VISUALIZING SPECIAL FUNCTIONS AND APPLICATIONS IN CALCULUS 1

Transcription

1 VISUALIZING SPECIAL FUNCTIONS AND APPLICATIONS IN CALCULUS 1 Rober E. Kowalczk and Adam O. Hausknech Universi of Massachuses Darmouh Mahemaics Deparmen, 85 Old Wespor Road, N. Darmouh, MA rkowalczk@umassd.edu and ahausknech@umassd.edu Modern calculus courses are incorporaing man more applicaions and special funcions ino heir conen. Using and appling hese special funcions is ofenimes a complee mser o our sudens. To help hem beer eplore real-life applicaions and he properies of special funcions, we have developed a special se of visual ools ha we use wih our sudens. In his paper, we presen a collecion of eamples ha demonsrae he use of hese visual ools. Defining a Special Funcion In he firs week of our calculus courses when we firs inroduce he concep of a funcion o our sudens, we alk abou inpu, oupu, and he rule describing he funcion. TEMATH s Recangular Tracker ool is used o help sudens visualize he concep of inpuing a value of, oupuing a value of, and using a graph o represen his process (see Figure 1). A few weeks laer, we inroduce he concep of he derivaive as a funcion. From a purel algebraic poin of view, his is quie bewildering o our sudens. We use TEMATH s Dnamic Tangen ool o help sudens visualize he derivaive as a funcion where he inpu is a value of and he oupu is he slope of he angen line o he curve. As he angen dnamicall moves along he curve of he funcion f (), he value of he slope of he angen line is ploed creaing he graph of he derivaive funcion f () (see Figure ). This visual demonsraion helps sudens develop an inuiion abou he derivaive as a funcion, and more imporanl, as a funcion ha measures he insananeous rae of change of f (). 4 (1.3, 3.37) 3 f() = sin() f()=^3 7^+1 f'() = cos() Figure 1 Definiion of a Funcion Figure The Derivaive Funcion Our calculus sudens readil accep funcions like f () = + and g() = 3sin(), bu when we use he Fundamenal Theorem of Calculus o sar consrucing anideriva- on Technolog in Collegiae Mahemaics", Addison-Wesle, 1.

2 ives like Si() = sin() d, and hen acuall call Si() a funcion ha has applicaions o opics, our sudens look on wih bewildermen. This is when we use echnolog and is powerful visual ools o help our sudens visualize he concep of his pe of funcion definiion. We use TEMATH s Dnamic Inegraor ool o visuall demonsrae he definiion of Si() and o eplore is properies. This ool dnamicall shades in he area so far and plos he value of ha area, hus, generaing he plo of Si() (see Figure 3). Si() 1 f() = sin()/ 15 Figure 3 The Sine-Inegral Figure 4 Nauilus Shell Daa Sampling Modeling he Spiral of he Chambered Nauilus Shell Naure presens man ecellen opporuniies for using mahemaics o describe he srucure of is various life forms. For eample, polar coordinaes can be used o model he spiral srucures found in man shells, in paricular, he Chambered Nauilus. We used a digial camera o ake a picure of a Chambered Nauilus shell, impored he image ino a compuer, and copied i ino TEMATH s Polar Plo Mode. Using TEMATH s Poin ool, we sampled poins along he shell s spiral a inervals of /4 radians (see Figure 4). Ne, we creaed a able of values for hese poins, found he leas squares r 1 6 Figure 5 Leas Squares Eponenial Fi Figure 6 Nauilus Shell Spiral Fi on Technolog in Collegiae Mahemaics", Addison-Wesle, 1.

3 eponenial fi r = e using recangular coordinaes (see Figure 5), and overlaid he polar plo of he fi on op of he image of he shell (see Figure 6). The fi is ecellen! Real-life applicaions rul make our sudens appreciae he modeling poenial of mahemaical funcions. Modeling a Hanging Chain In man radiional elemenar differenial equaion es, Bessel s differenial equaion, + + (1 v ) = where v is a parameer, is used o inroduce or illusrae power series echniques for solving differenial equaions around regular singular poins. The geomeric properies of is soluions are rarel discussed and is applicaions are onl menioned in passing. Thus, a suden s firs eposure o Bessel s equaion is almos alwas from he algebraic poin of view (see [5]). To moivae suden ineres in Bessel s equaion, we use TEMATH s differenial equaion solver o sud he equaion. For eample, if v =, hen we can visuall verif ha J () = k = ( 1) k (k!) k and Y () = π J () ( + ln( ) ) π k = ( 1) k (k!) (k) (where = is Euler s consan and (k) = L+ 1 ) are independen soluions (see Figure 7). I is eas o see ha he are independen k because k 1 Jo() L Hanging Chain Tension 1 d (,) Tension g 1 Yo() Figure 7 Plos of J () and Y () d Figure 8 Oscillaing Hanging Chain J () = 1 and lim Y () =. Small oscillaions of a hanging chain can be modeled b he parial differenial equaion = g where g is acceleraion due o gravi (see Figure 8). Because he parial derivaives on he lef are wih respec o alone and on Technolog in Collegiae Mahemaics", Addison-Wesle, 1.

4 he parial derivaives on he righ are wih respec o alone, sums of funcions of he form (,) = f ()cos( w + e) are soluions where f () is a differeniable funcion such ha f (L) = and w, e are consans. Making he subsiuions, = f ()cos( w + e) and = gu (w), reduces he model o Bessel s equaion u F + u F + F = where F(u) = ( f o )(u). SinceY () is undefined, he soluion of he reduced equaion mus be F(u) = cj (u). Transforming back, we find ha f () = cj (r L) where r is a roo of J. Consequenl, (,) = c k J k (r k L)cos( g L r k e k ) are soluions of he original model where r k is he k-h roo of J (), L is he lengh of he chain, and c k, e k are consans (see [1], [4], [5] for deails and hisor). A sequence of plos of he normal mode soluion (,) = J (r 1 )cos( 3 r 1 ) is shown in Figure 9. A sequence of plos for he muli-mode soluion (,) = 1 1. k =1 J (r k )cos( 3 r k ), is shown in Figure L = 1 L = 1.46 =.46.5 =.5 Figure 9 A Normal Mode Soluion Figure 1 A Muli-Mode Soluion Using TEMATH., we can save he overlaid plos of he hanging cable shown in Figure 9 (or Figure 1) as a sequence of separae picure files which hen can be convered ino an animaion using a varie of shareware sofware such as GraphicConverer. To moivae class discussion, ou can se up a hanging chain in class and compare is moion o he animaion generaed b he Bessel funcion model. Visualizing Newon s Mehod Newon s mehod converges rapidl o a roo if he iniial guess is nearb or he funcion near he roo is well-behaved. To help our sudens visualize he convergence and he pifalls of Newon s Mehod, we use TEMATH s Roo Finder Tool. We sar wih a sraigh forward eample like f () =. Afer an iniial guess is enered ino on Technolog in Collegiae Mahemaics", Addison-Wesle, 1.

5 TEMATH, each click of he Sep buon performs one ieraion, draws he angen o he curve, and displas he ne ierae. Sudens wach he angen line process converge rapidl (wihin si ieraions) o he roo (see Figure 11). Ne we presen he sudens wih a more challenging eample, f () = 3 3. Saring wih an iniial guess of =, Newon s ieraion process coninuousl ccles beween he values, 3, 1.96, and 1.15 (see Figure 1). As anoher eample ehibiing a differen behavior, we use he arcan funcion. We sar wih an iniial guess of = A he beginning, he ieraes appear o ccle back and forh beween wo values, bu hen afer abou en ieraions, he ieraes converge rapidl o he roo =. If we slighl increase he value of he iniial guess o = , he ieraes again appear o ccle back and forh beween wo values, bu hen afer abou en ieraions, he ieraes diverge owards boh ±. These visualizaions help our sudens develop a real undersanding of he convergence/divergence properies of Newon s mehod. 3 f() = ^ 3 3 f() = ^3 3 5 Figure 11 Newon s Mehod 7 Figure 1 Ccling wih Newon s Mehod Bibliograph [1] Frank Bowman, Inroducion o Bessel Funcions, 1958, Dover. [] Rober Kowalczk and Adam Hausknech, TEMATH - Tools for Eploring Mahemaics Version., [3] Rober Kowalczk and Adam Hausknech, Using TEMATH in Calculus [4] G.N. Wason, A Treaise on he Theor of Bessel Funcions, Cambridge Universi Press, [5] Dennis G. Zill, A Firs Course on Differenial Equaions, 5h Ediion, PWS, on Technolog in Collegiae Mahemaics", Addison-Wesle, 1.