[PDF] Fonctions vectorielles dune variables réelles

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8x2R;f(2x) = 2f(x)?

????? ??? ?????? ??b? ???? ???? ????x? ?? ???? ? D n(x) = x1 (0) x

2=2!x1

x

3=3!x2=2!x???

x n=n! x2=2!x S n=nX u(a)v(a)w(a) u(b)v(b)w(b) u

0(a)v0(a)w0(a)

= 0? u(a)v(a)w(a) u(b)v(b)w(b) u

00(c)v00(c)w00(c)

= 0? ??? ????? ??? ??????? 0p < n?? ? n X k=0(1)kn k k p= 0??nX k=0(1)kn k k n= (1)nn! 1h nn X k=0(1)kn k f(kh)? kf0k2

14kfk1kf00k1?

????f: [0;1]!E?? ??????C2????? ??? f(0) =f0(0) =f0(1) = 0?? f(1) = 1? x= cos(3t) y= sin(2t)? x= cos3t y= sin3t? x=tsin(t) y= 1cos(t)? (x=sin(t)1+cos 2(t) y=sin(t)cos(t)1+cos 2(t)?

F(1=p2;0)??F0(1=p2;0)?

MFMF0=12

x(t) = 2cos(t) + cos(2t) y(t) = 2sin(t) + sin(2t)? D t: (t3+ 3t)x2y=t3? x= 3t2 y= 2t3? x= 4t3 y= 3t4? ????D?? ?????? ?? ??????(0;0)?? ?? ?????R >0?? ????R2? f(x) =f(2x=2) = 2f(x=2)? f(x) = 2nf(x=2n)? ????f(x)x =f(x=2n)f(0)x=2n!f0(0) f(x) =f0(0)x? D

0n= det(C01;C2;:::;Cn) + det(C1;C02;:::;Cn) ++ det(C1;C2;:::;C0n)

D

0n= det(C1;C2;:::;C0n)?

D

0n(x) =Dn1(x)?

D f(x) =f(0) +x:f0(0) +x"(x) ????"!0+0? ???????f(0) = 0? ?? ??????? S n=1n 2n X k=1k:f

0(0) +1n

2n X k=1k:"kn 2 "kn 2 max]0;1=n]k"k nX k=1k=n(n+ 1)2

Snn+ 12nf0(0)

n+ 12nmax]0;1=n]k"k? ??"!0+0????max]0;1=n]k"k !0???? S n!12 f0(0)? f(x) = u(a)v(a)w(a) u(b)v(b)w(b) u(x)v(x)w(x) f

0(x) =

u(a)v(a)w(a) u(b)v(b)w(b) u

0(x)v0(x)w0(x)

f

00(c) =

u(a)v(a)w(a) u(b)v(b)w(b) u

00(c)v00(c)w00(c)

= 0? ????n= 1?Pn k=0(1)kn k= (11)n= 0??Pn k=0(1)kkn k=1? ????0p < n+ 1? ??p= 0?????Pn+1 k=0(1)kn+1 kkp= (11)n+1= 0? ??0< p < n+ 1?????Pn+1 k=0(1)kn+1 kkp= (n+ 1)Pn+1 k=1(1)kn k1kp1= (n+ 1)Pn k=0(1)k1n ??p=n+ 1?????Pn+1 k=0(1)kn+1 kkn+1= (n+ 1)Pn k=0(1)k1n k(k+ 1)n= (1)n+1(n+ 1)!? ? ??? ?????? ????? f(x) =Pn p=0f(p)(0)p!xp+ o(xn)????P n k=0(1)kn kf(kh) =Pn p=0f(p)(0)p!P n k=0(1)kn kkp+ o(hn)! (1)nf(n)(0)? f(x+h)f(x)hf0(x)h22 kf00k1 hf0(x)f(x+h)+f(x)+h22 kf00k1 ????f0(x)2kfk1h +hkf00k12 h= 2pkfk1=kf00k1>0? ?? ??????? f0(x)2qkfk1kf00k1 f12 f(0)12 f0(0) 18 kf00k1?? f12 f(1) +12 f0(1) 18 kf00k1? f12 18 kf00k1?? f12 f(1) 18 kf00k1 1 = f(1) f(1) f12 f12 14 kf00k1 ????kf00k14?

M(t+ 2) =M(t)?

t0=4=3=2x

0(t)0 0 + 3x(t)1& 1=p2& 1%0y(t)0%1&p3=2&0y

0(t)+ 0 2m(t)101 2=3?

(0;p3=2)? y

0(t) = 3costsin2t?

t0=4x

0(t)0 3=p8

x(t)1&23=2y(t)0%23=2y

0(t)0 + 3=p8

m(t)? 1? y(t)y(0)x(t)x(0)t!0t

33=2t2!0?

?? ?A(t)0 sint??B(t)cos(t)

0?? ????

A(t)B(t) = 1?

x0(t) = 1cos(t) y

0(t) = sin(t)?

t0x

0(t)0 + 2

x(t)0%y(t)0%2y

0(t)0 + 0?

y(t)y(0)x(t)x(0)t!0t 2=2t

3=6!+1?

?? ????t0= arccos(1=p3)? ?? ?cost0= 1=p3??sint0=p2=3? MF

2=sin2(1 + cos

2t)+12

p2 sint1 + cos

2t??MF02=sin2(1 + cos

2t)+12

+p2 sint1 + cos 2t MF

2MF02=14

x0(t) =2sint2sin2t y

0(t) = 2cost+ 2cos2t?

x0(t) = 0()t= 0;2=3; y

0(t) = 0()t==3;?

t0=3 2=3x

0(t)0 0 + 0x(t)3&1=2& 3=2% 1y(t)0%3p3=2&p3=2&0y

0(t)+ 0 0m(t)1 0 +1 ??

y(t)y()x(t)x()t!t!0? (t3+ 3t)x(t)2y(t) =t3(1)? ?? ???????f0(t)?????? ?? ??????Dt???? (t3+ 3t)x0(t)2y0(t) = 0 (2)? (3t2+ 3)x(t) = 3t2 x(t) =t21 +t2??y(t) =t31 +t2? t0 +1x

0(t)0 +

x(t)0%1y(t)0%+1y

0(t)0 +?

y(t)y(0)x(t)x(0)!t!00? ????? ?????t!+1 x(t)!1??y(t)!+1 ?????(Ox)x0(t) = 6t y

0(t) = 6t2?x0(t) = 0()t= 0

y

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

t0 +1x

0(t)0 +

x(t)0%+1y(t)0%+1y

0(t)0 +?

y(t)y(0)x(t)x(0)!t!00? t2(x3t2) +t(y2t3) = 0 txy=t3? t(x3t2)t2(y2t3) = 0 txt2y= 3t32t5? t+t22= 0 ??t6= 0?????6= 0??=1=t????3t +2t

3=t3????(t2+ 1)2(t22) = 0

?? ??? ?????t=p2;p2? y

0(t) = 12t3?

x0(t) = 0()t= 0 y

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

t0 +1x

0(t)0 +

x(t)0%+1y(t)0%+1y

0(t)0 +?

y(t)y(0)x(t)x(0)!t!00? txy=t4? 1t xy=1t 4? x=t3t+1t 1t 3 y=t2+ 11t 2? t2R x

0(t) = 3t211t

2+3t

4=(1+t2)(3t44t2+3)t

4 y

0(t) =2t+2t

3=2(1t4)t

3? t0 1 x

0(t)4x(t)+1 &0y(t)1 % 1y

0(t)+ 0?

????a?? ????? ?? ?????? ?????? ?? ??????(0;0)?? ?? ?????R >0? (0) =a? ??? ? ??????? ????D? ?? ?

8t2]";"[;

(t) 2R2? (0) (t)

2????? ?? ??????? ??t= 0?

2h 0(0); (0)i= 0? ?????? ?? ??????? ???????v= (0)?quotesdbs_dbs1.pdfusesText_1

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