Fvnctiones cvm radicalibvs: dominivm, signa et delineatio

Svmmarivm

Haec lectio methodvm tabvlarvm signorvm et divisionis polynomiorvm ad fvnctiones cvm radicalibvs extendit: earvm dominivm determinat cvm radices in nvmeratore et in denominatore svnt, earvm signvm et habitvm in finibvs dominii investigat, vbi delineatio in pvncto desinit avt ad asymptotvm verticalem accedit, et earvm asymptotos pro magnis variabilis valoribvs invenit, ex vnaqvaqve radice maximam potentiam ipsivs $x$ extrahendo, casibvs $x$ positivi et $x$ negativi diligenter distinctis. Desinit in copia imaginvm, qvae accvrate determinatvr cvm limites in finibvs id sinvnt, et in plena investigatione fvnctionvm qvarvm radicandvm est trinomivm gradvs $8$ .

Proposita discendi

Dominivm fvnctionis cvm radicalibvs in nvmeratore et in denominatore determinare per tabvlas signorvm et per resolvtionem in factores trinomiorvm gradvs $2n$ .
Signvm fvnctionis cvm radicalibvs et eivs habitvm in finibvs dominii investigare, terminos in qvibvs delineatio desinit ab asymptotis verticalibvs distingvendo.
Asymptotos fvnctionis cvm radicalibvs invenire cvm $x$ ad $\infty$ et ad $-\infty$ tendit, valores positivos et negativos variabilis separatim tractando.
Delineationem fvnctionis cvm radicalibvs dvcere et eivs copiam imaginvm determinare cvm investigatio id sinit, casvs agnoscendo qvi artes calcvli differentialis reqvirvnt.
Fvnctiones cvm radicalibvs investigare qvae magnitvdines adhibitas describvnt, vt differentiam distantiarvm a dvabvs antennis avt factorem Lorentzianvm relativitatis, et earvm dominivm, asymptotos et copiam imaginvm in contextv interpretari.

Dominivm fvnctionis cvm radicalibvs

In lectione de fvnctionibvs algebraicis regvlae statvtae svnt qvae dominivm fvnctionis per formvlam definitae figvnt: nvllvs denominator nihilo aeqvalis esse potest, et radicandvm radicis indicis paris negativvm esse non potest. Cvm radix in denominatore est, ambae regvlae simvl agvnt, et eivs radicandvm stricte positivvm esse debet; cvm in nvmeratore est, satis est vt non negativvm sit. Radices indicis imparis nvllam condicionem imponvnt, praeter eam ne evanescant si in denominatore svnt.

Si radicanda svnt polynomia gradvs maioris qvam $1$ , qvaeqve condicio est inaeqvatio polynomialis, qvae solvitvr in factores resolvendo et tabvlam signorvm constrvendo, vt in lectione de tabvlis signorvm et inaeqvationibvs. Factores qvadratici formvla qvadratica obtinentvr, et trinomia formae $a x^{2n} + b x^n + c$ ad qvadratica redvcvntvr mvtatione variabilis $u = x^n$ , vt in lectione de factorizatione polynomii qvadratici. Factor qvadraticvs discriminantis negativi signvm coefficientis svi principalis semper servat, ita vt nvlla pvncta critica afferat et e tabvla tolli possit.

Dominivm fvnctionis cvm radicalibvs obtinetvr omnes condiciones simvl conivngendo: radicandvm non negativvm pro vnaqvaqve radice indicis paris nvmeratoris, radicandvm positivvm pro vnaqvaqve radice indicis paris alicvivs denominatoris, et denominatores a nihilo differentes. Qvaeqve condicio tabvla signorvm solvitvr, et dominivm est intersectio copiarvm obtentarvm.