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Engineering Applications in Differentialand Integral Calculus*

ALAN HORWITZ

Mathematics Department, Delaware County Campus, Penn State University, Pennsylvania, USA

E-mail: [email protected]

ARYA EBRAHIMPOUR

College of Engineering, Civil Engineering Program, Idaho State University, Idaho, Pocatello 83209, USA.

E-mail: [email protected]

The authors describe a two-year collaborative project between the Mathematics and the Engin-eering Departments. The collaboration effort involved enhancing the first year calculus courseswith applied engineering and science projects. Two enhanced sections of the differential (firstsemester) and integral (second semester) calculus courses were offered during the duration of theproject. The application projects involved both teamwork and individual work, and we required use

of both programmable calculators and Matlab for these projects. Some projects involved use of realdata often collected by the involved faculty. The paper lists all the projects, including where they fitwithin the course topics. Some selected projects are described in detail for both the differential andthe integral calculus courses. The paper also summarizes the results of the survey questions given tothe students in two of the courses followed by the authors own critique of the enhancement project.

INTRODUCTION

AT PENN STATE, most of Math 140 coversdifferential calculus, while about 30% of thecourse is devoted to integral calculus. Among thetopics covered are: limits and rates of change,continuous functions, derivatives of polynomials,

rational functions, trigonometric functions, curvesketching and optimization, applied wordproblems, the Riemann integral and the Funda-mental Theorem of Calculus, areas betweencurves, and volumes of solids of revolution.Almost all of the topics covered in Math 141involve the integral calculus including: inversefunctions, derivatives and integrals of exponentialand logarithmic functions, techniques of integra-tion, infinite sequences and series, parametricequations, and polar coordinates.

From Fall 1997 to Spring 1999, we offeredenhanced sections of the Math 140 and Math141. The objectives were:

. to introduce team-based projects in engineeringand science,

. to convey to the students the importance ofmathematics in engineering and science,

. to use Matlab and graphics calculators to ana-lyze experimental data and perform mathemati-cal operations.

We offered one enhanced section of Math 140 inFall 1997 and also in Fall 1998. We also offeredone section of enhanced Math 141 in Spring 1998and in Spring 1999. In this paper, we will describe

the projects used, the grading system, and surveyresults of the students' experiences.

BRIEF DESCRIPTIONS OF THE PROJECTS

The following are the projects (with pertinent

mathematical topics in parentheses) we used in theenhanced section. In the next section, selectedprojects are described in detail.

Math 140 Projects, Fall 19971. Data on strength of basswood samples (fitting

data with linear and polynomial functions)2. Temperature data from State College, Pennsyl-

vania (fitting data with sine functions, compar-ing average and instantaneous rates of change)

3. Beam analysis in mechanics (piecewise linear,quadratic, and cubic functions, and differen-tiability of a function)

4. Electrical circuit analysis (exponential functions

and derivatives)5. Fitting a pipeline with minimal cost (optimiza-

tion of a function on a closed interval)6. Crankshaft design (optimization of a function

on a closed interval)

Math 141 Projects, Spring 19981. Analysis of beams in mechanics (polynomial

integration and optimization of a function ona closed interval)

2. Tuning a radio (integration of sine and cosinefunctions)

3. Application of parametric curves (Cubic Bezier

Curves)4. Wrecking ball (approximating an integrand* Accepted 25 June 2001.

78

Int. J. Engng Ed. Vol. 18, No. 1, pp. 7888, 2002 0949-149X/91 $3.00+0.00Printed in Great Britain. # 2002 TEMPUS Publications.

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using a Taylor polynomial and finding roots ofthe resulting polynomial equation)

Math 140 Projects, Fall 19981. Hydraulic engineering (fitting data with power

functions and properties of log and exponentialfunctions)

2. Bridge project (piecewise functions anddifferentiability of a function)3. Automobile velocity data (fitting polynomial

functions to velocity data and testing models)4. Optimization of an irrigation channel (plane

geometry, trigonometry, and minimization offunctions).

Math 141 Projects, Spring 19991. Exponential growth and decay (exponential

functions, the definite integral and averagevalue of a function)

2. Drag force effect on a skydiver free fall (integra-tion of rational functions and use of integration

tables)3. Automobile velocity data (fitting polynomialfunctions to velocity data and numericalintegration)

4. Application of parametric curves (Cubic BezierCurves).

SELECTED PROJECTS FROM FIRSTSEMESTER CALCULUS

Hydraulic Engineering (Torricelli's Principle)Let f denote the volume flow rate of a liquid

through a restriction, such as an opening or avalve, out of a tank. Torricelli's principle statesthat f is proportional to the square root of thevolume V of liquid in the tank. Thus

f r

Vp

where r is a constant. The idea is to use regressionto fit a curve of the form

f rVm

to the given data. To use linear regression, trans-form the variables using logarithms to base 10first, giving a linear equation between the variablesx

log10 V and b

log10 r.

The following problem, taken from [6], wascovered in class:

`A 15-cup coffee pot was placed under a water faucetand filled to the 15-cup line. With the outlet valveopen, the faucet's flow rate was adjusted until thewater level remained constant at 15 cups, and the timefor one cup to flow out of the pot was measured. Thisexperiment was repeated with the pot filled to thevarious levels shown in Table 1.'

If t equals the time for one cup to flow out of thepot, then the flow rate equals 1at. During class wefirst found the linear regression line f c1V c2to the given data, without transforming the vari-

ables. Later we compared this fit with the oneobtained by transforming the variables. Matlab

was used to find and plot the regression curves,along with the data, and then to extrapolate from

the data. It was emphasized to the class that it isimportant not to use software blindly, and tounderstand some of the underlying mathematicalprinciples. In this case, one must understand theproperties of logarithms and exponential func-tions. We also emphasized the importance ofknowing some theory as well as the given data.In this case, the theory is Torricelli's principle,which says that f r Vp . Of course, for experi-mental data, the relationship is approximatelyf rV1a2. Here is where theory meets practice.The fit obtained without transforming the vari-ables is f X0062V X0718, as shown by the linearregression of Fig. 1. If one transforms the variablesfirst, the Matlab commands are: L2 poly-fit(log10(cups),log10(flow),1), which gives L2 0.4331, 1.3019; So m % X4331, b % 1X3019.To obtain f, 10logf 10bmx 10b10mx 10b10log Vm 10bVm rVm, where r 10b. Forthis example, f `10.^(1.3019).*v. .4331', orf X0499VX4331. A plot of both fits together (seeFig. 1) shows that the line diverges from the powerfunction as V becomes larger. Hence the powerfunction is the better estimate as V grows large.

The faculty involved in the project obtained thedata given to the student teams for their project.We drilled several holes in a plastic container

(Fig. 2). For a given opening size, the time to filla cup, t, was recorded for constant water levels(V 6Y 9Y 12Y 15 cups) in the container. Theconstant water level was maintained by adjustingthe faucet water entering the container. Each teamreceived a set of four data values for t. The valuesremained the same as in the sample done in class.The teams were asked to do what was done for thesample in class, and were encouraged to use Matlab.

Designing a pipeline with minimum costThis project involved derivatives and global

extrema of functions. It required determining themost cost effective pipeline route in connecting

various wells in an oil fertile area [7]. Figure 3shows a schematic view of the wetland and thecorresponding simplified rectangular model. Inthis problem we are connecting a pipeline fromwell at point A to another well at point B. Costsare associated with material (cost of pipe of $1.50/foot) and terrain type (normal terrain installationcost of $1.20 per foot). Installation in the wetlandrequires an additional Track Hoe at a cost of $60/hour. In a 10 hour day, the Track Hoe can digapproximately 300 feet of trench, and thus there isan additional cost of

$60ahrX

30ftXahrX $2aftX

Table 1.

V (cups) 15 12 9 6t (s) 6 7 8 9

Engineering Applications in Differential and Integral Calculus 79

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This gives a wetland installation cost of $4.70 perfoot. Three routes were considered in class (labeled1, 2, and 3 in Fig. 3):

Route 1: Cost 2X7d1 d2 d3Route 2: Cost 4X7

d1 d32 d22

q

Route 3: Cost 2X7x 4X7d1 x2 d22

q,

0 x d1 d3The solution for the third route was explained indetail in class using geometry, differential calculus,and Matlab. Here is a brief summary of thecommands:

Define variable, x symHxHDefine cost function, C

x

Find the derivative of the cost function,dC diffCSet the derivative equal to zero, solve for x, andfind the global minimum cost.

For team assignments, we gave the teams differ-ent values for the rectangle representing thewetland and the coordinates for wells at points Aand B. Also, given the costs involved, we requiredthe students to find the total cost of the routesconsidered.

Student comments were generally very favorableabout the project. We feel that this project seems

very applicable to something in real life, and themathematics involved is not too difficult.Fig. 2. Hydraulic engineering experimental setup.

Fig. 1. Flow rate vrs. volume for the Hydraulic Engineering Project.

A. Horwitz and A. Ebrahimpour80

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Optimization of an irrigation channelThe following sample problem was covered in

class [6]. Figure 4 shows the cross-section of

an irrigation channel, which is a trapezoidalregion. To maintain a certain volume flowrate, we want the cross-sectional area, A, to befixed, to say A 100 sq. ft. The objective is tominimize the amount of concrete that must beused to line the channel. To do this one mini-mizes the length, L, of the channel's perimeter,excluding the top. For the classroom example,we assumed that 1 2 , h1 h2 h, ande1 e2 e. Using geometry and trigonometry,one can express L as a function of and dalone.

L 100

d d

tan 2d

sin

Minimizing L as a function of both and drequires multivariable calculus, something thatthe students had not yet learned. We first reducedthe problem to single variable calculus by specify-ing either or d.

Fix a value of , and minimize with respect to dExample: a3 A L fd 100ad

3

pd.

We want the global minimum of L for0 ` d ` I. Setting fHd 0 yields d 10a 3

4p 7X5984. Since fHHd 200ad3 b 0 on

0Y I, f7X5984 26X3215 is the globalminimum.

Fix a value of d, and minimize with respect to Example: d 1 A L g 100 1a tan

2a sin

100

cot

2csc . We want the

global minimum for 0 ` ` %a2. Setting yields cos11a2 %a3. The First Derivative Testfor local extrema shows that g%a3 101X73 is theglobal minimum.

The students were asked to work individuallyfor the assignment, though they were encouragedto consult with other members of their team.Each student, however, was given a differentproblem to answer. The faculty wanted thestudents to use some of the geometry and trigono-metry covered in class to derive certain formulason their own. Hence, the students could not assumethat 1 2 , h1 h2 h, and e1 e2 e.Each student was given a value for A, 1, and 2and asked to:

. obtain a formula for the perimeter v (excludingthe fourth side);

. minimize L with respect to d on 0Y I, andexplain why it's the global minimum;

. use their given value of 1, along with d 1.Then minimize L with respect to 2 on0Y %a2, and again explain why it's the globalminimum.

Some students commented that this project wastoo difficult, though most of the students did itcorrectly. Part of the reason they thought it was

difficult was the geometry and trigonometryinvolved with deriving the correct formulas, given

Fig. 3. Plan view of the wetland for the Pipeline Project.


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that the values of , h, and e were not assumed tobe equal. The instructors of the course feel that it isgood if the students are encouraged to obtainformulas on their own.

SELECTED PROJECTS FROM SECONDSEMESTER CALCULUS

Exponential decay (temperature data)The definition and calculus of exponential

functions are covered during second semestercalculus at PSU. This project was designed toshow where these kinds of functions might beuseful. The faculty collected experimental data ofthe temperature, T, versus time, t, of a heatedcontainer of water. The water was allowed tocool down to room temperature. Students wererequired to

. find an exponential fit to the data;

. predict the temperature using interpolation andextrapolation;

. compare estimated average temperature usingnumerical integration and the given data.

Outside of class we collected data by heatingwater in a cup to an initial temperature of approxi-mately 988C, with the surrounding room tempera-ture at approximately 248C. The temperature wasrecorded at one minute intervals, from t 0 tot 60. In class we used data corresponding tot 0 to t 15 and discussed how to fit anexponential function to the data using linearregression and the natural logarithm. We fitted acurve of the form Tt cekt 24 to the data, andplotted the data along with the graph of Tt.Matlab was used along with properties ofexponential functions. We also predicted thetemperature at a time not given in the data. Thelinear regression we obtained was Lt X0358t 4X2668, which gives the temperaturefunction Tt 71X293 1eX0358t 24. The resultinggraph is shown in Fig. 5.

Again, to involve the student with understand-ing properties of exponential functions, we set thetemperature equal to a given value and solved fort. Given a continuous function fx defined on aclosed interval [aY b], in integral calculus onedefines the average value of f to be

1

b ab

afxdxX

For this project, we wanted the average value ofTt on [0Y 15],

Tave 115

150

71X2931eX0358t 24dt 79X1627

Finally, we compared Tave with the average tem-perature over the time interval [0, 15] using thedata only, which gives 818C.

Each team was given a proper subset of thetemperature data to work with, and asked to dothe same work as with the sample problem in class.

Skydiver free fallThis project involved integration of rational

functions and use of integration tables, and theeffect of drag force on free fall velocity of askydiver [4] provided a good application (Fig. 6).A mini-lecture was given to explain the relation-ship between the position, velocity, and accelera-tion in rectilinear motion of a particle.

The mathematics of the following problem

was covered in class. Given the drag force actingon a 60 kg skydiver, FD cv2, where c 0X2088 Ns2am2, determine (a) the terminal velocity,vt, (b) the velocity after 300m of free fall; and (c)time for the skydiver to reach a speed of 160 km/h.

For Part (a), it was necessary to explain brieflyNewton's second law and that terminal velocity isobtained when acceleration is zero. That is,

F maY mg FD maY

a 9X81 0X00348v2Y a 0 A vt 53X1 masX

Part (b) involved the following relation amongacceleration, a, velocity, v, and displacement, x:adx vdv; dx dva9X81 X00348v2;

x0 dx vv0va9X81 X00348v2dv. Using the substitution

9X81 X00348v2 gives v 53X094

1 eX0069xp

;v300 53X094

1 eX00696300

p 49X7 m/s. Part (c)

of the problem used the following relation:dt dvaa; since 160 kmah 44X44 mas, we havet44X44

t0 dt 44X44

v0 dva9X81 X00348v2. Making asubstitution and using a table of integrals yieldst44X44

44X44v0 dva9X81 X00348v2 2X706 lnv

53X0 9 3 9a53X0 9 3 9 v4 4X4 40 2X7 0 6 l n 1 1X2 7 0 5 6X55s.

Team assignments involved completing the

above three parts using different values for themass and drag coefficients.

Fig. 4. Channel cross section for the Irrigation Project.


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Wrecking ballThis project combines three ideas from calculus

in an interesting way: Taylor polynomials, approx-imate integration, and numerical solution ofnonlinear equations are rarely used in conjunctionwith anything else.. The basic idea for this projectwas taken from [1]. We quote `A wrecking ballattached to a crane may be thought of as apendulum. The period of such a pendulum isgiven (in seconds) by:

P 4k%a2

0

11 a2 sin2 x

p dx

where a sina2, is the angle of the swing(measured in radians from the rest position),

k

rg

rY

r is the length of the cable holding the ball, and gisthe acceleration due to gravity.' The derivation ofthe formula for Pabove is beyond the scope of firstsemester calculus. Given the formula, the follow-ing sample problem was covered in class. Supposethat the construction engineer determines that aperiod of P 5 seconds is required for the wreck-ing ball to have its maximum impact. Also,suppose that the cable is 18 ft. long (thenk 3a4 A 4k 3). Then compute the angle required for the swing, rounded off to two decimalplaces. Note that, in the above calculation, theintegral cannot be computed exactly and mustsomehow be estimated. While it is possible to usenumerical integration for specific values of theparameter a, we require an answer in terms of a.This is an important point to emphasize with thestudents. Blindly using computer programs thatexecute numerical integration rules, such as theTrapezoidal or Simpson's Rule, does not workhere. We instead approximated the integrandusing a Taylor polynomial, T5x, of order fiveexpanded about zero. While all the techniquesnecessary for computing T5, have been covered in

class, the work is rather cumbersome. Hence weused Matlab for the computations, as well as forFig. 6. The free body diagram for the Skydiver Free Fall

Project.

Fig. 5. Temperature vrs. time for the Exponential Decay Project.


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computing%a2

0T5xdx. The resulting polynomial

equation in a was also solved using Matlab (givingapproximate solutions). Finally, we wanted tocheck how accurate the computed value of awas, by comparing 3

%a20

dxa

1aa2 sin2 xp

toP 5. Earlier in the semester, the students wereprovided with a program for the TI graphing

calculator that executes the Midpoint Rule. Weused that to estimate the integral, with 500subdivisions. The Taylor polynomial weobtained using Matlab was T5x 1 1a2a2x2 1a6a2 3a8a4x4. Setting 5 3%a2

0T5xdx yields the polynomial equation

3qa 5, where qa 1a2% 1a48%31a960%5a2 3a1280%5a4. Approximate solutionswere found using the roots command in Matlab.Only one of the solutions is a positive realnumber, a X4504. To check the accuracy ofthe computed value for a, we estimated3

%a2

0dxa

1 X45042 sin2 x

p using the Midpoint

Rule. This gave 4.9833, which is very close toP 5. Finally, 2sin1 a 2sin1 X4504 X9344radians 53X548.

Each team was given specific values for N, k, r,and P, and were asked to do work similar to thesample problem done in class using both Matlaband their graphing calculator.

Each team met with the mathematics professorwho led them through the steps necessary to do thework for this project. One student commented thatthey should have been required to do more of thework on their own. We agree, although thestudents were not asked to cover any of the otherprojects in this way. Other students commented

that they could do the work for this project, butwere not very sure what they were doing. That wasprobably because Taylor polynomials are difficultfor students to understand, as well as evaluating adefinite integral with a parameter involved. None-theless, we feel that the wrecking ball projectexposed the students to an interesting combinationof ideas and methods.

Automobile velocity data (integration)This project was first introduced in Math 140,

where given velocity data and polynomial regres-sion were used to predict velocity and accelerationat times not in the data. This also provided a link

with the regular syllabus, where one differentiatesa given formula for velocity to obtain acceleration.In Math 141, one learns that total distance traveledis the integral of the velocity function. When one isonly given data, there are two ways to estimate thetotal distanceeither use numerical integrationand the given data, or find a polynomial regressioncurve to the data, and then integrate. One hopesthat the two estimates are fairly close.

The faculty obtained and handed out automo-bile velocity data taken from [5]. The sampleproblem done in class used the data in Table 2for the Mercedes-Benz SLK230, taken from the

July, 1997 issue. The data were actually given witht as a function of v. We wanted to consider v as a

function of t, and we also added the data point

0Y 0.We estimated the total distance traveled, D,

using the Trapezoidal Rule with six subdivisions.The result was D 564a3600 X156667 miles. Wefitted the velocity data with a regression polyno-mial of degree three using Matlab. First we showedthat polynomials of degree one or two did not fitthe data well. We obtained the velocity functionvt X0483t3 1X2599t2 15X0979t X2883, andwe plotted the data along with vt using Matlabas shown in Fig. 7. We estimated the total distancetraveled using the regression polynomial.D 1a3600

11X4

0vtdt X1572, which is very

close to the other estimate. We also estimatedwhen (in seconds) the total distance traveledequals a tenth of a mile. We used Matlab toapproximate the roots of the correspondingpolynomial equation. Setting

1

3600

t0

v%d% X1 At

0

X0483(3 1X2599(2

15X0979( X2883d( 360A X0121t4 X4100t3 7X5490t X2833t 360 0.The Matlab commands p [.0121 .4100 7.5490.2833 360], roots(p) yielded only one positive realroot, t

8.5116 seconds.

Each team was given velocity data for thefollowing cars, taken from [5].

. Team 1: Jaguar XK8 Convertible

. Team 2: Porsche Boxster

. Team 3: 1998 Ford Escort ZX2

. Team 4: 1998 Pontiac Trans AM

. Team 5: Acura Integra Type R

. Team 6: Honda Prelude Type SH

. Team 7: Nissan Skyline GTR

. Team 8: Nissan Altima

. Team 9: Ford Ranger XLT.

They were asked to do work similar to what was

done in class.

GRADING SYSTEM

From 20% to 30% of each student's grade in theenhanced versions of both Math 140 and Math141, consisted of team-based projects in engineer-ing and science. Students carried out a smallnumber of projects individually. Students carriedout the work for each project outside class, andthis always involved the use of technologyeitherMatlab or programmable calculators. In addition,

each project had a direct connection to some areaof engineering and/or physics. Finally, the projects

Table 2.

t (s) 0 2.3 3.6 5.0 6.9 9.0 11.4v (mph) 0 30 40 50 60 70 80


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often involved data taken from real life situations.While individual and group take home problemswere also assigned in the regular versions of Math

140 and 141, the key differences with the enhancedversion were:

. A smaller percentage of a student's gradeinvolved the take home problems. In addition,some group take home problems were assignedin the enhanced version in addition to theapplied projects.

. The take home problems almost never had directapplications to science and engineering.

. The take home problems usually did not involveworking with data.

. While the take home problems often did requirethe use of programmable calculators, they

usually did not involve the use of computersoftware such as Matlab.

SURVEY QUESTIONS AND RESULTS

In the Fall 1997, Spring 1998, and Fall 1998students were asked to complete informal evalua-tions of Math 140 and Math 141 during the lastweek of the class. The questions are listed belowtogether with responses to questions for Fall 1997and Spring 1998. Some of the students took both

classes, while others only took one of the enhancedversions of Math 140 or Math 141.

Questions1. On the scale of 1 to 5, rate the projects in

MATH 140/141. You may rate them overall,

or rate specific ones if you wish. Scale: 1 worstto 5 best. Also, make any comments youwish about the projects in MATH 140/141.

2. Did the projects help you understand and/orappreciate the practical applications ofmathematics in science and engineering?

3. Were the explanations of your instructorand the engineering instructors helpful inunderstanding the projects?

4. What suggestions do you have for improvingfuture projects?

5. Did members of your assigned group have anydifficulty working as a team? Please explain.

6. Did your team manage to distribute the workequally among all members? Please explain.

7. On the scale of 1 to 5, rate MATLAB. Also,make any comments you wish aboutMATLAB.

8. Did you have any difficulty using MATLAB?Please explain.

9. Would you recommend this section of MATH140/141 to other students?

10. General Comments: You are welcome to makeany other comments which may not pertain toany of the questions asked.

Responses for Math 140, Fall 1997

Sixteen students filled out the evaluations out ofa total of 18 in class.

Fig. 7. Velocity vrs. time for the Automobile Velocity Project.


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Responses to questions1. The average of the overall rating was 3.59.

Some students rated individual projects. A fewindicated that they had more difficulty withsome projects than others.

2. Eleven students indicated yes; two answeredno due to the level of difficulty of some of the

projects; and the rest either did not answer oronly addressed specific projects. Of those whoanswered yes, typical comments were:. `the projects made practical use of informa-

tion that we learned in class and they helpedme understand the subject matter fully;'

. Ì really didn't understand what calculuswas all about until these projects broughtit together for me;'

. ìt showed us how math is used in otherfields and gave us reason to understand andnot just memorize for tests.'

3. The majority (12 out of 16) answered yes.

Some indicated that some of the more challen-ging projects should have been explained inmore detail.

4. The majority of answers had to do with moreexplanations about the projects. Some res-ponded that projects should be less technical.

5. About half of the students answered no. Theremaining students indicated some difficulty;reasons varied from having difficulty findingcommon meeting time to only one or two teammembers doing most of the work. Please noteour campus is a commuter campus and manystudents have to work to support their college

expenses.6. Over half of the students answered yes. Theremaining students indicated one or twopeople in the team did most of the work.

7. The average rating was 3.41. Typical com-ments included:. `sometimes it was hard to know what com-

mand to use; without the instructor, no onewould have known what to do';

. Ìt was enjoyable to work with MATLAB;the program was relatively easy to workonce you got used to it';

. `the program was easy to learn and theinstructor helped explaining it.'

8. Over half of the students indicated somedegree of difficulty using MATLAB. Mostcomments were very similar to responses toQuestion 7.

9. From those who responded to this question,almost everybody answered yes.

10. Response to General Comments: Only twostudents had comments; they were:. à separate course should be available for

people who are interested. I do not think itshould be included in the MATH 140course'; and

. `the only negative comment is that it dis-

turbed the flow of learning in the classroom,but overall very cool!'

Responses for Math 141, Spring 1998Twenty one students filled out the evaluations

out of a total of 27 in class.

Responses to questions1. The average of the overall rating was 4. Some

students rated individual projects. From the

ones that provided comments, most indicatedthat the projects addressed the practicalapplications of mathematics. Some commen-ted that working as teams helped them betterunderstand the projects.

2. Seventeen students indicated yes; twoanswered somewhat (one of the three preferredto see more practical applications in the reg-ular course work); and one did not respondto this question. Typical comments of thestudents who answered yes were:. Ì personally had more appreciation for the

projects since they related to my scheduled

courses in engineering;'. Èven though the projects were notapplicable to me as an education major(Math option), I enjoyed seeing first handsome of the calculus' theories used in reallife situations;'

. Ì grew a better understanding of thematerial we were learning.'

3. The majority (19 out of 21) answered yes. Ofthose who answered yes, typical commentswere:. Àbsolutely, if it were not for the explana-

tions, I would have been lost on certainparts;'

. `They helped us through the material whichcould get complicated;'

. `The handouts were helpful;'

. `The explanations made the projects easierto understand.'

4. The majority of the students answered eithernone or did not answer this question. Of thosewho provided comments, typical commentswere:. Ì suggest having a couple more projects

with practical applications;'. `More teaching of MATLAB, I was at a

disadvantage sometimes due to lack ofexperience (presumably not having theenhanced section of Math 140 in theprevious term);'

. `Perhaps assign projects individually andnot in groups.'

5. Fifteen students answered no; two answeredyes; three had some level of difficulty; and onedid not respond. Typical answers included:. `No, we got along well and working in

groups helped us understand the materialbetter;'

. `No but it was hard to find times we couldall meet;'

. `Sometimes it was hard to get the whole

team together, because of conflictingschedules.'


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. The comment of one of the students whoanswered yes was: `Yes, motivation levels ofmembers varied greatly.'

6. Over half of the students (11 out of 21)answered yes. The remaining students indi-cated one or two people in the team did mostof the work. From those who answered yes,

typical comments included:. `Yes, we each played equal roles in one way

or another. Everything worked out great;'. `Yes, we all did even amounts of work.'From those who answered no, typicalcomments were:. `Not at all, when work was done it was done

individually then compared to the group;'. `No, some were not at school at the time the

project was being done.'7. The average rating was 3.71. Typical comments

included:. `MATLAB is great and I really enjoyed

using it;'. `I never really learned how to use it, so it

was kind of confusing;'. `Good program, a little hard to use might be

from a lack of prior usewould have likedto have done more with it.'

8. Seven of the students responded no or minordifficulties; nine indicated yes; and the remain-ing students did not respond to this question.

9. From those who responded to this question,everyone answered yes.

10. Response to General Comments: Typical com-ments included:. `Thank you for a fun semester;'

. `This was a great class and I hate to see itend so soon.'

SUMMARY AND CONCLUSIONS

Most of the students seemed pleased to see someapplications of differential and integral calculus toother fields, such as engineering. While many ofthe students are engineering majors, a fair numberare in other fields or are unsure of their major.These students were less enthusiastic about manyof the projects, though several of the projectsinvolved applications to science (usually physics).

There was also some negative reaction to a few ofthe projectsmost particularly the crankshaftdesign project and the optimization of an irrigationchannel project. The major complaint was thatthe work was too difficult. Also, for some ofthe projects, the students could do the work

well, but did not fully understand what they weredoing. Here is our critique of the enhancementproject:

. While there was some enthusiasm for doing theprojects, this did not spill over into the regularpart of the calculus course. One of our objectiveswas to use the projects to motivate the studentsto want to do and understand calculus. We donot think that this was achieved. Part of thereason may be the nature of the assignments.

. The team and individual assignments weresometimes duplicated, and students could oftendo the work by following the steps done in thesample problem in class. While it was good thatthey saw some interesting applications, theywere not really asked to do enough thinkingon their own. This was partly due to the natureof the enhancement project.

. We were still required to cover the regularsyllabus from Math 140 and Math 141. While

there is some leeway in what and how certaintopics are covered, time restrictions make itdifficult to any project in depth. We alsowanted to give a breadth of applications thatused different areas of calculus and whichapplied to different areas of engineering, suchas electrical, mechanical, and civil. We did offerfewer projects the second year, which allowedfor [a little] more depth of coverage. However,the projects at times were grafted onto theregular course content, rather than interactingwith that content in a vibrant way. We suspect,however, that many of the students will have agreater appreciation of the projects after theyhave taken other courses in engineering and thesciences.

. Perhaps a better way to offer an enhancement ofcalculus is to add an optional extra credit to firstor second semester calculus. The extra timeprovided and the fact that the course is optionalshould improve the course.

. Finally, we should add that several students didexpress that they had fun doing the projects and/or learning that calculus can be useful in the realworld. Many also seemed to enjoy using thesoftware as well, though they expressed adesire to obtain a more thorough knowledge of

Matlab.

AcknowledgmentsThe authors wish to thank the College ofEngineering at Penn State University, the GE Fund, and theDelaware County Campus of Penn State University for itssupport of this paper.

REFERENCES

1. A. Wayne Roberts (Project Director) and Robert Fraga (Editor), Calculus Problems for a NewCentury, Page 158, Problem #8, MAA Notes Number 28, Vol. 2 of Resources for Calculus,(1993).

2. Clark Benson, How to tune a radio, in Applications of Calculus, MAA Notes Number 29, Vol. 3 ofResources for Calculus, (1993).


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3. Steve Boyce, Crankshaft design in Problems for Student Investigation, MAA Notes Number 30,Vol. 4 of Resources for Calculus, (1993).

4. A. Pytel and Jaan Kiusalaas, Engineering Mechanics: Dynamics, p84, Harper Collins, New York,(1994).

5. Motor Trend Magazine (1997).6. William J. Palm, Introduction to Matlab for Engineers, McGraw Hill, (1997).7. John R. Ramsay, Designing a Pipeline with Minimal Cost in Problems for Student Investigation,

MAA Notes Number 30, Vol. 4 of Resources for Calculus, (1993).

Alan Horwitz is a professor in the Mathematics Department at the Delaware CountyCampus of Penn State University, Pennsylvania. He received his Ph.D. in mathematicsfrom Temple University in 1984. He teaches calculus, differential equations, and linearalgebra. He is a member of the American Mathematical Society and the MathematicalAssociation of America. He has published over 20 journal articles in areas such aspolynomial interpolation, numerical integration, algebraic differential equations, andmeans.

Arya Ebrahimpour, formerly at the Delaware County Campus of Penn State University, isnow an Associate Professor of Civil Engineering at the Idaho State University, Idaho, andteaches structural engineering. His research interests include structural dynamics, bridgeanalysis, and applications of advanced composite materials in civil structures. He haspublished several articles in professional journals and conference proceedings. He is amember of the American Society of Civil Engineers, the American Institute of Aeronauticsand Astronautics, the American Society for Engineering Education, and the Sigma XiScientific Research Society.


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