Analyse TD 3 1 Fonctions dérivables - IMJ-PRG

3
1.
f(x) = x2cos( 1
x)x6= 0 f(0) = 0
f(x) = sin xsin( 1
x)x6= 0 f(0) = 0
f(x) = |x|√x2−2x+ 1
x−1x6= 1 f(1) = 1
f(x) = x
1 + |x|
2.
f(x) = p1 + x2sin2x
f(x) = log 1 + sin x
1−sin x!
f(x) = (x(x−3))
1
3
f(x) = ln cos(π+x2−1
x2+ 1)!
3.
f(x) = ln x
x
f(x) = sh x−x−x3
6
f(x) = cos x−1 + x2
f(x) = x4−x3+ 1
f(x) = 1 + xn
(1 + x)nn≥2x > 0
1. n 2a b Xn+aX +b
∗2. f R
f(x) = (e−1
1−x2|x|<1
0
C∞
φ:x7→ e−
1
x0+

3. f g n
fg
(fg)(n)=
n
X
k=0 n
kf(k)g(n−k).
n xn(1 −x)n
n
X
k=0 n
k2
1. P n n
P0n−1
2. f : [a, +∞[→R]a, +∞[ lim
x→∞
f(x) = f(a)∃x0∈
]a+∞[, f0(x0) = 0.
3. f [a, a + 2h]α]0,2[
f(a+ 2h)−2f(a+h) + f(a) = h2f00 (a+αh).
g(t) = f(a+t+h)−f(a+t)
4. n ∈N
Pn=1
2nn!
dn
dxn(x2−1)n.
Pn
Pn(1) Pn(−1)
Pnn]−1,1[