2.1 FONCTIONS À PLUSIEURS VARIABLE

(1)

cours 9

2.1 FONCTIONS À

PLUSIEURS VARIABLE

(2)

Aujourd’hui, nous allons voir

(3)

Aujourd’hui, nous allons voir

Fonctions à plusieurs variables

(4)

Aujourd’hui, nous allons voir

Fonctions à plusieurs variables Les courbes de niveau

(5)

Aujourd’hui, nous allons voir

Limite de fonction à deux variables

(6)

Aujourd’hui, nous allons voir

Limite de fonction à deux variables Continuité

(7)

Une fonction à plusieurs variable est une fonction de la forme

(8)

f<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> : Rⁿ ! R

(9)

Pour les cours qui suivent, nous allons surtout explorer les fonctions à deux variables.

(10)

f<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> : R² ! R

(11)

(x, y) 7 ! z = f (x, y)

(12)

(x, y) 7 ! z = f (x, y)

Le graphe d’une telle fonction est l’ensemble des points de la forme Une fonction à plusieurs variable est une fonction de la forme

(13)

(x, y) 7 ! z = f (x, y)

Le graphe d’une telle fonction est l’ensemble des points de la forme

⇣x, y, f (x, y)⌘

(14)

(x, y) 7 ! z = f (x, y)

⇣x, y, f (x, y)⌘

⇢<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> R³

(15)

(x, y) 7 ! z = f (x, y)

⇣x, y, f (x, y)⌘

⇢<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> R³

Nous regarderons aussi, mais dans une moindre mesure, les fonctions à trois variables.

(16)

Exemple

^f ^{(x, y}^{) =} ^xy ⁺ ³

2

(17)

Exemple

^f ^{(x, y}^{) =} ^xy ⁺ ³

2

f (1, 1) = 5 2

(18)

Exemple

^f ^{(x, y}^{) =} ^xy ⁺ ³

2

f (0, 1) = 3 2

f (1, 1) = 5 2

(19)

Exemple

^f ^{(x, y}^{) =} ^xy ⁺ ³

2

f (0, 1) = 3 2

f (1, 1) = 5 2

f

✓

1, 1 2

◆

= 1

(20)

Exemple

^{Le plan} ^2x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ^3y ^5z ⁼ ⁵

(21)

Exemple

^{Le plan} ^2x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ^3y ^5z ⁼ ⁵

z = 2x 3y + 5 5

(22)

Exemple

^{Le plan}

f (x, y) =

2x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> 3y 5z = 5

z = 2x 3y + 5 5

(23)

Exemple

^{Le plan}

f (x, y) =

2x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> 3y 5z = 5

z = 2x 3y + 5 5

(24)

Exemple

Trouver le domaine de ^f<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ^{(x, y}^{) =} ^p^x ⁺ ^y

(25)

Exemple

x + y > 0

(26)

Exemple

x + y > 0

y ><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> x

(27)

Exemple

x + y > 0

(28)

Exemple

x + y > 0

(29)

Exemple

Trouver le domaine _f _{(x, y}_{) =} ¹ xy

(30)

Exemple

xy<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 0 () x = 0 ou y = 0

(31)

Exemple

dom(f<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ) = {(x, y) 2 R²|x 6= 0, y 6= 0}

(32)

Exemple

dom(f<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ) = {(x, y) 2 R²|x 6= 0, y 6= 0}

(33)

Faites les exercices suivants

p.746 # 5 à 8

(34)

Exemple

Courbes de niveaux.

(35)

Exemple

Courbes de niveaux.

f (x, y) = x²

2 + y² 4

(36)

Exemple

Courbes de niveaux.

f (x, y) = x²

2 + y² 4

(37)

Exemple

Courbes de niveaux.

f (x, y) = x²

2 + y² 4

z = 1

(38)

Exemple

Courbes de niveaux.

f (x, y) = x²

2 + y² 4

z = 1

(39)

Exemple

Courbes de niveaux.

f (x, y) = x²

2 + y² 4

x²

2 + y²

4 = 1

z = 1

(40)

Exemple

Courbes de niveaux.

f (x, y) = x²

2 + y² 4

x²

2 + y²

4 = 1

z = 1

(41)

Exemple

Courbes de niveaux.

f (x, y) = x²

2 + y² 4

x²

2 + y²

4 = 1

z = 1 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 2

(42)

Exemple

Courbes de niveaux.

f (x, y) = x²

2 + y² 4

x²

2 + y²

4 = 1

z = 1 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 2 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 3

(43)

Exemple

Courbes de niveaux.

f (x, y) = x²

2 + y² 4

x²

2 + y²

4 = 1

z = 1 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 2 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 3 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 4

(44)

Exemple

Courbes de niveaux.

f (x, y) = x²

2 + y² 4

x²

2 + y²

4 = 1

z = 1 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 2 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 3 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 4 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 5

(45)

Exemple

_f _{(x, y}_{) =} ^x²

4 y² + 1

(46)

Exemple

_f _{(x, y}_{) =} ^x²

4 y² + 1

(47)

Exemple

_f _{(x, y}_{) =} ^x²

4 y² + 1

z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 2

(48)

Exemple

_f _{(x, y}_{) =} ^x²

4 y² + 1

z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 2 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 1

(49)

Exemple

_f _{(x, y}_{) =} ^x²

4 y² + 1

(50)

Exemple

_f _{(x, y}_{) =} ^x²

4 y² + 1

(51)

Exemple

_f _{(x, y}_{) =} ^x²

4 y² + 1

(52)

Exemple

_f _{(x, y}_{) =} ^x²

4 y² + 1

z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 1 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 2 z<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 3

(53)

On peut aussi considérer de fonction à trois variables

(54)

f<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> (x, y, z) = w

(55)

or le graphe d’une telle fonction est un sous-ensemble de . _R<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ⁴

(56)

or le graphe d’une telle fonction est un sous-ensemble de . _R<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ⁴

Par contre, on peut essayer de comprendre de telles fonctions en regardant ses surfaces de niveau.

(57)

f<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> (x, y, z) = x² + y² + z²

(58)

(59)

x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 1

(60)

x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 1

(61)

x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 1 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 4

(62)

x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 1 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 4

(63)

x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 1 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 4 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 9

(64)

x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 1 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 4 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 9

(65)

x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 1 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 4 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 9 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² + z² = 16

(66)

f<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> (x, y, z) = x² + y² z²

(67)

(68)

x² + y² z² = 1

(69)

x² + y² z² = 1

(70)

x² + y² z² = 1 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² z² = 2

(71)

x² + y² z² = 1 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² z² = 2

(72)

x² + y² z² = 1 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² z² = 2 x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² z² = 3

(73)

(74)

x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ² + y² z² = 4

(75)

Faites les exercices suivants

p. 746 # 9 à 20 et 35 à 40

(76)

Définition

La limite d’une fonction à deux variables est

(x,y)lim!(a,b) f (x, y) = L

(77)

Définition

8✏ > 0

(78)

Définition

9<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> > 0 8✏ > 0

(79)

Définition

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> k(x, y) (a, b)k < =) |f (x, y) L| < ✏

(80)

Définition

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> k(x, y) (a, b)k < =) |f (x, y) L| < ✏

(81)

Définition

L<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ✏ < f (x, y) < L + ✏

k(x, y) (a, b)k < =) |f (x, y) L| < ✏

(82)

Définition

L<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ✏ < f (x, y) < L + ✏

k(x, y) (a, b)k < =) |f (x, y) L| < ✏