CREATIVE USE OF MATHEMATICAL STRATEGIES IN PROOFS IN UNDERGRADUATE CALCULUS

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1 CREATIVE USE OF MATHEMATICAL STRATEGIES IN PROOFS IN UNDERGRADUATE CALCULUS VARGA Mrek (SK), NAŠTICKÁ Zuzn (SK) Abstrct Mthemticl theories could be developed neither without mthemticl proofs, nor without mthemticl cretivity Undergrdute students should be, thus, shown multiple proofs of fundmentl theorems of vrious mthemticl brnches, cretively pplying vrious techniques nd strtegies In this pper we present ltogether nine proofs of Lgrnge s Men Vlue Theorem, employing vrious strtegies which, in generl, re useful in problem solving Keywords: mthemticl proof, mthemticl cretivity, Lgrnge s theorem, Rolle s theorem, Cuchy s theorem Mthemtics Subject Clssifiction: 97 I 40 1 Introduction More thn hlf-century go the well-known mthemticin nd mthemtics eductor George Póly (1962) defined mthemticl knowledge s informtion nd know-how (in: Mnn, 2006, p 237), regrding the ltter one s the more importnt, defining it s the bility to solve problems requiring independence, judgement, originlity, nd cretivity (ibid) As for the cretivity in mthemticl thinking, it comprises the bility to see new reltionships between techniques nd res of ppliction, nd to mke ssocitions between possibly previously unrelted ides (Tmmdge, 1979, in: Hylock, 1987, p 60) Furthermore, mthemticl cretivity mnifests itself in the independent formultion of uncomplicted mthemticl problems, finding wys nd mens of solving these problems, the invention of proofs nd theorems, the independent deduction of formuls, nd finding originl methods of solving nonstndrd problems (Krutetskii, 1976, in: Hylock, 1987, p 60) In ccordnce with the bove presented ides developed by vrious mthemtics eduction reserchers, we suggest tht undergrdute clculus lectures nd seminrs should py more ttention to proofs of mthemticl theorems in multiple wys In ddition, encourging the students to devise their own proofs using vrious techniques nd strtegies my improve not only their understnding of the theorems nd theories, but lso their problem solving skills nd rise the level of their mthemticl cretivity 1083

2 2 Diversity in proofs of Lgrnge s Men Vlue Theorem Differentil clculus of one rel vrible plys n essentil role in undergrdute clculus courses its theoreticl prt llows for lter generliztion of the theory for multivrible functions, nd its ppliction prt is widely used in solutions of problems outside mthemtics Mny fundmentl propositions in differentil clculus ber the nmes of recognized mthemticins, such s Pierre de Fermt, Rolle, Jen Gston Drboux, Joseph Louis Lgrnge, Augustine Louis Cuchy, Guillume Frncois Antoine de l Hospitl, Brook Tylor In this pper we focus on Lgrnge s Theorem, more precisely Lgrnge s Men Vlue Theorem After introducing its wording, we present severl strtegies of proving the vlidity of the theorem Theorem (Lgrnge) Let function e continuous over closed intervl ; b nd differentible over n open intervl ; b Then there exists t lest one number ; b such tht f f Note Geometriclly speking, the theorem sys tht if the function f stisfies ll the ; b such tht the tngent to the grph of the function ssumptions, then there is point f t the point ; f is prllel to the line AB, where A ; f, B b; f Fig 1) (see Fig 1 Geometricl illustrtion of Lgrnge s Men Vlue Theorem In Proofs 1-6, other fundmentl theorems of differentil clculus nd the strtegy of using n uxiliry function re employed in order to prove Lgrnge s Theorem Proof 7 shows n ppliction of mtrix determinnt s function of rel vrible This proof might be of extrordinry vlue for university students, who, unfortuntely, often misunderstnd undergrdute lgebr nd clculus s very distnt brnches of mthemtics In Proof 8 the 1084

3 strtegy of specil cse s n opposite of generliztion is pplied Proof 9 ims to fcilitte students understnding of Lgrnge s Theorem nd its obvious vlidity through trnsition from bstrct clculus to concrete interprettion in physics (or other sciences) Proof 1 Let us consider n uxiliry function defined s follows x f x Kx, where K R The function is continuous over closed intervl ; b, nd lso differentible over n open intervl ; b Tking K such tht we obtin K f The function now stisfies the ssumptions of Rolle s theorem * It follows tht there must ; b for which 0 Since exist f K 0, it is esily seen tht f f Proof 2 A light tint of the previous proof cn be obtined if the rel number K is directly stted Then, the uxiliry function cn be immeditely defined s f x f x x, nd the rest of the proof is the sme s shown bove Proof 3 Let us consider n uxiliry function defined s follows x x f f b f x This function is continuous over ; b nd differentible over ; b It holds tht * Rolle s Theorem: Let function F be continuous over ; b, differentible over ; b, nd let F Then there exists t lest one number ; such tht F 0 F 1085

4 bf In other words, the function stisfies the ssumptions of Rolle s theorem, which implies ; b such tht 0 Obviously, tht there exists f f b f 0, which immeditely implies tht f f Proof 4 Indeed, the uxiliry function cn ssume vrious forms Let us now consider f bf x f x x b b Agin, the function is continuous over ; b nd differentible over ; b In ddition, it holds tht 0 Deriving from Rolle s theorem, there exists ; Tking x = ξ we get tht 0 such f f, nd immeditely f f Proof 5 Although n uxiliry function is pplied in this proof gin, let us refer bck to the geometricl interprettion of Lgrnge s Theorem (see Fig 1) The line AB is described by the eqution f yab f x We will investigte now the difference x f x yab, ie f x f x f x The function is continuous over ; b nd differentible over ; b Evluting t nd b we see tht 0, implying tht stisfies ll the three ssumptions of 1086

5 Rolle s Theorem Hence, there is t lest one ; such tht 0 obtin f f 0, Finlly, we nd consequently f f Proof 6 By tiny modifiction of the function from the previous proof we get new wy to prove Lgrnge s Theorem Suffice it to consider n uxiliry function f x f x x It cn be esily justified tht the function is lso continuous over ; b nd differentible over ; b In ddition, it holds tht f By Rolle s Theorem, there is ; such tht x 0 Hence, we obtin f f Proof 7 Besides pplying geometricl interprettions, lgebric concepts cn lso be wisely utilized Let us define determinnt f x g x h x x f g h g b h b This function is lso continuous over ; b nd differentible over ; b Following the rules of clcultions with determinnts it is immedite tht 0 Rolle s Theorem, there is ; such tht 0 Tking g x Thus, by More precisely, we obtin g g b f f f g h 0 h h b h h b g g b x nd hx 1, the finl eqution ssumes the form 0, ie f b f f f 1087

6 Proof 8 The proof cn be quite esily nd quickly done by pplying Cuchy s Theorem ** Lgrnge s Theorem is, in fct, its specil cse It is sufficient to tke g x x Proof 9 Lst but not lest, if not s mthemticl proof, then surely s n intuitive considertion of vlidity of Lgrnge s Theorem, let us employ one of its pplictions in elementry physics Let f f x denote function of point mss trjectory, the point chnging its position from while x ; b ; b; f A f, where the continuous vrible x denotes time, Then, there must exist n instnt t which the instntneous velocity, ie f f, is equl to the verge velocity, ie Let us consider the motion described by the function f f x over the time intervl ; b Roughly speking, the instntneous velocity of the point mss in motion cnnot be smller thn the verge velocity over the whole intervl in question Similrly, the instntneous velocity of the point mss in motion cnnot be greter thn the verge velocity t ll instnts of the considered time intervl Therefore, the instntneous velocity of the point mss must inevitbly ssume vlues greter s well s smller thn the verge velocity Since the chnge of velocity over time is continuous phenomenon, there must be n instnt within the intervl when the instntneous velocity of the point mss ssumes the vlue which is equl to f f the overll verge velocity, ie there is ; such tht f 3 Conclusion Every mthemticl theory uses fundmentl s well s less importnt theorems Trditionlly undergrdute clculus lectures im to build students knowledge of the theories grdully, systemticlly, including proofs of ll theorems We suggest this trdition be enriched by discussing with students multiple wys of proving the theorems Diversity nd cretivity in techniques nd strtegies in mthemticl proofs my stisfy students needs, ssuming tht different students might prefer nd better comprehend different pproches ** Cuchy s Theorem Let functions F, G be continuous over closed intervl ; b, differentible over n open intervl ; b, nd for ll x ; let gx 0 Then there exists t lest one ; Incidentlly, this theorem cn be esily proved if we tke 1 f f g b g g x such tht h x in Proof

7 References [1] BUDINSKÝ, B, CHARVÁT, J: Mtemtik I Prh: SNTL/Alf, s [2] FULIER, J, VRÁBEL, P: Diferenciálny počet Nitr: UKF, s [3] GREBENČA, M K, NOVOSELOV, S I: Kurs mtemtičeskogo nliz 1 Moskv: Učpedgiz, s [4] HAYLOCK, D W A frmework for ssessing mthemticl cretivity in school children In: Eductionl Studies in Mthemtics 1987, vol 18, no 1, p DOI: [5] IGNATJEVOVÁ, A V: Kurs vysšej mtemtiky Moskv: Vysšj škol, s [6] KLUVÁNEK, I, MIŠÍK, L, ŠVEC, M: Mtemtik I Brtislv: SVTL, s [7] KRUTETSKII, V A The Psychology of Mthemticl Abilities in Schoolchildren, 1976 (J Kilptrick nd I Wirzup, eds; J Teller, trns), University of Chicgo Press, Chicgo [8] KUDRJAVCEV, L D kol: Kurs mtemtičeskogo nliz 1 Moskv: Vysšj škol, s [9] MANN, E L Cretivity: The Essence of Mthemtics In: Journl for the Eduction of the Gifted 2006, Vol 30, No 2, p DOI: /jeg [10] POLYA, G Mthemticl Discovery: On Understnding, Lerning nd Teching Problem Solving 1962, Vol 1, Hoboken, NJ: John Wiley & Sons [11] TAMMADGE, A Cretivity: Presidentil ddress to the Mthemticl Assocition In: The Mthemticl Gzette 1979, vol 63, p DOI: / [12] TRENCH, W F: Introduction to rel nlysis NJ: Person, s ISBN [13] URL, Current ddress Vrg Mrek, PedDr, PhD Deprtment of Mthemtics, Fculty of Nturl Sciences Constntine the Philosopher University Tr A Hlinku 1, Nitr, Slovk Republic E-mil: mvrg@ukfsk Nštická Zuzn, Mgr Deprtment of Mthemtics, Fculty of Nturl Sciences Constntine the Philosopher University Tr A Hlinku 1, Nitr, Slovk Republic E-mil: zuznnstick@ukfsk 1089