cours expo ( PDF

n n0=Kn
y y0=y
y0=y y(0) = 1
Rf(0) = 1
f0(0) = 1 A0Cf(0 ; 1) T0Cf
y=x+ 1
T0A0
Cf
f(α)f0(α) = f(α)
y=f0(α)(x−α) + f(α)⇔y=f(α)(x−α+ 1)
h f(α+h)≈f(α)(h+ 1)
α= 0 f(0) = 1 f(h) = h+1 f(2h) = (h+1)2f(3h) = (h+1)3
h= 0,1f(0) = 1 f(0,1) = 1,1f(0,2) = 1,21 f(0,3) = 1,331
f(0,4) = 1,4641
∀fRy0=y y(0) = 1 f
fRf0=y f(0) = 1
ϕRϕ(x) = f(x)×f(−x)
ϕ0(x) = f0(x)f(x)−f(x)f0(x)x f0(x) = f(x)ϕ0(x) = f(x)f(x)−
f(x)f(x) = 0 ϕ
ϕ(0) = f(0) ×f(−0) = 1 ∀x∈Rϕ(x) = 1
α f(α) = 0 f(α)×f(−α) = 1
0×f(−α) = 1
∀x∈Rf(−x) = 1
f(x)
y0=y y(0) = 1
g g0=g g(0) = 1
x g(x)6= 0 ψ ψ(x) = f(x)
g(x)
ψ0(x) = f0(x)g(x)−f(x)g0(x)
g2(x)=f(x)g(x)−f(x)g(x) g2(x)= 0
ψRψ(0) = f(0)
g(0) = 1 ∀x∈Rψ(x)=1⇔f(x) = g(x)
exp y0=y
y(0) = 1
exp(0) = 1

∀x∈Rexp0(x) = exp(x)
∀x∈Rexp(x)6= 0
∀x∈Rexp(−x) = 1
exp(x)
exp(a+b) = exp(a)×exp(b)
ϕaRϕa(x) = exp(x+a)
exp(a)
ϕ0
a(x) = exp(x+a)
exp(a)ϕa(0) = exp(0 + a) exp(a)= 1 exp ϕa= exp
exp(x+a)
exp(a)= exp(x)⇐⇒ exp(x+a) = exp(x)×exp(a)
exp(a−b) = exp(a)
exp(b)
exp(a−b) = exp (a+ (−b)) = exp(a)×exp(−b) = exp(a)×1
exp(b)=exp(a) exp(b)
n∈Zexp(na) = [exp(a)]n
exp(2a) = exp(a+a) = exp(a)×exp(a) = [exp(a)]2
exp(3a) = exp(2a+a) = exp(2a)×exp(a) = [exp(a)]2×exp(a) = [exp(a)]3
∀x∈Rexp(x)>0
exp(x) = exp 2×x
2=exp x 22>0 exp x 26= 0
e
e e = exp(1)
e≈2,718
n∈Zexp(n) = exp(n×1) = [exp(1)]n=en
x ex= exp(x)
e0= 1
∀x∈Rex>0
∀a∈R∀b∈R
ea+b=ea×eb
e−a=1
ea
ea−b=ea
eb
ena = (ea)n
y0=ay R
f:x7−→ Keax K
=⇒fRf(x) = Keax f0(x) = aKeax =af(x)
⇐=g ϕ
Rϕ(x) = g(x)
eax ϕ0(x) = g0(x)eax −g(x)aeax
(eax)2=ag(x)eax −g(x)aeax
e2ax = 0
ϕ∃K∈R∀x∈Rϕ(x) = Kg(x)
eax =K⇔
g(x) = Keax
y0=ay +bR
f:x7−→ Keax −b
aK

=⇒fRf(x) = Keax −b
a
f0(x) = aKeax af(x) + b=aKeax −b
a+b=aKeax −b+b=aKeax f0(x) = af(x) + b
⇐=g ϕ
Rϕ(x) = g(x)+ b
aϕ0(x) = g0(x) = ag(x)+b=ag(x) + b a=aϕ(x)ϕ
y0=ay ϕ :x7−→ Keax
Keax =g(x) + b
a⇐⇒ g(x) = Keax −b a
R
∀x∈Rex>0
R
+∞
lim
x→+∞ex= +∞
ψ:x7→ ex−x ψ0(x) = ex−1 = ex−e0
x7→ exRx > 0ex> e0⇐⇒ ψ0(x)>0
x < 0ex< e0⇐⇒ ψ0(x)<0
x−∞ +∞
ψ
ψ0(x)−+
ψ∀x∈Rψ(x)>0⇐⇒ ex>x x →+∞
ex>xlim
x→+∞ex= +∞
−∞
lim
x→−∞ex= 0
lim
x→−∞ex= lim x→−∞ 1
e−x
X=−xlim
x→−∞ex= lim
X→+∞
1
eX= 0
x−∞ +∞
exp
exp0(x)
+∞
+