[PDF] dérivation.pdf La fonction g est dé

View - Download dérivation.pdf

Previous PDF

Next PDF

???x7!px

2x3???x7!(x21)arccos(x2)

???x7!xjxj???x7!xjxj+1 ???f:x7!( xsin(1=x)??x6= 0 x2sin(1=x)??x6= 0

0?????

???x7!arctanxx

2+1???x7!1(x+1)2???x7!sinx(cosx+2)4

f

1(x) = arctanex;f2(x) = arctan(shx)??f3(x) = arctan

thx2 ????f: [0;=2]!R?????? ??? f(x) =psinx+x?

8(x;y)2R2;f(x+y) =f(x) +f(y)?

12hf(a+h)f(ah)

??f??a? ???x7!x2(1 +x)n???x7!(x2+ 1)ex x7!11x;x7!11 +x????x7!11x2? x7!11x2? ????f:R!R?????? ???f(x) = exp3 sinx? ??????? ??? f (n)(x) = 2nexp3 sin x+n6 ??????? ??? ?? ??????? ???????n??xn1e1=x??? ????f:x7!arctanx? ??? ??????? ? ??????? n1 f (n)(x) = (n1)!cosn(f(x))sin(nf(x) +n=2)? k=0 n k 2 ????a;b;c2R? ??????? ????? ??????x2]0;1[??? ???

4ax3+ 3bx2+ 2cx=a+b+c?

????? ??? ???? ???I? ?? ????f:x7!(x21)n(n)? f00(d)? f(a) =f0(a) =:::=f(n1)(a) = 0??f(b) = 0 ????? ?? ??????c2]a;b[??? ???f(n)(c) = 0? lim

1f= lim+1f= +1?

??????? ????? ??????c2R??? ???f0(c) = 0? lim +1f=f(0)? f(0) = 0??f(a)f0(a)<0? ??????? ????? ??????c2]0;a[??? ???f0(c) = 0? f(0) =f(a) = 0??f0(0) = 0? f(a) =f(b) = 0??f0(a)>0;f0(b)>0? ??????? ????? ??????c1;c2;c32]a;b[???? ???c1< c2< c3?? f

0(c1) =f(c2) =f0(c3) = 0?

f(a) =f0(a)??f(b) =f0(b)? ??????? ????? ??????c2]a;b[??? ??? f(c) =f00(c)?

9c2]a;a+ 2h[;f(a+ 2h)2f(a+h) +f(a) =h2f00(c)

lim x!+1(x+ 1)e1x+1xe1x n+1pn+ 1npn lnnn 2?

8x >0;11 +x lim n!1kn X p=n+11p ????f2 C2(R+;R)????? ???limx!+1f(x) =a2R? ?? f00??? ??????? ??? ???? ??f0(x)?????x!+1? ???8x2]1;+1[;x1+xln(1 +x)x ???8x2R+;ex1 +x+x22 ????p2]0;1]? ??????? ??? ? ??????? t0? ?? ? (1 +t)p1 +tp? ?? ??????? ??? ? ??????? x;y0? (x+y)pxp+yp? f(x) =( x2lnx??x6= 0

0??x= 0

??? ?? ??????C1???R+? f n:x7!( xn+1??x0

0?????

??? ?? ??????Cn???R? ????f:R+!R?? ??????C2????? ???f0(0) = 0? ??????? ????? ??????g:R+!R?? ??????C1????? ???

8x2R+;f(x) =g(x2)?

???f(x) =px ?????h!0+? f(h)f(0)h =p1h!1 ?? ?????h!0? f(h)f(0)h ! 1 ?????h!0? f(1 +h)f(1)h =ph2h2h3h ! 1 ?????h!0? f(1 +h)f(1)h = (2 +h)arccos((1 +h)2)!0 ?????h!0+? f(h)f(0)h =h!0 ?? ?????h!0? f(h)f(0)h =h!0 ?????h!0? f(h)f(0)h =1jhj+ 1!1 ?????h!0? f(h)f(0)h = sin1h ?????h!0? g(h)g(0)h =hsin1h !0? ???x7!arctanxx arctanxx 2+ 1 0 =12xarctanx(x2+ 1)2?

1(x+ 1)2

0 =2(x+ 1)3? sinx(cosx+ 2)4 0 =cosx(cosx+ 2)4+4sin2x(cosx+ 2)5=4 + 2cosx3cos2x(cosx+ 2)5? (xx)0= (exlnx)0= (1 + lnx)xx? ((chx)x)0=exlnchx0= (lnchx+xthx)(chx)x? (lnjxj)0=1x f

01(x) =ex1 + e

2x;f02(x) =2ex1 + e

2x??f03(x) =ex1 + e

2x? f

1(x) =12

f2(x) +4 =f3(x) +4 f

0(x) =cosx2

psinx+ 1>0 ?????h!0+? ?? ??????x=f1(h)!0 f

1(h)f1(0)h

=xf(x)? xf(x)=xpsinx+x=xpx+ o(px) +xpx!0 f

0(x+y) =f0(x)?

12hf(a+h)f(ah)!h!012

f0d(a) +f0g(a)? x2(1+x)n(n)=n 0 x

2(1+x)n(n)+n

1 (x2)0(1+x)n(n1)+n 2 (x2)00(1+x)n(n2) (x2(1 +x)n)(n)=n!x2+ 2n:n!x(1 +x) +n(n1)n!2 (1 +x)2? (x2+ 1)ex(n)=nX k=0 n k (x2+ 1)(k)(ex)(nk)=x2+ 2nx+n(n1) + 1ex? 11x 0 =1(1x)2;11x 00 =1(1x)2 0 =2(1x)3 11x (n) =n!(1x)n+1? 11 +x (n) = (1)nn!(1 +x)n+1?

11x2=12

11x+12

11 +x 11x2 (n) =n!2(1x)n+1+(1)nn!2(1 +x)n+1?

11x2=12

11x+12

11 +x?

11x (n) =n!(1x)n+1??11 +x (n) = (1)nn!(1 +x)n+1 11x2 (n) =n!2(1x)n+1+(1)nn!2(1 +x)n+1? cos 3x=14 (3cosx+ cos3x)? (cosx)(n)= cos(x+n=2)??(cos3x)(n)= 3ncos(3x+n=2) (cos

3x)(n)=14

3cos(x+n=2) + 3ncos(3x+n=2)?

cos(t)et= Ree(1+i)t (cos(t)et)(n)=Re(e(1+i)t)(n)= Re(1 + i)ne(1+i)t? ??(1 + i)n= 2n=2ein=4???? (cos(t)et)(n)= 2n=2etcos(t+n=4)? ????n= 0? ?? f (n+1)(x) = 2 nexp3 sin x+n6 0 f (n+1)(x) = 2np3sin x+n6 + cos x+n6 e xp3 f (n+1)(x) = 2n+1sin x+(n+ 1)6 e xp3 f(x) = exp3 sinx= Ime(p3+i)x ????n= 0? ??? xne1=x(n+1)=x:xn1e1=x(n+1)=xxn1e1=x(n+1)+ (n+ 1)xn1e1=x(n) xne1=x(n+1)=x(1)nx(n+1)e1=x0+ (n+ 1)(1)nx(n+1)e1=x xne1=x(n+1)= (1)n+1x(n+2)e1=x? ????n= 1 f

0(x) =11 +x2??

cos(f(x))sin(f(x) +=2) = cos2(arctanx) =11 +x2? f (n+1)(x) =n!1 +x2" sin(f(x))sinnf(x) +n=2 +cos nf(x) +n=2cos(f(x))# cos n1(f(x))?

11 +x2= cos2(f(x))

f (n+1)(x) =n!"sin(f(x))cosnf(x) + (n+ 1)=2 +sin nf(x) + (n+ 1)=2cos(f(x))# cos n+1(f(x)) f (n+1)(x) =n!sin(n+ 1)f(x) + (n+ 1)=2cosn+1(f(x))? f (n)(x) = 0()sin(nf(x) +n=2) = 0 f (n)(x) = 0()f(x) =kn 2 ????k2 f1;:::;n1g? ?? ????? ??? ??????? ??f(n)???? ??? cot kn ????k2 f1;:::;n1g? ????? ????x2n(n)=(2n)!n!xn? x2n(n)=xnxn(n)=nX k=0 n k (xn)(k)(xn)(nk) x2n(n)=nX k=0 n k n!(nk)!n!k!xn=n!nX k=0 n k 2 x n? nX k=0 n k 2 =(2n)!(n!)2=2n n ????': [0;1]!R?????? ??? '(x) =ax4+bx3+cx2(a+b+c)x f????? ?? ??????? ???[a;b]??? ?? ???? ???? ?? ??a? ?? ??b? ?? ??????? ??f [ai1;ai]? f(ai1) = 0 =f(ai)? ???????b1< a1< b2<< an1< bn? ???b1;:::;bn???? ???? ? ???? ??f0+f? ???(X21)n??? ?? ?????2n????(X21)n(n)??? ?? ?????n? ?????x!1?? ? g(x) = (x+ 1)n(x1)n= 2n(x1)n+ o(x1)n? g(x) =g(n)(1)n!(x1)n+ o(x1)n f(1) =g(n)(1) = 2nn! ??????? ??g;g0;:::;g(n1)? g:x7!(xb)f(a) + (ax)f(b) + (ba)f(x)12 (ab)(bx)(xa)K f

0(c1) = 0?

f

00(c2) = 0?

c n2]a;cn1[??? ???f(n)(cn) = 0? ???????lim1f= +1??lim+1f= +1? ?? ??????a <0??b >0???? ??? f(a)> f(0) + 1??f(b)> f(0) + 1? ??b??????? ????? ?? ??????2]a;0[??2]0;b[???? ???f() =f(0) + 1 =f()? f

0(c) = 0?

?????? ?? ??????x02]0;+1[??? ???f(x0)6=f(0)? ??????b2]x0;x1]??? ???f(b) =y? ???????f0(a)<0? ?? ??????b2]0;a[??? ???f(b)> f(a)?

2]0;b[??? ???f() =f(a)?

???f0(c) = 0? ?????x!0? g(x)!f0(0) = 0? g

0(x) =xf0(x)f(x)x

2 ????g0(c) = 0?????cf0(c) =f(c)? y=f0(c)(xc) +f(c) =f0(c)x? ???????f(a) = 0??f0(a)>0? ?? ??????x12]a;b[??? ???f(x1)>0? ?? ????? ?? ???? ????x12]a;b[?f(x1)0????? ?????h!0+?f(a+h)f(a)h 0 ?? ????f0(a)0? ????? ?? ??????c22]a;b[??? ???f(c2) = 0? '(a) = 0 ='(b)?

0(c) = 0?

0(x) =f(x)f00(x)ex

????'0(c) = 0????? f(c) =f00(c)? k? ? ?? ?????? ?????y!x?? ???????f0(x)k? ??? ?????f0??? ??????? f(a+ 2h)2f(a+h) +f(a) ='(a+h)'(a)? ??????b2]a;a+h[??? ??? '(a+h)'(a) =h'0(b) =h(f0(b+h)f0(b))? c2]b;b+h[2]a;a+ 2h[??? ??? f

0(b+h)f0(b) =hf00(c)????f(a+ 2h)2f(a+h) +f(a) =h2f00(c)?

??x+ 1? ?? ??????cx2]x;x+ 1[??? ??? (x+ 1)e1=(x+1)xe1=x=cx1c x e 1c x(x+ 1x) =cx1c x e 1c x? ?????x!+1?cx!+1???cxx? cx1c x e 1c x!1 limx!+1(x+ 1)e1x+1xe1x = 1? n+1pn+ 1npn=1lncc 2c1=c ????c2]n;n+ 1[? ???????cn!+1?lnclnn?? ???????c1=c!1 n+1pn+ 1npn lnnn 2? ?? ??????c2]x;x+ 1[??? ??? ln(1 +x)lnx=1c ??x < c < x+ 1?????1x+ 1<1c <1x kn X p=n+1ln(p+ 1)lnpknX p=n+11p knX p=n+1lnpln(p1)????? ln kn+ 1n+ 1knX p=n+11p lnk? lim n!1kn X p=n+11p = lnk? ? ?????M2R+??? ???f00(x)M???? ????x2R+? ????" >0? ?? ?????(xn)?? ????? ??????? x n=n"M ??????? ????+1?? ???? f(xn+1)f(xn)!0? ??? ????? ?? ??????N2N??? ??? ???? ????nN f(xn+1)f(xn)"2M f0(cn)(xn+1xn)"2M f0(cn)"?

8u2[xn;xn+1];f0(u)f0(cn)Mjucnj "

8u2[xn;xn+1];f0(u))"+f0(cn)2"

??? ??????? ???? ???? ???? ????n2N? ?? ? ?? ??????A=xN?

8uA;f0(u)2"?

f(t) =cos(t2)t+ 1? ???? f:x7!xln(1 +x)?????? ?? ?? ??????C1???]1;+1[? f

0(x) =x1 +x?

x1 0 +1f(x)+1 &0%+1? ????g:x7!ln(1 +x)x=(1 +x)?????? ?? ?? ??????C1???]1;+1[? g

0(x) =x(1 +x)2?

x1 0 +1g(x)+1 &0%+1? ???? f:x7!ex1x12 x2?????? ?? ?? ??????C1???R+? f

000(x) = ex0?

x0 +1f

00(x)0%+1f

0(x)0%+1f(x)0%+1?

?? ?(0) = 0?? ????t >0?

0(t) =ptp1(1 +t)p1?

???????p10?tp1(1+t)p1?? ????0(t)0? ?? ?? ?????? ??? ???? (x+y)p=xp 1 +yx p xp 1 +yx p =xp+yp? f ????x >0?f0(x) = 2xlnx+x? f ????x6= 0?f0n+1(x) = (n+ 2)fn(x)? f

0n+1(0) = 0?

??????g:R+!R?????? ??? g(t) =f(pt)?

8x >0;g0(t) =f0(pt)2

pt g

0(t) =f0(pt)f0(0)2

ptquotesdbs_dbs33.pdfusesText_39

[PDF] fonction dérivable en 0

[PDF] progression spiralée maths première s

[PDF] dérivées usuelles

[PDF] dérivé de

[PDF] exercices dérivées 1ere sti2d

[PDF] derivee 1sti2d

[PDF] on souhaite que le raccordement se fasse sans cassure en i

[PDF] raccordement de deux fonctions

[PDF] le profil d un toboggan est constitué de deux parties

[PDF] raccordement de courbes représentatives de fonctions

[PDF] raccordement routier maths

[PDF] dérivée de 1/u^n

[PDF] polyploidie

[PDF] dérive génétique exemple animaux

[PDF] spéciation sans isolement géographique