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Full text of "Cours Math"



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. ^Vimo ot SjUc 1$jLj j^ M cj^ Sj^i-" /(x) = ax + b ^JW 

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(X jL-oJ (_jic j d LW^ CT^ '^alui-a f CllJlS IJJ /), 3 (J 4-°^ •^JC' o^ai-a-a f 4j|j (jj^J 

.(sjjxji^jus^wiii) lim x >^ a /(x) = /(a) £A£ IJJ Df 3 a u^i <Jc- Sj^iu^ y 7 3Jb jj£s 



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Jjk2i! j*j Jl jji^lj JIJ2 

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•lie JJ JjjJ Iglib La (j\£ bj JLijjJ ojcla JaxIuiJ j 



Jlla 

im r ^ +O0 x 2 + 5x - 7 lim^.ooX 2 + 5x - 7 lim^.™-^ — lim 



X-> + CO A T JA / x->-00 A T JA / X-> + O0 2_ x _g X->-2 x 2_ x _ 6 

3x-9 ,. 5x / -3 ,. x 3 -8 ,. x 3 -8 

im x ^ 3 — hm x ^ +00 — hm x ^ +O0 — hm x ^ 2 — 

x 6 x 2 -x-6 x + x 2 -x-6 x + x 2 -x-6 x z x 2 +x-6 

X 



5x 2 -3 

X— > + 00 7 - 

x / -x-6 


li 


lim^-^.oo Vx 2 - 


- 1 + X 



im x ^ +O0 Vx + 3 — x lim^-^.oo Vx 2 — 1 + x lim x ^_ O0 Vx 2 ~^T + 



im x ^ +O0 Vx 3 — 1 — x lim x ^ +O0 Vx 2 — 5x + 1 — x lim x ^ +O0 Vx 2 — 1 — 5 



X 

im x ^ +O0 V2x 2 + x - Vx 3 + 1 lim x ^ +O0 Vx 2 + x - 1 - Vx 3 + 5x 2 + 3 

Vx+l-2 ,. Vx+l-5+x ,. V4x+l-V5x 2 -ll ,. Vx 3 -1-V^-1 
x A x 2 -x-6 x 6 x 2 -9 x z Vx 3 +8-2x x y Vx 2 -1 

Ln(x) 



im x ^ +O0 Ln(x 3 + 2x - 1) -Vx lim,^ Vx - 1 ■ Ln{x 3 - 1) lim^i 



yx-i 



im x ^ +O0 x 2 + 5x + 3 - e^ \\rr\ x ^_ m e x {x - x 2 ) lim x ^ +00 3x - Ln{e 2x + Vx) 

im x ^ — : — - li m x ^ +O0 xLn( 1 + -) lim x ^ +O0 xLn(x + 1) —xLn(x) 

xsin(x 2 + l) .. e x -cos{x) ■■ xstn(x) 

IITI x ^ +0O - " m ^0 Ln(x+1) " m ^° C0 5(x)-l 

im x ^ xLn(x)sin(-) lim x ^ n ° 5 2 lim x ^ +O0 sin(x)Ln(x) 

\xj (sin{x)) 



]a , M JMI ^ JSVt J* 5U J£ /(x) = iial— »cii f{a)-f{b)<0 j^ bl 

c = Sup(A) c£J j A= {i £ [a , I)] : /"(x) < 0} <cj^l jU>VI u^ iiL 

/(c) = J^-^Wu^ 

/([a , &]) = [/(a) , /(£)] u^ [a , b] Jl^l ^ ^lji« j Sj-i-- *Jb / cjjIS bl 

/([a , 6]) = [f{b) , fia)] d* [a , b] JMI ^ ^-SLsi. j Sj-l*. ^ib / cjj\£ bl 

<-^ /(]a , M) = ]a , /?[ Oi* ]a , M JW-»1I Jfc sjjI jl«j Sj-lab« AJb / cjj\£ bl 

fa = lim^ a /(x) 
1)5 = \\m x ^ b f[x) 

£***> /(]a , M) = ]/? , a[ u^ ]a , M JW-»1I ^Jc sjjI jwj Sj-i— AJb f cA£. bl 

fa = lim^ a /(x) 
I/? = lim^ b /(x) 

/? ^*]| j lim x >^ a /U) ~ /(a) = /? G 5R c^i£ b] a u^j c> L3i£i^ ^Lii bi / *Jhli & J_£ 

a ij^ (Jt y 7 4JUI (jjA* t^-AJjj 

j cj Aic (jlaluJ^U Alii /" 4JUI Jliia Qi jUjj cs Jc 1 3 sj uj " (_jjLuij (j (jj-aj cs ic. f aJUI (Jjlii-a (jl£ bl 

d Aic /" 4JUI (jjii-o cs -ojaiJ 4 jlfrjll lilii <Lajii 

b* ^ ^i^ ilt (3lili!>U aIAI f aJU ciil£ bl I Q D f JW-a J& jlili!^ 4lii Uii / 4JIJI (> Jli 

JU^ll 






Jlla 



V /\(i)=i j 1 -<*a\ *c $&zyi mi f( x ) = y/x y^i(* 



lim^-, - — = lim^-, -^— = - 

x L x-l -Jx+l 2 



lim^of = lim x ^ ^= +00 



jV <-^l Xe. jlili!^ aIAI j^ f(x) = yfx ^ (* 



jV /\ (a) = 5 j a G 5R *-*i c?i ^ jUiiSSU *L15 /( x ) = 5x + B *JUI (* 

{Sx + B) -{Sa + B) Six -a) 
I im — = lim = S 

x->a x — a x^a x — a 

uV fH ) = cosia) j a£5K *-*i c?i ^ li*^ *Ui /"(x) = sinix) *JUI (* 

,. sin(x) -5in(a) ,. 2cos(— ^ )sin(— 2 J 5in(y) , . . . 

lim = lim — y _ „ — - — - = lim cos(y + a) = cos(a) 

x^a x — a x^a 2— — v^° y 



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^Jj Ld (jAULLyaj AaLLoJI 



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.AjJLill -Icljall LjjJ j (jUu*^U Alii AJb ^ aj-l*-a JJC- (jjjA* dlli j 

[/(x)+^(x)]\ =/Ux)+^(x) 
[/(x)-^(x)]\ =/Ux)^(x)+/(x)^(x) 



/(*) 



#(x) 



* = fHx)g{x)-f{x)gHx) 
g 2 {x) 



[g[f[x)]) x = g x (y) ■ f x (x) avec y = fix) 

f x = /(y) 



[/-Mx)]^^ m, e c P-/.^ 



f'(y) 



[f a (x)]\ = fHx) ■ f a ~Hx) 

f x (x) 

[Ln[f[x)W = [ 7 rr 
f(x) 

[efMf =fHx)-ef {x) 

fHx) = f[x) ■ [Ln[f[xW 

(sin[f(x)])\ = fHx) ■ cos[f(x)] 

{cos[f(x)W = -fHx) ■ sin[f(x)] 

(tg[f(x)W = j f , { *\„ 2 = fHx) ■ [1 + (tg[f(x)]) 2 ] 
[cos[f(x)\) 2 

(cotg[f(x)W = j 7{}, X \^ 2 = -fHx) ■ [1 + (cotg[f(x)]) 2 ] 
(sin[f(x)\) 2 

Aj.u£»JI Ajiillall Jlj^l 

*JWIj ]-| , |[ JMJ- t#(x) UUIj [0 , tt] JMU* co5(x) =Uhitj[-| , |] 
.l$i« jSi 4au£«J! aJW! jlik-VI Oi*4 ^Vl o^j ^Mj j aauj JIjj ^ ]0 , n[ JW-JI Jc- cotg{x) 

i4rcsin: 5H -> 5H 

x -> i4rcsin(x) = a tq sin{a) = x 
Arccos'.yi -» 91 

x -> i4rccos(x) = b tq cos[b) = x 
Arctg-.ft^ft . 

x -> Arctgix) = c tq tg[c) = x 
Arccotg:^,^^, . 

x -» Arccotgix) = d tq cotgid) = x 



Ijjli 4_ii*a£*JI 4JUI (jjliiwi (j->^ ' . '"'"*• 



[Arcsin(x) ] x = , . = — — = . , = -7==: . 

Urccos(x)] y = — r-— = — — — = , r f )17 = -7== 

cosHb) -sin(b) Jl-[cos(b)] 2 Vl-x 2 

[i4rct,g(x)] y = — -— = . , ... = - — -. 

u tg\[c) l+[tg(c)] 2 1+x 2 



[Arccotgix)]^ = ; 



cotg\(d) -(l+[cotg(d)] 2 ) 1+x 2 






Kcco S (/W)] ( - ^ 



[i4rccot,g(/(x))] = 



\ _ ~[/(x)]\ 



l + [/U)] 2 



(j^a*J JJ^-a £ 



(j^aJ 4jl La£ Ajjai^aJI 4_jlila]l JljJI dll jjajII (Jj.l> r-liijlui! (j^aJ 4_iiliLa]l (JljJI Cll! jjxj (Jj-li. (j-a 
, f(x) = y <=> / _1 (y) = X Ai^J«JI ijjSj tSUli j 4_iui5L*JI Ajdllall JIjaII AiajJjj SjJ^jill jjjsll 

. {x G Df tq — 1 < f{x) < 1} cr* i4rcsin[/(x)] i_sjj*j <c>a^ 

. (x G D^ tq — 1 < fix) < 1} csA Arccos[fix)] i_sjj*j 4c>a^ 

. D^ (^a i4rctg[/"(x)] i_jjj*j <cj^v> 

. D^ tsA i4rccot,g[/(x)] <-±ij*i <c.>«^ 



AjjolLuil (_ya\j^. 



QT ^Lajjj Ait ojji Jjii <Ub f (j£jl 

. /Ma) = u^ a Ak. (jlii^U aIAS / <JIa!I cjj\£ li! 



.]a,b[ J&- (jjlili!>U <L13 j [a , 6] ^Jb Sj^l^ 3Jb ^ jiU 
.]a , b[ JMi<^ f ^s /\ AiiullAJUloli /(a) = /(&) o^ lij 

]a , M ^ (jlili!>U Alii j [a , b] ^Je S j*!-- 4Jb / cjj\£ lil 
.celaJl'VW -/(a) =/\(c)-[fc-a] u^ 

(>-i«Jj V g (jli* jj£ ^ ]a , Z?[ ^ (3lili!>U jjjLli j [a , b] t^ OPJ^** 00* f ,g cjj\£ lij 
■ #M = f{ ,T f \ a \ Wx) - gla)] + f(a) -fix) + (3 UU\ J, Jjjj a,j^ J-i-3 



x-o IgJLaxiail Ait j* ku3 Ift-ljlja Lai j A^i.j-aJI (JLaE-VI ^ji A V) a*^ I jj '— » ; jjjLciuij SjJJ^ dlVt rtn'unl UjJ <"<ljj!Viti sift 

.■LjJj-aVl Jlj.l!l£ SjJuLva JJC- "lajjlaj Aijx-a JljJ 

(Jlij^aJ oAtla 4JA^)J (j^-<y *J^^ AjjJaiJI 4Jajail_aJ 

Aa.i jjSj iliia. ?-jjL. JL>- ^ jjUli^J j^IiS u^ f(x) , g(x) ^- h[x) = , , aJIa!I jSjJ 

/(x)-^(a) 

f \ { \ 

sHxl