Developpement Limité

Résumé du document

Formules des Développements limités:



-Fonctions trigonométriques

-Fonctions circulaires inverses

-Exponentielle et logarithme

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Extraits

[...] ` dl.tex page 2 Solutions Exercice e1/ + 90 x2 + o(x3 ) . [...]

[...] Calculs de DL Fonctions trigonom´triques e sin x cos x ln(sin a n DL + x2/6 + 7x4/ + x2/2 + 5x4/ x4/180 x6 /2835 exp(sin e(1 x2 + x4/45) tan x π/ + h + h2 sin(x + x2 + x3 x4 ) ln(x tan(1/x)) cos 1)2 sin((π cos cos x ln(1 + (sin x 1)/(cos x + ln(2 cos x + tan ecos x 4 + x + x2 + 5x3/6 3x4/ 1/2 x/2 + x2 / π 2x4 /32 + π2 x6/ + x/2 x2 / e(1 x2 + x4/6) 4 x x2/2 x3 / ln 2 + x/2 5x2/8 + 11x3/24 59x4/192 Fonctions circulaires inverses a n DL arcsin2 x x2 + x4 + 8x6/45 arcsin2 x arctan 1/3 x2 /15 + + 2 π/4 + arccos(sin x / 3(1 x2/90) arcsin(sin2 arctan(1 + arctan x + x/3 4x3/45 + 44x5/945 arcsin x π/6 + 3(2h 4h2/3 + 32h3/9) π/4 + x/2 x2/4 + x3 / x2 x4 + 19x6/90 107x8/630 arcsin x2) earcsin x + x + 7x2/6 eπ/6 + 2h/ 3 + 2(1 + 3)h2/(3 π 2 π + ( π + ( π + ( 12 1 e1/x arctan x 3 dl.tex lundi 20 aoˆt 2001 u Exponentielle et logarithme a n DL x/(ex x/2 + x2/12 ln x h h2 + 23h3/24 ln(1 + x + x2) x + x2/2 x3/6 ln((2 x2 ln(2/3) x/2 + 5x2 /24 ch ln(1 + + 1/2 + 5x/12 ln(ln(1 + + 5x2/24 x3/8 ln(ax + bx ) ln 2 + x ln ab + x2 ln2(a/b)/8 exp(1/x)/x e(1 3h + 13h2/2 73h3/6) Fonctions hyperboliques inverses a n DL argth(sin x + x3/6 + x5/24 argsh(ex ) ln(1 + + 2(x + x2/4) Formes exponentielles x + x ) 2 1/x a n DL 2 + x/2 + 19x2/24) + 2αx + 2α2x2 + 2α(2α2 + 1)x3 / x2 4 x2 /60 139x4/151200) 2 e(1 x/2 + 7x2/24) + x ln 2 + x2 (ln + D´velopper d'abord ln((1 + e + x))α (sin x/x)2/x (sin x/x)3/x + sin x)1/x + sin x + cos x)x (tan x)tan 2x π/ + 2h2/3 + 4h4 (sin x)sin x π/ h2/2 + 7h4/24 Radicaux x + 1 1 x2 ex 1 + 2x ( 3 x3 + x2 + 3 x3 x2 Exercice 1. [...]

[...] EIT 1999 Calculer le d´veloppement limit´ de e e tan x x a n DL 3(2 + 5h/3 + h3/54) 2(1 x/8 5x2 /128 21x3/1024) x / 2(1 + x2/8 + 7x4/128) x2 x3/3 + 2x4/3 13x5/15 1/x2 en 0 a l'ordre 3. [...]