• Aucun résultat trouvé

1.3 DÉRIVÉES DE FONCTIONS VECTORIELLES

N/A
N/A
Protected

Academic year: 2022

Partager "1.3 DÉRIVÉES DE FONCTIONS VECTORIELLES"

Copied!
30
0
0

Texte intégral

(1)

cours 3

1.3 DÉRIVÉES DE FONCTIONS

VECTORIELLES

(2)

Au dernier cours, nous avons vu

Fonctions vectorielles Comment les visualiser Limite

Surfaces paramétrées

(3)

Aujourd’hui, nous allons voir

Dérivées de fonctions vectorielles.

Règles de dérivation.

Vecteur tangent unitaire.

(4)

La dérivée d’une fonction est

f

0

(x) = lim

h!0

f (x + h) f (x) h

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

De manière analogue, on peut définir la dérivée d’une fonction vectorielle.

~ r

0

(t) = lim

h!0

~ r (t + h) ~ r (t) h

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

Voyons voir ce que ça représente géométriquement.

(5)
(6)

~ r

0

(a) = lim

h!0

~ r (a + h) ~ r(a) h

<latexit sha1_base64="5KjMuXLXwuJCwZOf7Sn4GS0WTPo=">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</latexit><latexit sha1_base64="5KjMuXLXwuJCwZOf7Sn4GS0WTPo=">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</latexit><latexit sha1_base64="5KjMuXLXwuJCwZOf7Sn4GS0WTPo=">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</latexit><latexit sha1_base64="5KjMuXLXwuJCwZOf7Sn4GS0WTPo=">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</latexit>

= lim

h!0

1

h ( ~ r (a + h) ~ r (a))

<latexit sha1_base64="2DI+ziauhSudH5S9IZrXXd2s2w4=">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</latexit><latexit sha1_base64="2DI+ziauhSudH5S9IZrXXd2s2w4=">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</latexit><latexit sha1_base64="2DI+ziauhSudH5S9IZrXXd2s2w4=">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</latexit><latexit sha1_base64="2DI+ziauhSudH5S9IZrXXd2s2w4=">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</latexit>

= lim

h!0

1

h ((f (a + h), g (a + h), k (a + h)) (f (a), g (a), k(a)))

<latexit sha1_base64="tEZh3qITBv3IsL85yfaVb0iKDNY=">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</latexit><latexit sha1_base64="tEZh3qITBv3IsL85yfaVb0iKDNY=">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</latexit><latexit sha1_base64="tEZh3qITBv3IsL85yfaVb0iKDNY=">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</latexit><latexit sha1_base64="tEZh3qITBv3IsL85yfaVb0iKDNY=">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</latexit>

= lim

h!0

✓ f (a + h) f (a)

h , g (a + h) g(a)

h , k (a + h) k(a) h

<latexit sha1_base64="DMltcSL9ytrzCq1opGHL60k+l+E=">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</latexit><latexit sha1_base64="DMltcSL9ytrzCq1opGHL60k+l+E=">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</latexit><latexit sha1_base64="DMltcSL9ytrzCq1opGHL60k+l+E=">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</latexit><latexit sha1_base64="DMltcSL9ytrzCq1opGHL60k+l+E=">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</latexit>

=

h

lim

!0

f (a + h) f (a)

h , lim

h!0

g (a + h) g (a)

h , lim

h!0

k (a + h) k (a) h

<latexit sha1_base64="2WOQufVV/MwMk/N3qrfNzfsYf9g=">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</latexit><latexit sha1_base64="2WOQufVV/MwMk/N3qrfNzfsYf9g=">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</latexit><latexit sha1_base64="2WOQufVV/MwMk/N3qrfNzfsYf9g=">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</latexit><latexit sha1_base64="2WOQufVV/MwMk/N3qrfNzfsYf9g=">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</latexit>

= (f

0

(a), g

0

(a), k

0

(a))

<latexit sha1_base64="E/2esIgTXMqXzsYLu5paU3yxlL0=">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</latexit><latexit sha1_base64="E/2esIgTXMqXzsYLu5paU3yxlL0=">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</latexit><latexit sha1_base64="E/2esIgTXMqXzsYLu5paU3yxlL0=">AAAC53icjVHLSsQwFD3W97vq0k1xEEcQaUXQjSC6cTmCo4Ij0ta0humLNBWGwb07d+LWH3CrfyL+gf6FN5kIPhBNaXJy7j0nublBkfBSuu5Ln9U/MDg0PDI6Nj4xOTVtz8welnklQtYM8yQXx4FfsoRnrCm5TNhxIZifBgk7Ctq7Kn50yUTJ8+xAdgp2mvpxxiMe+pKoM3thq5WwSNajpbq/vOLEvaWtFqcleHwhl8/smrvq6uH8BJ4BNZjRyO1ntHCOHCEqpGDIIAkn8FHSdwIPLgriTtElThDiOs5whTHSVpTFKMMntk1zTLsTw2a0V56lVod0SkK/IKWDRdLklCcIq9McHa+0s2J/8+5qT3W3Dq2B8UqJlbgg9i/dR+Z/daoWiQibugZONRWaUdWFxqXSr6Ju7nyqSpJDQZzC5xQXhEOt/HhnR2tKXbt6W1/HX3WmYtU+NLkV3tQtqcHe93b+BIdrqx7h/fXa9o5p9QjmsYA69XMD29hDA03yvsYDHvFkcevGurXueqlWn9HM4cuw7t8BCuSaCg==</latexit><latexit sha1_base64="E/2esIgTXMqXzsYLu5paU3yxlL0=">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</latexit>

(7)

~ r

0

(a) = lim

h!0

~ r (a + h) ~ r(a) h

<latexit sha1_base64="5KjMuXLXwuJCwZOf7Sn4GS0WTPo=">AAAC/nicjVHLShxBFD22ifGR6ESXbpoMkhlChh4REhcBMRuXCs4D7GGorqmZKewX1dWCNAP+SXbZBbf+gFuzFP/A/EVulT2gDsFU092nzr3nVN17gzSUmfa8uzln/tXrhTeLS8srb9+trlXer7ezJFdctHgSJqobsEyEMhYtLXUouqkSLApC0QlOv5t450yoTCbxsT5PRS9io1gOJWeaqH5l1z8T3FW++7HG6t/8UEb9YuzrxPUm/lAxXjzEa+zTuP55iuuTYjzpV6pew7PLnQXNElRRrsOkcgsfAyTgyBFBIIYmHIIho+cETXhIieuhIE4RkjYuMMEyaXPKEpTBiD2l74h2JyUb0954ZlbN6ZSQXkVKF1ukSShPETanuTaeW2fD/su7sJ7mbuf0D0qviFiNMbEv6aaZ/6sztWgM8dXWIKmm1DKmOl665LYr5ubuo6o0OaTEGTyguCLMrXLaZ9dqMlu76S2z8XubaViz52Vujj/mljTg5vNxzoL2dqNJ+GinurdfjnoRm/iAGs3zC/ZwgEO0yPsHrnGD386F89P55Vw+pDpzpWYDT5Zz9ReI8KTc</latexit><latexit sha1_base64="5KjMuXLXwuJCwZOf7Sn4GS0WTPo=">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</latexit><latexit sha1_base64="5KjMuXLXwuJCwZOf7Sn4GS0WTPo=">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</latexit><latexit sha1_base64="5KjMuXLXwuJCwZOf7Sn4GS0WTPo=">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</latexit>

= (f

0

(a), g

0

(a), k

0

(a))

<latexit sha1_base64="E/2esIgTXMqXzsYLu5paU3yxlL0=">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</latexit><latexit sha1_base64="E/2esIgTXMqXzsYLu5paU3yxlL0=">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</latexit><latexit sha1_base64="E/2esIgTXMqXzsYLu5paU3yxlL0=">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</latexit><latexit sha1_base64="E/2esIgTXMqXzsYLu5paU3yxlL0=">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</latexit>

(8)

Faites les exercices suivants

p.706 #1

(9)

Exemple

~ r(t) =

✓

2 + cos

✓ 3t 2

◆◆

cos t,

2 + cos

✓ 3t 2

◆◆

sin t, sin

✓ 3t 2

<latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit>

(10)

sin

✓ 3t 2

0

= 3

2 cos

✓ 3t 2

<latexit sha1_base64="8Twz5KoPMTu/oKPiT512tl5lNt8=">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</latexit><latexit sha1_base64="8Twz5KoPMTu/oKPiT512tl5lNt8=">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</latexit><latexit sha1_base64="8Twz5KoPMTu/oKPiT512tl5lNt8=">AAADGXicjVHLSgMxFL2Or/oedelmsIi6KdMq6EYounFZwVahU8pMzLTBeZFkhFL6Jf6JO3fi1oW4c6l+hTdpCmoRzTDJybn3nOTmBlnEhHTdlwlrcmp6ZrYwN7+wuLS8Yq+uNUSac0LrJI1Sfhn4gkYsoXXJZEQvM079OIjoRXB9ouIXN5QLlibnspfRVux3EhYy4kuk2va5F9FQNj3BEo12vJD7pL8nB/3KwOOs05W7w6W1fWRiOkRS8augbRfdkquHMw7KBhTBjFpqP4MHV5ACgRxioJCARByBDwK/JpTBhQy5FvSR44iYjlMYwDxqc8yimOEje41zB3dNwya4V55CqwmeEuHPUenAFmpSzOOI1WmOjufaWbG/efe1p7pbD9fAeMXISugi+5dulPlfnapFQgiHugaGNWWaUdUR45LrV1E3d75UJdEhQ07hK4xzxEQrR+/saI3Qtau39XX8TWcqVu2Jyc3hXd0SG1z+2c5x0KiUyojP9ovVY9PqAmzAJuxgPw+gCqdQgzp638Er+n5Yt9a99WA9DlOtCaNZh2/DevoEWT2yyw==</latexit><latexit sha1_base64="8Twz5KoPMTu/oKPiT512tl5lNt8=">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</latexit>

✓

2 + cos

✓ 3t 2

◆◆

sin t

0

<latexit sha1_base64="8JlfBe1OKCu/9cztmwIh+F6M8rg=">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</latexit><latexit sha1_base64="8JlfBe1OKCu/9cztmwIh+F6M8rg=">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</latexit><latexit sha1_base64="8JlfBe1OKCu/9cztmwIh+F6M8rg=">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</latexit><latexit sha1_base64="8JlfBe1OKCu/9cztmwIh+F6M8rg=">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</latexit>

✓

2 + cos

✓ 3t 2

◆◆

cos t

0

<latexit sha1_base64="qD8PQo22YcwxakYgGV/iO4J/WpI=">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</latexit><latexit sha1_base64="qD8PQo22YcwxakYgGV/iO4J/WpI=">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</latexit><latexit sha1_base64="qD8PQo22YcwxakYgGV/iO4J/WpI=">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</latexit><latexit sha1_base64="qD8PQo22YcwxakYgGV/iO4J/WpI=">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</latexit>

~ r(t) =

✓

2 + cos

✓ 3t 2

◆◆

cos t,

2 + cos

✓ 3t 2

◆◆

sin t, sin

✓ 3t 2

<latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit>

Exemple

= 3

2 sin

✓ 3t 2

cos t

2 + cos

✓ 3t 2

◆◆

sin t

<latexit sha1_base64="i1NygJC8EHmPTRXHdFFH36n9RJA=">AAADKHicjVFNS8QwEH3W7+9Vj16Ki6CI0l0FvQiiF48KrgpWpI3ZNdhtS5IKy7J/yH/izduyV0968qo3JzGCH4imtHnzZt5rJhPniVA6CHp9Xv/A4NDwyOjY+MTk1HRpZvZYZYVkvMayJJOncaR4IlJe00In/DSXPGrGCT+Jr/dM/uSGSyWy9Ei3cn7ejBqpqAsWaaIuSmx7NazLiLXXO+1qJ1QiDRNe10uO1JaVonGll0OWKV+vvuerKyb8vdZt5Ofri1I5WAvs8n+CigNluHWQlboIcYkMDAWa4EihCSeIoOg5QwUBcuLO0SZOEhI2z9HBGGkLquJUERF7Td8GRWeOTSk2nsqqGf0loVeS0sciaTKqk4TN33ybL6yzYX/zbltPc7YW7bHzahKrcUXsX7qPyv/qTC8adWzZHgT1lFvGdMecS2FvxZzc/9SVJoecOIMvKS8JM6v8uGffapTt3dxtZPOPttKwJmautsCTOSUNuPJ9nD/BcXWtQvhwo7yz60Y9gnksYInmuYkd7OMANfK+wzNe8Ordevde1+u9l3p9TjOHL8t7eANR97f5</latexit><latexit sha1_base64="i1NygJC8EHmPTRXHdFFH36n9RJA=">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</latexit><latexit sha1_base64="i1NygJC8EHmPTRXHdFFH36n9RJA=">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</latexit><latexit sha1_base64="i1NygJC8EHmPTRXHdFFH36n9RJA=">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</latexit>

= 3

2 sin

✓ 3t 2

sin t +

2 + cos

✓ 3t 2

◆◆

cos t

<latexit sha1_base64="ieWcIjtgN9IrAIH2ilTfkezP4Mc=">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</latexit><latexit sha1_base64="ieWcIjtgN9IrAIH2ilTfkezP4Mc=">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</latexit><latexit sha1_base64="ieWcIjtgN9IrAIH2ilTfkezP4Mc=">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</latexit><latexit sha1_base64="ieWcIjtgN9IrAIH2ilTfkezP4Mc=">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</latexit>

~r 0(t) =

3 2 sin

3t 2

cos t

2 + cos

3t 2

◆◆

sint, 3 2 sin

3t 2

sint +

2 + cos

3t 2

◆◆

cost, 3 2 cos

3t 2

<latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">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</latexit><latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">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</latexit><latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">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</latexit><latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">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</latexit>

(11)

sin

✓ 3t 2

0

= 3

2 cos

✓ 3t 2

<latexit sha1_base64="8Twz5KoPMTu/oKPiT512tl5lNt8=">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</latexit><latexit sha1_base64="8Twz5KoPMTu/oKPiT512tl5lNt8=">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</latexit><latexit sha1_base64="8Twz5KoPMTu/oKPiT512tl5lNt8=">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</latexit><latexit sha1_base64="8Twz5KoPMTu/oKPiT512tl5lNt8=">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</latexit>

✓

2 + cos

✓ 3t 2

◆◆

sin t

0

<latexit sha1_base64="8JlfBe1OKCu/9cztmwIh+F6M8rg=">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</latexit><latexit sha1_base64="8JlfBe1OKCu/9cztmwIh+F6M8rg=">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</latexit><latexit sha1_base64="8JlfBe1OKCu/9cztmwIh+F6M8rg=">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</latexit><latexit sha1_base64="8JlfBe1OKCu/9cztmwIh+F6M8rg=">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</latexit>

✓

2 + cos

✓ 3t 2

◆◆

cos t

0

<latexit sha1_base64="qD8PQo22YcwxakYgGV/iO4J/WpI=">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</latexit><latexit sha1_base64="qD8PQo22YcwxakYgGV/iO4J/WpI=">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</latexit><latexit sha1_base64="qD8PQo22YcwxakYgGV/iO4J/WpI=">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</latexit><latexit sha1_base64="qD8PQo22YcwxakYgGV/iO4J/WpI=">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</latexit>

~ r(t) =

✓

2 + cos

✓ 3t 2

◆◆

cos t,

2 + cos

✓ 3t 2

◆◆

sin t, sin

✓ 3t 2

<latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit>

Exemple

~r 0(t) =

3 2 sin

3t 2

cost

2 + cos

3t 2

◆◆

sint, 3 2 sin

3t 2

sint +

2 + cos

3t 2

◆◆

cos t, 3

2 cos

3t 2

<latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">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</latexit><latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">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</latexit><latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">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</latexit><latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">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</latexit>

= 3

2 sin

✓ 3t 2

cos t

2 + cos

✓ 3t 2

◆◆

sin t

<latexit sha1_base64="i1NygJC8EHmPTRXHdFFH36n9RJA=">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</latexit><latexit sha1_base64="i1NygJC8EHmPTRXHdFFH36n9RJA=">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</latexit><latexit sha1_base64="i1NygJC8EHmPTRXHdFFH36n9RJA=">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</latexit><latexit sha1_base64="i1NygJC8EHmPTRXHdFFH36n9RJA=">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</latexit>

= 3

2 sin

✓ 3t 2

sin t +

2 + cos

✓ 3t 2

◆◆

cos t

<latexit sha1_base64="ieWcIjtgN9IrAIH2ilTfkezP4Mc=">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</latexit><latexit sha1_base64="ieWcIjtgN9IrAIH2ilTfkezP4Mc=">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</latexit><latexit sha1_base64="ieWcIjtgN9IrAIH2ilTfkezP4Mc=">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</latexit><latexit sha1_base64="ieWcIjtgN9IrAIH2ilTfkezP4Mc=">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</latexit>

(12)

Exemple

~ r(t) =

✓

2 + cos

✓ 3t 2

◆◆

cos t,

2 + cos

✓ 3t 2

◆◆

sin t, sin

✓ 3t 2

<latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit><latexit sha1_base64="nDACV+qOj6HUtj90euVy2GNw9Wo=">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</latexit>

~r 0(t) =

3 2 sin

3t 2

cost

2 + cos

3t 2

◆◆

sint, 3 2 sin

3t 2

sint +

2 + cos

3t 2

◆◆

cos t, 3

2 cos

3t 2

<latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">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</latexit><latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">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</latexit><latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">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</latexit><latexit sha1_base64="h0BAGKqKI6VvttMMupMPN552yA4=">AAAD0HiclVHNTttAEB7jUmgoJcCxl1WjqkEkyE4rlQsSai89AiIkUhwhe9mEFY5tdteoKIoQ175Ar+1TVX2D9i06OywSFEWEtWx/881887OTFKnUJgh+e3P+s/nnC4svKksvl1+tVFfXjnReKi7aPE9z1U1iLVKZibaRJhXdQol4lKSik5x9tv7OhVBa5tmhuSxEfxQPMzmQPDZIHa96lehCcKYi9q5uNnaiVAxMrxkNVMzH7yfj1iTSMiO27khDrJLDU7MR8Vwz07zxtzatOT3W/TAfMw02ew0SbD6pBrXVYHdLPC7rH1drwVZAhz0EoQM1cGcvr/6CCE4gBw4ljEBABgZxCjFofHoQQgAFcn0YI6cQSfILmEAFtSVGCYyIkT3D7xCtnmMztG1OTWqOVVJ8FSoZvEVNjnEKsa3GyF9SZstOyz2mnLa3S/wnLtcIWQOnyD6mu42cVWdnMTCAbZpB4kwFMXY67rKUdCu2c3ZnKoMZCuQsPkG/QsxJeXvPjDSaZrd3G5P/D0Va1trcxZbw13aJCw7/X+dDcNTaChHvf6jtfnKrXoTX8AbquM+PsAtfYA/awL1z77v3w/vpH/hf/Sv/+iZ0znOadbh3/G//ABj38ik=</latexit>

Références

Documents relatifs

Son architecture, qui fournit le cadre ecclésial approprié à la célébration des obsèques avec sa dimension baptismale et eucharistique, en fait le lieu privilégié

Sans être passéiste, en vain je m’interroge Sur la vie d’aujourd’hui soulevant chaque jour Un scandale nouveau pour celui qui s’arroge Le droit d’empoisonner sans prendre

Grâce aux bénévoles , nous distribuons en effet des paniers repas chaque samedi à Saint-Joseph des Épinettes En cette période particulièrement difficile pour

UNEP MULTISELECTION PLUS (avec ou sans option Protection Majeur) est un contrat d’assurance vie de groupe proposé par Oradéa Vie Société Anonyme d’assurance sur la vie et

De plus en plus considérée comme un partenaire sexuel, elle souhaite aussi pour les femmes un mariage plus tardif : en moyenne 19,5 ans contre 21,4 ans, âge moyen au

La théorie de l’art moderniste a renoué avec une certaine forme de platonisme, reléguant l’ornement dans le domaine des apparences malignes, cherchant avant

Notre mission d’aumônier oscille bien souvent entre l’accompagnant spirituel parfois psychologique, mais aussi dans le rôle d’assistant social ou de visiteur de prison, car

Appellations Villages Premiers Crus 10% de la production totale 640 Climats classés en Premiers Crus ex : Chablis 1 er Cru, Montmains Appellations Villages 37% de la production