Chapitres Maths en ECG1
Chapitres Maths en ECG1
Exercices corrigés : Formules de Taylor, développements limités ECG1
Résumé de cours Exercices Corrigés
Cours en ligne de Maths en ECG1
Corrigés – Formules de Taylor et développements limités
Exercice 1 :
1) On pose On a :
.
2) On pose Faisons le développement limité de
à l’ordre
lorsque
tend vers
On a
![Rendered by QuickLaTeX.com \sin \left( \dfrac{\pi}{4} + h \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d3c4d7754800207ed8be90324c5acdf4_l3.png)
![Rendered by QuickLaTeX.com = \dfrac{\sqrt{2}}{2} \left( \cos \left( h \right) + \sin \left( h \right) \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d382ad525d12473d3ca6d7884c09b864_l3.png)
![Rendered by QuickLaTeX.com \underset{h \to 0}{=} \dfrac{\sqrt{2}}{2} ( 1 - \dfrac{h^2}{2!} + \dfrac{h^4}{4!} + o \left( h^5 \right) h](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ad104040a538a8002b638a1e445b572a_l3.png)
![Rendered by QuickLaTeX.com - \dfrac{h^3}{3!} + \dfrac{h^5}{5!} + o \left( h^5 \right))](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c9dfc14963efd80e6b6899f0029520f5_l3.png)
![Rendered by QuickLaTeX.com \underset{h \to 0}{=} \dfrac{\sqrt{2}}{2} ( 1 + h - \dfrac{h^2}{2} - \dfrac{h^3}{6}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-24335aa058aba16dd452dbac46abdf11_l3.png)
![Rendered by QuickLaTeX.com + \dfrac{h^4}{24} + \dfrac{h^5}{120} + o \left( h^5 \right) )](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a32ead0bce34f4925f8a4febb854fd37_l3.png)
![Rendered by QuickLaTeX.com h = x - \dfrac{\pi}{4},](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-74792e1a4aa6cd4b12d46b456a28b847_l3.png)
![Rendered by QuickLaTeX.com \sin \left( x \right) \underset{x \to \dfrac{\pi}{4}}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-32b15eddb429c0e167c7ab18a92d1f00_l3.png)
![Rendered by QuickLaTeX.com \dfrac{\sqrt{2}}{2} ( 1 + \left( x - \dfrac{x}{4} \right) - \dfrac{\left( x - \dfrac{x}{4} \right)^2}{2}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-0bbc23097ec2ae5fbc7505b037866d5f_l3.png)
![Rendered by QuickLaTeX.com - \dfrac{\left( x - \dfrac{x}{4} \right)^3}{6} + \dfrac{\left( x - \dfrac{x}{4} \right)^4}{24}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9fa07c0d0d6c625f273798b21a6e2417_l3.png)
![Rendered by QuickLaTeX.com + \dfrac{\left( x - \dfrac{x}{4} \right)^5}{120}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-38ffc2988d32f776319a1ad73ddc9d92_l3.png)
![Rendered by QuickLaTeX.com + o \left( \left( x - \dfrac{x}{4} \right)^5 \right) ).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-7da85134dbc5b8ba6b5c2a4b48bbb466_l3.png)
On ne développe pas, bien sûr !
3) On a
et
Ce qui donne :
![Rendered by QuickLaTeX.com e^{-x} \cos\left( 2 x \right) \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-545b7e0bb83d0f22a099728140eb7a01_l3.png)
![Rendered by QuickLaTeX.com \left( 1 - x + \dfrac{x^2}{2} - \dfrac{x^3}{6} + o \left( x^3 \right) \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-cd678bbe8028a6a2d39de423ec6d7ccb_l3.png)
![Rendered by QuickLaTeX.com \times \left( 1 - \dfrac{\left( 2 x \right)^2}{2!} + o \left( x^3 \right) \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-530aa20d6eb9935577604f8585c49f6d_l3.png)
Lorsque que l’on développe, les termes ayant
![Rendered by QuickLaTeX.com x^k](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-81e025d46b78edcab0363d17fe2192ae_l3.png)
![Rendered by QuickLaTeX.com k \ge 4](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-57adf94bad8541f284d03795925a411d_l3.png)
![Rendered by QuickLaTeX.com o \left( x ^3 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d4632d187bdad51f0b055ec61125de3e_l3.png)
![Rendered by QuickLaTeX.com e^{-x} \cos\left( 2 x \right) \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-545b7e0bb83d0f22a099728140eb7a01_l3.png)
![Rendered by QuickLaTeX.com 1 - 3 x + \dfrac52 x^2 - \dfrac76 x^3 + o \left( x^3 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-76966d9072969d18049dd08715a00448_l3.png)
![Rendered by QuickLaTeX.com = \dfrac12 \cos \left( x \right) \times \left( 1 - \dfrac{x}{2} \right)^{-1}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3f7d086c4c0c713bf52836b727e16b6d_l3.png)
![Rendered by QuickLaTeX.com \underset{x \to 0}{=} \dfrac12 \left( 1 - \dfrac{x^2}{2} + o \left( x^3 \right) \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-0df28b3f41f26520061522ff18ff7f84_l3.png)
![Rendered by QuickLaTeX.com \times \left( 1 + \dfrac12 x + \dfrac14 x^2 + \dfrac18 x^3 + o \left( x^3 \right) \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3073386ade55fbbdce808cf5d3416b1b_l3.png)
![Rendered by QuickLaTeX.com \underset{x \to 0}{=} \dfrac12 ( 1 + \dfrac12 x + \dfrac14 x^2 + \dfrac18 x^3](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ea9f0a0d3dc68befe2b3b2836a7756a5_l3.png)
![Rendered by QuickLaTeX.com - \dfrac12 x^2 - \dfrac14 x^3 + o \left( x^3 \right) )](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-02954a5bdf65bf106f63d9e0a7dec7f9_l3.png)
![Rendered by QuickLaTeX.com \underset{x \to 0}{=} \dfrac12 \left( 1 + \dfrac12 x - \dfrac14 x^2 - \dfrac18 x^3 + o \left( x^3 \right) \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-7e928dd97862fe865d4407b86fde3175_l3.png)
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Exercice 2 :
1) On a
Ainsi
soit
Ainsi En particulier,
2) Trouvons un équivalent du numérateur et du dénominateur.
![Rendered by QuickLaTeX.com e^x \underset{x \to 0}{=} 1 + x + o \left( x \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-f172f12e8e587a76938bd6c88f7d8d87_l3.png)
![Rendered by QuickLaTeX.com \left( x - 1 \right) e^x + 1 \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9c4c6a4013a24a91ce87522c5a55b0ae_l3.png)
![Rendered by QuickLaTeX.com \left( x - 1 \right) \left( 1 + x + \dfrac12 x^2 + o \left( x^2 \right) \right) + 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-bf97871862cfcc77291e2d808c894459_l3.png)
![Rendered by QuickLaTeX.com \underset{x \to 0}{=} x^2 + o \left( x^2 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e7bfa4577b90772bf0fe6e0431eeeb57_l3.png)
Ce qui donne
![Rendered by QuickLaTeX.com e^x \underset{x \to 0}{=} 1 + x + o \left( x \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5cc2a1351c052d2e006cd81a68bcaa7a_l3.png)
![Rendered by QuickLaTeX.com e^x - 1 \underset{x \to 0}{\sim} x .](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-8374c2700c39c1e7c3bb9f2d0dd14368_l3.png)
Finalement
![Rendered by QuickLaTeX.com \dfrac{\left( x - 1 \right) e^x + 1}{x \left( e^x - 1 \right)}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-54148b1e619e5e4238806930fe39c2c3_l3.png)
![Rendered by QuickLaTeX.com \underset{x \to 0}{\sim} \dfrac{x^2}{x^2}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b5e1d79e349b9449e1d2e500ed75f3eb_l3.png)
![Rendered by QuickLaTeX.com \underset{x \to 0}{\sim} 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2835a461509e96fc26890157e7608974_l3.png)
![Rendered by QuickLaTeX.com \lim_{x \to 0} \dfrac{\left( x - 1 \right) e^x + 1}{x \left( e^x - 1 \right)} = 1.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3e0220ad2ae3d747de6da0cf873e2626_l3.png)
3) Trouvons un équivalent du numérateur et du dénominateur.
![Rendered by QuickLaTeX.com e^x - 1 - \sin \left( x \right) \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-f2d9793f502d2216f5d2675cf3adb79d_l3.png)
![Rendered by QuickLaTeX.com \left( 1 + x + \dfrac12 x^2 + o \left( x^2 \right) \right) - 1 - \left( x + o \left( x^2 \right) \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4cf11c6e5a8ab485edbbf63dc87d17d1_l3.png)
![Rendered by QuickLaTeX.com \underset{x \to 0}{=} \dfrac12 x^2 + o \left( x^2 \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c04f12b353945c740259f312d52c3d45_l3.png)
![Rendered by QuickLaTeX.com \cos \left( x \right) \underset{x \to 0}{=} 1 - \dfrac12 x^2 + o \left( x^2 \right),](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-7c80b76c21743590e6024c6b1ad6cc4d_l3.png)
Ce qui donne finalement :
![Rendered by QuickLaTeX.com \dfrac{e^x - 1 - \sin \left( x \right)}{\cos \left( x \right) - 1}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-789f6d0482e1844691c78add5d0bc02e_l3.png)
![Rendered by QuickLaTeX.com \underset{x \to 0}{\sim} \dfrac{\dfrac12 x^2}{- \dfrac12 x^2}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-71e2fb79e53487f0fb84be319f408069_l3.png)
![Rendered by QuickLaTeX.com \underset{x \to 0}{\sim} - 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3dbd7f19715f9b9868bf7a53a2f5dbd5_l3.png)
Faisons un développement limité à l’ordre
![Rendered by QuickLaTeX.com 4](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-6111a899fd636b7a5238708f8679f6ec_l3.png)
![Rendered by QuickLaTeX.com \sqrt{1 + 2x} \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-bb651d443e8aca5932faf48a3ef1f1eb_l3.png)
![Rendered by QuickLaTeX.com 1 + x - \dfrac12 x^2 + \dfrac12 x^3 - \dfrac58 x^4 + o \left( x^4 \right),](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-234a7f10e0e553992563ef56b17658b5_l3.png)
![Rendered by QuickLaTeX.com \cos \left( x \right) \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-57126fed168b977bf0bc3377fb32dcea_l3.png)
![Rendered by QuickLaTeX.com 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} + o \left( x^4 \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-8db984a6bd93c20a21d31b99dec4ad5c_l3.png)
![Rendered by QuickLaTeX.com = 1 - \dfrac{x^2}{2} + \dfrac{x^4}{24} + o \left( x^4 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-306a85d89fac80eb71d1f3c576d3c1b6_l3.png)
Ainsi
![Rendered by QuickLaTeX.com \sqrt{1 + 2x } - x - \cos \left( x \right) \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e57dc045a5fc98fccfc9340a918911bf_l3.png)
![Rendered by QuickLaTeX.com \dfrac12 x^3 - \dfrac{7}{12} x^4 + o \left( x^4 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b1123abb5ba11ef42185af3b00ef71c5_l3.png)
Ce qui donne finalement :
![Rendered by QuickLaTeX.com \dfrac{\sqrt{1 + 2x } - x - \cos \left( x \right)}{x^3} \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9f84141ea3100123b9c472c2b75f5e6d_l3.png)
![Rendered by QuickLaTeX.com \dfrac12 - \dfrac{7}{12} x + o \left( x \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4519a86a55d161c2425df0deba883b9f_l3.png)
On a donc montré que
![Rendered by QuickLaTeX.com f](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png)
![Rendered by QuickLaTeX.com 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png)
![Rendered by QuickLaTeX.com 0,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-cf4770355d567cab8db571d879e923cb_l3.png)
![Rendered by QuickLaTeX.com f](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png)
![Rendered by QuickLaTeX.com 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png)
![Rendered by QuickLaTeX.com f ' \left( 0 \right) = - \dfrac{7}{12}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4a3e92d22dd0643e4a740bf277a9e4d1_l3.png)
![Rendered by QuickLaTeX.com f](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png)
![Rendered by QuickLaTeX.com 0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d077f1202cae43a1855e4e1bb5939948_l3.png)
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Exercice 4 :
On utilise la formule de Taylor-Young à à l’ordre
en
:
![Rendered by QuickLaTeX.com f \left( x \right) \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-054acaeab32584ff651bce9725e16d03_l3.png)
![Rendered by QuickLaTeX.com f \left( 0 \right) + f ' \left( 0 \right) x + \dfrac{f'' \left(0 \right)}{2} x^2 + o \left( x^2 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-10ea82c665e16f9116c35c9bf3df478f_l3.png)
Ainsi,
![Rendered by QuickLaTeX.com \dfrac{f \left( x \right) - f \left( 0 \right)}{x} \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2f1dac6cd32b0e56fea274cfbb3a8740_l3.png)
![Rendered by QuickLaTeX.com f ' \left( 0 \right) + \dfrac{f'' \left( 0 \right)}{2} x + o \left( x \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-91448a4445385a98d35693b29c52cb7d_l3.png)
De plus,
![Rendered by QuickLaTeX.com f ' \left( x \right) \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-8cc3c2e3ce79844f64b3a53031537b22_l3.png)
![Rendered by QuickLaTeX.com f ' \left( 0 \right) + f'' \left( 0 \right) x + o \left( x \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-f0eda349e987350761ab9a08daeb281f_l3.png)
Soit finalement
![Rendered by QuickLaTeX.com f ' \left( x \right) - \dfrac{f \left( x \right) - f \left( 0 \right)}{x} \underset{x \to 0}{=}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-85d33ac2c63830349108c61ef06e6829_l3.png)
![Rendered by QuickLaTeX.com \dfrac12 f '' \left( 0 \right) x + o \left( x \right),](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-dbcb40495e8172693e1b9e8ca3e7e3e6_l3.png)
ce qui donne
N’attendez pas le dernier moment pour réviser les maths en ECG1. Faites la différence avec vos camarades en prenant de l’avance sur les chapitres de maths à venir :