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Cvrvatvra et radivs cvrvatvrae

Svmmarivm

Haec lectio metitvr qvantvm delineatio fvnctionis in pvncto cvrvetvr. Tangens est recta qvae in pvncto evndem valorem eandemqve derivatam habet ac fvnctio; circvmferentia oscvlans est ea qvae praeterea eandem derivatam secvndam habet. Aeqvatione circvmferentiae bis derivata efficitvr vt in qvoqve eivs pvncto combinatio qvaedam derivatae primae et secvndae radio eivs exacte aeqvalis sit; hinc lectio ostendit circvmferentiam oscvlantem, si derivata secvnda fvnctionis non evanescit, exsistere et vnicam esse, eivsqve centrvm et radivm dat, radivm cvrvatvrae. Eivs inversvm est cvrvatvra, qvae evanescit vbi derivata secvnda evanescit. Deinde ostendit centrvm cvrvatvrae esse limitem pvnctorvm in qvibvs normales vicinae se secant, vt Hvgenivs vidit, et cvrvatvram axibvs permvtatis non mvtari, qvod sinit eam metiri vbi tangens verticalis est. Problemata decernvnt vtrvm circvmferentia oscvlans delineationem transeat, invenivnt pvnctvm cvrvatvrae maximae exponentialis, pvncta cvrvatvrae maximae et minimae ellipsis et radivm cvrvatvrae catenariae, et saltvm cvrvatvrae inter rectam et arcvm perscrvtantvr.

Proposita discendi

  1. Circvmferentiam oscvlantem ad delineationem fvnctionis in pvncto definire, ostendere eam exsistere et vnicam esse cvm derivata secvnda non evanescit, eivsqve centrvm et radivm compvtare.
  2. Cvrvatvram et radivm cvrvatvrae delineationis in pvncto compvtare, cvrvatvram nvllam interpretari et pvncta cvrvatvrae maximae vel minimae delineationis invenire.
  3. Centrvm cvrvatvrae vt limitem intersectionvm normalivm obtinere, et cvrvatvram vbi tangens verticalis est metiri mvneribvs axivm permvtatis.
  4. Criterio derivatarvm svccessivarvm decernere vtrvm circvmferentia oscvlans delineationem transeat an ab vna eivs parte maneat.

Circvmferentia oscvlans

Tangens ad delineationem ipsivs ff in (a,f(a))(a, f(a)) est recta qvae in aa evndem valorem eandemqve derivatam habet ac ff. Lectio de concavitate, convexitate et pvnctis inflexionis ostendit, si f′′(a)≠0f''(a) \neq 0, delineationem prope aa ab vna tantvm parte tangentis svae manere: svpra eam si f′′(a)>0f''(a) > 0 et infra eam si f′′(a)<0f''(a) < 0. Sed non dixit qvantvm ab ea recedat, id est qvantvm cvrvetvr. Vt hoc metiamvr, convenit eam cvm cvrvis comparare qvae semper aeqve cvrvantvr, circvmferentiis: parva mvltvm cvrvatvr, magna parvm. Circvmferentia non est delineatio fvnctionis, qvia qvaeqve recta verticalis per interiora eivs transiens eam bis secat; sed vtrvmqve eivs dimidivm, sine extremis svis, est.

LemmaDerivatae semicircvli

Sint (h,k)(h, k) pvnctvm plani, r>0r > 0, σ\sigma aeqvale 11 vel −1-1, et u(x)=k+σr2−(x−h)2u(x) = k + \sigma \sqrt{r^2 - (x - h)^2} pro xx in (h−r,h+r)\ioo{h - r}{h + r}, cvivs delineatio est semicircvlvs svperior si σ=1\sigma = 1, vel inferior si σ=−1\sigma = -1, centro (h,k)(h, k) et radio rr, sine extremis svis. Tvm uu, u′u' et u′′u'' in (h−r,h+r)\ioo{h - r}{h + r} derivabiles svnt, u(x)≠ku(x) \neq k, et in qvoqve pvncto xx intervalli:

(x−h)+(u(x)−k)u′(x)=0,1+(u′(x))2+(u(x)−k)u′′(x)=0(x - h) + \left(u(x) - k\right) u'(x) = 0, \qquad 1 + \left(u'(x)\right)^2 + \left(u(x) - k\right) u''(x) = 0

Qvam ob rem u′′(x)u''(x) non evanescit et signvm ipsivs −σ-\sigma habet, ita vt semicircvlvs inferior stricte convexvs sit et svperior stricte concavvs, et in omnibvs eivs pvnctis:

(1+(u′(x))2)32∣u′′(x)∣=r\frac{\left(1 + \left(u'(x)\right)^2\right)^{\frac{3}{2}}}{\abs{u''(x)}} = r

Demonstratio

  1. (x−h)2+(u(x)−k)2=r2,x∈(h−r,h+r)(x - h)^2 + \left(u(x) - k\right)^2 = \dato{circ}{r^2}, \qquad x \in \ioo{h - r}{h + r}

    Si xx in intervallo est, r2−(x−h)2>0r^2 - (x - h)^2 > 0, ita vt u(x)−k=σr2−(x−h)2u(x) - k = \sigma \sqrt{r^2 - (x - h)^2} non evanescat, et uu derivabilis est per corollarivm de derivatis compositionvm vsitatarvm, ex lectione de regvla catenae. Qvadratis svmptis, delineatio ipsivs uu in circvmferentia iacet:

  2. 2(x−h)+2(u(x)−k)u′(x)=0,1+(u′(x))2+(u(x)−k)u′′(x)=02(x - h) + 2\left(u(x) - k\right) u'(x) = 0, \qquad 1 + \left(u'(x)\right)^2 + \left(u(x) - k\right) u''(x) = \dato{dos}{0}

    Ambo membra eadem fvnctio ipsivs xx in intervallo svnt, et derivatae eorvm congrvvnt: haec est derivatio implicita lectionis de derivata fvnctionis inversae et derivatione implicita, ad fvnctionem adhibita qvam iam derivabilem esse scimvs. Inde u′(x)=−x−hu(x)−ku'(x) = -\frac{x - h}{u(x) - k} est qvotiens fvnctionvm derivabilivm denominatore non nvllo, derivabilis per regvlam qvotientis, ex lectione de regvlis derivandi; itervm derivando obtinetvr secvnda aeqvalitas, ex qva u′′(x)=−1+(u′(x))2u(x)−ku''(x) = -\frac{1 + \left(u'(x)\right)^2}{u(x) - k} qvoqve est qvotiens fvnctionvm derivabilivm.

  3. 1+(u′(x))2=r2(u(x)−k)2,∣u′′(x)∣=r2∣u(x)−k∣3→(1+(u′(x))2)32∣u′′(x)∣=r1 + \left(u'(x)\right)^2 = \frac{r^2}{\left(u(x) - k\right)^2}, \qquad \abs{u''(x)} = \frac{r^2}{\abs{u(x) - k}^3} \rightarrow \resaltar{\frac{\left(1 + \left(u'(x)\right)^2\right)^{\frac{3}{2}}}{\abs{u''(x)}} = r}

    Cvm 1+(u′(x))2>01 + \left(u'(x)\right)^2 > 0, derivata secvnda non evanescit et signvm habet oppositvm signo ipsivs u(x)−ku(x) - k, qvod est signvm ipsivs σ\sigma; corollarivm de convexitate et derivata secvnda convexitatem strictam vel concavitatem strictam dat. Per primam aeqvalitatem, 1+(u′(x))21 + \left(u'(x)\right)^2 est svmma ipsorvm (x−h)2(x - h)^2 et (u(x)−k)2\left(u(x) - k\right)^2, qvae r2r^2 valet, per (u(x)−k)2\left(u(x) - k\right)^2 divisa.

In circvmferentia, declivitas tangentis et derivata secvnda a pvncto ad pvnctvm mvtantvr, sed combinatio lemmatis non mvtatvr: semper radivm valet. Hoc svadet vt ea metiamvr qvantvm qvaelibet delineatio cvrvetvr. Antea convenit circvmferentiam invenire qvae delineationi in pvncto optime accommodatvr, eam qvae cvm ea vsqve ad derivatam secvndam congrvit.

DefinitioCircvmferentia oscvlans

Sit ff fvnctio in intervallo aperto II derivabilis et in pvncto aa ipsivs II bis derivabilis. Circvmferentia centro (h,k)(h, k) et radio r>0r > 0 oscvlans est ad delineationem ipsivs ff in (a,f(a))(a, f(a)) si per id pvnctvm transit neqve id est pvnctvm eivs maxime sinistrvm neqve maxime dextrvm, id est si (a−h)2+(f(a)−k)2=r2(a - h)^2 + \left(f(a) - k\right)^2 = r^2 et f(a)≠kf(a) \neq k, et si semicircvlvs qvi id continet, delineatio fvnctionis uu lemmatis cvm σ\sigma aeqvali signo ipsivs f(a)−kf(a) - k, in aa evndem valorem, eandem derivatam eandemqve derivatam secvndam habet ac ff:

u(a)=f(a),u′(a)=f′(a),u′′(a)=f′′(a)u(a) = f(a), \qquad u'(a) = f'(a), \qquad u''(a) = f''(a)

Centrvm eivs centrvm cvrvatvrae delineationis in eo pvncto appellatvr.

Recta tangens dvas primas aeqvalitates explet; circvmferentia oscvlans etiam tertiam. Nomen ex verbo Latino oscvlari dvcitvr: circvmferentia non solvm delineationem tangit, vt ceterae circvmferentiae tangentes, sed etiam vt ea cvrvatvr. Figvra ostendit, cvm f(x)=x32−x+1f(x) = \frac{x^3}{2} - x + 1 et a=1,2a = 1{,}2, tres circvmferentias delineationem tangentes et, inter eas, oscvlantem.

TheoremaExistentia et vnicitas circvmferentiae oscvlantis

Sit ff fvnctio in intervallo aperto II derivabilis et in pvncto aa ipsivs II bis derivabilis, et sint m=f′(a)m = f'(a) et q=f′′(a)q = f''(a). Si q≠0q \neq 0, delineatio ipsivs ff in (a,f(a))(a, f(a)) vnam prorsvs circvmferentiam oscvlantem habet, centro (h,k)(h, k) et radio ρ\rho:

h=a−m(1+m2)q,k=f(a)+1+m2q,ρ=(1+m2)32∣q∣h = a - \frac{m\left(1 + m^2\right)}{q}, \qquad k = f(a) + \frac{1 + m^2}{q}, \qquad \rho = \frac{\left(1 + m^2\right)^{\frac{3}{2}}}{\abs{q}}

Centrvm eivs in recta normali ad delineationem in pvncto iacet, svpra pvnctvm si q>0q > 0 et infra id si q<0q < 0. Si q=0q = 0, delineatio in (a,f(a))(a, f(a)) circvmferentiam oscvlantem non habet.

Demonstratio

  1. (a−h)+(f(a)−k)m=0(a - h) + \left(f(a) - k\right) m = \dato{normal}{0}

    Ponamvs circvmferentiam centro (h,k)(h, k) et radio rr oscvlantem esse, cvm semicircvlo svo uu. Prima aeqvalitas lemmatis, in x=ax = a, cvm u(a)=f(a)u(a) = f(a) et u′(a)=mu'(a) = m, dicit centrvm aeqvationem rectae normalis ad delineationem in pvncto explere, secvndvm definitionem rectae tangentis et rectae normalis ex lectione de derivata vt limite, tam si m≠0m \neq 0 qvam si normalis verticalis est.

  2. 1+m2+(f(a)−k)q=0→f(a)−k=−1+m2q,a−h=m(1+m2)q,r2=(1+m2)3q21 + m^2 + \left(f(a) - k\right) q = 0 \rightarrow \resaltar{f(a) - k = -\frac{1 + m^2}{q}}, \qquad a - h = \frac{m\left(1 + m^2\right)}{q}, \qquad r^2 = \frac{\left(1 + m^2\right)^3}{q^2}

    Secvnda aeqvalitas lemmatis, in x=ax = a, cvm u′′(a)=qu''(a) = q, fieri non potest si q=0q = 0, qvia 1+m2>01 + m^2 > 0: circvmferentia oscvlans nvlla est. Si q≠0q \neq 0, determinat f(a)−kf(a) - k; prima aeqvalitas tvm dat a−ha - h, et aeqvatio circvmferentiae radivm. Ergo vna ad svmmvm est circvmferentia oscvlans, eivsqve centrvm et radivs svnt ea envntiati.

  3. u′(a)=−a−hf(a)−k=m,u′′(a)=−1+m2f(a)−k=qu'(a) = -\frac{a - h}{f(a) - k} = \resaltar{m}, \qquad u''(a) = -\frac{1 + m^2}{f(a) - k} = \resaltar{q}

    Vicissim, sint hh, kk et ρ\rho ea envntiati. Circvmferentia centro (h,k)(h, k) et radio ρ\rho per (a,f(a))(a, f(a)) transit, per calcvlvm svpra factvm, et f(a)−kf(a) - k non evanescit et signvm ipsivs −q-q habet, ita vt σ=−1\sigma = -1 si q>0q > 0, et centrvm svpra pvnctvm iaceat, et σ=1\sigma = 1 si q<0q < 0. Fvnctio uu lemmatis explet u(a)=k+σ∣f(a)−k∣=f(a)u(a) = k + \sigma\abs{f(a) - k} = f(a), et dvae aeqvalitates lemmatis, in x=ax = a, derivatas eivs dant: circvmferentia oscvlans est.

Si q>0q > 0, circvmferentia oscvlans delineationi desvper semicircvlo svo inferiore, qvi convexvs est, innititvr; si q<0q < 0, svperiore, qvi concavvs est. Vtroqve casv, propositio de delineatione et tangente eivs, ex lectione de concavitate et convexitate, cvm n=2n = 2, ad ff et ad uu adhibita, qvae eandem tangentem et derivatas secvndas eivsdem signi habent, delineationem et circvmferentiam, prope aa, ad idem latvs eivs tangentis ponit. Si f′′(a)=0f''(a) = 0, nvlla circvmferentia convenit, qvia derivata secvnda semicircvli nvmqvam evanescit; recta tangens, cvivs derivata secvnda nvlla est, tres aeqvalitates explet, et vicem circvmferentiae radii infiniti gerit.