY 1: SPEED Name

Susan wants to get from Bloomington, Indiana, to Columbus, Indiana. To do

so, she must pass through Nashville, Indiana. Between cities the speed limit is fifty-five miles per hour. In Nashville, she must slow down to thirty-five

miles per hour. If s represents Susan's speed at time ?, then a graph of s versus t would look something like the figure below.

One curve that can be used to reproduce this graph qualitatively is a sixth

degree polynomial, that is, an equation of the form

Using a curve of this form as a starting point, find a function (equation) that is close in shape to the graph above. Choose the equation so that s = 0 when

Indicate in your write-up of this project the strategy you used to find the values of the coefficients and what you learned about the effect of these coefficients on the shape of your graph. Write in correct, complete sentences.

LABORATORY 2: ACCELERATION

1. The derivative (instantaneous rate of change) of the velocity with

respect to time is called the acceleration. For the equation you used in

laboratory 1, find the acceleration at the beginning and end of the trip. Speed and velocity are identical for this problem, since the velocity is always positive. On the graph of the function itself, show the sequence of

approximations (secant lines) that you used to find the tangent line.

2. Find all places where the acceleration is zero. Your answers should be accurate to one decimal place. Magnify the graph if necessary.

s

s = at6 + bt5 + et4 + dt3 + et2 + ft + g.

t = 0.

From the Mathematics Teacher, October 1989



LABORATORY 3: MAXIMA and MINIMA Name

1. Find all maxima, minima, and inflection points for your speed function s(t) from laboratory 1 by graphing s(t), s'(?), and s"(t) on the same axes using the appropriate option on your calculus disk. Find where s'(t) and s"(t) equal zero by seeing where they cross the x-axis. Magnify, if necessary, to get two

decimal-place accuracy.

2. Find s'(?) and s"(t) and check your answers to question 1 by calculating s'(t) at the maxima and minima and s"(t) at the inflection points. Also find s"(t) at the maxima and minima. Is this the result you expected?

3. On a graph, show where s(t) is increasing and decreasing. On another

graph show where s(t) is concave up and concave down. Explain what these

shapes mean in terms of your trip from Bloomington to Columbus.

4. What is the largest speed obtained on the trip and when did it occur?

LABORATORY 4: Name

THE FUNDAMENTAL THEOREM OF CALCULUS 1. We are going to investigate the

speed function s(t) from your previous laboratories. Fill out table 1 for the values of s(t) for t given in increments of 0.1. A(t) represents the area under s(t) from 0 to t. For example, A(0.2) is the area under s(t) between t = 0 and t = 0.2. To find each of these areas, use the numerical integration graphic from your software disk. Finally, to find A'(?), use the approximation formula

A'{t) = [A(t + h)-A(t)]/h with h = 0.1. Graph the functions A(t) and A'(t) on the same axes as s(t). What do you notice?

2. How can you express the

relationship that you found above? Use this relationship to find a formula for A(t). Does the formula

give you the same values for A(t) as the computer?

3. Interpret the function A(t) in terms of the trip from Bloomington interpretation makes sense.

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

Table 1

sit) A{t) Ait)

to Nashville. Explain why this

From the Mathematics Teacher, October 1989



-< -1 .3>A6+2*< -? .2> 4

< -.9>A2+?.5

Fig. 1. The speed-versus-time function s(t)

magnify the graph to obtain a specified amount of accuracy. (Fig. 2 shows an exam

ple of the computer screen the students see

for laboratory 2.) The third laboratory is designed to ac

company the sections that cover maxima, minima, and inflection points. The student must find these three kinds of points by inspecting graphs of the first and second

derivatives, magnifying where necessary. The software graphs both the function and its first or second derivative together, visu

ally helping to drive home the relationship between a function and its derivatives. These results can be checked against the results from the second laboratory. Finding these points by setting derivatives equal to zero and solving equations would be impos sible with a sixth-degree polynomial. How

ever, the students are asked to check their answers by substituting them into the first and second derivatives. (Figs. 3 and 4 show

examples of the computer screens the stu

dents see for laboratory 3. The first two in

flection points indicated by the graph of s"(t) are very difficult to see on the graph of s(t), even when magnified, but can be verified by calculator.)

The fourth and final laboratory is given after the concept of area is discussed but

DELIA X 1 ,5 ,1 ,ei ,881

I

H-SECANT

1.482878 3.895327 7.918245 9.99137 18.22785 18.25892

Fig. 2. The acceleration at the start of the trip

Fig. 3. Speed s(t) and its derivative s'(t)

October 1989 561



Fig. 4. Speed s(t) and its second derivative s"(t)

before the fundamental theorem of calculus is stated. The students find the values of their speed function s(t) from 0 to 2 in incre

ments of 0.1. Then the area function A(t) (i.e., the area under the curve between 0 and the given i-value) is found using the soft ware. The approximation of areas using rectangles, trapezoide, and parabolas has been discussed previously so that the stu dents understand how the areas are calcu lated by the computer. Then the value of the derivative of A(f) is computed by using a

calculator to find the difference quotient. Fi

nally, all three functions are graphed by hand by the student on the same coordinate

system. The students are then asked to ex

plain what they see. Antiderivatives have

already been discussed, though not in terms of areas. Hence, after the students make the connection that s(t) and A'(f) coincide, they can find an exact formula for A(t) and check that it gives the same values as calculated.

The software for finding areas displays the graph of the function and shades the

corresponding area, as well as finds the nu merical value of the area. This approach helps the student to understand visually what the area function represents. Students see that as they calculate the area for each new increment, they are adding in the area of a tall, skinny rectangle. Then, when they try to find A'(t) using a difference quotient, they are just dividing this rectangle by its base and getting back the height s(i). The students learn a rudimentary proof of the fundamental theorem at the same time that

they discover it. (Fig. 5 shows an example of one of the computer screens seen by the students for laboratory 4.)

The class discussion following the first

laboratory was quite lively. The students seemed to enjoy explaining what they dis covered as they varied each of the parame ters in the polynomial. Just about every

Ualue 2.531561

Fig. 5. The area under the speed function s(t)

562 _ Mathematics Teacher



group used a different approach. Only a few

groups discovered that they needed to work

exclusively with the coefficients of the even

powers of t; none realized that the coefficient of the sixth power of t could be zero. Some

groups found unusual ways of translating the graph so that it went through the origin, even though we had just covered the trans lation of graphs in class. I suppose that the students had trouble realizing that the tech

niques they used on textbook problems worked on functions they made up them selves!

Laboratories 2 and 3 were not as condu cive to class discussion, but the written re

ports were satisfactory. Most students made the correct connections between the alge braic and geometric approaches.

To begin the discussion of the final labo

ratory, I selected a student who had a good grasp of the important points covered in it. The student showed the graphs of s(t), A(t), and A'(f) to the class and explained why s(i) and A'(f) were about the same. (Fig. 6 shows this student's graph of s(t), A(t), and A'(?).) The ensuing discussion replaced my usual lecture on the fundamental theorem. Since the students were working with a function

they created themselves, they realized that

they were discovering a general principle rather than solving an artificial problem.

On the student evaluations, most stu dents were very positive about the computer

2.60

2.17

1.73

1.30

0.87 s(t) A(t) A'(t).

0.43 /

0 0.5 1.0 1.5 2.0 t

Fig. 6. Behold: the fundamental theorem of calculus

projects, saying that they understood the

concepts better ?s a result. A few felt that the laboratories were too time-consuming and taught computer operation rather than calculus. I believe that this perspective can

be interpreted as an indication that al

though computers can be a help in making connections in mathematics, they are no

panacea. Students who have trouble with the standard calculus course may get bogged down by using the computer and not see the

point of the projects. It is desirable to keep to a minimum the amount of computer knowl

edge needed to complete the laboratories. In

particular, I believe that the use of software is preferable to teaching programming.

REFERENCE Fraleigh, John. Calculus with Analytic Geometry. 2d ed.

Reading, Mass.: Addison-Wesley Publishing Co., 1985. m

ANSWERS TO CALENDAR?(Continued from page 533) Alternate solution: irr2 =

1; therefore r = / / The area of the three circles is 3.1/6 area of 3 circles = 1/2 (area I + II + III). Area ABC = r2Vs = (1/V^2 V3 = VS/tt. Shaded area = area AABC - (area I + II + III) -V3/7t- 1/2.

^6) ^r^nSs with two A's: Z~y AAB or AAC, each of which

can permute 3 ways x 2, or 6.

Strings with one A: ABB or ABC, the first of which permutes 3 ways and the second, 6 ways, for a total of 9. Strings with no A's: BBC, which has 3 permutations. Or write them out: AAB, ABA, BAA, AAC, ACA, CAA, ABB, BAB, BBA, ABC, ACB, CAB, CBA, BAC, BCA, BBC, BCB, CBB.

@EG and FO are diagonals of

a rectangle and hence equal. Since FO is a radius equal to 1, so

does EG equal 1.

2ab

a+ b

We label the segments as

shown, since the ones marked ck and dk are proportional to the ones marked c and d. By similar

triangles,

c d

(Continued on page 580)

October 1989 563



discovering calculus

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