Rationes mvtationis et rationes mvtationis conexae

Svmmarivm

Haec lectio derivatam legit vt id qvod in applicationibvs metitvr: qvam celeriter vna magnitvdo mvtetvr cvm altera mvtatvr. Rationem mvtationis mediam, momentaneam et relativam distingvit, cvm earvm vnitatibvs, et incrementvm ratione relativa constanti describit. Motvm rectilinevm examinat, cvm positione, velocitate, celeritate et acceleratione, et demonstrat qvid signvm derivatae in instanti dicat, et qvid non dicat. Deniqve, cvm plvres magnitvdines qvae a tempore pendent relatione aliqva conivngvntvr, regvla catenae etiam earvm rationes mvtationis conivngit: hae svnt rationes mvtationis conexae, qvae methodo explicita solvvntvr.

Proposita discendi

Derivatam vt rationem mvtationis momentaneam interpretari, cvm eivs vnitatibvs, et eam a ratione mvtationis media et a relativa distingvere.
Motvm rectilinevm positione, velocitate, celeritate et acceleratione describere, et demonstrare qvid signvm derivatae in instanti de valoribvs vicinis dicat.
Problemata rationvm mvtationis conexarvm methodo explicita solvere, ostendendo magnitvdines qvae derivantvr derivabiles esse.
Rationes relativas mvtationis prodvctorvm et potentiarvm cvm derivata logarithmica conivngere, et accelerationes cvm derivata secvnda relationis.

Derivata vt ratio mvtationis

Lectio de derivata vt limite ad eandem definitionem a dvabvs qvaestionibvs pervenit: a declivitate tangentis et a velocitate momentanea. Ambae svnt casvs qvaestionis generalioris. Qvotiens magnitvdo vna ab altera pendet, area circvli a radio svo, positio mobilis a tempore, massa frvsti virgae a longitvdine sva, ivvat scire qvam celeriter prima mvtetvr cvm altera mvtatvr.

DefinitioRatio mvtationis

Sit $f$ fvnctio in intervallo aperto $I$ definita, qvae magnitvdinem $y = f(x)$ vt fvnctionem alterivs magnitvdinis $x$ exprimit, et sint $a$ et $a + h$ pvncta ex $I$ , cvm $h \neq 0$ . Ratio mvtationis media ipsivs $y$ secvndvm $x$ inter $a$ et $a + h$ est qvotiens

\frac{\Delta y}{\Delta x} = \frac{f(a + h) - f(a)}{h}

Si $f$ derivabilis est in $a$ , ratio mvtationis momentanea (vel simpliciter ratio mvtationis) ipsivs $y$ secvndvm $x$ in $a$ est $f'(a)$ ; si praeterea $f(a) \neq 0$ , ratio mvtationis relativa in $a$ est qvotiens $\frac{f'(a)}{f(a)}$ .

Ratio media est declivitas secantis, et momentanea eivs limes: declivitas tangentis. Ambae metivntvr vnitatibvs ipsivs $y$ in singvlas vnitates ipsivs $x$ : metris in secvndo, si $y$ longitvdo est et $x$ tempvs; chiliogrammatis in metro, si $y$ est massa frvsti virgae et $x$ eivs longitvdo. In hoc casv, si $M(x)$ est massa frvsti inter extremvm altervm et pvnctvm distantia $x$ positvm, ratio mvtationis $M'(x)$ est densitas linearis virgae in eo pvncto: massa in vnitatem longitvdinis qvam frvstvm brevissimvm circa ipsvm haberet. Ratio relativa contra solvm vnitatibvs ipsivs $x$ inversis metitvr (in secvndo, in die): non pendet ab vnitate qva $y$ metitvr, et ideo mvtationem magnitvdinvm diversae natvrae vel diversae magnitvdinis comparare sinit. Est derivata logarithmica lectionis de variabili in exponente, et saepe per centesimas exprimitvr: ratio relativa $0{,}03$ in die est 3 per centvm in die.

Exemplvm geometricvm ostendit qvomodo ratio mvtationis legatvr. Area circvli radii $r$ est $A = \pi r^2$ , et eivs ratio mvtationis secvndvm radivm est $\frac{\mathrm{d}A}{\mathrm{d}r} = 2\pi r$ , perimetrvm, vt problema propositvm 1 lectionis de derivata vt limite confirmare ivbet. Cavsa clara est: radio avcto per $h$ , area avgetvr corona circvlari latitvdinis $h$ , cvivs area fere aeqvat perimetrvm per $h$ mvltiplicatvm.

PropositioIncrementvm ratione relativa constanti

Sint $y_0 > 0$ et $k \neq 0$ nvmeri reales, et sit $y(t) = y_0 e^{kt}$ pro omni reali $t$ . Tvm ratio mvtationis relativa ipsivs $y$ valet $k$ in omni instanti, et pro omni $t$ et omni $T$ :

y(t + T) = e^{kT} \, y(t)

Praesertim, si $k > 0$ , magnitvdo dvplicatvr in qvoqve intervallo temporis dvrationis $\frac{\ln 2}{k}$ , qvocvmqve instanti incipiat; et si $k < 0$ , ad dimidivm redigitvr in qvoqve intervallo dvrationis $\frac{\ln 2}{\abs{k}}$ .

Demonstratio

$y'(t) = y_0 e^{kt} \cdot k = k \, y(t), \qquad \frac{y'(t)}{y(t)} = \resaltar{k}$
Per corollarivm de compositionibvs cvm exponentiali et logarithmo, ex lectione de variabili in exponente, cvm $u(t) = kt$ , fvnctio $y$ derivabilis est in toto $\mathbb{R}$ et eivs derivata est $k$ vicibvs ipsa fvnctio, qvae non evanescit.
$\frac{y(t + T)}{y(t)} = \frac{y_0 e^{kt} e^{kT}}{y_0 e^{kt}} = \resaltar{e^{kT}}$
Per legem exponentivm, qvotiens inter valores intervallo dvrationis $T$ separatos non pendet a $t$ .
$e^{kT} = 2 \Leftrightarrow T = \frac{\ln 2}{k}, \qquad e^{kT} = \frac{1}{2} \Leftrightarrow T = \resaltar{\frac{\ln 2}{-k}}$
Cvm exponentialis iniectiva sit, qvotiens valet $2$ si et solvm si $kT = \ln 2$ , et valet $\frac{1}{2}$ si et solvm si $kT = -\ln 2$ .

Popvlatio bacteriorvm in medio sine limitibvs vel massa materiae radioactivae mvtantvr, bona approximatione, ratione relativa constanti, et propositio explicat cvr tempvs dvplicationis vel tempvs dimidiationis fixvm habeant. Fvnctiones $y_0 e^{kt}$ solas esse cvm ratione relativa constanti $k$ vervm est, sed eivs demonstratio scire postvlat fvnctionem cvm derivata nvlla in intervallo constantem esse, qvod e theoremate valoris medii partis proximae conseqvitvr.