From e5a430acb821897a0ba8379362a6a01d6933cdf6 Mon Sep 17 00:00:00 2001 From: Pier-Luc St-Onge Date: Mon, 13 May 2024 13:01:12 -0400 Subject: [PATCH] =?UTF-8?q?Chap=204=20-=20=C3=89quations=20LaTeX=20et=20gr?= =?UTF-8?q?aphique=20de=2010=20rectangles?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 4-collectives.ipynb | 22 +++++++++++++++++++--- images/integration-rectangles.png | Bin 0 -> 17748 bytes 2 files changed, 19 insertions(+), 3 deletions(-) create mode 100644 images/integration-rectangles.png diff --git a/4-collectives.ipynb b/4-collectives.ipynb index f8f77b3..80d8826 100644 --- a/4-collectives.ipynb +++ b/4-collectives.ipynb @@ -425,18 +425,34 @@ }, { "cell_type": "markdown", - "id": "780f805a-0702-4bbd-8895-fa9789e78ce1", + "id": "ceb94aaa-b702-4490-a433-adeb4964d90c", "metadata": {}, "source": [ "### Exemple - Intégration par la méthode des rectangles\n", "Soit la fonction :\n", "\n", - "$$f(x) = \\sin^2{x \\mathrm{e}^{-x}}$$\n", + "$$f(x) = \\sin^2(x) \\ \\mathrm{e}^{-x}$$\n", "\n", "Calculons en parallèle une approximation de l'intégrale $I$\n", "de cette fonction dans l'intervalle allant de $0$ à $\\pi$ :\n", "\n", - "$$I = \\int_{0}^{\\pi} \\sin^2{x \\mathrm{e}^{-x}} \\mathrm{d}x$$" + "$$I = \\int_{0}^{\\pi} \\sin^2(x) \\ \\mathrm{e}^{-x} \\mathrm{d}x\n", + " = 2(1 - \\mathrm{e}^{-\\pi}) / 5$$" + ] + }, + { + "cell_type": "markdown", + "id": "b8743ecd-5427-46e9-b488-c3ea24f55ea8", + "metadata": {}, + "source": [ + "Pour ce faire, nous calculerons $N$ rectangles de largeur\n", + "$\\Delta{}x = \\pi / N$ et d'abscisse $x_i = i\\Delta{}x$,\n", + "où $i$ varie de $0$ à $N - 1$ :\n", + "\n", + "$$I \\approx \\Sigma_{i=0}^{N-1} \\sin^2(x_i) \\ \\mathrm{e}^{-x_i} \\Delta{}x$$\n", + "$$I \\approx \\Sigma_{i=0}^{N-1} \\sin^2(i\\pi/N) \\ \\mathrm{e}^{-i\\pi/N} \\pi / N$$\n", + "\n", + "![Intégration par la méthode des rectangles](images/integration-rectangles.png)" ] }, { diff --git 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