Liste d`exercices 5 Di érentiabilité Exercice 7 - IMJ-PRG

f:R2→Rf(x, y) = x3+y2
fR2(x0, y0)
R2
f(x0, y0)R2
g, h :R2→R
g(x, y) = exsin y h(x, y) = (x2+y2)e−xy
f:R2→R3f(x, y) = (x2y, xy, ey)
fR2
f(x0, y0)R2
f(x0, y0)R2
f:R2→R
f(x, y) = 
(x2+y2) sin 1
√x2+y2(x, y)6= (0,0)
0 (x, y) = (0,0)
R2
fR2
f(0,0)
f:R2→R
f(x, y) = (x3−y3
x2+y2(x, y)6= (0,0)
0 (x, y) = (0,0)
f(0,0)
f(0,0)
g:R2→R
g(x, y) = (xy sin π
2¡x+y x−y¢x6=y
0x=y
g(0,0)
C R | |
z7→ z2z7→ z
E F
f∈ L(E, F )f a E fa

f:R3→R2f(x, y, z) = (x+y2, xy2z)
(x, y, z)R3
g:R2→R3g(u, v) = (u2+v, uv, ev)
f◦g(u, v)∈R2
f◦g
URE
f:U−→ E
f f a ∈U
f a U fa
ϕ:U−→ L(R, E)
a7−→ fa
E F G
f∈ L(E, F ;G)
f(a+h, b +k)f(a, b)f(a, k)f(h, b)f(h, k)
f(a, b)E×F f(a,b)
E
g E Rg(x) =kxk2
E
(u◦v)u v
Ek k
fR2R2
f(x, y) = ¡sin(x+ 2y),cos(2x+y)¢.
fR2
aR2fa
fR2
Rnk.k2aRn
a λ ∈R∗
f:Rn\ {a} → Rn
x7→ f(x) = a+λx−a
kx−ak2.
f(Rn\{a})⊂Rn\{a}fRn\{a} → Rn\{a}
f x ∈Rn\ {a}
x∈Rn\ {a}kx−ak2
λdfx
dfx