Ad argvmentvm transi
Topos Uranos
Latine

Fvnctiones exponentiales: synthesis et provocationes

Svmmarivm

Haec lectio partem de fvnctionibvs exponentialibvs, logarithmicis et hyperbolicis clavdit, qvae vltima cvrsvs est. Eivs eventvs in vnam catenam dedvctivam colligit, qvae a completvdine nvmerorvm realivm proficiscitvr et ad descriptionem parametricam hyperbolae pervenit; ostendit qvomodo haec catena in partibvs antecedentibvs cvrsvs nitatvr, ab indvctione et binomio Newtoniano vsqve ad aeqvationem hyperbolae, et qvomodo cvrsvm Algebrae et Geometriae II praeparet, qvi a fvnctionibvs trigonometricis et coordinatis polaribvs incipit. Desinit in decem provocationibvs, difficilioribvs qvam problemata in lectionibvs proposita, qvae plvres earvm conivngere et ideam propriam invenire postvlant.

Proposita discendi

  1. Catenam definitionvm et theorematvm partis restitvere, in qvoqve anvlo eventvm antecedentem agnoscendo a qvo pendet.
  2. Eventvs partis cvm eventibvs partivm antecedentivm conferre et eorvm mvnvs in fvnctionibvs trigonometricis cvrsvs Algebrae et Geometriae II praevidere.
  3. Problemata solvere qvae monotoniam exponentialis, leges logarithmorvm, inaeqvalitatem ex≥1+xe^x \geq 1 + x et formvlas additionis hyperbolicas conivngvnt.
  4. Svo arbitrio svbstitvtionem, inaeqvalitatem vel fvnctionem avxiliarem eligere qvae problema non tritvm ad notvm redvcit.

Qvae vnitas consecvta est

Pars a defectv profecta est: in fine partis de generibvs nvmerorvm, potentia axa^x sensvm habebat pro xx rationali, non avtem pro x=2x = \sqrt{2}. Eivs eventvs finalis est familia fvnctionvm in tota recta definitarvm, cvm legibvs algebraicis completis, inversis explicitis et interpretatione geometrica. Anvli catenae hi svnt.

Potentia exponentis realis

Lectio de fvnctione exponentiali et potentia exponentis realis definivit axa^x, pro a>1a > 1, vt svpremvm potentiarvm ara^r cvm rr rationali et r≤xr \leq x. Completvdo exsistentiam svpremi praestitit; densitas nvmerorvm rationalivm inclvsionem ar<ax<asa^r < a^x < a^s pro r<x<sr < x < s dedit, et cvm ea monotoniam strictam; inaeqvalitas Bernovlliana longitvdinem intervallorvm inclvsionis circvmscripsit et, cvm proprietate Archimedea, leges exponentivm demonstrare sivit, qvarvm vltimam casv per casvm secvndvm basem et signvm exponentis:

ax+y=axay,(ab)x=axbx,(ax)y=axya^{x + y} = a^x a^y, \qquad (ab)^x = a^x b^x, \qquad \left( a^x \right)^y = a^{xy}

His legibvs, fvnctio exponentialis positiva, iniectiva et stricte monotona evasit; novvm argvmentvm svpremorvm, cvm inaeqvalitate Bernovlliana, demonstravit eam omnes valores positivos svmere, ita vt biiectio ex R\mathbb{R} in (0,∞)\ioo{0}{\infty} sit; et demonstratvm est eam tandem qvadrata svperare.

Nvmervs e

Lectio de nvmero e et exponentiali natvrali basem praestantem elegit. Binomio Newtoniano et inaeqvalitate Bernovlliana demonstravit (1+1n)n\left( 1 + \frac{1}{n} \right)^n crescere et (1+1n)n+1\left( 1 + \frac{1}{n} \right)^{n + 1} decrescere, et ee definivit vt nvmervm qvem ambae seqventiae inclvdvnt; demonstravit evm etiam svpremvm esse svmmarvm inversorvm factorialivm, errore minore qvam 1n! n\frac{1}{n! \, n}, et irrationalem esse, et, inaeqvalitate fvndamentali et potentia potentiae, demonstravit svpremvm sortivm (1+rn)n\left( 1 + \frac{r}{n} \right)^n esse ere^r, sine notione limitis. Eventvs fecvndissimvs lectionis est inaeqvalitas fvndamentalis, valida pro omni reali xx, aeqvalitate solvm in x=0x = 0, qvae in omnibvs fere demonstrationibvs posterioribvs redit:

ex≥1+xe^x \geq 1 + x

Logarithmvs et aeqvationes

Lectio de fvnctione logarithmo log⁡a\log_a vt inversam exponentialis obtinvit et qvamqve legem exponentivm in legem logarithmorvm vertit; mvtatio basis ostendit omnes logarithmos mvltiplices vnivs esse, et lemma Evclidis demonstravit log⁡23\log_2 3 irrationalem esse. Lectio de aeqvationibvs et inaeqvationibvs exponentialibvs et logarithmicis dvas harvm fvnctionvm proprietates in methodos convertit: iniectivitas exponentes et argvmenta aeqvare sinit, et monotonia inaeqvalitates secvndvm basem servat vel invertit. Difficvltates harvm qvaestionvm propriae, dominivm, solvtiones alienae et valores non positivi variabilis avxiliaris, omnes inde orivntvr qvod exponentialis solvm valores positivos svmit. Vnvs tantvm eventvs illivs lectionis in aliqvo nititvr qvod pars non demonstrat: exsistentia solvtionis aeqvationis sine methodo algebraica, vt x+2x=5x + 2^x = 5, in theoremate valorvm intermediorvm nititvr, in lectione de fvnctionibvs rationalibvs admisso, et in continvitate exponentialis, qvae in lectione de potentia exponentis realis admittitvr; demonstratio vtrivsqve facti ad calcvlvm pertinet. Eivs vnicitas avtem, et bisectio qvae eam locat, ex ordine dedvcvntvr.

Exemplaria et fvnctiones hyperbolicae

Lectio de exemplaribvs incrementi et decrementi demonstravit legem P(t+h)=ekhP(t)P(t + h) = e^{kh} P(t) exemplar P0ektP_0 e^{kt} proprie designare, ideoqve omnem magnitvdinem monotonam qvae eadem proportione in intervallis aeqvalibvs mvtatvr exponentialem esse; inde tempvs dvplicationis et tempvs dimidiationis, datatio per carbonivm 14, lex Newtoniana, linearizatio semilogarithmica et exemplar logisticvm. Postremo lectio de fvnctionibvs hyperbolicis exe^x et e−xe^{-x} in cosinvm et sinvm hyperbolicos conivnxit, qvorvm identitas fvndamentalis conseqventia directa legis exponentivm est:

cosh⁡2x−sinh⁡2x=(cosh⁡x+sinh⁡x)(cosh⁡x−sinh⁡x)=exe−x=1\cosh^2 x - \sinh^2 x = \left( \cosh x + \sinh x \right) \left( \cosh x - \sinh x \right) = e^x e^{-x} = 1

Ex ea secvtae svnt descriptio parametrica rami dextri hyperbolae x2−y2=1x^2 - y^2 = 1, cvivs parameter logarithmo recvperatvr, formvlae additionis et fvnctiones inversae, qvae logarithmis exprimvntvr aeqvationes qvadraticas in eye^y solvendo. Catena sic in se ipsa clavditvr: exponentialis fvnctiones hyperbolicas gignit, et logarithmvs eas invertit.

Pictvra qvae seqvitvr qvattvor segmenta in vnam catenam colligit. Qvisqve nodvs colvmnae est definitio avt theorema, secvndvm ordinem logicvm nvmeratvs et arcv cvm anvlis antecedentibvs 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 in qvo stabilitvr, exceptis iis qvae sine demonstratione admittvntvr avt ex partibvs antecedentibvs 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

  • ACompletvdo. Omnis copia non vacva nvmerorvm realivm qvae svperne circvmscripta est svpremvm habet; envntiata est in parte de generibvs nvmerorvm. In hac vnitate sine demonstratione admittitvr.
  • BDensitas et proprietas Archimedea. Inter dvos reales distinctos semper est rationalis, et omnis realis ab aliqvo natvrali svperatvr; demonstratae in parte de generibvs nvmerorvm. In hac vnitate sine demonstratione admittitvr.
  • CInaeqvalitas Bernovlliana. Si x≥−1x \geq -1 et nn natvralis est, tvm (1+x)n≥1+nx(1 + x)^n \geq 1 + nx; per indvctionem demonstrata in parte de nvmeris natvralibvs. In hac vnitate sine demonstratione admittitvr.
  • DBinomivm Newtonianvm. Pro omni natvrali nn valet (x+y)n=∑k=0n(nk)xn−kyk(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n - k} y^k; demonstratvm in parte de nvmeris natvralibvs. In hac vnitate sine demonstratione admittitvr.
  • ELemma Evclidis. Si primvs prodvctvm natvralivm dividit, aliqvem factorvm dividit; demonstratvm in parte de nvmeris natvralibvs. In hac vnitate sine demonstratione admittitvr.
  • FLeges exponentivm. Pro a>0a > 0, b>0b > 0 et xx, yy realibvs valent ax+y=axaya^{x + y} = a^x a^y, (ab)x=axbx(ab)^x = a^x b^x et (ax)y=axy\left( a^x \right)^y = a^{xy}. Constitvitvr in: 3. Leges exponentivm realivm.
  • GMonotonia exponentialis. Si a>1a > 1, pro realibvs uu et vv habetvr u<vu < v si et solvm si au<ava^u < a^v; basi inter 00 et 11, inaeqvalitas invertitvr. Constitvitvr in: 2. Inclvsio et monotonia.
  • HLogarithmvs vt inversa. Pro x>0x > 0, log⁡ax=y\log_a x = y si et solvm si ay=xa^y = x; vnde alog⁡ax=xa^{\log_a x} = x et log⁡a(ay)=y\log_a \left( a^y \right) = y. Constitvitvr in: 15. Logarithmvs.

  1. Segmentvm 1 Potentia exponentis realis

    • axa^x pro omni exponente reali vt svpremvm potentiarvm rationalivm definire et confirmare eam definitionem iam notam prodvcere.
    • axa^x inter potentias rationales inclvdere et inde monotoniam strictam dedvcere.
    • Leges exponentivm realivm demonstrare, potentia potentiae inclvsa, inclvsionibvs qvarvm longitvdinem inaeqvalitas Bernovlliana circvmscribit.
    • Fvnctionem exponentialem describere: dominivm, signvm, iniectivitatem, incrementvm sine margine et asymptotvm horizontalem.
    • Completvdine et inaeqvalitate Bernovlliana demonstrare exponentialem omnes valores positivos svmere.

    1. Potentia exponentis realis. Si a>1a > 1, potentia axa^x est svpremvm potentiarvm ara^r cvm rr rationali et r≤xr \leq x; svpremvm exsistit et, si xx rationalis est, cvm potentia iam nota congrvit.

      Nititvr: A (Completvdo), B (Densitas et proprietas Archimedea). Demonstratvr in: Fvnctio exponentialis et potentia exponentis realis.

    2. Inclvsio et monotonia. Si a>1a > 1 et rr, ss rationales svnt cvm r<x<sr < x < s, tvm ar<ax<asa^r < a^x < a^s. Itaqve, si x<yx < y, tvm ax<aya^x < a^y cvm a>1a > 1 et ax>aya^x > a^y cvm 0<a<10 < a < 1; praeterea ax>0a^x > 0 pro omni basi positiva.

      Vtitvr: 1. Potentia exponentis realis. Nititvr: B (Densitas et proprietas Archimedea). Demonstratvr in: Fvnctio exponentialis et potentia exponentis realis.

    3. Leges exponentivm realivm. Pro a>0a > 0, b>0b > 0 et xx, yy realibvs valent ax+y=axaya^{x + y} = a^x a^y et (ab)x=axbx(ab)^x = a^x b^x; prima lege, vnicitate radicis principalis et nova inclvsione, etiam (ax)y=axy\left( a^x \right)^y = a^{xy}, in omnibvs casibvs signorvm.

      Vtitvr: 2. Inclvsio et monotonia. Nititvr: B (Densitas et proprietas Archimedea), C (Inaeqvalitas Bernovlliana). Demonstratvr in: Fvnctio exponentialis et potentia exponentis realis.

    4. Fvnctio exponentialis. Si a>0a > 0 et a≠1a \neq 1, fvnctio axa^x in R\mathbb{R} definita est, positiva et iniectiva; si a>1a > 1, crescit, omnem marginem svperat et ad 00 sinistrorsvm accedit, et delineatio fvnctionis (1a)x\left( \frac{1}{a} \right)^x est eivs reflexio respectv axis yy.

      Vtitvr: 1. Potentia exponentis realis. Nititvr: C (Inaeqvalitas Bernovlliana), G (Monotonia exponentialis). Demonstratvr in: Fvnctio exponentialis et potentia exponentis realis.

    5. Copia imaginvm exponentialis. Si a>0a > 0 et a≠1a \neq 1, pro omni y>0y > 0 exsistit vnicvs realis xx cvm ax=ya^x = y: si a>1a > 1, est svpremvm rationalivm rr cvm ar<ya^r < y. Exponentialis est biiectio ex R\mathbb{R} in (0,∞)\ioo{0}{\infty}.

      Vtitvr: 4. Fvnctio exponentialis. Nititvr: A (Completvdo), B (Densitas et proprietas Archimedea), C (Inaeqvalitas Bernovlliana), F (Leges exponentivm), G (Monotonia exponentialis). Demonstratvr in: Fvnctio exponentialis et potentia exponentis realis.

  2. Segmentvm 2 Nvmervs e

    • Binomio Newtoniano et inaeqvalitate Bernovlliana demonstrare (1+1n)n\left( 1 + \frac{1}{n} \right)^n crescere et (1+1n)n+1\left( 1 + \frac{1}{n} \right)^{n + 1} decrescere, et illam inaeqvalitate k!≥2k−1k! \geq 2^{k - 1}, per indvctionem demonstrata, circvmscribere.
    • ee definire vt nvmervm qvem ambae seqventiae inclvdvnt.
    • Demonstrare ee svpremvm esse svmmarvm inversorvm factorialivm, errorem circvmscribere et dedvcere ee irrationalem esse.
    • Inaeqvalitatem fvndamentalem ex≥1+xe^x \geq 1 + x demonstrare et, ea adhibita, svpremvm sortivm (1+rn)n\left( 1 + \frac{r}{n} \right)^n esse ere^r.

    1. Incrementvm seqventiae. Seqventia an=(1+1n)na_n = \left( 1 + \frac{1}{n} \right)^n stricte crescens est.

      Nititvr: D (Binomivm Newtonianvm). Demonstratvr in: Nvmervs e et exponentialis natvralis.

    2. Factorialia et potentiae nvmeri dvo. Pro omni natvrali kk valet k!≥2k−1k! \geq 2^{k - 1}, aeqvalitate solvm si k=1k = 1 avt k=2k = 2; per indvctionem demonstratvr.

      Demonstratvr in: Nvmervs e et exponentialis natvralis.

    3. Margo seqventiae. Pro omni natvrali nn valet an≤sn<3a_n \leq s_n < 3, vbi sn=∑k=0n1k!s_n = \sum_{k=0}^{n} \frac{1}{k!}: qvisqve terminvs svmmae sns_n termino progressionis geometricae rationis 12\frac{1}{2} circvmscribitvr.

      Vtitvr: 7. Factorialia et potentiae nvmeri dvo. Nititvr: D (Binomivm Newtonianvm). Demonstratvr in: Nvmervs e et exponentialis natvralis.

    4. Seqventia comes. Seqventia bn=(1+1n)n+1b_n = \left( 1 + \frac{1}{n} \right)^{n + 1} stricte decrescens est.

      Nititvr: C (Inaeqvalitas Bernovlliana). Demonstratvr in: Nvmervs e et exponentialis natvralis.

    5. Nvmervs e. Nvmervs ee, svpremvm terminorvm ana_n, satisfacit an<e<bna_n < e < b_n pro omni natvrali nn.

      Vtitvr: 6. Incrementvm seqventiae, 8. Margo seqventiae, 9. Seqventia comes. Nititvr: A (Completvdo). Demonstratvr in: Nvmervs e et exponentialis natvralis.

    6. Series nvmeri e eivsqve error. Svpremvm svmmarvm sns_n est ee, et pro omni natvrali nn valet 0<e−sn<1n! n0 < e - s_n < \frac{1}{n! \, n}.

      Vtitvr: 8. Margo seqventiae, 10. Nvmervs e. Nititvr: A (Completvdo), B (Densitas et proprietas Archimedea), C (Inaeqvalitas Bernovlliana). Demonstratvr in: Nvmervs e et exponentialis natvralis.

    7. Irrationalitas nvmeri e. Nvmervs ee irrationalis est.

      Vtitvr: 11. Series nvmeri e eivsqve error. Demonstratvr in: Nvmervs e et exponentialis natvralis.

    8. Inaeqvalitas fvndamentalis. Pro omni reali xx valet ex≥1+xe^x \geq 1 + x, et aeqvalitas solvm in x=0x = 0 habetvr.

      Vtitvr: 10. Nvmervs e. Nititvr: B (Densitas et proprietas Archimedea), C (Inaeqvalitas Bernovlliana), F (Leges exponentivm), G (Monotonia exponentialis). Demonstratvr in: Nvmervs e et exponentialis natvralis.

    9. Capitalizatio continva. Si r>0r > 0, sortes (1+rn)n\left( 1 + \frac{r}{n} \right)^n cvm nn crescvnt, nvlla ere^r attingit et earvm svpremvm est ere^r; pro t>0t > 0, svpremvm nvmerorvm (1+rn)nt\left( 1 + \frac{r}{n} \right)^{nt} est erte^{rt}.

      Vtitvr: 1. Potentia exponentis realis, 6. Incrementvm seqventiae, 10. Nvmervs e, 13. Inaeqvalitas fvndamentalis. Nititvr: D (Binomivm Newtonianvm), F (Leges exponentivm), G (Monotonia exponentialis). Demonstratvr in: Nvmervs e et exponentialis natvralis.

  3. Segmentvm 3 Logarithmvs et aeqvationes

    • log⁡a\log_a vt inversam exponentialis obtinere et qvamqve legem exponentivm in legem logarithmorvm vertere.
    • Omnes logarithmos ad vnvm per mvtationem basis redvcere, et eorvm monotoniam signvmqve perscrvtari.
    • Lemmate Evclidis demonstrare log⁡23\log_2 3 irrationalem esse.
    • Iniectivitatem et monotoniam in methodos convertere ad aeqvationes et inaeqvationes solvendas.

    1. Logarithmvs. Cvm exponentialis basis aa biiectio ex R\mathbb{R} in (0,∞)\ioo{0}{\infty} sit, pro qvoqve x>0x > 0 nvmervs log⁡ax\log_a x est vnicvs realis yy talis vt ay=xa^y = x, et fvnctio log⁡a\log_a est eivs inversa.

      Vtitvr: 5. Copia imaginvm exponentialis. Demonstratvr in: Fvnctio logarithmvs eivsqve proprietates.

    2. Leges logarithmorvm. Pro xx, yy positivis et rr reali valent log⁡a(xy)=log⁡ax+log⁡ay\log_a(xy) = \log_a x + \log_a y, log⁡axy=log⁡ax−log⁡ay\log_a \frac{x}{y} = \log_a x - \log_a y et log⁡a(xr)=rlog⁡ax\log_a \left( x^r \right) = r \log_a x.

      Nititvr: F (Leges exponentivm), H (Logarithmvs vt inversa). Demonstratvr in: Fvnctio logarithmvs eivsqve proprietates.

    3. Mvtatio basis. Valet log⁡ax=log⁡bxlog⁡ba\log_a x = \frac{\log_b x}{\log_b a}; omnis logarithmvs est natvralis per constantem divisvs, et log⁡ab⋅log⁡ba=1\log_a b \cdot \log_b a = 1.

      Vtitvr: 16. Leges logarithmorvm. Nititvr: H (Logarithmvs vt inversa). Demonstratvr in: Fvnctio logarithmvs eivsqve proprietates.

    4. Monotonia et signvm logarithmi. Fvnctio log⁡a\log_a stricte crescens est si a>1a > 1 et stricte decrescens si 0<a<10 < a < 1, et log⁡ax>0\log_a x > 0 si et solvm si (a−1)(x−1)>0(a - 1)(x - 1) > 0.

      Nititvr: G (Monotonia exponentialis), H (Logarithmvs vt inversa). Demonstratvr in: Fvnctio logarithmvs eivsqve proprietates.

    5. Logarithmi irrationales. Si aa et bb natvrales inter se primi maiores qvam 11 svnt, nvmervs log⁡ab\log_a b irrationalis est; praesertim log⁡23\log_2 3.

      Vtitvr: 18. Monotonia et signvm logarithmi. Nititvr: F (Leges exponentivm), H (Logarithmvs vt inversa), E (Lemma Evclidis). Demonstratvr in: Fvnctio logarithmvs eivsqve proprietates.

    6. Iniectivitas et aeqvationes. Habetvr au=ava^u = a^v si et solvm si u=vu = v, et log⁡au=log⁡av\log_a u = \log_a v si et solvm si u=vu = v; aeqvatio ax=ca^x = c vnicam solvtionem log⁡ac\log_a c habet si c>0c > 0, et nvllam si c≤0c \leq 0.

      Vtitvr: 4. Fvnctio exponentialis. Nititvr: H (Logarithmvs vt inversa). Demonstratvr in: Aeqvationes et inaeqvationes exponentiales et logarithmicae.

    7. Monotonia et inaeqvationes. Si a>1a > 1, habetvr au<ava^u < a^v si et solvm si u<vu < v, et log⁡au<log⁡av\log_a u < \log_a v si et solvm si u<vu < v; si 0<a<10 < a < 1, aeqvivalentiae sensv inverso valent.

      Vtitvr: 18. Monotonia et signvm logarithmi. Nititvr: G (Monotonia exponentialis). Demonstratvr in: Aeqvationes et inaeqvationes exponentiales et logarithmicae.

  4. Segmentvm 4 Exemplaria et fvnctiones hyperbolicae

    • Exemplar P0ektP_0 e^{kt} per rationem constantem in intervallis aeqvalibvs designare et tempvs dvplicationis et tempvs dimidiationis obtinere.
    • Data exponentialia in plano semilogarithmico linearizare et exemplar logisticvm perscrvtari.
    • Cosinvm et sinvm hyperbolicos ex exe^x et e−xe^{-x} definire, et eorvm monotoniam perscrvtari.
    • Identitatem fvndamentalem demonstrare, ramvm dextrvm hyperbolae parametrice describere, et formvlas additionis fvnctionesqve inversas obtinere.

    1. Ratio constans. Si P(t)=P0ektP(t) = P_0 e^{kt} cvm P0>0P_0 > 0, tvm P(t+h)=ekhP(t)P(t + h) = e^{kh} P(t) pro omnibvs realibvs tt et hh; temporibvs integris est exemplar discretvm cvm k=ln⁡(1+r)k = \ln(1 + r), et fvnctio crescit si k>0k > 0 et decrescit si k<0k < 0.

      Nititvr: F (Leges exponentivm), G (Monotonia exponentialis), H (Logarithmvs vt inversa). Demonstratvr in: Exemplaria incrementi et decrementi exponentialis.

    2. Conversvm rationis constantis. Si PP positiva et monotona est et qvotiens P(t+h)P(t)\frac{P(t + h)}{P(t)} solvm a hh pendet, tvm P(t)=P0ektP(t) = P_0 e^{kt} pro aliqvo reali kk: aeqvatio f(t+h)=f(t)f(h)f(t + h) = f(t) f(h) et monotonia exponentialem determinant.

      Vtitvr: 2. Inclvsio et monotonia. Nititvr: B (Densitas et proprietas Archimedea), F (Leges exponentivm), H (Logarithmvs vt inversa). Demonstratvr in: Exemplaria incrementi et decrementi exponentialis.

    3. Dvplicatio et tempvs dimidiationis. Si k>0k > 0, exemplar P0ektP_0 e^{kt} in omni intervallo dvrationis T=ln⁡2kT = \frac{\ln 2}{k} dvplicatvr, et in his solis; si k=−λ<0k = -\lambda < 0, ad dimidivm redigitvr in omni intervallo dvrationis ln⁡2λ\frac{\ln 2}{\lambda}.

      Vtitvr: 22. Ratio constans, 20. Iniectivitas et aeqvationes. Nititvr: H (Logarithmvs vt inversa). Demonstratvr in: Exemplaria incrementi et decrementi exponentialis.

    4. Linearizatio semilogarithmica. Pvnctvm (t,y)(t, y) cvm y>0y > 0 in delineatione fvnctionis y=Cekty = C e^{kt} iacet si et solvm si (t,ln⁡y)(t, \ln y) in recta Y=ln⁡C+ktY = \ln C + kt iacet; per dvo pvncta abscissarvm distinctarvm vnica delineatio hvivs formae transit.

      Vtitvr: 16. Leges logarithmorvm. Nititvr: H (Logarithmvs vt inversa). Demonstratvr in: Exemplaria incrementi et decrementi exponentialis.

    5. Exemplar logisticvm. Fvnctio P(t)=K1+Ae−rtP(t) = \frac{K}{1 + A e^{-rt}}, cvm KK, AA et rr positivis, stricte crescit inter 00 et KK, K2\frac{K}{2} in t0=ln⁡Art_0 = \frac{\ln A}{r} attingit et eivs delineatio respectv illivs pvncti symmetrica est.

      Vtitvr: 4. Fvnctio exponentialis. Nititvr: G (Monotonia exponentialis), H (Logarithmvs vt inversa). Demonstratvr in: Exemplaria incrementi et decrementi exponentialis.

    6. Fvnctiones hyperbolicae. Cvm cosh⁡x=ex+e−x2\cosh x = \frac{e^x + e^{-x}}{2} et sinh⁡x=ex−e−x2\sinh x = \frac{e^x - e^{-x}}{2}, cosinvs hyperbolicvs par est, sinvs impar, et ex=cosh⁡x+sinh⁡xe^x = \cosh x + \sinh x, e−x=cosh⁡x−sinh⁡xe^{-x} = \cosh x - \sinh x.

      Vtitvr: 4. Fvnctio exponentialis. Demonstratvr in: Fvnctiones hyperbolicae.

    7. Monotonia hyperbolica. Sinvs et tangens hyperbolici stricte crescentes svnt, cvm −1<tanh⁡x<1-1 < \tanh x < 1; cosinvs hyperbolicvs decrescit pro x≤0x \leq 0, crescit pro x≥0x \geq 0 et eivs minimvm est cosh⁡0=1\cosh 0 = 1.

      Vtitvr: 27. Fvnctiones hyperbolicae. Nititvr: F (Leges exponentivm), G (Monotonia exponentialis). Demonstratvr in: Fvnctiones hyperbolicae.

    8. Identitas fvndamentalis. Pro omni reali xx valet cosh⁡2x−sinh⁡2x=1\cosh^2 x - \sinh^2 x = 1.

      Vtitvr: 27. Fvnctiones hyperbolicae. Nititvr: F (Leges exponentivm). Demonstratvr in: Fvnctiones hyperbolicae.

    9. Descriptio parametrica hyperbolae. Fvnctio t↦(cosh⁡t,sinh⁡t)t \mapsto \left( \cosh t, \sinh t \right) est biiectio ex R\mathbb{R} in ramvm dextrvm cvrvae x2−y2=1x^2 - y^2 = 1, et eivs inversa pvncto (x,y)(x, y) nvmervm ln⁡(x+y)\ln(x + y) attribvit.

      Vtitvr: 20. Iniectivitas et aeqvationes, 27. Fvnctiones hyperbolicae, 28. Monotonia hyperbolica, 29. Identitas fvndamentalis. Nititvr: H (Logarithmvs vt inversa). Demonstratvr in: Fvnctiones hyperbolicae.

    10. Formvlae additionis. Valent cosh⁡(a+b)=cosh⁡acosh⁡b+sinh⁡asinh⁡b\cosh(a + b) = \cosh a \cosh b + \sinh a \sinh b et sinh⁡(a+b)=sinh⁡acosh⁡b+cosh⁡asinh⁡b\sinh(a + b) = \sinh a \cosh b + \cosh a \sinh b.

      Vtitvr: 27. Fvnctiones hyperbolicae. Nititvr: F (Leges exponentivm). Demonstratvr in: Fvnctiones hyperbolicae.

    11. Fvnctiones hyperbolicae inversae. Valent arcsinh⁡x=ln⁡(x+x2+1)\operatorname{arcsinh} x = \ln \left( x + \sqrt{x^2 + 1} \right), arccosh⁡x=ln⁡(x+x2−1)\operatorname{arccosh} x = \ln \left( x + \sqrt{x^2 - 1} \right) pro x≥1x \geq 1, et arctanh⁡x=12ln⁡1+x1−x\operatorname{arctanh} x = \frac{1}{2} \ln \frac{1 + x}{1 - x} pro −1<x<1-1 < x < 1.

      Vtitvr: 28. Monotonia hyperbolica. Nititvr: H (Logarithmvs vt inversa). Demonstratvr in: Fvnctiones hyperbolicae.

Ad fvnctiones trigonometricas. Catena in se ipsa clavditvr: exponentialis fvnctiones hyperbolicas gignit, et logarithmvs eas invertit. Cvrsvs proximvs cvm circvmferentia x2+y2=1x^2 + y^2 = 1 faciet qvod haec pars cvm hyperbola x2−y2=1x^2 - y^2 = 1 fecit.