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zgHTV6ybt+A9)59qD)UEDnN7!Tj>0f9bKl!WHQ{qBq9NG^;X4LWQbKIyf2+!s;IJ$@ z_gv4*(aTBUs3h*Zh|qGHgkC^&Z)v1VDBxhOXunrv~a7f5gw_~dKB@IcuR-G zc}dORi8*~;;Ms|J^{Q*}O>2hRG{mw2GN@YDG9!h5L{~yMyqw~*5RQf+Nq$3zbX<$M zj2|Yq5J-OVkt)*+JCh!f3x=g=6XKyDiC(mcAYx9|PKUe-L+*m1KLZd#04VxG|D@pI zn8VVD%;s2;wP@3TLPFmrb>0fir&{(hZJMe*ZKu4dJ)N@;^ehBZx>Sj87;Wd-f#d(Nswtj(5SzkrpnGNP`Uk@(E%wG0CdS GVf_bY4{F5# diff --git a/TST_sti2d/01_Derivation/exercises.tex b/TST_sti2d/01_Derivation/exercises.tex index a0063aa..95d7e98 100644 --- a/TST_sti2d/01_Derivation/exercises.tex +++ b/TST_sti2d/01_Derivation/exercises.tex @@ -85,22 +85,55 @@ \item $g(x) = 0.1x^5 + 2x^4 + x$ \item $h(x) = 5x^8 + 4x^4 + \frac{1}{x}$ - \item $f(x) = (2x + 1) + (4x^2 - 1)$ - \item $f(x) = -3x^4 + \frac{1}{3}x^3 - \frac{3}{4}x^2 + 10$ - \item $f(x) = 5x + \frac{3x^2}{2}$ + \item $i(x) = (2x + 1) + (4x^2 - 1)$ + \item $j(x) = -3x^4 + \frac{1}{3}x^3 - \frac{3}{4}x^2 + 10$ + \item $k(x) = 5x + \frac{3x^2}{2}$ - \item $f(x) = (2x + 1)(4x^2 - 1)$ - \item $f(x) = x^2(x-1)$ - \item $f(x) = 5x(x^4 + x)$ + \item $l(x) = (2x + 1)(4x^2 - 1)$ + \item $m(x) = x^2(x-1)$ + \item $n(x) = 5x(x^4 + x)$ \end{enumerate} \end{multicols} \end{exercise} +\begin{exercise}[subtitle={Démonstrations}, step={2}, origin={Création}, topics={Dérivation}, tags={Dérivation, Démonstration}] + Dans cet exercice, nous allons démontrer quelques formules de dérivations (toutes les autres formules se démontrent de la même manière, les calculs sont juste un peu plus long). + \begin{enumerate} + \item Formule de dérivation de $f(x) = 1$. + + On veut connaître la dérivée de $f(x)$ au point $x$. Pour cela, on définit $x_1 = x$ et $x_2 = x + h$ avec $h$ un nombre que l'on rendra très petit. + \begin{enumerate} + \item Calculer $\dfrac{\Delta f}{\Delta x} = \dfrac{f(x_2) - f(x_1)}{x_2 - x_1}$. + \item En rendant $h$ très petit (proche de 0) déterminer $f'(x) = \dfrac{df}{dx}$. + \end{enumerate} + + \item Formule de dérivation de $g(x) = 2x$. + + On veut connaître la dérivée de $g(x)$ au point $x$. Pour cela, on définit $x_1 = x$ et $x_2 = x + h$ avec $h$ un nombre que l'on rendra très petit. + \begin{enumerate} + \item Calculer $\dfrac{\Delta g}{\Delta x}$. + \item En rendant $h$ très petit (proche de 0) déterminer $g'(x) = \dfrac{dg}{dx}$. + \end{enumerate} + + \item De la même façon que dans les deux questions précédentes, démontrer que la dérivée de $h(x) = x^2$ est + + $h'(x) = \dfrac{dh}{dx} = 2x$. + \item Démonstration de formule dérivation d'un produit: $f(x) = u(x)\times v(x)$ se dérive en $f'(x) = u'(x)\times v(x) + u(x)\times v'(x)$. + + Comme précédement, on pose $x_1 = x$ et $x_2 = x + h$ avec $h$ un nombre que l'on rendra très petit. + \begin{enumerate} + \item Exprimer en fonction de $u$ et $v$ la quantité $\dfrac{\Delta f}{\Delta x}$ + \item Simplifier l'expression $\dfrac{\Delta u}{\Delta x}\times v(x+h) + u(x) \times \dfrac{\Delta v}{\Delta x}$. + \item En déduire la formule de dérivation du produit. + \end{enumerate} + \end{enumerate} +\end{exercise} + \begin{exercise}[subtitle={Cercle et rayon}, step={2}, origin={Création}, topics={Dérivation}, tags={Dérivation, technique}] On travaille avec un cercle de rayon $R$. On définit les deux formules suivantes \[ \mbox{Périmètre: } P(R) = 2\pi R \qquad \qquad \qquad - \mbox{Aire } A(R) = \pi R^2 + \mbox{Aire: } A(R) = \pi R^2 \] \begin{enumerate} \item Calculer $\dfrac{dP}{dR}$. @@ -119,29 +152,7 @@ \end{enumerate} \end{exercise} -\begin{exercise}[subtitle={Démonstrations}, step={2}, origin={Création}, topics={Dérivation}, tags={Dérivation, Démonstration}] - Dans cet exercice, nous allons démontrer quelques formules de dérivations (toutes les autres formules se démontrent de la même manière, les calculs sont juste un peu plus long). - \begin{enumerate} - \item Formule de dérivation de $f(x) = 1$. - - On veut connaître la dérivée de $f(x)$ au point $x$. Pour cela, on définit $x_1 = x$ et $x_2 = x + h$ avec $h$ un nombre que l'on rendra très petit. - \begin{enumerate} - \item Calculer $\dfrac{\Delta f}{\Delta x}$. - \item En rendant $h$ très petit (proche de 0) déterminer $f'(x) = \dfrac{df}{dx}$. - \end{enumerate} - - \item Formule de dérivation de $g(x) = 2x$. - - On veut connaître la dérivée de $g(x)$ au point $x$. Pour cela, on définit $x_1 = x$ et $x_2 = x + h$ avec $h$ un nombre que l'on rendra très petit. - \begin{enumerate} - \item Calculer $\dfrac{\Delta g}{\Delta x}$. - \item En rendant $h$ très petit (proche de 0) déterminer $g'(x) = \dfrac{dg}{dx}$. - \end{enumerate} - - \item De la même façon que dans les deux questions précédentes, démontrer que la dérivée de $h(x) = x^2$ est - - $h'(x) = \dfrac{dh}{dx} = 2x$. - \end{enumerate} +\begin{exercise}[subtitle={}, step={3}, origin={Création}, topics={Dérivation}, tags={Dérivation, Physique}] \end{exercise} \collectexercisesstop{banque} diff --git a/TST_sti2d/01_Derivation/index.rst b/TST_sti2d/01_Derivation/index.rst index 8f7f88b..8d76103 100644 --- a/TST_sti2d/01_Derivation/index.rst +++ b/TST_sti2d/01_Derivation/index.rst @@ -2,7 +2,7 @@ Dérivation Tsti2d ################# :date: 2020-08-26 -:modified: 2020-08-26 +:modified: 2020-08-28 :authors: Benjamin Bertrand :tags: Dérivation, Trigonométrie, Physique :category: TST_sti2d @@ -40,6 +40,12 @@ Cours: Définition taux de variation, fonction dérivée, tangente et parallèle :height: 200px :alt: Vision mathématique de la dérivation. +Exercices techniques avec la démonstration de quelques formules de dérivation. + +.. image:: ./2E_derivation.pdf + :height: 200px + :alt: Exercices pure math sur la dérivation. + Étape 3: Problèmes physiques ============================