Colle du 16 septembre : Rappels sur les fonctions de la variable réelle

f:R→R
x∈Rfk(x) = x f
f, g : [0,1] →[0,1]
f◦g=g◦f x ∈[0,1] f(x) = g(x)
]1,+∞[×]1,+∞[→R,(x, y)7→ ln(x+y)−pln xln y.
a∈Rf:R→Rf(2x)−f(x)
x→x→0a f
f: [a, b]→RC3
c∈]a, b[
f(b)−f(a) = b−a
2(f0(b) + f0(a)) −(b−a)3 12 f(3)(c).
f: [−1,1] →RC3f(−1) = f(0) =
f(1) = 0
b∈[−1,1] c∈[−1,1] f(b) = b(b+1)(b−1)f000 (c)
6
C > 0R1
−1|f(t)|dt ≤Ckf(3)k∞
f:R→Rf(n)(0) = 0 n∈N
r > 0|f(n)(x)| ≤ n!rnn∈Nx∈Rf
a∈RR1
0(f00(t))2dt f
C2[0,1] Rf(0) = f(1) = 0 f0(0) = a
g C5
|g(x)−x
3(g0(x)+2g0(0))| ≤ |x|5 180 sup
t∈[0,x]|g(5)(t)|.
f C4[a, b]
|Zb
a
f(t)dt −b−a
6(f(a) + f(b)+4f(a+b 2))| ≤ (b−a)5 2880 sup
t∈[a,b]|f(4)(t)|.
C4

limx→0+sin xsin x−1
xx−1
f(t) = R1
0
dx
(1+x+x2)tt→+∞
A B AB = 2 R > 1f(R)
{M|AM ≤R, BM ≤R}f(R)R→1+
a > 0u0>0 (un)un+1 =un
1+ua
n
un
limn→∞ 1
n√nPn
k=1 E(√k)
limn→∞ Pn
k=1 arcsin(k/n2)
n > 0ex=n−x
xnxn
xn∈R+xn
n+xn−1 n+... +xn= 1
xn
f(R)
0R f(R)R→+∞
0< a < b Rbx
ax
u−u
u4x→0+
f: [0,1] →Rf(0) 6= 0
+∞g(t) = R1
0
f(x)
1+tx dx
g f C1
f: [a, b]→R+
∗
a=x0< x1< ... < xn=b
k n −1Rxk+1
xkf(t)dt =1
nRb
af(t)dt
1
nPn−1
k=0 f(xk)