[PDF] CLASSE de MATHÉMATIQUES PHYSIQUE et SCIENCE de l

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1 ???????(fg)0=f0g+fg0 fg 0 =f0gfg0g 2 0(x)x x 11 x 2 2x 3x

44x3px

=12 1 2 px sin(x)cos(x)cos(x)sin(x)tan(x)1 + tan

2(x) =1cos

2(x)exp(x)exp(x)ln(x)1

x f(x)f

0(x)u(x)u

0(x)u(x)1sin(u(x))u

0(x)exp(u(x))ln(u(x))u

sin(x)cos(x)0= cos(x)2sin(x)2 ln(1 +x2)0=2x1 +x2exp(sin(x))0= cos(x)exp(sin(x))pexp(x)0=exp(x)2 pexp(x)=12 0(x)x 2+1x x px sin(x)x exp x22 ln 1 +1x x exp(x)cos(x)3??????? sin(x)x 2 0 =sin0(x)x2sin(x)(x2)0(x2)2(1) x2cos(x)2xsin(x)x 4(2) xxcos(x)2sin(x)x 4(3) xcos(x)2sin(x)x 3(4) ab c 2ab c 2abc 2bac 2ac2b c 2b a bc 2a bc 2a abc 2 ????a;b2R a=b)a2=ab(1) )a2b2=abb2(2) )(a+b)(ab) =b(ab) (3) )a+b=b(4) )a= 0 (5) ??11 +ex+nx= 1 +ex+nxq q

2 + sin(x)2

= 1 + sin(x)q q ??ln(px) + ln(px) = ln(x)q q ?? x2h 2 ;0i ?????sin(x) =p1cos(x)2q q e3xe x=lne3xln(ex)=3xx=3q q ??exp(2ln(x)) =1x

2q q???????

? ???x2]0;1[?f(x) =ln(1 +x)ln(x) ? ???x2Rnf1g?g(x) =x2+x1x1 ? ???x2R+?h(x) =xln 1 +1x

8x2I; f(x)g(x)

h:x7!g(x)f(x) ??8x2R;1xex ??8x2 12 ;+1 ; xx2ln(1 +x)x??????? ?? ????? ????n2N?n! = 12 (n1)n ????? ?? ??????? ??? ??????? ?? ? ?n? n p=n!p!(np)!=n(n1)(n2)(np+ 1)p! n p=n np n 0= 1n 1=nn

2=n(n1)2

12 10=12

2=12112

= 66 ??5

3=:::7

2=::: 10

8=:::4

3=::: n n1=:::n+ 1 n1=::: ??(n+ 1)! =::: n! ??(n+ 1)! =:::(n+ 2)! (2n+ 2)!(2n)!=::: (n+ 1)!(n+ 2)!(n!)2=::: ??(3!)2= (32)!q q ??n! =n(n1)!q q (2n)!n!=n!q q ??(np)! =(n+ 1)(p+ 1))!q q??????? ?? ????? ????? ???R? exp

0= exp

???? ???? ????x? ?? ? ?ex>0? ????0??1?? ????+1??+1? ???? ???? ?????a??b?? ? ? e a+b=eaeb; eab=eae b; eb=1e b e0= 14 2 0244
2 024

ClnCexp??????? ?? ?????

???? ????x2R?lnex=x ?? ???? ????x >0?eln(x)=x?

8x >0;ln0(x) =1x

lim

0+ln =1lim+1ln = +1

???? ???? ?????a??b??R+? ?? ? ? ln(ab) = ln(a) + ln(b) ln ab = ln(a)ln(b);ln1b =ln(b) ln(e) = 1 ??ln8 = ln2 + ln2 + ln2q q ??e1a+b=e1a e1b q q ??exp ln1t 2 =t2q q qlnalnb=lnalnb qln(a+b) = lna+ lnb q lnalnb= lnab ???????limq] 1;1]]1;1[1]1;+1[lim A\B=; ???P( ) = 1 ?????P(A[B) =P(A) +P(B)

P(A[B) =P(A) +P(B)P(A\B)

P(A\B) =P(A)P(B)

P

A(B) =P(A\B)P(A)

P(A) =Card(A)Card(

?? ??q q ?? ??q q ?? ?????q q ????? ??????q q X( ) =fx1;x2;:::;xng ?? ???? ????i2J1;nK?P(X=xi) =pi? ?? ???? ??? ??

8 i2J1;nK; pi2[0;1]

nX i=1p i=p1+p2++pn= 1

E(X) =nX

i=1x ipi=x1p1+x2p2++xnpn

V(X) =nX

i=1 x2ipiE(X)2

P(X2)?

?????? Y( ????? ??? ??????? ???? ?? ???? ?????? ??Ri=V i? ??? ????? ???? ???? ?????? ???????nn0? ?? ?????? ???P(n+ 1)??? ?????? ????nn0? N

P(n) :nX

k=1k= 1 + 2 ++n1 +n=n(n+ 1)2 k=1k= 1??

1(1 + 1)2

n+1X k=1k= 1 + 2 ++n|{z} +n+ 1 n(n+ 1)2 +n+ 1 n+ 12 =n+ 12 (n+ 2) ????P(n+ 1)??? ?????? k=1k=n(n+ 1)2 ? ??????? n2N?

P(n) :nX

k=1k

2= 1+22++(n1)2+n2=n(n+ 1)(2n+ 1)6

? ??????? nn0????n02N??q2Cnf1g?

P(n) :nX

k=n0q k=qn0+qn0+1++qn=qn01qnn0+11q u0=1u1= 3

8n2N; un+2= 4u+13un

8n2N;P(n) :un=3 + 23n

n X k=1knX k=1k 2nX k=n0q k a2b2=::: a2+ 2ab+b2=::: a22ab+b2=::: (ab)2=::: (ab)(a+b) =::: (a+b)2=::: ??(x+ 1)(x+ 1)4(x+ 2)2=::: ??(x2)(x+ 2) + 2x+ 5 =::: ??2x2bax+ab=::: ??(x+y)24xy=::: ??(p21)(p2 + 1) =::: ??(x+ 1)(x+ 2)(x+ 3) =::: ??(a+b)2+ (ab)2=::: ???? ????x >0??2R? ?? ????x=elnx? ?? ??????? ??x7!x???x7!x1? ???? ????(x;y)2]0;+1[2??(;)2R2? ?? ? ? x =1x ; xy= (xy);xy =xy x x=x+;(x)=x;ln(x) =lnx: ??p2N? ?? ???? ??????x7!xp???R???? ??????? x7!x2?? ??p2Z? ?? ???? ??????x7!xp???R???? ??????? x7!x3?? ??q2N? ?? ???? q px=x1q ??q??? ?????x7!qpx??? ?????? ???R+???? ??????? ? ?? ?????? ??????x7!px?? ??q??? ???????x7!qpx??? ?????? ???R???? ??????? ? ?? ?????? ???????x7!3px?? ???? ????x >0?p2Z??q2N? ?? ? ? x pq =qpx p=qpx p

012345012345

xx xp x xx2 xx3 f:x7!(x+ 1)x ?? ?? ???????g:x7!ln(x+ 1) +xx+ 1? ??????? ??? ??p6p2 p2 2n+14 n12

3n2n16

n+1=:::

4n3n+12

2n13n1=:::

1 +p2p2

= 1 +p2 2 q q 1p3p2 =p2 + p3q q p23p2 =13p2p2 q q??????? ?? ????? lim x!0ln(1 +x)x = 1 limx!0e x1x = 1 limx!0sin(x)x = 1

86= 0;limx!0(1 +x)1x

??limx!0p1 +x1x ??limx!+1xe1x 1=::: ??limx!1ln(x)x1=::: ??limx!1x2sin1x 2 ??limx!01x

11 +x1

??limx!0xsin(2x)=::: ??limx!01p1 +xx p1 +x=::: q?????? ?????? lim x!0(1 +x)1x ??? ??????? ? ??????? x2R? 1x22 cos(x)1x22 +x424 ???? ??????? ??f:x7!cos(x)1 +x22 ???? ??????? ??g:x7!cos(x)1 +x22 x424 lim x!0cos(x)1x ??????? ? ?????? ??limx!0+ln(1 +px)sin(x) ln(1 +px)px !x!0+1 ln(1 + px)sin(x)=1px xsin(x)|{z} !1ln(1 + px)px |{z} !1!x!0++1 ??limx!0 epx 1 3x ??limx!1x2ln(2 +x2)2ln(x)=::: ??limx!1ln(x)x

21=:::

??limx!0sin(2x2)ln(1 +x)2=::: ??limx!+1x e1x 21
n+1X k=1k=::: nX k=3k 2=::: 12X k=12 k=::: lim x!+1e xx = +1limx!1xex= 0 limx!+1ln(x)x = 0 ??limx!0+xln(x) = 0 ? ??? >0?limx!+1e xx = +1 ? ??? >0?limx!+1xex= 0 ? ??? >0?limx!+1ln(x)x = 0 ? ??? >0?limx!+1xln(x) = 0 ????? ???? ??????? ?limx!+1e2xx4= +1 ???? ??????? ?limx!1e

3xln(x)5= 0???????

x ln(x)e x x? lim x!+1e2x= +1 limx!1e

3xln(x)5=

lim x!1e3x= 0 limx!01px + ln(x) = lim x!01px = +1 ??limx!+1exln(x)2= limx!+1ex= +1q q ??limx!+1xln(x) +ex= limx!+1ex=1q q ??limx!0+1pxln(x3)= limx!0+1ln(x3)= 0q q ??limx!0+1pxln(x3)= limx!0+1px = +1q q ??limx!1(x1)ln(x1) = limx!1(x1) = 0q q ??limx!+1ln(x)3x

2= limx!+1:::::::::=:::

??limx!0+1x + ln(x) = limx!0+:::::::::=::: ??limx!1x5e3x= limx!1:::::::::=::: ??limx!1x 5e

3x= limx!1:::::::::=:::

??limx!+1e3x3x4+ ln(x2) = limx!+1:::::::::=:::??????? ??limx!+1ln(x2+ 1)x epx ????? ??? ?? ????e x? xe px = exp pt+ ln(x) ln(x2) = 2ln(x)??? ???? ?? ????ln(x)? ????ln(x2+

1)?? ????? ??? ?

ln(x2+ 1)x =lnx21 +1x 2x =ln(x2) + ln1 +1x 2x =ln(x2)x +ln1 +1x 2x |{z} ln(x2)x ?????F??Zb a f(t)dt=F(b)F(a) x7!Z x x

0f(t)dt

??????? ?Z 1 0 t3dt=14 t4 1 0 =14 1414
04=14 ??Z 2

0sin(t)dt=:::

Z 4

1ptdt=:::

Z ln(2) ln(2)etdt=::: Z 10 51t
dt=:::?? Z 1 0 t11dt=::: f0gg0 x7!5xcos(x2) cos(x) ?? ??????? ??g?g0(x) = 2x ?? ???? ?? ?????(f0g)g0?

5xcos(x2) =52

2x|{z}

g

0(x)cos(x2)|{z}

f

0(g(x))

sin(x2) ??a:x7!exp(2x1) ??b:x7!1(3x+ 2)3 ??c:x7!sin(x)cos(x)4 ??d:x7!2xexp(x2) ;P(

P(A1\ \An1)6= 0

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