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Asymptoti et artes repraesentationis graphicae

Svmmarivm

Haec lectio stvdivm limitvm hvivs partis conclvdit prima eorvm applicatione geometrica: asymptotis. In Algebra et Geometria asymptoti verticales, horizontales et obliqvae fvnctionvm rationalivm et fvnctionvm cvm radicalibvs ope tabvlae signorvm et divisionis polynomiorvm inventae svnt, et habitvs cvrvae ivxta eas vt factvm intvitivvm admissvs est. Hic per limites laterales, limites ad infinitvm et limites infinitos definivntvr, et ea facta admissa demonstrantvr. Ostenditvr rectam asymptotvm esse tvm demvm cvm distantia cvrvae ab ea ad nihilvm tendit, formvla declivitatis et ordinatae in origine asymptoti obliqvae invenitvr, et dedvcitvr in vtroqve fine vnam ad svmmvm esse. Lectio praeterea opinionem vvlgarem corrigit, qva cvrva asymptotvm svam secare non posse pvtatvr, methodvm proponit fvnctionis sine derivatis repraesentandae, eamqve ad fvnctiones Algebrae et Geometriae adhibet, qvarvm tabvlae signorvm et divisiones polynomiorvm nvnc limitibvs complentvr.

Proposita discendi

  1. Asymptotos verticales, horizontales et obliqvas per limites definire, easqve cvm distantia cvrvae a recta conferre.
  2. Formvlas declivitatis et ordinatae in origine asymptoti obliqvae demonstrare, et vnicitatem asymptoti non verticalis in vtroqve fine.
  3. Facta in Algebra et Geometria admissa de habitv fvnctionis rationalis ivxta nihilvm svi denominatoris demonstrare, et positionem cvrvae respectv asymptoti svae decernere.
  4. Fvnctiones algebraicas et trigonometricas repraesentare dominivm, symmetrias, intersectiones cvm axibvs, tabvlam signorvm et asymptotos conivngendo.

Definitio

In lectione de fvnctionibvs rationalibvs Algebrae et Geometriae, asymptotvs erat recta ad qvam cvrva magis magisqve accedit dvm per aliqvem ex ramis svis recedit. Limitibvs hvivs partis haec notio accvrate scribitvr. Dvo modi recedendi svnt: variabilis ad nvmervm accedit dvm valores sine margine crescvnt, avt variabilis sine margine crescit. Primo respondent asymptoti verticales; alteri, ceterae.

DefinitioAsymptoti

Sit ff fvnctio cvivs dominivm est AA.

  • Recta x=ax = a est asymptotvs verticalis delineationis ipsivs ff si saltem vnvs ex limitibvs lateralibvs ipsivs ff in aa est ∞\infty avt −∞-\infty.
  • Si AA svperne circvmscripta non est, recta y=mx+by = mx + b est asymptotvs delineationis ipsivs ff in ∞\infty si lim⁡x→∞(f(x)−(mx+b))=0\lim_{x \to \infty} \left(f(x) - (mx + b)\right) = 0. Appellatvr horizontalis si m=0m = 0 et obliqva si m≠0m \neq 0. Eodem modo asymptotvs in −∞-\infty definitvr.

Tres animadversiones definitionem accvratiorem reddvnt. Prima: asymptotvs horizontalis y=Ly = L in ∞\infty nihil alivd est qvam assertio lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L, ex lectione de limitibvs cvm variabilis ad infinitvm tendit. Altera: asymptoti non verticales in vtroqve fine separatim definivntvr, et in ∞\infty et in −∞-\infty diversae esse possvnt, sicvt rectae y=π2y = \frac{\pi}{2} et y=−π2y = -\frac{\pi}{2} arcvs tangentis. Tertia: si f(x)−g(x)f(x) - g(x) ad 00 tendit pro fvnctione gg qvae primi gradvs non est, sicvt polynomivm qvotiens divisionis cvm gradvs nvmeratoris gradvm denominatoris dvobvs avt plvribvs svperat, cvrva y=g(x)y = g(x) asymptotvs cvrvilinea dicitvr; omnia qvae de asymptotis obliqvis seqvvntvr de ea qvoqve valent, praeter formvlas declivitatis et ordinatae in origine.

Definitio differentiam ordinatarvm metitvr, id est distantiam verticalem inter cvrvam et rectam. Notio intvitiva contra de distantia cvrvae a recta loqvitvr, qvae in perpendicvlari metitvr. Propositio qvae seqvitvr ostendit vtramqve simvl ad nihilvm tendere.

PropositioDistantia ab asymptoto

Sit Px=(x,f(x))P_x = (x, f(x)) pvnctvm delineationis cvivs abscissa est xx, et sit rr recta y=mx+by = mx + b. Tvm rr est asymptotvs delineationis ipsivs ff in ∞\infty si et solvm si lim⁡x→∞d⁡(Px,r)=0\lim_{x \to \infty} \operatorname{d}(P_x, r) = 0. Distantia a recta est distantia verticalis per constantem positivam mvltiplicata. In figvra cvrva est f(x)=x2+1+1x−1f(x) = \frac{x}{2} + 1 + \frac{1}{x - 1}, cvivs asymptotvs est recta y=x2+1y = \frac{x}{2} + 1.

Demonstratio

  1. d⁡(Px,r)=∣mx−f(x)+b∣m2+1=∣f(x)−(mx+b)∣m2+1\operatorname{d}(P_x, r) = \frac{\abs{mx - f(x) + b}}{\sqrt{m^2 + 1}} = \resaltar{\frac{\abs{f(x) - (mx + b)}}{\sqrt{m^2 + 1}}}

    Recta rr aeqvationem mx−y+b=0mx - y + b = 0 habet. Per formvlam distantiae pvncti a recta ex lectione de vectoribvs in plano, distantia ipsivs PxP_x a rr est distantia verticalis per m2+1\sqrt{m^2 + 1} divisa, qvae ab xx non pendet.

  2. f(x)−(mx+b)→0⇔d⁡(Px,r)→0f(x) - (mx + b) \to 0 \Leftrightarrow \resaltar{\operatorname{d}(P_x, r) \to 0}

    Per lemma de limite vt distantia ad nihilvm tendente ex lectione de algebra limitvm, ad infinitvm translatvm, f(x)−(mx+b)f(x) - (mx + b) ad 00 tendit si et solvm si eivs valor absolvtvs ad 00 tendit; et hic ad 00 tendit si et solvm si eivs prodvctvm per constantem positivam 1m2+1\frac{1}{\sqrt{m^2 + 1}} eodem tendit, per regvlam combinationis linearis.