Fonctions usuelles

f:R→R R f(0) = 1
f0=f
fR∗
+
exp : R→R∗
+
x7→ exp(x).
Bexp(x)
exp(x) =
+∞
X
k=0
xk
k!= 1 + x+x2 2! +x3 3! +···
d
dxxk k!=xk−1
(k−1)!
∀x∈R∀x0∈Rexp(x+x0) = exp(x) exp(x0).
∀x∈Rexp(−x) = 1
exp(x)
∀x∈R∀x0∈Rexp(x−x0) = exp(x)
exp(x0)
∀x∈R∀p∈Zexp(px) = exp(x)p
e:= exp(1) 2,7 exp(x)
ex∀x, x0∈Rex+x0=exex0

0
∀x∈Rex≥x+ 1.
∀n∈N∀x∈R+exp(x)≥xn
n!+xn−1 (n−1)! +. . . +x+ 1.
exp R R∗
+
ln
ln : R∗
+→R
x7→ ln(x).
R∗
+R
e0= 1 ln(1) = 0 ln(2) ≈0,69 ln(10) ≈2,3

ln R∗
+
∀y∈R∗
+ln0(y) = 1
y.
∀y∈R∗
+∀y0∈R∗ +ln(yy0) = ln(y) + ln(y0).
∀y∈R∗
+ln(1
y) = −ln(y)
∀p∈Z∀y∈R∗
+ln(yp) = pln(y)
ln(1010) = 10 ln(10) ≈23

a∈R∗
+aloga
loga:(R∗
+→R
x7→ loga(x) := ln(x)
ln(a)
.
log10 log
64
X
k=1
2k−1=
63
X
k=0
2k=264 −1
2−1= 264 −1
N
10N≤264 <10N+1
10
N≤log10(264)< N + 1.
log10(264) = 64 ln(2)
ln(10) ≈19.1 264 >1019 = 10 ×109×109
:
R→R
x7→ ex+e−x
2
:
R→R
x7→ ex−e−x
2
.
∀x∈R2(x)−2(x)=1.
{(x, y)∈R2:x2−y2= 1}
R
∀x∈R0(x) = (x)0(x) = (x)

n x xnx n
xn:= x×x×. . . ×x. x n
n1n2N∗
xn1·xn2=xn1+n2.
x0= 1 xn+1 =xn·x n = 0
n x
x−n:= 1
xn
.
xaa
p x ∈R∗
+
xp=eln(x)p=
(∗)exp (pln(x)) ,(♦)
(∗)
a∈Ra
·a:R∗
+→R
x7→ xa:= exp(aln(x)) .
π3π×π×π π√2
exp √2 ln(π).
p∈N∗x∈R∗
+(♦)xp