Exponential function and its derivative revisited

Transcription

1 Expoetial fuctio a its erivative revisite Weg Ki Ho, Foo Him Ho Natioal Istitute of Eucatio, Sigapore {wegki,foohim}.ho@ie.eu.sg Tuo Yeog Lee NUS High School of Math & Sciece hsleety@us.eu.sg February 3, 01 Abstract Most of the available proofs for x ex ) = e x rely o results ivolvig either power series, uiform covergece or a rou-about efiitio of the atural logarithm fuctio l x) by the efiite itegral x 1 t, 1 t a are thus ot reaily accessible by high school teachers a stuets. Eve istructors of calculus courses avoi showig the complete proof to their uergrauate stuets because a irect a elemetary approach is missig. This short paper fills i this gap by supplyig a simple proof of the aforemetioe basic calculus fact. 1 Itrouctio The efiitio of the expoetial fuctio e x x ) := lim 1 + origiate from Leohar Euler [1, p.363]). Apart from Euler himself, several authors base their proofs of x ex ) = e x o this efiitio; the ufortuate state of affairs beig that most of these are spawe with gaps. It is impossible to be ecyclopeic i ocumetig all such icomplete but oetheless publishe) proofs. But for illustratio s sake, we quickly supply some eviece of such iaequacy. For istace, i [3], apart from the icomplete proof of the mootoicity of 1 + x ) whe x 0, the justificatio for e x+y = e x e y, crucial i the proof of the result x ex ) = e x, is missig. I aother work [5], the result e x = lim 1 + x ) was prove oly for 0 x 1. However, it oes ot appear to work i geeral ue to the limitatio of the fuctioal equatio 10) o [5, p.843]. Furthermore, a careful reaig of the works [5] a [] reveals that the esity of Q i R has bee exploite this approach, i our view, caot be cosiere as elemetary. 1

2 Our paper uses a simple proof of x ex ) = e x, free of the aforemetioe eficiecies, that ca be emostrate eve to a easily uerstoo by) a freshma calculus auiece. To the best of the authors kowlege, this proof is ew. Such a emostratio, we believe, is beeficial to a calculus stuet as it weaves together may previously acquire otios a results ito oe meaigful mathematical fabric. I the esuig evelopmet, the oly two pre-requisites we assume of the reaer are the Squeeze Theorem a the Archimeea property of real umbers. Some crucial lemmata I a typical calculus or real aalysis course, the Beroulli s a Arithmetic-Mea Geometric-Mea iequalities AM-GM iequality, for short) are ofte iclue i fouatioal materials. For self-cotaimet, we recor them below: Propositio.1 Beroulli s iequality). For ay iteger 0 a ay real x 1, it hols that 1 + x) 1 + x. Propositio. Arithmetic-Geometric iequality). For ay o-egative x 1,, x R, it hols that x1+x+ +x x 1 x x. These famous iequalities collaborate to justify the followig crucial lemmata: Lemma.3. If N N the N ) N 4. Proof. The above iequality follows from the observatio that for N 3, 1 N 1 4 = N 1 ) 1N 1 1 N + 1 ) + N ) 1 N N + 1, owig to AM-GM iequality, a that the first two terms are less tha 4. Lemma.4. If x 0 a N, the 1 + x ) 4 1+ x. Proof. By the Archimeea property there exists N N so large that if N the 1 + x ) 1 + x 1 + ) x N1 + x ) )N1+ x ) 1 + 1N ) 1+ x )N The proof is complete. = 4 1+ x by Lemma.3).

3 Lemma.5. If c R, the lim 1 + c ) = 1. Proof. I view of Lemma.4 a the Squeeze Theorem, it suffices to prove that 1 + c 1 + c ) c ) for all sufficietly large itegers. Iee, if is ay positive iteger satisfyig 1 + c > 0 thaks to the Archimeea property), the Beroulli s iequality immeiately justifies the first iequality: 1 + c ) c 1 + = 1 + c. The seco iequality follows from the AM-GM iequality: 1 + c ) = c ) c ) ). k=1 As a cosequece of Lemma.5, we obtai the followig: Corollary.6. If x ) =1 is a boue sequece of real umbers, the lim 1 + x ) = 1. 3 Elemetary properties of x e x ) This sectio is evote to some elemetary properties of the expoetial fuctio x e x ), a to o so we must first establish its fuctioal status. Propositio 3.1. If x R, the sequece 1 + x ) ) =1 coverges. ) Proof. Base o the observatio that for ay N, lim 1 x = 1 by virtue of Lemma.5, oce the covergece of the sequece 1) o x 0 is prove it the follows that lim 1 x = lim ) 1 x 1 + x ) ) 1) 3

4 exists, thus accoutig for the covergece of the sequece 1) o x < 0. It thus remais to prove that the sequece 1) coverges o x 0. Now for ay N, we have 1 + x ) = x ) x )) = 1 + x , where the seco-to-last iequality is the AM-GM iequality. Thus, the sequece 1) is mootoe. Because every boue mootoe sequece coverges, the proof is complete. Crucially, the preceig propositio justifies the existece of lim 1 + x ) which is eote by e x. Itereste reaers ca fi a alterative but more complicate) proof of this covergece o [4, p.193] usig power series. We ow establish the followig for the expoetial fuctio x e x ): Theorem 3.. i) e 0 = 1. ii) If a, b R, the e a e b = e a+b. I particular, e x > 0 for all x R. iii) If a, b R, the e a e b = e a b. iv) If a, b R, the e a b a) e b e a e b b a). v) The fuctio x e x is strictly icreasig; i particular, it is ijective. vi) The fuctio x e x is cotiuous o R. Proof. i) is obvious. To prove ii), we observe that if N the 1 + ) a 1 + ) b = 1 + a + b ) ) ab a+b ). Cosequetly, ii) follows from Lemma.6 a i). It is also clear that iii) follows from ii). We ow establish iv). Without loss of geerality, we suppose that b a. The for ay N satisfyig 1 + a > 0, we have 1 + b 1 + a a hece b a) 1 + a ) 1 b a b ) 1 k 1 + a ) k = 1 + ) b b a 1 = b a) 1 + a ) 1 + b ) b ) 1. 4

5 Lettig completes the argumet. It is easy to check that v) is a immeiate cosequece of iv) a ii). Also, vi) follows from iv). 4 A elemetary proof of x ex ) = e x Theorem 4.1. The expoetial fuctio x e x ) is ifferetiable o R a x ex ) = e x x R). Proof. Let x R be give. The, for ay real umber h 0, we use Theorem 3.iv) with a = x a b = x + h to get e x h e x+h e x e x+h h or equivaletly, e x+h e x e x e x+h e x. h Lettig h 0 a ivokig vi) complete the argumet. 5 Cocluig remarks I a curretly o-goig) survey 1 coucte i 011 amog Sigapore secoary school a juior college teachers, prelimiary fiigs reveale that over 80% of the participats has either o or icomplete i.e., poorly or wrogly justifie) kowlege of the efiitio of the expoetial fuctio x e x ). As a result, a umber of classroom approaches base the proof of x ex ) = e x o the e ujustifie fact lim h 1 h 0 h = 1. There were also a umber of teachers who coveietly chose a sufficietly covicig but iaequately justifie proof of x ex ) = x x k k! = ) x k = x k! k=1 x k 1 k 1)! = x k k! = ex. The rest of the participats, formig the majority, supplie o proofs at all. A irect isaster of this eficiecy i the mathematical peagogical cotet kowlege mpck, for short) is the emergece of recet geeratios of mathematics stuets memorizig x ex ) = e x as a meaiglessly isolate fact. Repercussios abou; for istace, there is a wiesprea lack of uerstaig as to why e is calle the atural base especially whe the ecimal represetatio 1 The results of this survey will be aalyze a reporte i greater etail i a upcomig report. 5

6 is far from beig atural) sice the special role of y = e x as a fixe poit of the aturally occurrig ifferetial equatio y x = y is completely remove from the stuet s learig experiece. Respoig to this gap i the teachers mpck, our preset paper elivers a elemetary proof of that the erivative of expoetial fuctio is itself startig from Euler s origial efiitio of the expoetial fuctio x e x ). Because our simple proof oes ot rely o ay results of power series a uiform covergece, it is easily accessible by both istructors a learers of calculus. Refereces [1] L. Debath, The Legacy of Leohar Euler, Iteratioal Joural of Mathematical Eucatio i Sciece a Techology, vol ), p [] B.J. Garer, The atural expoetial comes before the atural logarithm, Iteratioal Joural of Mathematical Eucatio i Sciece a Techology, vol ), p [3] E.V. Hutigto, A Elemetary Theory of the Expoetial a Logarithmic Fuctios, America Mathematical Mothly, vol 3 o ), p [4] K. Kopp, Theory a applicatio of ifiite series, Dover Publicatios, Ic., New York, [5] Z.A. Melzak, O the expoetial fuctio, America Mathematical Mothly, vol 8 o ), p