COMPUTER LABORATORY IMPLEMENTATION ISSUES AT A SMALL LIBERAL ARTS COLLEGE Richard A. Weida Lycomig College Williamsport, PA 17701 weida@lycomig.edu Abstract: Lycomig College is a small, private, liberal arts college i cetral Pesylvaia. The Departmet of Mathematical Scieces has recetly added weekly computer labs usig Derive to PreCalculus, Calculus I, ad Calculus II. This presetatio will discuss various issues that have arise i the implemetatio of these labs. Overview: Lycomig College has a studet body of approximately 1500 studets. The Departmet of Mathematical Scieces cosists of seve full-time faculty ad several part-time adjucts. Each semester the departmet offers, amog other courses, oe sectio each of PreCalculus, Calculus I, ad Calculus II. Util 1994 these courses were taught i the traditioal lecture format. I 199 the departmet decided to iclude computer laboratories to these courses. A committee of three faculty members was established to evaluate Computer Algebra Systems (CAS's). The author was chose as chair of this committee. Oce Derive was chose ad purchased, I was put i charge of implemetig weekly laboratories i the three courses. Weekly labs were added to PreCalculus startig i the Fall semester of 1993, ad i Calculus I ad II startig i the Fall semester of 1994. The past academic year was used to "fie-tue" these labs. Evaluatio of CAS's: The departmet iitially cosidered the use of graphig calculators. However, it was felt that the additioal cost would be a problem for may of our studets. With this i mid we decided to ivestigate differet Computer Algebra Systems for use o the college computer etwork. Our first step was to see what had bee accomplished at other schools. I atteded the Fifth Aual Iteratioal Coferece o Techology i Collegiate Mathematics i 199, where I atteded several extremely useful workshops usig Derive, Mathematica, ad Maple V. I also atteded various short presetatios o CAS applicatios at that coferece. The committee read software reviews foud i PC Magazie ad other sources. We the cotacted various compaies for evaluatio copies (some of which we had to purchase). After "hads-o" experiece with several differet software packages we chose Derive for our labs. While ot as powerful as some of the other CAS's, Derive's meu system made learig the package much easier ad faster tha most of the other systems. Furthermore, its ability to factor polyomials i a otatio uderstadable to a first-year college studet made Derive a better choice for PreCalculus. The departmet hopes to add Maple V at some future poit for use i upper-level mathematics courses. 1
Descriptio of Lab Purpose: We iteded these weekly labs as a additioal way for studets to lear the material. That is, the labs would add to, ot replace, the regular lectures. As much as possible, the labs should be exploratory, leadig the studets to discover facts "by themselves." Durig labs, studets would work i small groups of two or three. Each lab would iclude a writig compoet. The cocetratio would be o the mathematics ivolved, ot o the software beig used. Effort should also be take to demostrate both the stregths ad weakesses of Computer Algebra Systems. Writig Labs: While there were may fie laboratory mauals available from various publishers (with eve more available ow), at that time oe seem desiged for weekly labs. For example, oe maual had a total of 19 labs coverig all of Calculus I, II, ad III. With a 14-week semester we wated a miimum of 14 labs for each course. Also, the lab mauals are to some degree tied to particular textbooks ad the oes we were curretly usig did ot have ay lab mauals i Derive. I order to get the best "fit" with our studets, usig our texts, it was decided that oe of us would write the labs for these courses. Durig the summer of 1993 I prepared labs for PreCalculus. I the taught the course usig these labs i both the fall ad sprig semesters of that year. Mior adjustmets were made durig the sprig semesters, ad more major chages were made over the summer of 1994. A lab from this course is icluded as a example at the ed of this paper. Durig that same summer, alog with aother faculty member, I prepared labs for Calculus I ad II. I the fall I taught Calculus I ad the other faculty member taught Calculus II, both usig the appropriate labs from this collaboratio. A lab from these courses is also icluded at the ed of this paper. Durig the sprig semester, while o sabbatical, I cotiued revisig the labs for all three courses while others taught the courses usig the labs I prepared. A ufortuate result of switchig textbooks i Calculus that semester was the eed to reorgaize the order of the Calculus labs. The fiished labs have ow bee used for oe full year. Reactios: Studet reactio to the labs varied greatly. Most egative commets were directed at the additio of two cotact hours per week. (This complait was also made by some of the faculty.) Iitially, studets questioed the value of what they saw as a repetitio of material from the lectures. However, as time wet o, most studets came to appreciate the ew approach iheret i the more graphical methods employed i the labs. Surprisigly, the studets who resisted the logest were those from ear the "top" of the class. The best studets were already used to lookig at material i more the oe way ad approached the lab with a almost playful attitude. Some of the studets just "below" this level seemed cotet with their level of uderstadig, ad were aoyed at beig asked to approach the material i a differet way. The laboratory format seemed to have the greatest impact o studets who were strugglig i the lecture format. The most dazzlig example of this was oe studet who was extremely umotivated i class. O lecture days she would arrive late, seldom tured i
homework, ad always seemed bored with the class ad subject matter. O lab days she arrived early, always completed each lab, ad frequetly stayed after the lab to cotiue "playig" with Derive. Clearly, for this studet the graphical approach preseted a much more uderstadable way to look at mathematics. As a educator I was particular impressed with the studet ivolvemet durig the laboratories. Workig i small groups of two or three makes it possible for eve shy studets to take part. I the begiig of the courses, studets sometimes sped too much time cocetratig o the "keystrokes" istead of the mathematics. However, this problem decreases as the studets get more familiar with the software package. Freeig studets from the "drudgery" of calculatios really does lead to more discussio o the "how" ad "why" of the mathematics ivolved. As author of the labs, I was extremely pleased with some of the labs ad slightly disappoited with some of the others. For example, it was virtually impossible to write a PreCalculus lab o simplifyig expressios ivolvig radicals that actually gaied aythig from the use of the computer. The power of Derive actually added to this problem sice the program does its simplificatio of such expressios over the complex umbers. That is, if a simplificatio holds for the real umbers, but ot for complex umbers, the Derive must be "forced" to make the simplificatio. First-year college studets are usually ot ready for such distictios. A PreCalculus lab I am extremely pleased with is a lab that ivolved comparig the graphs of f(x), f(x)+c, f(x+c), Cf(x), ad f(cx). (Please see Lab 6 "Graphs of Fuctios" located at the ed of this paper.) Not oly did the lab go well at the time, but it made a tremedous impact later i the course whe we studied the trigoometric fuctios. It has always bee my experiece that studets have a hard time whe they first ecouter the topics of amplitude, period, ad phase shift. I was amazed at how quickly they picked up these ideas whe expressed i terms of the earlier lab. Eve after six weeks they still remembered the lab ad could apply it i a "ew" situatio! I Calculus I was pleased with Derive's ability to create ad simplify sums. I took advatage of this durig the presetatio of the defiitio of itegratio i Calculus I, ad i may of the applicatios of itegratio preseted i Calculus II. (Please see Lab 17 "The Area Betwee Two Curves" located at the ed of this paper.) Beig able to (relatively) quickly calculate Reima sums helped studets feel comfortable with the cocept. Furthermore, i may cases, Derive could evaluate the limit as wet to ifiity of the Reima sum. The fact that this aswer agreed with the defiite itegral coviced studets of the coectio. I experieced two difficulties i the labs. The first was dealig with various hardware problems. Experiece ad lots of calls to the Computer ceter have helped with this problem. Nevertheless, the first few weeks of the semester ofte provide extra "excitemet," particularly if there have bee ay chages to the etwork betwee semesters. The secod problem is maitaiig the fie lie betwee helpig studets move i the right directio ad "givig away" the aswers. This problem is particularly exacerbated at the ed of a lab whe the slower 3
workers are ruig out of time. If possible, I would suggest schedulig so as to alleviate the eed to rush off immediately after the lab. Oe uexpected problem with the implemetatio of these labs was the icreased resetmet of studets ot i the courses. The more classes scheduled i the computer labs, the less time there was available for other studets to use the computer labs for their ow assigmets. As other courses, icludig courses offered by other departmets, add computer labs ad/or computer assigmets, the demad for these computer labs cotiues to rise. Studets who would ever thik to iterrupt a class seem to have o problem iterruptig a lab ad demadig use of oe of the computers. Coclusios: I am defiitely glad that we added the computer labs to PreCalculus, Calculus I, ad Calculus II. I hope we expad the use of Computer Algebra Systems to our other courses, at least o a iformal basis. Studets foud havig a secod approach to the material aided their comprehesio. Some studets showed drastic improvemet. However, the workload o the istructor/coordiator is greatly icreased. This process should ot be etered ito without a firm commitmet from both the Departmet ad the College i geeral. Sources: [1] Larso, Hostetler, ad Edwards, Calculus Of a Sigle Variable, 5th Ed., D. C. Heath ad Compay, 1994. [] Leithold, L., Before Calculus: Fuctios, Graphs, ad Aalytic Geometry, d Ed., HarperCollis Publisher, 1989. [3] Weida, R., PreCalculus Computer Labs Usig Derive, Davis-Murphy Publishig, 1996. [4] Weida, R., Calculus Computer Labs Usig Derive, Davis-Murphy Publishig, 1996. 4
PreCalculus Lab 6 Graphs of Fuctios Objective I this lab you will compare the graphs of related fuctios ad ivestigate the effect of addig or multiplyig a costat value to the expressio f(x). Specifically, we wat to study the relatioship betwee the graph of f(x) ad the graphs of f(x)+c, f(x+c), Cf(x), ad f(cx), where C is ay real umber. New Derive Commads I this lab you will have more freedom the i previous labs. You should create your ow examples to ivestigate the properties we are iterested i. You should study eough differet examples to be sure that you completely uderstad what is happeig. It is importat that you use several fuctios ad that you try may values for the costat C. I particular, make sure that you use both positive ad egative umbers ad values whose magitudes are both larger ad smaller tha oe. Whe you have completely ivestigated oe area, write a summery of the results i the most geeral form while beig as precise as possible. We will do the first oe together ad you will do the others as part of the lab exercises. The first relatioship to study is the oe betwee the graph of f(x)+c ad the graph of f(x). Let's start with a simple fuctio. Declare the fuctio f(x) = x. The eter the equatio y = f(x) ad plot it. Now o the same graph we wat to plot y = f(x) + C for various values of C. Remember that each graph will be a differet color ad keep track of which graph is which. A few "ice" values to try first might be 1,, ad ½. Switch back to the Algebra scree ad eter the three equatios, y = f(x) + 1, y = f(x) +, ad y = f(x) + ½. Highlight the first equatio, switch to the Graphics scree, ad plot the equatio. Switch back to the Algebra scree, highlight the ext equatio, switch back to the Graphics scree, ad plot it. Do the same for the third equatio. To see the four graphs redraw i order, press the F9 ad F10 fuctio-keys simultaeously. The fuctios will be redraw i the order i which Figure 1 you etered them. Your scree should look like Figure 1. 5
Do you begi to see a patter? Let's try some egative values. Addig! would be the same as subtractig. Istead of eterig y = f(x) + (!) we'll use the equatio y = f(x)!. Eter this equatio i the Algebra scree ad plot it i the Graphics scree. Do the same for the equatios y = f(x)! 1 ad y = f(x)! ½. OK, it looks like, whe C is a positive umber, the resultig graph is the origial graph moved up C uits. Whe C is a egative umber, the graph is moved dow C uits. Whoops! What does that last setece mea? For example, if C is!3, the graph is "moved dow!3 uits." How do you move egative uits? OK, we better use absolute values so that the meaig is clear. Whe C is a egative umber, the graph is moved dow *C* uits. We eed to check this coclusio by tryig other fuctios. Delete all the graphs i the 3 Graphics scree. I the Algebra scree declare the fuctio f(x) = x. Now, oe at a time, plot each equatio o the same graph. Remember you do't have to retype the equatios sice Derive will oly use the latest versio of the declared fuctio. The "rule" we guessed above still seems to work. Maybe it oly works for polyomials. Let's try a ratioal fuctio. Delete all the graphs, declare the fuctio f (x) ' 1 % x, ad plot the equatios. You'll probably wat x to zoom out a couple of times to get a better view. The rule still seems to hold. Next try the fuctio f ( x) ' x! 3. We've tried a few differet fuctios ad we used differet values for C so we've probably covered all the cases. Now we eed to write a coclusio for this case. The graph of the equatio y = f(x) + C is exactly the graph of the equatio y = f(x) moved vertically. If C is a positive umber the the graph is moved C uits up the y-axis. If C is a egative umber the the graph is moved *C* uits dow the y-axis. As you complete this lab you will wat to use a variety of fuctios. The followig list of fuctios, while far from complete, provides some further examples you might cosider. f (x ) ' 1 f (x ) ' 1 3 f(x) = x f(x) = x x x f (x ) ' x ( x! )(x % 1) f (x) ' x f(x) = si x f(x) = cos x f (x) ' si x x f (x ) ' x % si x x 6
Exercises 1. Explore the relatioship betwee the graphs of the equatios y = f(x+c) ad y = f(x). Be sure to use several fuctios ad may possible values for the costat C icludig both positive ad egative umbers ad values whose magitudes are both larger ad smaller tha oe. Whe you have completely ivestigated this relatioship, write a summery of the results i the most geeral form while beig as precise as possible.. Explore the relatioship betwee the graphs of the equatios y = Cf(x) ad y = f(x). Be sure to use several fuctios ad may possible values for the costat C icludig both positive ad egative umbers ad values whose magitudes are both larger ad smaller tha oe. (I strogly suggest you specifically look at the value C =!1.) Whe you have completely ivestigated this relatioship, write a summery of the results i the most geeral form while beig as precise as possible. 3. Explore the relatioship betwee the graphs of the equatios y = f(cx) ad y = f(x). Be sure to use several fuctios ad may possible values for the costat C icludig both positive ad egative umbers ad values whose magitudes are both larger ad smaller tha oe. (I strogly suggest you specifically look at the value C =!1.) Whe you have completely ivestigated this relatioship, write a summery of the results i the most geeral form while beig as precise as possible. 4. Describe the relatioship betwee the graphs of the equatios y = 3f(x!1) + 4 ad y = f(x) as a combiatio of the above "movemets." (I strogly suggest that you test your aswer for specific fuctios usig Derive.) 5. Let g(x) = x! 4x! 5. Use the techique of completig the square to rewrite this fuctio i the form A(x+B) + C. Describe the relatioship of the graph of the equatios y = g(x) ad y = x. Sketch the graph of the equatio y = g(x) by had. Quick Review I today's lab you studied the relatioship betwee the graph of f(x) ad the graphs of f(x)+c, f(x+c), Cf(x), ad f(cx), where C is ay real umber. You the saw that you could rewrite "messy" fuctios i terms of icer fuctios. This will allow you to move ad stretch the graphs of these icer fuctios to get the graphs of the "messy" fuctios. 7
Calculus Lab 17 The Area Betwee Two Curves Objective I this lab you will review the formula for fidig the area betwee two curves ad use Derive to calculate these areas. New Derive Commads Recall that whe we looked at the area uder a curve, we first approximated the regio by a collectio of rectagles. The sum of the areas of these rectagles was approximately the area of the regio. The actual area of the regio is the the limit of this sum as the widths of all the rectagles go to zero. Luckily, this limit is othig more tha the defiite itegral ad, usually we ca use the Fudametal Theorem of Calculus to evaluate it. Let's repeat this process for the area betwee two curves. Declare the two fuctios f(x) = x ad g(x) = x! ad plot them o the same graph. It appears that the two graphs itersect at the poits x =!1 ad x =. Verify these itersectios by solvig the equatio f(x) = g(x). The area betwee these two curves is thus the area of the regio betwee x =!1 ad x = that lies below the curve f(x) ad above the curve g(x). For our first approximatio of the area of this regio, let's partitio the iterval [!1, ] ito six equal pieces. Each subiterval will have legth ) x ' b! a '! (!1) ' 1. Thus, 6 the six subitervals are [!1,!½], [!½, 0], [0, ½], [½, 1], [1, 1½], ad [1½, ]. We must choose a poit t k i each subiterval. The "top" ad th "bottom" of the k rectagle are the determied by f(t k) ad g(t k), respectively. Figure 1 shows the resultig rectagles whe t k is chose as the th midpoit of the k subiterval. The midpoit of the first subiterval is!¾. Therefore, the first rectagle has legth f(!¾)! g(!¾) ad width is )x = ½. Use Derive to compute the area of this rectagle. You should 11 get. The secod rectagle has legth 3 f(!¼)! g(!¼) ad width ½. Compute its area. Figure 1 8
7 You should get 3 areas together. You should get. Cotiue this process for the other four rectagles ad the add the six 73 16 or 4.565 as the sum of the areas of the rectagles. If we had chose a differet poit for t k the our approximatio would have a differet value. To see this, use the same partitio as above but choose t k to be the right edpoit of the 5 iterval. The first rectagle has legth f(!½)! g(!½) ad width ½. Therefore, its area is. 8 35 Compute ad add the areas of all six rectagles. This time the sum of the areas is or 4.375. 8 Which of these two approximatios do you thik is better? Why? To isure a better approximatio of the area of the regio we must decrease the width of the subitervals ad therefore icrease their umber. Istead of cotiuig this process "by had," let's automate as much of this process as possible. If we have subitervals of equal legth, the each has width ) x ' b! a '! (!1) ' 3. Therefore, the first subiterval is!1,!1 % 3!1 % 3,!1 % 3 ; the secod subiterval is ; the third subiterval is 3!1 %,!1%3 3 th 3!1 %(k!1),!1 %k 3 ; ad the k subiterval is. Sice it would be "messy" to deote the midpoit of this iterval, we will choose t k to be the right 3 edpoit of the iterval. That is, we will choose t k '!1%k. )x ' 3 th With this choice we see that the k rectagle has legth f(t k)! g(t k) ad width. f!1% 3k! g!1 % 3k 3 Therefore, it has area. Addig the areas of the j f!1% 3k! g!1% 3k 3 rectagles, we get as the approximatio of the area k'1 of the regio. To create this sum, first eter the expressio (f(!1+3k/)!g(!1+3k/))(3/). The choose Calculus, Sum ad press Eter three times. The first time is for the correct expressio, the secod time is for the correct variable, ad the third time is for the correct limits. Fially, use the F3 fuctio-key ad declare this summatio as the fuctio h(). The last three expressios o your scree should look like Figure. The fuctio h() represets the sum of the areas of the rectagles with equal width ad usig the right edpoit to determie the height. 9
To check our computatios evaluate h(6). You should get the same aswer as whe we did this 35 "by had," amely or 4.375. 8 Now evaluate h(5), the sum of the areas usig 5 rectagles. Similarly, evaluate h(100), ad h(1000). As you ca see, as we icrease the umber of rectagles, the area Figure approaches a value of 4.5. (If you icrease the umber of digits of accuracy, you will see that it ever quite reaches this value.) To get the exact value of the area of the regio betwee the two curves you must lim evaluate 64 h (). Use Derive to do so. (Type i the fuctio ame ad choose Calculus, lim Limit... ) The area betwee the curves is 4.5. Remember that 64 h() really represets lim 64 j f (t k )! g(t k ) )x. This is just the limit of a Riema Sum ad is therefore equal to k'1 f (x)! g(x) dx. Use Derive to evaluate this itegral. (Type i the expressio ad choose m!1 Calculus, Itegrate... ) Of course, you will get the same aswer. You're probably woderig why we wet to all the trouble to evaluate various sums of areas whe the itegral was so much easier to compute. The first reaso to do so is to reiforce the defiitio of itegratio. Too may studets cofuse the itegral ad the shortcut provided by the Fudametal Theorem. Secodly, this approach will help you uderstad (ad remember) the various formulas i this chapter. Thirdly, this process should give you some isight ito umerical itegratio. 3 For our ext example, cosider the two fuctios f(x) = x + x! x! 1 ad g(x) = x! 1. Declare ad graph these fuctios. The verify that the graphs itersect at the poits x =!1, x = 0, ad x = 1. It might be temptig to thik that the area betwee these two curves is equal to m!1 1 f (x)! g(x) dx, but it's ot. (Evaluate this itegral. What do you get? Why?) Remember, i the formula for area betwee two curves, the fuctio f was the fuctio that formed the "top" of the rectagle ad the fuctio g formed the "bottom" of the rectagle. Which of the two give fuctios i this example is the "top" oe? From the graph we see that f is the "top" fuctio from x =!1 to x = 0, but g is the "top" fuctio from x = 0 to x = 1. We'll 10
have to break this regio ito two pieces ad use two itegrals. The area of the regio is give by 0 m! 1 1 f (x )! g ( x) dx % m 0 g(x)!f (x) dx. Evaluate these itegrals to verify that the area of the regio is ½. Exercises 3 1. Cosider the regio betwee the curves y = x ad y = x. a) Approximate the area of this regio by partitioig the appropriate iterval ito 4 equal parts. b) Approximate the area of this regio by partitioig the appropriate iterval ito 0 equal parts. c) Approximate the area of this regio by partitioig the appropriate iterval ito 100 equal parts. d) Fid the exact area of the regio by takig the limit of a appropriate Riema Sum. e) Use itegratio to fid the exact area of the regio. 3. Fid the area of the regio betwee the two curves give by f(x) = x! x ad g(x) = x! 3x. 4 3 3. Fid the area of the regio betwee the two curves give by f(x) = x! 3x! x + 3x + 1 ad g(x) =! x. 4. Write a paragraph (at least five seteces) describig how to fid the area betwee two curves. 5. Fid the area of the regio bouded by the curves y = x +, y = 6! 3x, ad 3y =! x. 6. Fid a horizotal lie, y = k, which divides the area betwee y = x ad y = 9 ito two equal parts. Quick Review I this lab you worked with the defiitio of itegratio to fid the area of a regio bouded by two or more curves. The you used the Fudametal Theorem of Calculus to evaluate these itegrals. Hopefully, this reiforced both the defiitio of itegratio ad the "formula" for fidig the area betwee two curves. 11