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Forum Studenti •determinare l'ordine di infinitesimo
Pagina 1 di 1

determinare l'ordine di infinitesimo

Inviato: giovedì 31 dicembre 2015, 22:18
da Help!
Buongiorno

Io mi sono bloccato in un esercizio che chiedeva la determinazione dell'ordine di infinitesimo e la parte principale p(x) della seguente funzione f(x):

sin( ­­­Pi * cosx ) per x che tende a Pi

nella soluzione l'ordine di infinitesimo di f(x) è 2
mentre la parte principale p(x)= (-Pi/2)*(x-Pi)^2

ho provato senza riuscirci usando principalmente limiti notevoli e o piccolo

Grazie in anticipo!

Re: determinare l'ordine di infinitesimo

Inviato: venerdì 1 gennaio 2016, 11:39
da GIMUSI
allego un possibile svolgimento...probabilmente migliorabile...che ne pensi?

colgo l'occasione per augurare un BUON NUOVO ANNO a tutto il popolo del forum!

Re: determinare l'ordine di infinitesimo

Inviato: sabato 2 gennaio 2016, 1:12
da Help!
Buon anno 2016 anche a Lei e a coloro che hanno collaborato alla creazione e organizzanione di questo sito.

Per quanto riguarda l'esercizio, data la funzione in questioneImmagine

non capisco il passaggio trigonometrico per arrivare a determinare l'uguaglianza Immagine

Thank you very much!

Re: determinare l'ordine di infinitesimo

Inviato: sabato 2 gennaio 2016, 10:37
da GIMUSI

Re: determinare l'ordine di infinitesimo

Inviato: lunedì 4 gennaio 2016, 21:06
da Massimo Gobbino
Io lo avrei fatto così (che poi è praticamente la stessa soluzione di GIMUSI). Parto con cambio di variabili e precorso

[tex]\sin(\pi\cos(\pi+h))=\sin(-\pi\cos h)=-\sin(\pi\cos h)[/tex]

Poi sviluppo il coseno (per semplicità non metto gli [tex]o(h^2)[/tex] che andrebbero messi ovunque)

[tex]-\sin(\pi\cos h)=-\sin\left[\pi\left(1-\dfrac{h^2}{2}\right)\right]=-\sin\left[\pi-\dfrac{\pi h^2}{2}\right][/tex]

Ora di nuovo precorso e sviluppo del seno (sempre trascurando i soliti o piccolo)

[tex]-\sin\left[\pi-\dfrac{\pi h^2}{2}\right]=\sin\left(-\dfrac{\pi h^2}{2}\right)=-\dfrac{\pi h^2}{2}[/tex]

ed il gioco è fatto.

Segnalo anche due approcci alternativi. Il primo è sviluppare la funzione data di ordine due con centro in pi greco, calcolando i coefficienti facendo le derivate (approccio orribile, ma in questo caso praticabile). Il secondo è molto a posteriori, cioè verificare che

[tex]\displaystyle\lim_{x\to\pi}\frac{\sin(\pi\cos x)}{(x-\pi)^2}=-\frac{\pi}{2}[/tex]

cosa che si può fare con solo cambi di variabili, precorso e limiti notevoli, senza mai nemmeno nominare o piccolo.

Re: determinare l'ordine di infinitesimo

Inviato: martedì 5 gennaio 2016, 20:30
da Help!
Grazie infinite!