cours 1
1.1 INTRODUCTION
ET RÉVISION
Aujourd’hui, nous allons voir
Survol des cours calcul différentiel, calcul intégral et algèbre linéaire.
Introduction des idées que nous verrons cette session.
y x
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f 0(a) = lim
x!0
y
x = lim
x!0
f (a + x) f (a) x
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= pente de
= f (a + x) f (a) x
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f 0(x) = lim
x!0
f (x + x) f (x) x
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Les règles de dérivations
d
dx (kf (x)) = k d
dx f (x)
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d
dx (f (x) + g(x)) = df (x)
dx + dg(x) dx
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d
dx (f (x)g(x)) = df (x)
dx g(x) + f (x) dg(x) dx
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d dx
✓ f (x) g(x)
◆
=
df(x)
dx g(x) f (x) dgdx(x) g2(x)
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d
dx (f (g(x))) = df (g(x)) dg(x)
dg(x) dx
<latexit sha1_base64="JhC5STm4lqXBBN+KnPq+/1Omz5U=">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</latexit><latexit sha1_base64="JhC5STm4lqXBBN+KnPq+/1Omz5U=">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</latexit><latexit sha1_base64="JhC5STm4lqXBBN+KnPq+/1Omz5U=">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</latexit><latexit sha1_base64="JhC5STm4lqXBBN+KnPq+/1Omz5U=">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</latexit>
= df (u) du
du dx
<latexit sha1_base64="yJQumpQo/BeZ0smmvhKDeUfFadc=">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</latexit><latexit sha1_base64="yJQumpQo/BeZ0smmvhKDeUfFadc=">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</latexit><latexit sha1_base64="yJQumpQo/BeZ0smmvhKDeUfFadc=">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</latexit><latexit sha1_base64="yJQumpQo/BeZ0smmvhKDeUfFadc=">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</latexit>
Exemple
f (x) = 3x sin(x2)ln 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>
= (3x ln 3 sin(x2) + 3x cos(x2)(x2)0) ln x (3x sin(x2)) x1 ln2 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>
= (3x ln 3 sin(x2) + 3x cos(x2)2x) ln x (3x sin(x2)) x1 ln2 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>
= (3x0 sin(x2) + 3x sin(x2)0) ln x (3x sin(x2)) x1 ln2 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>
f 0(x) = (3x sin(x2))0 ln x (3x sin(x2)) ln x0 ln2 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>
Faites les exercices suivants
Calculer le dérivée des fonctions suivantes:
1)
2)
3)
4)
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) = 4 tan3(xex)
f (x) = log5(p
4x7 3) sin 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>
f (x) = 4x p5
x3 sin x + csc3 x log6 ⇡2
<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) = cot 4cos(x2+1) ln(log3(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>
différentielle
dy
dx = f 0(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>
Le facteur multiplicatif qui permet de changer la longueur dx en la longueur dy.
dy = f 0(x)dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Lorsque <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 ⇡ 0
y
x ⇡ dy dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
f 0(x)dx = f (x) + C
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Intégrale indéfinie
Propriétés Z
f (x)dx = F (x) + C
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
où 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> 0(x) = f (x)
Exemple
Techniques d’intégration.
Changement de variable Z
x sin(x2)dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
= Z
x sin(u) du 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>
= 1 2
Z
sin(u)du
<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
2 cos(u) + C
<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
2 cos(x2) + C
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
u<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = x2
du<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = 2xdx
Exemple
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> = 3 sin ✓dx<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 cos ✓d✓
p<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> 9 x2 = 3 cos ✓ (p
9 x2)3 = 33 cos3 ✓
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z 1
(9 x2) 32 dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
=
Z 3 cos ✓d✓
33 cos3 ✓
<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 9
Z
sec2 ✓d✓
<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
9 tan ✓ + C
<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
9p
9 x2 tan ✓ + C
<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<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<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
p<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> 9 x2
✓<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Exemple
Technique d’intégration Z
udv = uv
Z
vdu
<latexit sha1_base64="uNj9+2Y9kHJPBQf/TCu6VS4WtbQ=">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</latexit><latexit sha1_base64="uNj9+2Y9kHJPBQf/TCu6VS4WtbQ=">AAAC23icjVHLSsNAFD2Nr1pfVcGNm2AR3FhSEXQjFN24rGAf0JaSpNM6NE1CMhMo1ZU7cesPuNX/Ef9A/8I7YwpqEZ2Q5Nxz7zkzd64TejwWlvWaMWZm5+YXsou5peWV1bX8+kYtDmTksqobeEHUcOyYedxnVcGFxxphxOyh47G6MzhT+XrCopgH/qUYhaw9tPs+73HXFkR18lst7gtTdpMTUyb7OkjMruzkC1bR0sucBqUUFJCuSpB/QQtdBHAhMQSDD0HYg42YniZKsBAS18aYuIgQ13mGG+RIK6mKUYVN7IC+fYqaKetTrDxjrXZpF4/eiJQmdkkTUF1EWO1m6rzUzor9zXusPdXZRvR3Uq8hsQJXxP6lm1T+V6d6EejhWPfAqadQM6o7N3WR+lbUyc0vXQlyCIlTuEv5iLCrlZN7NrUm1r2ru7V1/k1XKlbFblor8a5OSQMu/RznNKgdFEuELw4L5dN01FlsYwd7NM8jlHGOCqrkfY1HPOHZaBu3xp1x/1lqZFLNJr4t4+EDug+XyQ==</latexit><latexit sha1_base64="uNj9+2Y9kHJPBQf/TCu6VS4WtbQ=">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</latexit><latexit sha1_base64="uNj9+2Y9kHJPBQf/TCu6VS4WtbQ=">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</latexit>
Par partie Z
x cos x dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
u<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 du<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = dx
dv<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = cos x dx v<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> = sin x
= x sin x
Z
sin x dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
=<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 sin x + cos x + C
Exemple
Technique d’intégration Fractions partielles
Z 3x + 1
(x 2)(2x + 3) dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
3x + 1
(x 2)(2x + 3) = A
x 2 + B
2x + 3
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
= A(2x + 3) + B(x 2) (x 2)(2x 3)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
= (2A + B)x + (3A 2B) (x 2)(2x 3)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⇢ 2A + B = 3 3A 2B = 1
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
A =
3 1
1 2
2 1
3 2
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
B =
2 3 3 1
2 1
3 2
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
= 7 7
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
= 7 7
<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<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<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
=
Z 1
x 2 + 1
2x + 3 dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
= ln |x 2| + 1
2 ln |2x + 3| + C
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
= lim
max xi!0
X f (xi) xi
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z b
a
f (x)dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Intégrale définie
Théorème fondamental du calcul.
Z b
a
f 0(x)dx = f (b) f (a)
<latexit sha1_base64="evVJPk8SSDfRCQgmvU4gqa632+s=">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</latexit><latexit sha1_base64="evVJPk8SSDfRCQgmvU4gqa632+s=">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</latexit><latexit sha1_base64="evVJPk8SSDfRCQgmvU4gqa632+s=">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</latexit><latexit sha1_base64="evVJPk8SSDfRCQgmvU4gqa632+s=">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</latexit>
d dx
Z x
a
f (t)dt = f (x)
<latexit sha1_base64="q1lZG6cojbuFyIYuFnWWnDhieRw=">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</latexit><latexit sha1_base64="q1lZG6cojbuFyIYuFnWWnDhieRw=">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</latexit><latexit sha1_base64="q1lZG6cojbuFyIYuFnWWnDhieRw=">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</latexit><latexit sha1_base64="q1lZG6cojbuFyIYuFnWWnDhieRw=">AAAC5HicjVHLTttAFD1xeTVQCGXZRa1GSGET2QgJNkiIblimUvOQCETj8ZiOcGxrPEaJoizZsUPd8gNs4Vsq/oD+Re8MRoIiBGPZPnPuPWfm3htkscy1591VnA8zs3PzCx+ri0uflldqq587eVooLto8jVPVC1guYpmItpY6Fr1MCTYMYtENTr+bePdMqFymyU89zsTRkJ0kMpKcaaIGta/9SDE+CaeTcDTty0QP2PEoauiNUO9GjdHGoFb3mp5d7kvgl6COcrXS2h/0ESIFR4EhBBJowjEYcnoO4cNDRtwRJsQpQtLGBaaokragLEEZjNhT+p7Q7rBkE9obz9yqOZ0S06tI6WKdNCnlKcLmNNfGC+ts2Ne8J9bT3G1M/6D0GhKr8YvYt3SPme/VmVo0IuzYGiTVlFnGVMdLl8J2xdzcfVKVJoeMOINDiivC3Cof++xaTW5rN71lNn5vMw1r9rzMLfDX3JIG7P8/zpegs9n0/ab/Y6u+t1+OegFf8A0Nmuc29nCAFtrkfY5r3ODWiZwL59L5/ZDqVErNGp4t5+ofKamcNw==</latexit>
Faites les exercices suivants
Calculer les intégrales suivantes:
1)
2)
3)
4) Z cos(ln x)
x dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
eeee
x
eeex eex ex dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
x2e4x dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z p
x2 1
x dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
cos(ln x) dx
<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)
6)
Z x2 + 4x 1
x3 x dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
~u = (a, b, c)
<latexit sha1_base64="XXKki1O8Fv9VDqqXRK8xBzXZd8I=">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</latexit><latexit sha1_base64="XXKki1O8Fv9VDqqXRK8xBzXZd8I=">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</latexit><latexit sha1_base64="XXKki1O8Fv9VDqqXRK8xBzXZd8I=">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</latexit><latexit sha1_base64="XXKki1O8Fv9VDqqXRK8xBzXZd8I=">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</latexit>
~v = (d, e, f )
<latexit sha1_base64="vpibZzrpPFMArbnJ1QS8NhVuNJc=">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</latexit><latexit sha1_base64="vpibZzrpPFMArbnJ1QS8NhVuNJc=">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</latexit><latexit sha1_base64="vpibZzrpPFMArbnJ1QS8NhVuNJc=">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</latexit><latexit sha1_base64="vpibZzrpPFMArbnJ1QS8NhVuNJc=">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</latexit>
k~u = (ka, kb, kc)
<latexit sha1_base64="l0HScfhlnJIQzD/RafFp7O5lxes=">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</latexit><latexit sha1_base64="l0HScfhlnJIQzD/RafFp7O5lxes=">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</latexit><latexit sha1_base64="l0HScfhlnJIQzD/RafFp7O5lxes=">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</latexit><latexit sha1_base64="l0HScfhlnJIQzD/RafFp7O5lxes=">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</latexit>
~u<latexit sha1_base64="4v9ZNFRSjB7pvYhzZ4tXf/BN0Uk=">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</latexit><latexit sha1_base64="4v9ZNFRSjB7pvYhzZ4tXf/BN0Uk=">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</latexit><latexit sha1_base64="4v9ZNFRSjB7pvYhzZ4tXf/BN0Uk=">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</latexit><latexit sha1_base64="4v9ZNFRSjB7pvYhzZ4tXf/BN0Uk=">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</latexit> + ~v = (a + d, b + e, d + f ) k~uk = p
a2 + b2 + c2
<latexit sha1_base64="hLXna5CHD1zeV8PfEpLoDRGv/5w=">AAAC4nicjVHLSsNAFD2Nr1pfVZciBIsgCCUpgm6EohuXCtYWmlaS6ajBNImTiSBtV+7ciVt/wK1+jPgH+hfeGVPwgeiEmZw5954zc+d6ceAn0rJecsbI6Nj4RH6yMDU9MztXnF84SqJUMF5jURCJhucmPPBDXpO+DHgjFtztegGve+e7Kl6/5CLxo/BQXsW81XVPQ//EZ64k6ri47PSdS87M1OlvO8mFkD23XVn3aLJ2ZXBcLFllSw/zJ7AzUEI29qPiMxx0EIEhRRccISThAC4S+pqwYSEmroUecYKQr+McAxRIm1IWpwyX2HNaT2nXzNiQ9soz0WpGpwQ0BSlNrJImojxBWJ1m6niqnRX7m3dPe6q7XdHfy7y6xEqcEfuXbpj5X52qReIEW7oGn2qKNaOqY5lLql9F3dz8VJUkh5g4hTsUF4SZVg7f2dSaRNeu3tbV8VedqVi1Z1luijd1S2qw/b2dP8FRpWzbZftgo1TdyVqdxxJWsEb93EQVe9hHjbyv8YBHPBkd48a4Ne4+Uo1cplnEl2HcvwNTJ5q1</latexit><latexit sha1_base64="hLXna5CHD1zeV8PfEpLoDRGv/5w=">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</latexit><latexit sha1_base64="hLXna5CHD1zeV8PfEpLoDRGv/5w=">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</latexit><latexit sha1_base64="hLXna5CHD1zeV8PfEpLoDRGv/5w=">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</latexit>
~u<latexit sha1_base64="XXKki1O8Fv9VDqqXRK8xBzXZd8I=">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</latexit><latexit sha1_base64="XXKki1O8Fv9VDqqXRK8xBzXZd8I=">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</latexit><latexit sha1_base64="XXKki1O8Fv9VDqqXRK8xBzXZd8I=">AAAC1XicjVHLSsNAFD2Nr1pfUZdugkWoUEoigm6EohuXFewD2lKS6bQG8yKZFErpTtz6A271l8Q/0L/wzpiCWkQnJDlz7j1n5t7rRJ6bCNN8zWkLi0vLK/nVwtr6xuaWvr3TSMI0ZrzOQi+MW46dcM8NeF24wuOtKOa273i86dxeyHhzxOPEDYNrMY5417eHgTtwmS2I6ul6Z8SZkZ6V7LLhlA122NOLZsVUy5gHVgaKyFYt1F/QQR8hGFL44AggCHuwkdDThgUTEXFdTIiLCbkqzjFFgbQpZXHKsIm9pe+Qdu2MDWgvPROlZnSKR29MSgMHpAkpLyYsTzNUPFXOkv3Ne6I85d3G9HcyL59YgRti/9LNMv+rk7UIDHCqanCppkgxsjqWuaSqK/LmxpeqBDlExEncp3hMmCnlrM+G0iSqdtlbW8XfVKZk5Z5luSne5S1pwNbPcc6DxlHFsirW1XGxep6NOo897KNE8zxBFZeooU7eIzziCc9aU5tqd9r9Z6qWyzS7+La0hw/iwJRI</latexit><latexit sha1_base64="XXKki1O8Fv9VDqqXRK8xBzXZd8I=">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</latexit> = (a, b, c) ~v<latexit sha1_base64="vpibZzrpPFMArbnJ1QS8NhVuNJc=">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</latexit><latexit sha1_base64="vpibZzrpPFMArbnJ1QS8NhVuNJc=">AAAC1XicjVHLSsNAFD2Nr1pfUZdugkWoUEoigm6EohuXFewD2lKS6bQG0yQkk0Ap3Ylbf8Ct/pL4B/oX3hlTUIvohCRnzr3nzNx7ndBzY2GarzltYXFpeSW/Wlhb39jc0rd3GnGQRIzXWeAFUcuxY+65Pq8LV3i8FUbcHjkebzq3FzLeTHkUu4F/LcYh747soe8OXGYLonq63kk5M9KzUr9s8LIxOOzpRbNiqmXMAysDRWSrFugv6KCPAAwJRuDwIQh7sBHT04YFEyFxXUyIiwi5Ks4xRYG0CWVxyrCJvaXvkHbtjPVpLz1jpWZ0ikdvREoDB6QJKC8iLE8zVDxRzpL9zXuiPOXdxvR3Mq8RsQI3xP6lm2X+VydrERjgVNXgUk2hYmR1LHNJVFfkzY0vVQlyCImTuE/xiDBTylmfDaWJVe2yt7aKv6lMyco9y3ITvMtb0oCtn+OcB42jimVVrKvjYvU8G3Uee9hHieZ5giouUUOdvFM84gnPWlObanfa/Weqlss0u/i2tIcP+q6UUg==</latexit><latexit sha1_base64="vpibZzrpPFMArbnJ1QS8NhVuNJc=">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</latexit><latexit sha1_base64="vpibZzrpPFMArbnJ1QS8NhVuNJc=">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</latexit> = (d, e, f )
~u<latexit sha1_base64="8mp6mzfiEO2gPmk0r6x9TQAbD1s=">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</latexit><latexit sha1_base64="8mp6mzfiEO2gPmk0r6x9TQAbD1s=">AAAC4XicjVHLSsNAFD2Nr1pfUZe6CBZBKJREBN0IRTcuK9gHtKUkk2kNpklIJoVS3LhzJ279Abf6M+If6F94Z0xBLaITkpw5954zc+91It9LhGm+5rSZ2bn5hfxiYWl5ZXVNX9+oJ2EaM15joR/GTcdOuO8FvCY84fNmFHN74Pi84VydynhjyOPEC4MLMYp4Z2D3A6/nMVsQ1dW320POjLTN3FAoODy23ZLh8JLh9rp60SybahnTwMpAEdmqhvoL2nARgiHFABwBBGEfNhJ6WrBgIiKugzFxMSFPxTmuUSBtSlmcMmxir+jbp10rYwPaS89EqRmd4tMbk9LALmlCyosJy9MMFU+Vs2R/8x4rT3m3Ef2dzGtArMAlsX/pJpn/1claBHo4UjV4VFOkGFkdy1xS1RV5c+NLVYIcIuIkdikeE2ZKOemzoTSJql321lbxN5UpWblnWW6Kd3lLGrD1c5zToL5ftqyydX5QrJxko85jCzvYo3keooIzVFEj7xs84gnPGtNutTvt/jNVy2WaTXxb2sMHHGyZ0w==</latexit><latexit sha1_base64="8mp6mzfiEO2gPmk0r6x9TQAbD1s=">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</latexit><latexit sha1_base64="8mp6mzfiEO2gPmk0r6x9TQAbD1s=">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</latexit> · ~v = ad + be + df =<latexit sha1_base64="ml/reNgQrJzs74/QCVCUbGC2NuI=">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</latexit><latexit sha1_base64="ml/reNgQrJzs74/QCVCUbGC2NuI=">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</latexit><latexit sha1_base64="ml/reNgQrJzs74/QCVCUbGC2NuI=">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</latexit><latexit sha1_base64="ml/reNgQrJzs74/QCVCUbGC2NuI=">AAAC5HicjVHLSsNAFD2Nr1pfUZcuDBbBVUlE0I1QdOOygn1AW0oynbahaRKSSaFUl+7ciVt/wK1+i/gH+hfemaagFtEJmTlz7j1n5s51Qs+NhWm+ZbS5+YXFpexybmV1bX1D39yqxEESMV5mgRdENceOuef6vCxc4fFaGHF74Hi86vTPZbw65FHsBv6VGIW8ObC7vttxmS2Iaum7p43rxpAzI6F1goa0siBuiB4XdkvPmwVTDWMWWCnIIx2lQH9FA20EYEgwAIcPQdiDjZi+OiyYCIlrYkxcRMhVcY4b5EibUBanDJvYPs1d2tVT1qe99IyVmtEpHv0RKQ3skyagvIiwPM1Q8UQ5S/Y377HylHcb0eqkXgNiBXrE/qWbZv5XJ2sR6OBE1eBSTaFiZHUsdUnUq8ibG1+qEuQQEidxm+IRYaaU03c2lCZWtcu3tVX8XWVKVu5ZmpvgQ96SGmz9bOcsqBwWLKtgXR7li2dpq7PYwR4OqJ/HKOICJZTJ+xZPeMaL1tHutHvtYZKqZVLNNr4N7fETqMGccg==</latexit> k~ukk~vk cos ✓ Produit scalaire
~ u<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
~v<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
✓<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
~u<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ? ~v () ~u · ~v = 0
