Planche 2 - Institut de Mathématiques de Marseille

R
a b c a −b a −c b −c
a
(a−b)(a−c)+b (b−c)(b−a)+c (c−a)(c−b)
(b−c)2
(a−b)(a−c)+(c−a)2 (b−c)(b−a)+(a−b)2 (c−a)(c−b).
a, b ∈R, a < b, I =]a, b[
c∈Qa<c<b
d∈R\Qa<d<b
I∩QI∩(R\Q)
2xy ≤x2+y2xy +yz +xz ≤x2+y2+z2x, y, z ∈R
2n−1≤n!≤nnn∈N
(1 + x)n≥1 + nx x ∈R, x ≥ −1, n ∈N
limn→∞ an=∞a∈R, a > 1
a∈R, a > 1k∈Nxn=an
nk, n ∈Nb > 1xn+1 ≥bxn
nlim
n→∞
an
nk=∞.
x∈R, x 6= 1
n
X
k=0
xk=1−xn+1
1−xn∈N
∞
X
n=0
xn=1
1−xx∈R,|x|<1
a b 1
x(x+ 1) =a x+b x+ 1
n Sn=1
1.2+1 2.3+1 3.4+. . . +1 n.(n+ 1)
a b a 6b x
x∈[−1,5], x ∈]2,5[, x ∈[−5,−3], x ∈]a, b[.
Rx|x−1|64|x+|x|| >2

|x+y| ≥ 
|x|−|y| x, y ∈R
x
y
=|x|
|y|x, y ∈R∗, y 6= 0
f:R→R, x 7→ x
1 + |x|
u:R→R, u(x)=[x2]
v: [0,1] →R, v(x) = √1−x2
w:R+→R, w(x) = sin(√x)
x∈]−1,+∞[x
x+ 1 6ln(1 + x)6x
I={x∈R;−2< x +1
2x≤2}
x x +1
2x≤2
x−2< x +1
2x
I
I
x
x=
∞
X
n=−n0
an10−n, an∈ {0,1,...,9}
N, T ∈Nan+T=ann≥N
ln(x2−1) = ln(2 −x) + ln(3 −x)
exp(2x)−3 exp(x) + 2 = 0
α > 0 lim
x7→+∞
xα
exp(x)lim
x7→+∞
ln(x)
xαlim
x7→0+xαln(x) lim x7→0+xx
α= 1
f1(x) = x−1
x+ 1, f2(x) = 1 1−x−2 1−x2, f3(x) = px2−2x−5,
f4(x) = r2+3x
5−2x, f5(x) = exp(sin(x) + cos(x)), f6(x) = x2cos 1 x,
f7(x) = 1
xln ex+e−x 2, f8(x) = sin(x) sin 1 x.

f f(x) = ln(x)
x
f
f
f f
f
g(x) = xexp−x2
arcsin
sin : −π
2,π 2→[−1,1]
arcsin
arcsin ] −1,1[
arcsin
arctan
tan : −π
2,π 2→R
arctan
arctan R
arctan
cos(x) = 1
2sin(x) = −√3 2tan(x)=1
cos(3x+ 1) = √3
2cos x+π 3= cos π 6−2xsin(2x) = cos π 3−x