diff --git a/5 Derive Numerique.ipynb b/5 Derive Numerique.ipynb
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+++ b/5 Derive Numerique.ipynb	
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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# 2 Dérivée numérique"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Soit $f:[a,b]\\rightarrow \\mathbf R$, de classe $C^1$ et $x_0\\in[a,b]$.\n",
+    "La dérivée $f'(x_o)$ est donnée par\n",
+    "\n",
+    "$$\\begin{array}{lcl}\n",
+    "f'(x_0)&=&\\displaystyle\\lim_{h\\rightarrow 0^+}\\frac{f(x_0+h)-f(x_0)}{h},\\\\[2mm]\n",
+    "&=&\\displaystyle\\lim_{h\\rightarrow 0^+}\\frac{f(x_0)-f(x_0-h)}{h},\\\\[2mm]\n",
+    "&=&\\displaystyle\\lim_{h\\rightarrow 0}\\frac{f(x_0+h)-f(x_0-h)}{2h}.\n",
+    "\\end{array}$$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "Soient $x_0\\in[a,b]$, $(Dy)$ une approximation de $f'(x_0)$\n",
+    "et $(D^2y)$ une approximation de $f\"(x_0)$.\n",
+    "\n",
+    "On appelle\n",
+    "\n",
+    "* **différence finie progressive** l'approximation\n",
+    "  $$(Dy)^{P} = \\frac{f(x_0+h) - f(x_0)}{h}$$\n",
+    "\n",
+    "* **différence finie rétrograde** l'approximation\n",
+    "  $$(Dy)^{R} = \\frac{f(x_0) - f(x_0-h)}{h}$$\n",
+    "\n",
+    "* **différence finie centrée** l'approximation\n",
+    "  $$(Dy)^{C} = \\frac{f(x_0+\\frac12h) - f(x_0-\\frac12h)}{h}$$\n",
+    "\n",
+    "* **différence finie centrée d'ordre 2** l'approximation\n",
+    "  $$(D^2y)^{C} = \\frac{f(x_0+h) - 2f(x_0) + f(x_0-h)}{h^2}$$\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# importing libraries used in this book\n",
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Les différences finies progressive, rétrograde et centrée approchent la dérivée de la fonction en le point $x_0$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define a function and a point x0\n",
+    "f = lambda x : 0.25*(x-4)**3 - 4*(x-4)**2 +6\n",
+    "x0 = 5 \n",
+    "\n",
+    "# Choosing finite difference size\n",
+    "h = 4\n",
+    "\n",
+    "\n",
+    "# Defininig first order finite differences\n",
+    "DP = ( f(x0 + h) - f(x0) ) / h\n",
+    "DR = ( f(x0) - f(x0-h) ) / h\n",
+    "DC = ( f(x0 + h/2) - f(x0-h/2) ) / h\n",
+    "\n",
+    "# defining the straight lines with slope equal to the FDs\n",
+    "dfp = lambda x : f(x0) + DP*(x-x0)\n",
+    "dfr = lambda x : f(x0) + DR*(x-x0)\n",
+    "dfc = lambda x : f(x0-h/2) + DC*(x-x0+h/2)\n",
+    "\n",
+    "# Drowing points\n",
+    "z = np.linspace(-5, 22, 100)\n",
+    "\n",
+    "plt.plot(z, dfp(z), 'g:', z, dfr(z), 'b:', z, dfc(z), 'r:' )\n",
+    "plt.plot(z, f(z), 'k', x0,f(x0),'ro')\n",
+    "\n",
+    "plt.xlabel('x'); plt.ylabel('$f(x)$');\n",
+    "plt.legend(['$D^P$', '$D^R$', '$D^C$', '$f$'])\n",
+    "plt.show()\n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "La différence finie d'ordre 2 approche la dérivée deuxième de la fonction en le point $x_0$. \n",
+    "\n",
+    "Pour le voir graphiquement, on peut dessiner une fonction quadratique qui a la forme\n",
+    "\n",
+    "$$ p_2(x) = f(x_0) + (Dy)^{C}\\, (x-x_0) + \\frac12 (D^2y)\\, (x-x_0)^2$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define a function and a point x0\n",
+    "f = lambda x : 0.25*(x-4)**3 - 4*(x-4)**2 +6\n",
+    "x0 = 5 \n",
+    "\n",
+    "# Choosing finite difference size\n",
+    "h = 4\n",
+    "\n",
+    "\n",
+    "# Defininig first order centered finite differences\n",
+    "DC = ( f(x0 + h/2) - f(x0-h/2) ) / h\n",
+    "\n",
+    "# Defininig second order centered finite differences\n",
+    "D2 = ( f(x0 + h) - 2* f(x0) + f(x0-h) ) / (h**2)\n",
+    "\n",
+    "\n",
+    "# defining the parabola going through (x_0, y_0)\n",
+    "parabola = lambda x : f(x0) + DC*(x-x0) + 0.5 * D2 * (x-x0)**2\n",
+    "\n",
+    "# Drowing points\n",
+    "z = np.linspace(-5, 22, 100)\n",
+    "\n",
+    "plt.plot(z, parabola(z), 'g:' )\n",
+    "plt.plot(z, f(z), 'k', x0,f(x0),'ro')\n",
+    "\n",
+    "plt.xlabel('x'); plt.ylabel('$f(x)$');\n",
+    "plt.legend(['parabola', '$f$'])\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Exercice\n",
+    "\n",
+    "Il s'agit de vérifier numériquement que\n",
+    "\n",
+    "$$\\left\\vert f'(x_0)- (Dy)^{P} \\right\\vert=O(h^1).$$\n",
+    "\n",
+    "On pose $x_0=1$, $f(x)=\\sin(x)$ $\\forall x\\in\\mathbb R$\n",
+    "\n",
+    "Calculez l'erreur commise en utilisant la différence finie progressive.\n",
+    "\n",
+    "Quelle est la valeur de l'erreur pour h=0.1?\n",
+    "\n",
+    "Ensuite verifiez numériquement l'ordre pour $(Dy)^{R}$ et $(Dy)^{C}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def DPerror(f,df, x0,h) :\n",
+    "    # formule de differences finies \n",
+    "    DP = ( f(x0 + h) - f(x0) ) / h\n",
+    "    err = abs(DP-df(x0));\n",
+    "    return err\n",
+    "\n",
+    "# code :\n",
+    "# def DRerror(f,df, x0,h) :\n",
+    "    # formule de differences finies \n",
+    "\n",
+    "# def DCerror(f,df, x0,h) :\n",
+    "    # formule de differences finies \n",
+    "\n",
+    "\n",
+    "# This function just provides a line to compare the convergence in a log-log plot.\n",
+    "from FiniteDifferenceLib import sampleConvergence\n",
+    "# sampleConvergence(h,order,ref) :\n",
+    "#   computes a pseudo order of convergence on h\n",
+    "#   ref is a reference maximum value \n",
+    "\n",
+    "\n",
+    "x0 = 0.2\n",
+    "\n",
+    "h = 2**np.linspace(-10,-1,10)\n",
+    "plt.plot(h, DPerror(np.sin, np.cos, x0, h) , 'o' )\n",
+    "#plt.plot(h, DRerror(np.sin, np.cos, x0, h) , '*' )\n",
+    "#plt.plot(h, DCerror(np.sin, np.cos, x0, h) , '*' )\n",
+    "\n",
+    "\n",
+    "plt.plot(h, sampleConvergence(h, 1, 1e-1) , ':',\n",
+    "        h, sampleConvergence(h, 2, 1e-1) , ':')\n",
+    "\n",
+    "plt.xlabel('h'); plt.ylabel('BDF2 error');\n",
+    "#plt.legend(['DP error','DR error', 'DC error', '$O(h^1)$', '$O(h^2)$'])\n",
+    "plt.xscale('log') \n",
+    "plt.yscale('log')\n",
+    "plt.show()\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Exercice\n",
+    "\n",
+    "Il s'agit de vérifier numériquement que\n",
+    "\n",
+    "$$\\left\\vert f\"(x_0)- (D^2y) \\right\\vert=O(h^2).$$\n",
+    "\n",
+    "On pose $x_0=1$, $f(x)=\\sin(x)$ $\\forall x\\in\\mathbb R$\n",
+    "\n",
+    "Calculez l'erreur commise par ce schéma.\n",
+    "\n",
+    "Que vaut l'erreur pour h=0.1?\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Code\n",
+    "# def D2error(f,df, x0,h) :\n",
+    "    # formule de differences finies BDF2\n",
+    "\n",
+    "x0 = 0.2\n",
+    "\n",
+    "df = lambda x : -np.sin(x)\n",
+    "\n",
+    "h = 2**np.linspace(-10,-1,10)\n",
+    "#plt.plot(h, D2error(np.sin, df, x0, h) , 'o' )\n",
+    "plt.plot(h, sampleConvergence(h, 1, 1e-2) , ':',\n",
+    "        h, sampleConvergence(h, 2, 1e-2) , ':')\n",
+    "\n",
+    "plt.xlabel('h'); plt.ylabel('D2 error');\n",
+    "#plt.legend(['D2 error', '$O(h^1)$', '$O(h^2)$'])\n",
+    "plt.xscale('log') \n",
+    "plt.yscale('log')\n",
+    "plt.show()\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Exercice\n",
+    "\n",
+    "Il s'agit de vérifier numériquement que\n",
+    "\n",
+    "$$\\left\\vert f'(x_0)-\\dfrac{3f(x_0)-4f(x_0-h)+f(x_0-2h)}{2h}\\right\\vert=O(h^2).$$\n",
+    "\n",
+    "Cette formule de différences finies est à l'origine du schéma \"BDF2'' pour résoudre numériquement des équations différentielles (Chapitre 9 du livre).\n",
+    "\n",
+    "On pose $x_0=1$, $f(x)=\\sin(x)$ $\\forall x\\in\\mathbb R$\n",
+    "\n",
+    "Calculez l'erreur commise par ce schéma.\n",
+    "\n",
+    "Que vaut l'erreur pour h=0.1?<br/><br/>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.7.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/FiniteDifferenceLib.py b/FiniteDifferenceLib.py
new file mode 100644
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--- /dev/null
+++ b/FiniteDifferenceLib.py
@@ -0,0 +1,20 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Created on Thu Jan 30
+
+@author: Simone Deparis
+"""
+
+import numpy as np
+
+# This function just provides a line to compare the convergence in a log-log plot.
+def sampleConvergence(h,order,ref) :
+    # computes a pseudo order of convergence on h
+    # ref is a reference maximum value 
+    return h**order * (ref/np.max(np.abs(h))**order)
+
+
+   
+    
+