Applicationes derivatarvm: synthesis et provocationes

Svmmarivm

Haec lectio partem de applicationibvs derivatarvm clavdit et, cvm ea, cvrsvm. In vnam catenam dedvctivam colligit qvae novem lectiones partis demonstravervnt: theorema Weierstrassii, ex axiomate completvdinis vt theorema Bolzani obtentvm, et theorema Fermatii, ex signo derivatae in pvncto obtentvm, theorema Rollii praebent; ex hoc orivntvr theoremata valoris medii Lagrangii et Cavchii, et ex illis regvla Hospitalii, monotonia ex signo derivatae, criteria ad extrema, convexitas, optimizatio et margo erroris methodi Newtoni et Raphsoni. Explicat qvomodo pars dvabvs partibvs svperioribvs et Algebrae et Geometriae innitatvr, et qvid cvrsvi posteriori relinqvat: integrale, svccessiones et series, et polynomivm Taylorianvm. Desinit in probatione finali, initiali parallela, et in decem provocationibvs, difficilioribvs qvam problemata lectionvm, qvarvm vnaqvaeqve effectvs plvrivm earvm et partivm svperiorvm conivngit.

Proposita discendi

Theoremata praecipva partis ordine logico exponere, indicando ex qvibvs effectibvs svperioribvs vnvmqvodqve pendeat et in qvibvs demonstrationibvs theorema Weierstrassii, theorema valoris medii et monotonia ex signo derivatae itervm adhibeantvr.
Discernere qvid pars admittat et qvid demonstret, hypotheses cvivsqve effectvs agnoscere et indicare vbi axioma completvdinis intercedat.
Effectvs partis cvm partibvs de limitibvs et de derivatis, cvm Algebra et Geometria et cvm iis qvae Calcvlvs II tractat conferre.
Problemata non vsitata solvere qvae exsistentiam extremorvm, theorema valoris medii, regvlam Hospitalii, monotoniam, convexitatem, cvrvatvram, optimizationem et methodvm Newtoni et Raphsoni conivngvnt.

Qvae vnitas consecvta est

Dvae partes svperiores in pvncto laboravervnt: limes in pvncto, continvitas in pvncto, derivata in pvncto. Haec pars a re pvnctvali ad rem globalem transiit, et pons fvit axioma completvdinis. Theorema Weierstrassii, svpremo demonstratvm vt theorema Bolzani, confirmat fvnctionem in intervallo clavso et circvmscripto continvam maximvm et minimvm svvm attingere; theorema Fermatii, qvod ex signo derivatae in pvncto oritvr, dicit vbi extrema interiora esse possint; et conivncta theorema Rollii praebent, ex qvo dvo theoremata valoris medii flvvnt. Theorema Lagrangii sic scribitvr:

f(b) - f(a) = f'(c)(b - a), \qquad c \in \ioo{a}{b}

Haec aeqvalitas notitiam de derivata in omnibvs pvnctis intervalli in notitiam de valoribvs fvnctionis convertit, et fere omnia cetera partis eivs interpretationes svnt: monotonia, criteria ad extrema, convexitas, regvla Hospitalii, per formam Cavchii, et margo erroris methodi Newtoni et Raphsoni. Demonstrationes haec theoremata citant potivs qvam argvmentvm itervm conficiant.

Existentia et valor medivs

Lectio de theoremate Weierstrassii primvm theorema circvmscriptionis demonstravit, svpremvm pvnctorvm svmendo vsqve ad qvae fvnctio circvmscripta est et vltra id progrediendo per circvmscriptionem localem qvam continvitas praebet; deinde svpremvm valorvm attingi, qvia, si non attingeretvr, inversvm distantiae valorvm ab eo continvvm et non circvmscriptvm esset. Exemplis contrariis ostendit nvllam hypothesim svperflvam esse (intervallvm apertvm, intervallvm non circvmscriptvm, fvnctionem discontinvam et nvmeros rationales, in qvibvs completvdo deest), theoremate valoris intermedii collegit imaginem intervalli clavsi et circvmscripti alivd intervallvm eivsdem generis esse, et fvnctionem valoribvs aeqvalibvs in terminis intervalli praeditam extremvm in eivs interiore attingere. Lectio de theorematis Fermatii, Rollii, Lagrangii et Cavchii extrema localia vt envntiata qvantificata definivit, signo derivatae in pvncto demonstravit derivatam in extremo locali interiore evanescere, et theorema Rollii, theorema valoris medii Lagrangii et theorema valoris medii Cavchii inter se conexvit, necessitatem hypothesivm eorvm ostendens. Eivs corollaria primae conclvsiones globales fvervnt: fvnctio derivata nvlla in intervallo praedita constans est, dvae fvnctiones eadem derivata praeditae constante differvnt, margo derivatae incrementa circvmscribit, inter dvas radices fvnctionis radix derivatae eivs iacet, et limes derivatae, si exstat, derivata est. Lectio de regvla Hospitalii eam theoremate Cavchii demonstravit, pro forma $\frac{0}{0}$ et, maiore cvra, pro forma $\frac{\infty}{\infty}$ , ex vno latere pvncti, in pvncto et ad infinitvm; ad eas formas $0 \cdot \infty$ , $\infty - \infty$ , $1^\infty$ , $0^0$ et $\infty^0$ redegit, ostendit regvlam nihil decernere cvm qvotiens derivatarvm limitem non habet, monvit eam non valere ad limites qvibvs derivatae compvtatae svnt, et ex ea corollarivm de derivatis nvllis in pvncto obtinvit, qvod incrementvm fvnctionis cvm potentia ipsivs $x - a$ comparat.

Monotonia, extrema et forma delineationis

Lectio de monotonia theoremate valoris medii demonstravit derivatam non negativam in intervallo fvnctionem crescentem praebere, et positivam stricte crescentem, et monotoniam strictam cvm derivata non negativa descripsit: derivata in toto intervallo evanescere non debet. Sic promissa partis de derivatis impleta svnt, vt illvd, derivatam positivam in intervallo svfficere vt inversa exstet et derivabilis sit. Pvncta critica definivit, criterivm derivatae primae et theorema Darbovxii demonstravit, secvndvm qvod derivata omnes valores intermedios attingit etiamsi continva non sit, ita vt saltvs non habeat, et collegit derivatam sine nihilis in intervallo signvm non mvtare. Lectio de maximis et minimis extrema absolvta et stricta definivit; criterivm derivatae secvndae demonstravit et, regvla Hospitalii et sine polynomio Tayloriano, criterivm derivatarvm svccessivarvm, qvod per paritatem primi ordinis cvm derivata non nvlla decernit; et extrema absolvta in intervallo clavso et circvmscripto inter terminos intervalli et pvncta critica qvaesivit, et in intervallo aperto vel non circvmscripto limitibvs in terminis et monotonia, qvae etiam praebet extremvm locale in vnico pvncto critico absolvtvm esse. Lectio de concavitate et convexitate convexitatem per positionem delineationis respectv chordarvm svarvm definivit, cvm conventione partis (convexa, infra chordas svas), et demonstravit, pro fvnctione derivabili, convexam esse idem valere ac derivatam crescentem habere et delineationem svpra tangentes svas manere, et pro fvnctione bis derivabili, derivatam secvndam negativam non esse; pvncta inflexionis definivit, de eis per mvtationem signi derivatae secvndae vel per derivatam tertiam decrevit, fvnctione $x^4$ ostendit derivatam secvndam nvllam non svfficere, et inaeqvalitatem Iensenianam et eam mediarvm geometricae et arithmeticae obtinvit. Lectio de cvrvatvra circvmferentiam oscvlantem definivit vt eam qvae in pvncto evndem valorem, eandem derivatam et eandem derivatam secvndam habet ac fvnctio; derivatione implicita demonstravit eam exstare et vnicam esse cvm derivata secvnda non evanescit, radio $\rho = \frac{\left(1 + f'(a)^2\right)^{\frac{3}{2}}}{\abs{f''(a)}}$ et centro in recta normali, ab eo latere ad qvod delineatio cvrvatvr, et ostendit hoc centrvm limitem esse concvrsvvm rectarvm normalivm vicinarvm.

Optimizatio et approximatio

Lectio de problematis optimizationis methodvm qvinqve articvlorvm statvit (variabilis, fvnctio, dominivm, extrema absolvta cvm ratione sva et conclvsio), cvivs pvnctvm decretorivm est confirmare extremvm inventvm absolvtvm esse: theoremate Weierstrassii, si dominivm clavsvm et circvmscriptvm est, et sin minvs, signo derivatae, convexitate vel limitibvs in terminis. Demonstravit fvnctionem eivsqve compositionem cvm fvnctione stricte crescente extrema sva in eisdem pvnctis attingere, pvnctvm delineationis pvncto dato proximvm exstare et in recta normali iacere, et ex tempore minimo leges refractionis et reflexionis dedvxit. Lectio de methodo Newtoni et Raphsoni delineationi rectam tangentem eivs svbstitvit ad radicem approximandam; iterata vt fvnctionem in nvmeris natvralibvs definitam tractavit, definitione limitis ad infinitvm et sine theoria svccessionvm; convexitate et infimo iteratorvm demonstravit ea, a latere recto, ad radicem accedere neqve eam praeterire et ad eam convergere, et a latere pravo primam iterationem ad latvs rectvm transilire; errorem theoremate valoris medii circvmscripsit, defectvs methodi ostendit (derivatam nvllam, cyclvm, iterata qvae effvgivnt) et eam cvm bisectione comparavit, qvae singvlis bisectionibvs digitvm binarivm lvcratvr, cvm methodvs Newtoni et Raphsoni, prope radicem simplicem, in exemplis nvmervm digitorvm rectorvm dvplicet.

Qvae admittvntvr, qvae demonstrantvr, qvae restant

Ratio partis sic svmmatim reddi potest:

Admittvntvr axioma completvdinis, qvo pars directe vtitvr in theoremate circvmscriptionis, in theoremate Weierstrassii et in convergentia iteratorvm Newtoni et Raphsoni; theoremata partivm de limitibvs et de derivatis, praesertim theorema Bolzani, conservatio signi, indoles localis derivatae et signvm derivatae in pvncto; et effectvs Algebrae et Geometriae, vt formvlae trigonometricae, inaeqvalitas $\sin x < x < \tan x$ et leges exponentivm et logarithmorvm.
Demonstrantvr ea qvae pars de derivatis nvntiaverat neqve demonstrare potverat: derivatam positivam in intervallo fvnctionem crescentem praebere, fvnctionem derivata nvlla in intervallo praeditam constantem esse et errorem approximationis per differentiale circvmscribi posse; et recta tangens instrvmentvm fit ad radices approximandas, ad delineationem secvm ipsa comparandam et ad metiendvm qvantvm cvrvetvr.
Tria instrvmenta itervm adhibentvr potivs qvam argvmenta repetantvr: theorema Weierstrassii, qvod extrema praebet; theorema valoris medii, qvod signvm derivatae in notitiam de valoribvs convertit; et monotonia ex signo derivatae, qvae criteria ad extrema et convexitatem svstinet.
Calcvlo II relinqvitvr qvod plvs qvam derivatam postvlat: integrale, longitvdo arcvs, qva cvrvatvra vt celeritas legitvr qva recta tangens vertitvr, continvitas vniformis, svccessiones et series, et polynomivm Taylorianvm; et Calcvlo III, extrema fvnctionvm plvrivm variabilivm et cvrvatvra cvrvarvm parametricarvm.

Pictvra infra posita tria segmenta in vnam catenam colligit. Qvisqve nodvs colvmnae est definitio avt theorema, secvndvm ordinem logicvm nvmeratvs et arcv cvm anvlis svperioribvs conivnctvs qvibvs eivs demonstratio vtitvr; circvm flvitant fvlcra, id est principia et instrvmenta ad qvae catena itervm atqve itervm recvrrit, qvodqve linea continva cvm anvlis qvi ei nitvntvr conivnctvm et linea pvnctata cvm anvlo vbi statvitvr, exceptis iis qvae ex partibvs svperioribvs vel ex Algebra et Geometria provenivnt, qvae vt circvli vacvi pingvntvr. Fvlcro pvlsato omnes anvli qvi eo vtvntvr illvstrantvr, et qvisqve anvlvs ad lectionem remittit in qva demonstratvr.

Fvlcra catenae

AAxioma completvdinis. Ex Algebra et Geometria I admittitvr reales corpvs ordinatvm completvm efficere: qvaevis copia realivm non vacva svperne circvmscripta svpremvm habet, et qvaevis copia non vacva inferne circvmscripta infimvm habet. In hac vnitate sine demonstratione admittitvr.
BHereditas vnitatis 1. Effectvs in parte de limitibvs et de continvitate demonstrati: algebra limitvm; limites laterales, ad infinitvm et infiniti; conservatio inaeqvalitatvm; continvitas operationvm et compositionis; circvmscriptio localis et conservatio signi, pro limitibvs et pro fvnctionibvs continvis; et limites notabiles. In hac vnitate sine demonstratione admittitvr.
CHereditas vnitatis 2. Effectvs in parte de derivatis demonstrati: definitio derivatae et derivatae laterales, fvnctionem derivabilem continvam esse, indoles localis derivatae, regvlae derivandi et regvla catenae, derivatae fvnctionvm elementarivm, derivata fvnctionis inversae et derivatio implicita, et derivatae svccessivae. In hac vnitate sine demonstratione admittitvr.
DValor intermedivs. Svpremo in parte de derivatis demonstratvm: si $f$ in $\icc{a}{b}$ continva est et $f(a)$ et $f(b)$ signa contraria habent, $f$ in aliqvo pvncto ipsivs $\ioo{a}{b}$ evanescit, qvod est theorema Bolzani; et imago intervalli per fvnctionem continvam intervallvm est. In hac vnitate sine demonstratione admittitvr.
ESignvm derivatae in pvncto. In parte de derivatis demonstratvm: si $f$ in intervallo aperto definita est et $f'(a) > 0$ valet, exstat $\delta > 0$ ita vt sit $f(x) < f(a)$ si $a - \delta < x < a$ , et $f(x) > f(a)$ si $a < x < a + \delta$ ; non efficitvr vt $f$ prope $a$ crescens sit. In hac vnitate sine demonstratione admittitvr.
FExtrema attacta. Fvnctio in intervallo clavso et circvmscripto continva maximvm et minimvm svvm in eo attingit: instrvmentvm qvod extremis et pvnctis intermediis theorematvm posteriorvm exsistentiam praebet. Constitvitvr in: 2. Theorema Weierstrassii.
GValor medivs. Si $f$ in $\icc{a}{b}$ continva est et in qvoqve pvncto ipsivs $\ioo{a}{b}$ derivabilis, exstat $c$ in $\ioo{a}{b}$ cvm $f(b) - f(a) = f'(c)(b - a)$ : notitia de derivata in omnibvs pvnctis in notitiam de valoribvs convertitvr. Constitvitvr in: 7. Theorema valoris medii Lagrangii.
HMonotonia ex derivata. In intervallo derivata non negativa fvnctionem crescentem praebet, et positiva stricte crescentem: instrvmentvm qvo de extremis et de convexitate decernitvr. Constitvitvr in: 12. Monotonia ex signo derivatae.

Segmentvm 1 Existentia et valor medivs
- Svpremo demonstrare fvnctionem in intervallo clavso et circvmscripto continvam circvmscriptam esse et maximvm et minimvm svvm attingere, et ostendere nvllam hypothesim svperflvam esse.
- Extrema localia interiora theoremate Fermatii invenire et ex eo et theoremate Weierstrassii theoremata Rollii, Lagrangii et Cavchii dedvcere.
- Ex theoremate valoris medii obtinere derivatam nvllam fvnctionem constantem praebere, incrementa circvmscribere et radices nvmerare.
- Theoremate Cavchii regvlam Hospitalii demonstrare, ceteras formas indeterminatas ad eam redigere et ea non vti ad limites qvibvs derivatae compvtatae svnt.
1. Theorema circvmscriptionis. Si $f$ in intervallo clavso et circvmscripto $\icc{a}{b}$ continva est, exstat nvmervs $K$ ita vt $\abs{f(x)} \leq K$ pro omni $x$ ipsivs $\icc{a}{b}$ .
  Nititvr: A (Axioma completvdinis), B (Hereditas vnitatis 1). Demonstratvr in: Theorema Weierstrassii de valoribvs extremis.
2. Theorema Weierstrassii. Si $f$ in intervallo clavso et circvmscripto $\icc{a}{b}$ continva est, exstant $x_1$ et $x_2$ in $\icc{a}{b}$ ita vt $f(x_1) \leq f(x) \leq f(x_2)$ pro omni $x$ ipsivs $\icc{a}{b}$ ; nvlla hypothesivm omitti potest.
  Vtitvr: 1. Theorema circvmscriptionis. Nititvr: A (Axioma completvdinis), B (Hereditas vnitatis 1). Demonstratvr in: Theorema Weierstrassii de valoribvs extremis.
3. Imago intervalli clavsi et circvmscripti. Si $f$ in $\icc{a}{b}$ continva est, et $m$ et $M$ valores eivs minimvs et maximvs in eo svnt, imago ipsivs $\icc{a}{b}$ est intervallvm $\icc{m}{M}$ .
  Vtitvr: 2. Theorema Weierstrassii. Nititvr: F (Extrema attacta), D (Valor intermedivs). Demonstratvr in: Theorema Weierstrassii de valoribvs extremis.
4. Extremvm in interiore. Si $f$ in $\icc{a}{b}$ continva est, cvm $a < b$ et $f(a) = f(b)$ , maximvm vel minimvm svvm in aliqvo pvncto ipsivs $\ioo{a}{b}$ attingit.
  Vtitvr: 2. Theorema Weierstrassii. Nititvr: F (Extrema attacta). Demonstratvr in: Theorema Weierstrassii de valoribvs extremis.
5. Theorema Fermatii. Si $f$ , in $A$ definita, extremvm locale in pvncto interiore $a$ ipsivs $A$ attingit, id est, si exstat $\delta > 0$ ita vt $f(x) \leq f(a)$ pro omnibvs $x$ ipsivs $A$ cvm $\abs{x - a} < \delta$ , vel $f(a) \leq f(x)$ pro omnibvs illis, et $f$ in $a$ derivabilis est, tvm $f'(a) = 0$ .
  Nititvr: E (Signvm derivatae in pvncto), C (Hereditas vnitatis 2). Demonstratvr in: Theoremata Fermatii, Rollii, Lagrangii et Cavchii.
6. Theorema Rollii. Si $f$ in $\icc{a}{b}$ continva est, cvm $a < b$ , in qvoqve pvncto ipsivs $\ioo{a}{b}$ derivabilis et $f(a) = f(b)$ , exstat $c$ in $\ioo{a}{b}$ cvm $f'(c) = 0$ .
  Vtitvr: 4. Extremvm in interiore, 5. Theorema Fermatii. Nititvr: F (Extrema attacta), C (Hereditas vnitatis 2). Demonstratvr in: Theoremata Fermatii, Rollii, Lagrangii et Cavchii.
7. Theorema valoris medii Lagrangii. Si $f$ in $\icc{a}{b}$ continva est, cvm $a < b$ , et in qvoqve pvncto ipsivs $\ioo{a}{b}$ derivabilis, exstat $c$ in $\ioo{a}{b}$ cvm $f(b) - f(a) = f'(c)(b - a)$ : aliqva recta tangens chordae parallela est.
  Vtitvr: 6. Theorema Rollii. Nititvr: B (Hereditas vnitatis 1), C (Hereditas vnitatis 2). Demonstratvr in: Theoremata Fermatii, Rollii, Lagrangii et Cavchii.
8. Theorema valoris medii Cavchii. Si $f$ et $g$ in $\icc{a}{b}$ continvae svnt, cvm $a < b$ , et in qvoqve pvncto ipsivs $\ioo{a}{b}$ derivabiles, exstat $c$ in $\ioo{a}{b}$ cvm $\left(f(b) - f(a)\right) g'(c) = \left(g(b) - g(a)\right) f'(c)$ ; si praeterea $g'$ in $\ioo{a}{b}$ non evanescit, tvm $g(b) \neq g(a)$ et $\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)}$ .
  Vtitvr: 6. Theorema Rollii. Nititvr: B (Hereditas vnitatis 1), C (Hereditas vnitatis 2). Demonstratvr in: Theoremata Fermatii, Rollii, Lagrangii et Cavchii.
9. Conseqventiae theorematis valoris medii. Sit $f$ in intervallo $I$ continva et in qvoqve pvncto interiore derivabilis. Si $f'$ in eis evanescit, $f$ constans est; dvae fvnctiones eadem derivata praeditae constante differvnt; si $\abs{f'(x)} \leq K$ in eis, tvm $\abs{f(x) - f(y)} \leq K \abs{x - y}$ pro omnibvs $x$ , $y$ ipsivs $I$ ; et si $f$ in $I$ derivabilis est et $n \geq 2$ radices distinctas in eo habet, $f'$ saltem $n - 1$ habet.
  Vtitvr: 6. Theorema Rollii, 7. Theorema valoris medii Lagrangii. Nititvr: G (Valor medivs), C (Hereditas vnitatis 2). Demonstratvr in: Theoremata Fermatii, Rollii, Lagrangii et Cavchii.
10. Regvla Hospitalii. Si $f$ et $g$ in $\ioo{a}{b}$ derivabiles svnt, cvm $g'(x) \neq 0$ in eo, si $f(x)$ et $g(x)$ ad $0$ tendvnt, vel $g(x)$ ad $\infty$ vel ad $-\infty$ tendit, cvm $x \to a^+$ , et si $\frac{f'(x)}{g'(x)}$ ad nvmervm realem $L$ , ad $\infty$ vel ad $-\infty$ tendit, tvm $\frac{f(x)}{g(x)}$ ad $L$ tendit; idem valet a sinistra, ab vtroqve latere pvncti et ad infinitvm.
  Vtitvr: 8. Theorema valoris medii Cavchii. Nititvr: B (Hereditas vnitatis 1), C (Hereditas vnitatis 2). Demonstratvr in: Regvla Hospitalii.
11. Derivatae nvllae in pvncto. Sint $n \geq 2$ nvmervs natvralis, $I$ intervallvm apertvm, $a$ pvnctvm ipsivs $I$ et $f$ fvnctio $n - 1$ vicibvs derivabilis in $I$ et $n$ vicibvs derivabilis in $a$ , cvm $f'(a) = \dots = f^{(n - 1)}(a) = 0$ . Tvm $\frac{f(x) - f(a)}{(x - a)^n}$ ad $\frac{f^{(n)}(a)}{n!}$ tendit cvm $x \to a$ .
  Vtitvr: 10. Regvla Hospitalii. Nititvr: C (Hereditas vnitatis 2). Demonstratvr in: Regvla Hospitalii.
Segmentvm 2 Monotonia, extrema et forma delineationis
- De monotonia per signvm derivatae decernere, monotoniam strictam describere et theorema Darbovxii demonstrare.
- Pvncta critica criteriis derivatae primae et secvndae et derivatarvm svccessivarvm distingvere, et extrema absolvta in intervallis clavsis, apertis vel non circvmscriptis invenire.
- Convexitatem per chordas, derivatam crescentem, rectas tangentes et derivatam secvndam describere, et pvncta inflexionis agnoscere.
- Cvrvatvram delineationis circvmferentia oscvlante metiri.
1. Monotonia ex signo derivatae. Sit $f$ in intervallo $I$ continva et in qvoqve pvncto interiore derivabilis. Si $f'(x) \geq 0$ in eis, $f$ in $I$ crescens est, et si $f'(x) > 0$ , stricte crescens; cvm $f'(x) \leq 0$ vel $f'(x) < 0$ , decrescens vel stricte decrescens. Vicissim, si $f$ in $I$ crescens est et in pvncto interiore $a$ derivabilis, tvm $f'(a) \geq 0$ .
  Vtitvr: 7. Theorema valoris medii Lagrangii. Nititvr: E (Signvm derivatae in pvncto), G (Valor medivs), C (Hereditas vnitatis 2). Demonstratvr in: Monotonia et criterivm derivatae primae.
2. Monotonia stricta cvm derivata non negativa. Si $f$ in intervallo $I$ continva est et $f'(x) \geq 0$ in qvoqve pvncto interiore, $f$ in $I$ stricte crescens est si et solvm si $f'$ non in omnibvs pvnctis vllivs intervalli $\ioo{p}{q}$ in $I$ contenti evanescit; praesertim, si in nvmero finito pvnctorvm tantvm evanescit.
  Vtitvr: 9. Conseqventiae theorematis valoris medii, 12. Monotonia ex signo derivatae. Nititvr: H (Monotonia ex derivata), C (Hereditas vnitatis 2). Demonstratvr in: Monotonia et criterivm derivatae primae.
3. Criterivm derivatae primae. Si $f$ in $\ioo{a - r}{a + r}$ continva est et in qvoqve pvncto a $a$ diverso derivabilis, cvm $f'(x) \geq 0$ a sinistra ipsivs $a$ et $f'(x) \leq 0$ a dextra, in $a$ maximvm locale attingit, strictvm si inaeqvalitates strictae svnt; inaeqvalitatibvs conversis, minimvm locale; et si $f'(x)$ ab vtroqve latere positiva est, vel ab vtroqve latere negativa, nvllvm extremvm in $a$ est.
  Vtitvr: 12. Monotonia ex signo derivatae. Nititvr: H (Monotonia ex derivata). Demonstratvr in: Monotonia et criterivm derivatae primae.
4. Theorema Darbovxii. Si $f$ in intervallo aperto $I$ derivabilis est et $a < b$ pvncta ipsivs $I$ svnt, $f'$ in $\ioo{a}{b}$ omnem valorem stricte inter $f'(a)$ et $f'(b)$ comprehensvm attingit; qvare $f'$ in $I$ neqve discontinvitates evitabiles neqve saltvs habet.
  Vtitvr: 2. Theorema Weierstrassii, 5. Theorema Fermatii. Nititvr: E (Signvm derivatae in pvncto), F (Extrema attacta), B (Hereditas vnitatis 1), C (Hereditas vnitatis 2). Demonstratvr in: Monotonia et criterivm derivatae primae.
5. Derivata sine nihilis. Si $f$ in intervallo $I$ plvs qvam vno pvncto praedito continva est et in qvoqve pvncto interiore derivabilis, cvm $f'(x) \neq 0$ in vnoqvoqve, $f'$ idem signvm in omnibvs eis habet et $f$ in $I$ stricte monotona est.
  Vtitvr: 12. Monotonia ex signo derivatae, 15. Theorema Darbovxii. Nititvr: H (Monotonia ex derivata). Demonstratvr in: Monotonia et criterivm derivatae primae.
6. Criterivm derivatae secvndae. Si $f$ in intervallo aperto $I$ derivabilis est et in pvncto $a$ ipsivs $I$ bis derivabilis, cvm $f'(a) = 0$ , in $a$ minimvm locale strictvm attingit si $f''(a) > 0$ , et maximvm locale strictvm si $f''(a) < 0$ .
  Vtitvr: 14. Criterivm derivatae primae. Nititvr: E (Signvm derivatae in pvncto), C (Hereditas vnitatis 2). Demonstratvr in: Maxima et minima fvnctionis.
7. Criterivm derivatarvm svccessivarvm. Si $f$ est fvnctio $n - 1$ vicibvs derivabilis in intervallo aperto $I$ et $n$ vicibvs derivabilis in pvncto $a$ ipsivs $I$ , cvm $n \geq 2$ , $f'(a) = \dots = f^{(n - 1)}(a) = 0$ et $f^{(n)}(a) \neq 0$ , tvm, si $n$ par est, $f$ in $a$ minimvm locale strictvm attingit si $f^{(n)}(a) > 0$ , et maximvm locale strictvm si $f^{(n)}(a) < 0$ ; et si $n$ impar est, in $a$ nvllvm extremvm locale attingit.
  Vtitvr: 11. Derivatae nvllae in pvncto. Nititvr: B (Hereditas vnitatis 1). Demonstratvr in: Maxima et minima fvnctionis.
8. Extrema absolvta in intervallo. Si $f$ in $\icc{a}{b}$ continva est, maximvm et minimvm sva absolvta attingit, et qvodqve pvnctvm in qvo ea attingit est $a$ , $b$ vel pvnctvm criticvm ipsivs $\ioo{a}{b}$ ; si in intervallo aperto continva est, circvmscripto vel non, et aliqvis valor limites eivs in vtroqve termino svperat, maximvm svvm absolvtvm attingit, in pvnctis criticis.
  Vtitvr: 2. Theorema Weierstrassii, 5. Theorema Fermatii. Nititvr: F (Extrema attacta), B (Hereditas vnitatis 1). Demonstratvr in: Maxima et minima fvnctionis.
9. Extrema absolvta per monotoniam. Si $f$ in intervallo $I$ continva est, $c$ pvnctvm interivs est, $f'(x) \geq 0$ in pvnctis interioribvs ipso $c$ minoribvs et $f'(x) \leq 0$ in maioribvs, $f$ in $c$ maximvm svvm absolvtvm in $I$ attingit; et si $c$ vnicvm pvnctvm criticvm ipsivs $f$ in $I$ est et $f$ in eo extremvm locale attingit, hoc extremvm absolvtvm et strictvm in $I$ est.
  Vtitvr: 12. Monotonia ex signo derivatae, 16. Derivata sine nihilis. Nititvr: H (Monotonia ex derivata). Demonstratvr in: Maxima et minima fvnctionis.
10. Convexitas et derivata crescens. Fvnctio $f$ in intervallo $I$ convexa est si $f\left(tx + (1 - t)y\right) \leq t f(x) + (1 - t) f(y)$ pro omnibvs $x$ , $y$ ipsivs $I$ et omni $t$ ipsivs $\icc{0}{1}$ . Si $f$ in $I$ continva est et in qvoqve pvncto interiore derivabilis, convexa est si et solvm si $f'$ in pvnctis interioribvs crescens est, et stricte convexa si et solvm si $f'$ in eis stricte crescens est.
  Vtitvr: 7. Theorema valoris medii Lagrangii. Nititvr: G (Valor medivs), B (Hereditas vnitatis 1), C (Hereditas vnitatis 2). Demonstratvr in: Concavitas, convexitas et pvncta inflexionis.
11. Delineatio svpra tangentes svas. Si $f$ in intervallo $I$ plvs qvam vno pvncto praedito continva est et in qvoqve pvncto interiore derivabilis, convexa est si et solvm si $f(x) \geq f(a) + f'(a)(x - a)$ pro omni pvncto interiore $a$ et omni $x$ ipsivs $I$ ; praesertim, si convexa est et $f'(a) = 0$ , in $a$ minimvm svvm absolvtvm in $I$ attingit.
  Vtitvr: 21. Convexitas et derivata crescens. Demonstratvr in: Concavitas, convexitas et pvncta inflexionis.
12. Convexitas et derivata secvnda. Si $f$ in intervallo $I$ plvs qvam vno pvncto praedito continva est et in qvoqve pvncto interiore bis derivabilis, convexa est si et solvm si $f''(x) \geq 0$ in eis, et stricte convexa est si praeterea $f''$ non in omnibvs pvnctis vllivs intervalli aperti in $I$ contenti evanescit.
  Vtitvr: 12. Monotonia ex signo derivatae, 13. Monotonia stricta cvm derivata non negativa, 21. Convexitas et derivata crescens. Nititvr: H (Monotonia ex derivata), C (Hereditas vnitatis 2). Demonstratvr in: Concavitas, convexitas et pvncta inflexionis.
13. Pvncta inflexionis. Si $f$ prope $a$ continva est, in $a$ derivabilis et $f''$ in $a$ signvm mvtat, pvnctvm $a$ inflexionis est; in pvncto inflexionis in qvo $f$ bis derivabilis est, $f''(a) = 0$ , sed id non svfficit, vt $x^4$ in $0$ ostendit; et si $f''(a) = 0$ et $f'''(a) \neq 0$ , $a$ pvnctvm inflexionis est et delineatio tangentem svam transit.
  Vtitvr: 5. Theorema Fermatii, 18. Criterivm derivatarvm svccessivarvm, 21. Convexitas et derivata crescens, 23. Convexitas et derivata secvnda. Nititvr: E (Signvm derivatae in pvncto), B (Hereditas vnitatis 1). Demonstratvr in: Concavitas, convexitas et pvncta inflexionis.
14. Inaeqvalitas Ienseniana. Si $f$ in intervallo $I$ continva et convexa est et in qvoqve pvncto interiore derivabilis, tvm $f\left(t_1 x_1 + \dots + t_n x_n\right) \leq t_1 f(x_1) + \dots + t_n f(x_n)$ pro pvnctis $x_i$ ipsivs $I$ et ponderibvs $t_i \geq 0$ svmmae $1$ ; logarithmo, qvi concavvs est, media geometrica nvmerorvm positivorvm arithmeticam non svperat.
  Vtitvr: 22. Delineatio svpra tangentes svas, 23. Convexitas et derivata secvnda. Nititvr: C (Hereditas vnitatis 2). Demonstratvr in: Concavitas, convexitas et pvncta inflexionis.
15. Circvmferentia oscvlans. Si $f$ in intervallo aperto derivabilis est et in pvncto $a$ bis derivabilis, cvm $m = f'(a)$ et $q = f''(a) \neq 0$ , exstat vna omnino circvmferentia qvae in $a$ evndem valorem, eandem derivatam et eandem derivatam secvndam habet ac $f$ ; eivs radivs est $\rho = \frac{\left(1 + m^2\right)^{\frac{3}{2}}}{\abs{q}}$ , inversvm cvrvatvrae $\kappa$ , et eivs centrvm in recta normali iacet, svpra pvnctvm si $q > 0$ et infra id si $q < 0$ .
  Nititvr: C (Hereditas vnitatis 2). Demonstratvr in: Cvrvatvra et radivs cvrvatvrae.
Segmentvm 3 Optimizatio et approximatio
- Problema geometricvm vel physicvm in qvaestionem extremi absolvti fvnctionis convertere, in dominio accvrate definito, et confirmare extremvm inventvm absolvtvm esse.
- Pvnctvm delineationis pvncto dato proximvm invenire et ex tempore minimo leges reflexionis et refractionis dedvcere.
- Radices methodo Newtoni et Raphsoni approximare, convergentiam eivs convexitate confirmare et errorem theoremate valoris medii circvmscribere.
1. Distantia pvncti a delineatione. Si $f$ continva est in intervallo plvs qvam vno pvncto praedito qvod terminos svos continet, si qvos habet, aliqvod pvnctvm delineationis eivs minima distantia a pvncto $P = (x_1, y_1)$ abest; si abscissa $c$ hvivs pvncti interior est et $f$ in ea derivabilis, $P$ in recta normali iacet: $(x_1 - c) + f'(c)\left(y_1 - f(c)\right) = 0$ .
  Vtitvr: 2. Theorema Weierstrassii, 5. Theorema Fermatii. Nititvr: F (Extrema attacta), B (Hereditas vnitatis 1), C (Hereditas vnitatis 2). Demonstratvr in: Problemata optimizationis.
2. Lex Snellii. Dvratio $T(x) = \frac{\sqrt{a^2 + x^2}}{v_1} + \frac{\sqrt{b^2 + (d - x)^2}}{v_2}$ viae qvae a $(0, a)$ ad $(x, 0)$ celeritate $v_1$ et inde ad $(d, -b)$ celeritate $v_2$ dvcit minimvm svvm absolvtvm strictvm in vnico pvncto attingit, vbi $\frac{\sin \theta_1}{v_1} = \frac{\sin \theta_2}{v_2}$ ; inde via brevissima inter dvo pvncta qvae rectam tangit angvlos aeqvales cvm perpendicvlari ad eam efficit, qvae est lex reflexionis.
  Vtitvr: 22. Delineatio svpra tangentes svas, 23. Convexitas et derivata secvnda. Nititvr: D (Valor intermedivs), C (Hereditas vnitatis 2). Demonstratvr in: Problemata optimizationis.
3. Convergentia methodi Newtoni et Raphsoni. Si $f$ in intervallo aperto $I$ derivabilis et convexa est, cvm $f'(x) \neq 0$ in toto $I$ , et $f(r) = 0$ valet, iterata $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ ab aliqvo $x_0$ ipsivs $I$ cvm $f(x_0) \geq 0$ ad $r$ modo monotono accedvnt, neqve eam praeterevnt, et ad $r$ convergvnt; ab aliqvo $x_0$ cvm $f(x_0) < 0$ , si $x_1$ in $I$ est, prima iteratio ad latvs rectvm dvcit.
  Vtitvr: 16. Derivata sine nihilis, 21. Convexitas et derivata crescens, 22. Delineatio svpra tangentes svas. Nititvr: A (Axioma completvdinis), B (Hereditas vnitatis 1). Demonstratvr in: Methodvs Newtoni et Raphsoni.
4. Margo erroris. Si $f$ in intervallo $J$ continva est et in qvoqve pvncto interiore derivabilis, cvm $\abs{f'(x)} \geq m > 0$ in eis, et $f(r) = 0$ pro aliqvo $r$ ipsivs $J$ , tvm $\abs{x - r} \leq \frac{\abs{f(x)}}{m}$ pro omni $x$ ipsivs $J$ .
  Vtitvr: 7. Theorema valoris medii Lagrangii. Nititvr: G (Valor medivs). Demonstratvr in: Methodvs Newtoni et Raphsoni.

Ad Calcvlvm II. Catena ibi consistit vbi cvrsvs desinit. In Calcvlo II corollarivm secvndvm qvod dvae fvnctiones eadem derivata praeditae constante differvnt primitivas fvndat, et integrale cvm derivata in theoremate fvndamentali calcvli conivngitvr; longitvdine arcvs, qvae integrale est, cvrvatvra vt celeritas legitvr qva recta tangens vertitvr. Svccessiones, qvarvm iterata Newtoni et Raphsoni primvm exemplvm fvervnt, et series fvnctionibvs in nvmeris natvralibvs definitis theoriam propriam praebent. Et polynomivm Taylorianvm, cvivs residvvm theorematis Rollii et Cavchii circvmscribitvr, rectam tangentem derivatis svccessivis prodvcit, criterivm derivatarvm svccessivarvm generalivs reddit et, hypothesibvs de derivata secvnda positis, convergentiam qvadraticam methodi Newtoni et Raphsoni demonstrat, qvam haec pars in exemplis tantvm observavit.