LICENCE 1 2005-2006 SFE1 Mathématiques

cos(2x) cos(x) cos(x) sin(x) cos(3x) sin(x)3
√2√2√2
f(x) = 5x7+ 3x3−4x2+ 9x+ 5 g(x) = xex
h(x) = 6x+ 5
x2+ 1 i(x) = cos √x
j(x) = sin(x) log(x)k(x) = qp1 + x2
l(x) = 6 sin x+ 5
(sin x)2+ 1 m(x) = 1/√x√x
f(x) = xx+x−1x > 0f(0) = 0 fR+
E
1
f(x) = E(x)2
g(x) = (x−1)E(x)
h(x) = (x−1)2E(x)
f(x) = (x−E(x)) sin πx
2x∈R
f
n f n
f
f:R→R0f(0) = 0 f0(0) = 2
f(sin x)
xx0
nlimx→1f(xn−1)−f(x−1)
x−1
limx→0f(2x)2−f(3x2)
x2
x∈R∗
+\ {1}log x < x −1
u(x) = xev(x) = exx > 0
u v R∗
+log y=x/e
u v
Py=x2
y=ax +bP4b+a2= 0
D1D2Pa16=a2
a1a2D1D2
Py=−1/4

f:R→Rf(x) = 2x3+ 4x2−10x+ 9
f(2) f(−3) f0
6x2+ 8x−10 = 0 ] −2,3[
g:x7→ e√xsin(x)R+
x g(x) = 0 g0
sin(x) = −2√xcos(x) [0,4]
P P (x) = xn+ax +b n ∈N\ {0,1}a b ∈R
P0(x) = 0 n a
f:R→Rk k ≥2
f0k−1
P n n
a f : [a, +∞[→R[a, +∞[ ]a, +∞[
limx→∞ f(x) = f(a)c∈]a, +∞[f0(c) = 0
g: [0, e−a]→Rg(0) = f(a)g(x) = f(−log x)x∈]0, e−a]
g[0, e−a] ]0, e−a[
g0g(0) g(e−a)
c∈]a, +∞[f0(c) = 0
f: ]0,1[→R
limx→0f(x) = limx→1f(x) = +∞
a < b f : [a, b]→RGff
x∈RPx(x, 0)
a= 0 b= 1 f(x) = x(x−1) f(a) = f(b) = 0 Gf
GfP−1
2P1 2P3
f f(a) = f(b) = 0 c /∈[a, b]
GfPc
f[a, b]x0∈[a, b]
Gf(x0, f(x0)) Pcf0(x0)(x0−c)−f(x0) = 0
f(a) = f(b) = 0
f0(x)(x−c)−f(x)a b
c /∈[a, b]c=a
f: [0,2π]→Rf(x) = ecos(2x)sin(x)
c∈RGfPc
Hn=Pn
k=1
1
kRn=Pn
k=1
1
2√kn∈N∗
x > 01
x+ 1 <log(x+ 1) −log x < 1 x
log n≤Hn≤log n+ 1
x≥1√x+ 1 −√x < 1
2√x<√x−√x−1
Rn2
a b

c a3−b3= 3c2(a−b)
a2+ab +b2a=b= 0
f:I→RK∈R|f(x)| ≤ K x ∈I
f:R→Rf0
K≥0|f(x)−f(0)| ≤ K|x|x∈R
K≥ |f(0)|
x7→ f(x)
1 + |x|