Derivando.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\LARGE{ \\textbf{LABORATORIO  VIRTUAL CON PYTHON}}$\n",
    "\n",
    "$\\LARGE{ \\textbf{LA DERIVADA Y SUS APLICACIONES}}$\n",
    "\n",
    "$\\Large{Autor: Jesús Edilberto Espinola Gonzales}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Parte I: DERIVADA DE UNA FUNCIÓN EN UN PUNTO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Presentación Geométrica"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Aproximaciones mediante secantes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dada una función $f(x): I \\rightarrow \\mathbb{R}$, un valor $x_0 \\in I$ y un valor $h$ trazamos la recta secante que pasa por los puntos $(x_0,f(x_0))$ y $(x_0+h,f(x_0+h))$. Conforme el valor de $h$ se va haciendo más pequeño las rectas secantes se irán aproximando a la recta tangente a la gráfica de la función en el punto $(x_0,f(x_0))$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Ejemplos}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Para un valor de $\\textit{h}$ determinado}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sea la función $f(x)=3x^2+5$, graficar la recta secante, que pasa por los puntos $(x_0,f(x_0))$ y $(x_0+h,f(x_0+h))$ para los valores $x_0$ y $h$ que se asignen, como ejemplo inicial $x_0=-5$ y $h=8$.\n",
    "\n",
    "En el Módulo 1 se puede graficar una función dada y la recta secante para $x_0$ y $h$ dados.\n",
    "\n",
    "En las primeras líneas del siguiente bloque de código se puede definir la función, por ejemplo $f=3x^2+5$, así mismo se pueden asignar distintos valores para $x_0$ y $h$, por ejemplo $x_0=-5$ y $h=8$.\n",
    "\n",
    "Para obtener la gráfica hay que posicionarse en la celda de código y hacer click en la opción 'Run' del menú de la parte superior."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Módulo 1: Grafica función f dada y secante que pasa por $(x_0,f(x_0))$ y $(x_0+h,f(x_0+h))$, para $x_0$ y $h$ dados"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[1mSOLUCIÓN:RECTA SECANTE\n",
      "\u001b[0m\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ***************** CONSIDERACIONES PARA LA DEFINICIÓN DE FUNCIONES*********************************\n",
    "# El producto hay que indicarlo con asterisco (*).Las potencias se indican con doble asterisco (**).\n",
    "# La función seno(x) se define como sin(x), la funcion arcoseno(x) como arcsin(x).\n",
    "# La función logaritmo neperiano como log(x). La función exponencial como exp(x).\n",
    "# La función valor absoluto como abs(x).\n",
    "# Definir la función f después de la línea \"AQUÍ SE DEFINE LA FUNCIÓN\"\n",
    "# Asingar valores para x0 y h después de la línea \"AQUÍ SE ASIGNAN LOS VALORES x0 y h \"\n",
    "# ***************************************************************************************************\n",
    "\n",
    "# *******************DEFINICIÓN DE LA FUNCIÓN********************************************************\n",
    "def f(x):\n",
    "    #Llamada de funciones de numpy\n",
    "    from numpy import sin, cos, tan, arcsin, arccos, arctan, log, exp\n",
    "    \n",
    "    # **************************************************************************** \n",
    "    #            AQUÍ SE DEFINE LA FUNCIÓN    \n",
    "    f=3*x**2+5\n",
    "    \n",
    "    # ****************************************************************************\n",
    "    \n",
    "    return f\n",
    "\n",
    "# ********************************************************************************\n",
    "#                 AQUÍ SE ASIGNAN LOS VALORES x0 Y h \n",
    "x0=-5\n",
    "h=8\n",
    "\n",
    "# ********************************************************************************\n",
    "\n",
    "\n",
    "%matplotlib inline\n",
    "def Graficasecante(x0,h):\n",
    "    #De matplotlib se importa pyplot como plt\n",
    "    import matplotlib.pyplot as plt\n",
    "    #Se importa numpy como np\n",
    "    import numpy as np\n",
    "\n",
    "    \n",
    "    # Se evalua la función en el rango de [x0-h,x0+2h]\n",
    "    x = np.linspace(x0-h,x0+2*h,101)\n",
    "    #Se le pasa a la funcion los valores definidos en x.\n",
    "    y = f(x)\n",
    "\n",
    "    #Se calcula la recta secante, de la forma y=(y1-y0)/(x1-x0)*(x-x0)+y0\n",
    "    \n",
    "    rsec = (f(x0+h)-f(x0))/h*(x-x0)+f(x0)  \n",
    "    \n",
    "    #Para el valor h se grafica la recta secante que pasa por (x0,f(x0)) y (x0+h,f(x0+h))\n",
    "    \n",
    "    # Se grafica la funcion y la secante.\n",
    "    plt.title('Secante con h  = '+str(h))\n",
    "    plt.plot(x,y,'b',x0,f(x0),'om',x0+h,f(x0+h),'om',x,rsec,'r')\n",
    "    \n",
    "    # Indicar sólo los valores requeridos en los ejes\n",
    "    plt.xticks([x0,x0+h])\n",
    "    plt.yticks([f(x0),f(x0+h),max(y)])\n",
    "    # Se mueven los ejes coordenados con intersección en el origen\n",
    "    ax = plt.gca()  # 'obtener el eje actual'\n",
    "    ax.spines['right'].set_color('none')\n",
    "    ax.spines['top'].set_color('none')\n",
    "    ax.xaxis.set_ticks_position('bottom')\n",
    "    ax.spines['bottom'].set_position(('data',0))\n",
    "    ax.yaxis.set_ticks_position('left')\n",
    "    ax.spines['left'].set_position(('data',0))\n",
    "    \n",
    "    #Incluir anotación en la gráfica\n",
    "    plt.annotate(r'('+str(x0)+','+str(f(x0))+')',xy=(x0, f(x0)), xycoords='data',\n",
    "            xytext=(+10, +30), textcoords='offset points', fontsize=10,\n",
    "            arrowprops=dict(arrowstyle=\"->\", connectionstyle=\"arc3,rad=.2\"))\n",
    "    \n",
    "        #Incluir anotación en la gráfica\n",
    "    plt.annotate(r'('+str(x0+h)+','+str(f(x0+h))+')',xy=(x0+h, f(x0+h)), xycoords='data',\n",
    "            xytext=(+10, +30), textcoords='offset points', fontsize=10,\n",
    "            arrowprops=dict(arrowstyle=\"->\", connectionstyle=\"arc3,rad=.2\"))\n",
    "    \n",
    "    #Se muestra la grafica.\n",
    "    plt.show()\n",
    "\n",
    "print('\\033[1m'+'SOLUCIÓN:RECTA SECANTE\\n'+'\\033[0m')\n",
    "#Llamada a la función Graficasecante para pintar la función y la secante\n",
    "Graficasecante(x0,h)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Para  diferentes valores de $\\textit{h}$}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sea la función $f(x)=3x^2+5$, graficar las rectas secantes para los valores iniciales $x_0$ y $h$ que se asignen. A partir de allí graficar de forma sucesiva las secantes que pasan por los puntos $(x_0,f(x_0))$ y $(x_0+h,f(x_0+h))$, para valores de $h$ cada vez más pequeños.\n",
    "\n",
    "En el Módulo 2 de código se puede graficar una función dada, con valores iniciales para $x_0$ y $h$, a partir de allí de forma interactiva se grafica la recta secante respectiva para cuando el valor de $h$ se aproxima a cero.\n",
    "\n",
    "Como en el Módulo 1, en las primeras líneas de programación se puede definir la función, por ejemplo $f(x)=3x^2+5$; así mismo se pueden asignar  valores para $x_0$ y $h$ iniciales, por ejemplo $x_0=-5$ y $h=8$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Módulo 2: Grafica función f dada y secantes que pasan por  $(𝑥_0,𝑓(𝑥_0))$  y  $(𝑥_0+ℎ,𝑓(𝑥_0+ℎ))$, para $x_0$ y $h$ dados, variando  $h$ hacia cero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Using matplotlib backend: Qt5Agg\n",
      "\u001b[1mSOLUCIÓN: RECTA SECANTE CON VARIACIÓN DE h (INTERACTIVO)\n",
      "\u001b[0m\n"
     ]
    },
    {
     "data": {
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       "    /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
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       "  }\n",
       "\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
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       "        // switch from oninput to onchange because IE <= 11 does not conform\n",
       "        // with W3C specification. It ignores oninput and onchange behaves\n",
       "        // like oninput. In contrast, Mircosoft Edge behaves correctly.\n",
       "        slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
       "        slider.setAttribute('oninput', null);\n",
       "    }\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<style>\n",
       ".animation {\n",
       "    display: inline-block;\n",
       "    text-align: center;\n",
       "}\n",
       "input[type=range].anim-slider {\n",
       "    width: 374px;\n",
       "    margin-left: auto;\n",
       "    margin-right: auto;\n",
       "}\n",
       ".anim-buttons {\n",
       "    margin: 8px 0px;\n",
       "}\n",
       ".anim-buttons button {\n",
       "    padding: 0;\n",
       "    width: 36px;\n",
       "}\n",
       ".anim-state label {\n",
       "    margin-right: 8px;\n",
       "}\n",
       ".anim-state input {\n",
       "    margin: 0;\n",
       "    vertical-align: middle;\n",
       "}\n",
       "</style>\n",
       "\n",
       "<div class=\"animation\">\n",
       "  <img id=\"_anim_img0cbdeea308834af59e43cbd11c8c2eb6\">\n",
       "  <div class=\"anim-controls\">\n",
       "    <input id=\"_anim_slider0cbdeea308834af59e43cbd11c8c2eb6\" type=\"range\" class=\"anim-slider\"\n",
       "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
       "           oninput=\"anim0cbdeea308834af59e43cbd11c8c2eb6.set_frame(parseInt(this.value));\"></input>\n",
       "    <div class=\"anim-buttons\">\n",
       "      <button title=\"Decrease speed\" onclick=\"anim0cbdeea308834af59e43cbd11c8c2eb6.slower()\">\n",
       "          <i class=\"fa fa-minus\"></i></button>\n",
       "      <button title=\"First frame\" onclick=\"anim0cbdeea308834af59e43cbd11c8c2eb6.first_frame()\">\n",
       "        <i class=\"fa fa-fast-backward\"></i></button>\n",
       "      <button title=\"Previous frame\" onclick=\"anim0cbdeea308834af59e43cbd11c8c2eb6.previous_frame()\">\n",
       "          <i class=\"fa fa-step-backward\"></i></button>\n",
       "      <button title=\"Play backwards\" onclick=\"anim0cbdeea308834af59e43cbd11c8c2eb6.reverse_animation()\">\n",
       "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
       "      <button title=\"Pause\" onclick=\"anim0cbdeea308834af59e43cbd11c8c2eb6.pause_animation()\">\n",
       "          <i class=\"fa fa-pause\"></i></button>\n",
       "      <button title=\"Play\" onclick=\"anim0cbdeea308834af59e43cbd11c8c2eb6.play_animation()\">\n",
       "          <i class=\"fa fa-play\"></i></button>\n",
       "      <button title=\"Next frame\" onclick=\"anim0cbdeea308834af59e43cbd11c8c2eb6.next_frame()\">\n",
       "          <i class=\"fa fa-step-forward\"></i></button>\n",
       "      <button title=\"Last frame\" onclick=\"anim0cbdeea308834af59e43cbd11c8c2eb6.last_frame()\">\n",
       "          <i class=\"fa fa-fast-forward\"></i></button>\n",
       "      <button title=\"Increase speed\" onclick=\"anim0cbdeea308834af59e43cbd11c8c2eb6.faster()\">\n",
       "          <i class=\"fa fa-plus\"></i></button>\n",
       "    </div>\n",
       "    <form title=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_select0cbdeea308834af59e43cbd11c8c2eb6\"\n",
       "          class=\"anim-state\">\n",
       "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_0cbdeea308834af59e43cbd11c8c2eb6\"\n",
       "             checked>\n",
       "      <label for=\"_anim_radio1_0cbdeea308834af59e43cbd11c8c2eb6\">Once</label>\n",
       "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_0cbdeea308834af59e43cbd11c8c2eb6\"\n",
       "             >\n",
       "      <label for=\"_anim_radio2_0cbdeea308834af59e43cbd11c8c2eb6\">Loop</label>\n",
       "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_0cbdeea308834af59e43cbd11c8c2eb6\"\n",
       "             >\n",
       "      <label for=\"_anim_radio3_0cbdeea308834af59e43cbd11c8c2eb6\">Reflect</label>\n",
       "    </form>\n",
       "  </div>\n",
       "</div>\n",
       "\n",
       "\n",
       "<script language=\"javascript\">\n",
       "  /* Instantiate the Animation class. */\n",
       "  /* The IDs given should match those used in the template above. */\n",
       "  (function() {\n",
       "    var img_id = \"_anim_img0cbdeea308834af59e43cbd11c8c2eb6\";\n",
       "    var slider_id = \"_anim_slider0cbdeea308834af59e43cbd11c8c2eb6\";\n",
       "    var loop_select_id = \"_anim_loop_select0cbdeea308834af59e43cbd11c8c2eb6\";\n",
       "    var frames = new Array(20);\n",
       "    \n",
       "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbAAAAEgCAYAAADVKCZpAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90\\\n",
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       "AABJRU5ErkJggg==\\\n",
       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        anim0cbdeea308834af59e43cbd11c8c2eb6 = new Animation(frames, img_id, slider_id, 299.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0x25c1a833520>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ***************** CONSIDERACIONES PARA LA DEFINICIÓN DE FUNCIONES*********************************\n",
    "# El producto hay que indicarlo con asterisco (*).Las potencias se indican con doble asterisco (**).\n",
    "# La función seno(x) se define como sin(x), la funcion arcoseno(x) como arcsin(x).\n",
    "# La función logaritmo neperiano como log(x). La función exponencial como exp(x).\n",
    "# La función valor absoluto como abs(x).\n",
    "# Definir la función f después de la línea \"AQUÍ SE DEFINE LA FUNCIÓN\"\n",
    "# Asingar valores para x0 y h después de la línea \"AQUÍ SE ASIGNAN LOS VALORES x0 y h \"\n",
    "# ***************************************************************************************************\n",
    "\n",
    "# *******************DEFINICIÓN DE LA FUNCIÓN********************************************************\n",
    "def f(x):\n",
    "    #Llamada de funciones de numpy\n",
    "    from numpy import sin, cos, tan, arcsin, arccos, arctan, log\n",
    "    \n",
    "    # **************************************************************************** \n",
    "    #            AQUÍ SE DEFINE LA FUNCIÓN\n",
    "    f=3*x**2+5\n",
    "    # ****************************************************************************\n",
    "    \n",
    "    return f\n",
    "\n",
    "# ********************************************************************************\n",
    "#                 AQUÍ SE ASIGNAN LOS VALORES x0 Y h \n",
    "x0=-5\n",
    "h=8\n",
    "\n",
    "# ********************************************************************************\n",
    "\n",
    "%matplotlib auto\n",
    "# IMPORTAMOS LIBRERÍAS\n",
    "import matplotlib.pyplot as plt\n",
    "#Se importa numpy como np\n",
    "import numpy as np\n",
    "from mpl_toolkits.mplot3d import axes3d\n",
    "import matplotlib.animation as animation\n",
    "from matplotlib import rc\n",
    "\n",
    "rc('animation', html='jshtml')\n",
    "\n",
    "#datos, para la gráfica de la curva de f(x)\n",
    "x = np.linspace(x0-h,x0+2*h,101)\n",
    "y=f(x)\n",
    "\n",
    "#grafica\n",
    "fig=plt.figure()\n",
    "ax=fig.gca()\n",
    "t=np.linspace(h,0.001,20)\n",
    "\n",
    "#actualizar\n",
    "def actualizar(i):\n",
    "    ax.clear()\n",
    "    h=t[i]\n",
    "    rsec = (f(x0+h)-f(x0))/h*(x-x0)+f(x0)      \n",
    "    ax.plot(x,y,'b',x0,f(x0),'om',x0+h,f(x0+h),'om',x,rsec,'r')        \n",
    "    plt.title('h='+str(np.round(t[i],3)))\n",
    "    plt.xlim(min(x),max(x))\n",
    "    plt.ylim(min(y)-np.mean(y),max(y))\n",
    "    \n",
    "\n",
    "print('\\033[1m'+'SOLUCIÓN: RECTA SECANTE CON VARIACIÓN DE h (INTERACTIVO)\\n'+'\\033[0m')\n",
    "#animación\n",
    "\n",
    "ani=animation.FuncAnimation(fig,actualizar,range(len(t)),interval=300,repeat=False)\n",
    "\n",
    "#Mostrar el gráfico animado interactivo\n",
    "ani"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Explicación de las secantes para cada valor de $h$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para cada valor de $h$ la recta secante que pasa   $(x_0,f(x_0))$ y $(x_0+h,f(x_0+h))$ tiene por ecuación $y=\\frac{f(x_0+h)-f(x_0)}{h}(x-x_0)+f(x_0)$, cuya pendiente es $\\frac{f(x_0+h)-f(x_0)}{h}$.\n",
    "Las gráficas anteriores muestran que conforme $h$ toma valores más pequeños, la recta secante se aproxima a la recta tangente a la gráfica en el punto $(x_0,f(x_0))$.\n",
    "La pregunta que surge es ¿Cuál es la pendiente de la recta tangente?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Definición de la derivada de una función $f$ en un punto $x_0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La derivada de la función $f(x)$ en un punto abscisa $x_0$ se escribe con el símbolo $f'(x)$ y se define como\n",
    "\n",
    "$$f'(x_0)=lim_{h \\to 0}\\frac{f(x_0+h)-f(x_0)}{h}$$\n",
    "\n",
    "a) Si el límite existe entonces diremos que la función $f$ es derivable en el punto $x_0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b) La derivada de una función en un punto es un número real, que puede ser positivo, negativo o cero."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c) La derivada de una función en un punto coincide con la pendiente de la recta tangente en ese punto"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Ejemplos  de funciones con sus tangentes en un punto }}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Probamos con algunas funciones y sus rectas tangentes mediante la derivada en un punto"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "i) $f(x)=3x^2+5, \\qquad x_0=-5$\n",
    "La derivada para esta función en $x_0=-5$, según la definición es \n",
    "\n",
    "$$f'(-5)=lim_{h \\to 0}\\frac{f(-5+h)-f(-5)}{h}=lim_{h \\to 0}\\frac{3(-5+h)^2+5-(3(-5)^2+5)}{h}=-30$$\n",
    "\n",
    "Teniendo en cuenta del apartado $c)$ de la definición se tiene que la pendiente de la recta tangente a la gráfica de la ecuación en el punto $(-5,f(-5))=(-5,80)$ es $-30$. Por lo tanto la recta tangente en el punto $(-5,80)$ tiene por ecuación $y=-30(x+5)+80$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el Módulo 3 se grafica la recta tangente a la gráfica de la función $f$ en el punto $(x_o,f(x_0))$ con pendiente $m$. Se da como ejemplo $f(x)=3x^2+5$, $x_0=-5$ y pendiente $m=-30$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se recomienda probar con la función $f(x)=sen (\\frac{x}{2})+3$, en $x_0=0.7854$, $m=0.462$, pues $f'(0.7854)=0.462$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Módulo 3: Grafica función f dada y tangente que pasa por  $(𝑥_0,𝑓(𝑥_0))$  para $x_0$ y pendiente $m$ dados"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[1mSOLUCIÓN: RECTA QUE PASA POR UN PUNTO Y PENDIENTE DADOS\n",
      "\u001b[0m\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ***************** CONSIDERACIONES PARA LA DEFINICIÓN DE FUNCIONES*********************************\n",
    "# El producto hay que indicarlo con asterisco (*).Las potencias se indican con doble asterisco (**).\n",
    "# La función seno(x) se define como sin(x), la funcion arcoseno(x) como arcsin(x).\n",
    "# La función logaritmo neperiano como log(x). La función exponencial como exp(x).\n",
    "# La función valor absoluto como abs(x).\n",
    "# Definir la función f después de la línea \"AQUÍ SE DEFINE LA FUNCIÓN\"\n",
    "# Asingar valores para x0 y h después de la línea \"AQUÍ SE ASIGNAN LOS VALORES x0 y m \"\n",
    "# ***************************************************************************************************\n",
    "\n",
    "# *******************DEFINICIÓN DE LA FUNCIÓN********************************************************\n",
    "def f(x):\n",
    "    #Llamada de funciones de numpy\n",
    "    from numpy import sin, cos, tan, arcsin, arccos, arctan, log\n",
    "    \n",
    "    # **************************************************************************** \n",
    "    #            AQUÍ SE DEFINE LA FUNCIÓN    \n",
    "    f=3*x**2+5\n",
    "    \n",
    "    # ****************************************************************************\n",
    "    \n",
    "    return f\n",
    "\n",
    "# ********************************************************************************\n",
    "#                 AQUÍ SE ASIGNAN LOS VALORES x0 Y m\n",
    "x0=-5\n",
    "m=-30\n",
    "\n",
    "# ********************************************************************************\n",
    "\n",
    "\n",
    "%matplotlib inline\n",
    "def Graficatangente(x0,h):\n",
    "    #Se importa numpy como np\n",
    "    import numpy as np\n",
    "    #De matplotlib se importa pyplot como plt\n",
    "    import matplotlib.pyplot as plt\n",
    "\n",
    "    # Se evalua la función en el rango de [x0-h,x0+2h]\n",
    "    x = np.linspace(x0-h,x0+2*h,101)\n",
    "    #Se le pasa a la funcion los valores definidos en x.\n",
    "    y = f(x)\n",
    "\n",
    "    #Se calcula la recta tangente, de la forma y=m*(x-x0)+y0. Donde m es la pendiente \n",
    "    ytangente = m*(x-x0)+f(x0)  \n",
    "    \n",
    "    # Se grafica la funcion y la secante.\n",
    "    plt.plot(x,y,'b',x0,f(x0),'om',x,ytangente,'r')\n",
    "    \n",
    "    # Indicar sólo los valores requeridos en los ejes\n",
    "    plt.xticks([x0])\n",
    "    plt.yticks([f(x0),max(y)])\n",
    "    # Se mueven los ejes coordenados con intersección en el origen\n",
    "    ax = plt.gca()  # 'obtener el eje actual'\n",
    "    ax.spines['right'].set_color('none')\n",
    "    ax.spines['top'].set_color('none')\n",
    "    ax.xaxis.set_ticks_position('bottom')\n",
    "    ax.spines['bottom'].set_position(('data',0))\n",
    "    ax.yaxis.set_ticks_position('left')\n",
    "    ax.spines['left'].set_position(('data',0))\n",
    "    \n",
    "    #Incluir anotación en la gráfica\n",
    "    plt.annotate(r'('+str(x0)+','+str(f(x0))+')',xy=(x0, f(x0)), xycoords='data',\n",
    "            xytext=(+10, +30), textcoords='offset points', fontsize=10,\n",
    "            arrowprops=dict(arrowstyle=\"->\", connectionstyle=\"arc3,rad=.2\"))\n",
    "    \n",
    "    plt.title('Grafica de la función f y la recta en (x0,f(x0)) y pendiente m='+str(m))\n",
    "    \n",
    "    #Se muestra la grafica.\n",
    "    plt.show()\n",
    "\n",
    "#Llamada a la función Graficasecante para pintarlas\n",
    "print('\\033[1m'+'SOLUCIÓN: RECTA QUE PASA POR UN PUNTO Y PENDIENTE DADOS\\n'+'\\033[0m')\n",
    "Graficatangente(x0,m)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Práctica}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{I}$. Usando el Módulo 1 grafica las funciones $f(x)$ y sus rectas secantes en los  $x_0$ y $h$ indicados."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a)  $𝑓(𝑥)=x^3+4x^2-1, \\qquad x_0= -2, \\qquad h=3$.\n",
    "\n",
    "b) $f(x) = x^2-7x+3, \\qquad x_0=6, \\qquad h=4$.\n",
    "\n",
    "c) $f(x) = \\frac{10}{(x-2)^2}, \\qquad x_0=5, \\qquad h=1$.\n",
    "\n",
    "d) $f(x)=\\frac{-1}{x^2+6}, \\qquad x_0= 1, \\qquad h=3$.\n",
    "\n",
    "e) $f(x)=4^x, \\qquad x_0=0, \\qquad h=2$.\n",
    "\n",
    "f) Probar con diferentes funciones y diferentes valores $x_0$ y $h$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{II}$. Usando el módulo 2 grafíca las mismas funciones de ($\\textbf{I}$.) y los mismos valores $x_0$ y $h$ para visualizar las rectas secantes conforme $h$ tiende a cero."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{III}$. Usando el módulo 3 grafica las mismas funciones de ($\\textbf{I}$.) con los valores de $x_0$ y $m$ que se indican, respectivamente, y luego variando valores de $m$\n",
    "\n",
    "a) $x_0=-1$, $m=15$\n",
    "\n",
    "b) $x_0=5$, $m=3$\n",
    "\n",
    "c) $x_0=4$, $m=-\\frac{5}{2}$\n",
    "\n",
    "d) $x_0=2$, $m=\\frac{1}{25}$\n",
    "\n",
    "e) $x_0=1$, $m=5.55$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Parte II: FUNCIÓN DERIVADA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dada una función $f: I \\rightarrow \\mathbb{R}$, la derivada de $f$ en x está definida por $f'(x)=\\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{f}$ siempre que exista el límite. Para todos los valores $x$ para los que exista el límite, $f'$ es una función de $x$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Ejemplo}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sea $f(x)=4x^2-3$, calcula la función derivada $f'(x)$.\n",
    "\n",
    "Por la definición se tiene\n",
    "\n",
    "$$f'(x)=\\lim_{h \\to 0} \\frac{4(x+h)^2-3-(4x^2-3)}{h}=\\lim_{h \\to 0}\\frac{8xh+h^2}{h}=8x$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el Móculo 4 se halla y grafica la funcion derivada para una función $f$ dada. Se grafica la función inicial en color azul y la función derivada en color rojo.\n",
    "\n",
    "Como ejemplo inicial se toma $f(x)=4x^2-3$. Pero se pueden definir otras funciones."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Módulo 4: Calcula la función derivada de una función $f$ dada y muestra las gráficas de la función inicial y de la función derivada."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[1mSOLUCIÓN: FUNCIÓN DERIVADA\n",
      "\u001b[0m\n",
      "Función dada, en color azul, f(x)= 4*x**2 - 3\n",
      "\n",
      "Función derivada, en color rojo, f´(x)= 8*x\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ***************** PARA LA DEFINICIÓN DE FUNCIONES*************************************************\n",
    "# El producto hay que indicarlo con asterisco (*).Las potencias se indican con doble asterisco (**).\n",
    "# La función seno(x) se define como sin(x),# la funcion arcoseno(x) SE DEFINE COMO asin(x),\n",
    "# La función arcoscosno(x) como acos(x), la funíon arcotangente(x) como atan(x)\n",
    "# La función logaritmo neperiano como log(x). La función exponencial como exp(x).\n",
    "# La función valor absoluto como abs(x).\n",
    "# Definir la función f después de la línea \"AQUÍ SE DEFINE LA FUNCIÓN\"\n",
    "# **************************************************************************************************\n",
    "#Llamada de funciones de sympy\n",
    "from sympy import sin, cos, tan, asin, acos, atan, log, exp\n",
    "\n",
    "# ****************************************************************************\n",
    "import sympy as sp\n",
    "x=sp.Symbol('x')\n",
    "# **************************************************************************** \n",
    "#            AQUÍ SE DEFINE LA FUNCIÓN\n",
    "f=4*x**2-3\n",
    "\n",
    "# ****************************************************************************\n",
    "\n",
    "%matplotlib inline\n",
    "from sympy.plotting import plot\n",
    "\n",
    "fp=sp.diff(f,x)\n",
    "p=plot(f,title='Gráfica de la función f y de la función derivada',line_color='blue',show=False)\n",
    "p1=plot(fp,line_color='red',show=False)\n",
    "p.append(p1[0])\n",
    "\n",
    "print('\\033[1m'+'SOLUCIÓN: FUNCIÓN DERIVADA\\n'+'\\033[0m')\n",
    "print('Función dada, en color azul, f(x)=',str(f)+'\\n')\n",
    "print('Función derivada, en color rojo, f´(x)=',str(fp)+'\\n')\n",
    "\n",
    "p.show()\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Observación}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Es importante notar que en la gráfica del Bloque de código 4 se presenta la gráfica de la función $f$ y de su función dericada $f'$, que no es lo mismo que la recta tangente en punto alguno."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algunas reglas de derivación"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f(x)=C, \\qquad f'(x)=0$\n",
    "\n",
    "$f(x)=x, \\qquad f'(x)=1$\n",
    "\n",
    "$f(x)=x^n,  \\  \\  n \\in \\mathbb{R}, \\qquad f'(x)=nx^{n-1}$\n",
    "\n",
    "$f(x)=ln (x), \\qquad f'(x)=\\frac{1}{x}$\n",
    "\n",
    "$f(x)=e^x, \\qquad f'(x)=e^x$\n",
    "\n",
    "$f(x)=a^x, \\ \\ a>0, \\qquad f'(x)=a^x ln (a)$\n",
    "\n",
    "$f(x)=sen(x), \\qquad f'(x)=cos(x)$\n",
    "\n",
    "$f(x)=cos(x), \\qquad f'(x)=-sen(x)$\n",
    "\n",
    "$f(x)=tan(x), \\qquad f'(x)=1+ tan^2(x)$\n",
    "\n",
    "$f(x)=arcsen(x), \\qquad f'(x)=\\frac{1}{\\sqrt{1-x^2}}$\n",
    "\n",
    "$f(x)=arccos(x), \\qquad f'(x)=\\frac{-1}{\\sqrt{1-x^2}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Propiedades de la derivada"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si $f$ y $g$ son funciones derivables en $x$, también lo son $cf$, $f+g$, $fg$, $\\frac{f}{g}$ y $f \\circ g$, además se verifica\n",
    "\n",
    "$[cf(x)]'=cf'(x),$\n",
    "\n",
    "$[f(x)+g(x)]' = f'(x)+g'(x),$\n",
    "\n",
    "$(f(x)g(x))' = f'(x)g(x)+f(x)g'(x)$\n",
    "\n",
    "$(\\frac{f}{g})'(x) = \\frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2}, \\qquad si \\ \\ \\ g(x) \\neq 0$\n",
    "\n",
    "$(g \\circ f)'(x) = g'(f(x))f'(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Ejemplo}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dada la función $f(x)=2x^3+1$, calcular la función derivada y la derivada en el punto $x_0=3$, dibujar la gráfica de la función $f$ y la recta tangente en el punto $(x_0,f(x_0))$\n",
    "\n",
    "Aplicando las reglas de derivación se tiene $f'(x)= 6x^2$, por otro lado, evaluando la función $f$, y $f'$ en $x_0=3$ se tiene $f(3)=55$ y $f'(3)=54$, por lo que la recta tangente tiene por ecuación $y=54(x-3)+55$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el Módulo 5 se halla la función derivada $f'$ para la función $f$ dada, se evaúa la derivada en un punto $x_0$ dado, se obtiene la ecuación de la recta tangente que pasa por $(x_0,f(x_0))$ usando la pendiente como $f'(x_0)$; y se grafíca la función f y la recta tangente.\n",
    "\n",
    "Como ejemplo inicial se toma $f(x)=2x^3+1$ y $x_0=3$. Pero se pueden definir otras funciones y otros puntos $x_0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Módulo 5: Calcula la función derivada de una función dada $f$, su valor en un punto indicado $x_0$, así mismo grafíca la funcion $f$ y la recta tangente a la gráfica de $f$ en el punto $(x_0,f(x_0))$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[1mSOLUCIÓN: RECTA TANGENTE A UNA CURVA\n",
      "\u001b[0m\n",
      "Función dada, en color azul, f(x)= x**2 + 3\n",
      "\n",
      "Función derivada f´(x)= 2*x\n",
      "\n",
      "El valor de la función derivada evaluada en x0=3   es  f´(3) = 6\n",
      "\n",
      "Recta tangente a f, en color rojo, en el punto (x0,f(x0))=(3,12)\n",
      "\n",
      "Ecuación de la recta tangente: y = 6(x-3)+(12)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ***************** PARA LA DEFINICIÓN DE FUNCIONES*************************************************\n",
    "# El producto hay que indicarlo con asterisco (*).Las potencias se indican con doble asterisco (**).\n",
    "# La función seno(x) se define como sin(x),# la funcion arcoseno(x) SE DEFINE COMO asin(x),\n",
    "# La función arcoscosno(x) como acos(x), la funíon arcotangente(x) como atan(x)\n",
    "# La función logaritmo neperiano como log(x). La función exponencial como exp(x).\n",
    "# La función valor absoluto como abs(x).\n",
    "# Definir la función f después de la línea \"AQUÍ SE DEFINE LA FUNCIÓN\"\n",
    "# Asingar valores para x0 después de la línea \"AQUÍ SE ASIGNAN EL VALOR x0\"\n",
    "# **************************************************************************************************\n",
    "#Llamada de funciones de sympy\n",
    "from sympy import sin, cos, tan, asin, acos, atan, log, exp\n",
    "import sympy as sp\n",
    "x=sp.Symbol('x')\n",
    "\n",
    "# ****************************************************************************\n",
    "# **************************************************************************** \n",
    "#            AQUÍ SE DEFINE LA FUNCIÓN\n",
    "f=2+x**2+1\n",
    "\n",
    "# ****************************************************************************\n",
    "\n",
    "# ****************************************************************************\n",
    "#                 AQUÍ SE ASIGNAN EL VALOR x0\n",
    "x0=3\n",
    "\n",
    "# ****************************************************************************\n",
    "\n",
    "\n",
    "%matplotlib inline\n",
    "from sympy.plotting import plot\n",
    "import numpy as np\n",
    "\n",
    "fp=sp.diff(f,x) #Cálculo de la derivada como función\n",
    "\n",
    "if str(fp.subs(x,x0))=='zoo':\n",
    "    print('Función derivada f`(x) = '+str(fp))\n",
    "    print('MENSAJE: No existe la derivada en x0 = '+str(x0))\n",
    "else:\n",
    "    # Preparar datos\n",
    "    x0=np.round(x0,3)\n",
    "    m=np.round(float(fp.subs(x,x0)),3)\n",
    "    y0=np.round(float(f.subs(x,x0)),3)\n",
    "\n",
    "    if x0==0:\n",
    "        x0=int(x0)\n",
    "    else:\n",
    "        if int(x0) % x0==0:\n",
    "            x0=int(x0)        \n",
    "    if m==0:\n",
    "        m=int(m)\n",
    "    else:\n",
    "        if int(m) % m==0:\n",
    "            m=int(m)\n",
    "    if y0==0:\n",
    "        y0=int(y0)\n",
    "    else:\n",
    "        if int(y0) % y0==0:\n",
    "            y0=int(y0)\n",
    "\n",
    "    y=m*(x-x0)+y0\n",
    "\n",
    "    p=plot(f,(x,x0-10,x0+5),title='Grafica de la función f y la recta tangente',line_color='blue',show=False)\n",
    "    p1=plot(y,(x,x0-10,x0+5),line_color='red',show=False)\n",
    "    p.append(p1[0])\n",
    "    print('\\033[1m'+'SOLUCIÓN: RECTA TANGENTE A UNA CURVA\\n'+'\\033[0m')\n",
    "    print('Función dada, en color azul, f(x)=',str(f)+'\\n')\n",
    "    print('Función derivada f´(x)=',str(fp)+'\\n')\n",
    "    print('El valor de la función derivada evaluada en x0='+str(x0)+'   es  f´('+str(x0)+') = '+str(m)+'\\n')\n",
    "    print('Recta tangente a f, en color rojo, en el punto (x0,f(x0))=('+str(x0)+','+str(y0)+')'+'\\n')\n",
    "\n",
    "    if x0<0:\n",
    "        x0=-x0\n",
    "        print('Ecuación de la recta tangente: y = '+str(m)+'(x+'+str(x0)+')+'+'('+str(y0)+')')\n",
    "    else:\n",
    "        print('Ecuación de la recta tangente: y = '+str(m)+'(x-'+str(x0)+')+'+'('+str(y0)+')')\n",
    "\n",
    "    p.show()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Práctica}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{IV}$. Halla la función derivada para las siguientes funciones que se dan a continuación y compara el resultado con los que se obtienen usando el módulo 4.\n",
    "\n",
    "a) $f(x)=x^5-2x^3+1$\n",
    "\n",
    "b) $f(x)=3x^7-5x^6+2x^3-4x^2-9$\n",
    "\n",
    "c) $f(x)=\\frac{x^4-x+1}{2}$\n",
    "\n",
    "d) $f(x)=\\frac{x^2-1}{(x+1)^2}$\n",
    "\n",
    "e) $f(x)=\\sqrt{x^3-2}=(x^3-2)^{\\frac{1}{2}}$\n",
    "\n",
    "f) $f(x)=\\sqrt[5]{x^5-x^2+3}=(x^5-x^2+3)^{\\frac{1}{5}}$\n",
    "\n",
    "g) $f(x)=3^{2x+1}$\n",
    "\n",
    "h) $f(x)=sen(2x^3+3x^2)^2$\n",
    "\n",
    "i) $f(x)=\\sqrt{sen(2x+1)}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{V}$. Con las mismas funciones de ($\\textbf{IV}$.) y los valores $x_0$ que se indican, respectivamente, halla el valor de la derivada de $f$ en $x_0$ y grafica la recta tangente correspondiente. Compara los resultados obtenidos con los que se obtienen usando el módulo 5.\n",
    "\n",
    "a) $x_0= 2$\n",
    "\n",
    "b) $x_0= 1$\n",
    "\n",
    "c) $x_0= 0$\n",
    "\n",
    "d) $x_0= 0$\n",
    "\n",
    "e) $x_0= 2$\n",
    "\n",
    "f) $x_0= 1$\n",
    "\n",
    "g) $x_0= 2$\n",
    "\n",
    "h) $x_0= 0.5$\n",
    "\n",
    "i) $x_0= \\frac{1}{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Parte III: DERIVADAS SUCESIVAS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dada una función $f$, se define la $n-ésima$ derivada de $f$ a la función que se obtiene como resultado de haber derivado $n-veces$ la función $f$, y se denota por $f^{n)}(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Ejemplo}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dada la función $f(x)=sen(x)$ calcular la quinta $(n=5)$ derivada de la función.\n",
    "\n",
    "$f'(x)=(sen(3x))'= 3 cos(3x)$\n",
    "\n",
    "$f''(x)=(3 cos(3x))'= -9 sen(3x)$\n",
    "\n",
    "$f'''(x)=(-9 sen(3x))'= -27 cos(3x)$\n",
    "\n",
    "$f^{iv)}(x)=(-27 cos(3x))'= 81 sen(3x)$\n",
    "\n",
    "$f^{v)}(x)=(81 sen(3x))'= 243 cos(3x)$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el módulo 6 se halla la $n-ésima$ función derivada de una función $f$ dada, para un valor de $n$ dado."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Módulo 6: Calcula la $n-ésima$ función derivada de una función dada $f$, para un valor dado $n$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[1mSOLUCIÓN: CÁLCULO DE LA k-ésima DERIVADA\n",
      "\u001b[0m\n",
      "La 5-ésima derivada de f es: 243*cos(3*x)\n"
     ]
    }
   ],
   "source": [
    "# ***************** PARA LA DEFINICIÓN DE FUNCIONES*************************************************\n",
    "# El producto hay que indicarlo con asterisco (*).Las potencias se indican con doble asterisco (**).\n",
    "# La función seno(x) se define como sin(x),# la funcion arcoseno(x) SE DEFINE COMO asin(x),\n",
    "# La función arcoscosno(x) como acos(x), la funíon arcotangente(x) como atan(x)\n",
    "# La función logaritmo neperiano como log(x). La función exponencial como exp(x).\n",
    "# La función valor absoluto como abs(x).\n",
    "# Definir la función f y asignar el valor para n después de la línea \"AQUÍ SE DEFINE LA FUNCIÓN Y\n",
    "# SE ASIGNA EL VALOR DE n\"\n",
    "# *************************************************************************************************\n",
    "#Llamada de funciones de sympy\n",
    "from sympy import sin, cos, tan, asin, acos, atan, log, exp\n",
    "import sympy as sp\n",
    "x=sp.Symbol('x')\n",
    "\n",
    "# ****************************************************************************\n",
    "#            AQUÍ SE DEFINE LA FUNCIÓN Y SE ASIGNA EL VALOR DE n\n",
    "f=sin(3*x)\n",
    "n=5\n",
    "\n",
    "# ****************************************************************************\n",
    "\n",
    "%matplotlib inline\n",
    "from sympy.plotting import plot\n",
    "\n",
    "fp=sp.diff(f,x,n) #Cálculo de la n-ésima derivada como función\n",
    "\n",
    "print('\\033[1m'+'SOLUCIÓN: CÁLCULO DE LA k-ésima DERIVADA\\n'+'\\033[0m')\n",
    "print('La '+str(n)+'-ésima derivada de f es: '+str(fp))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Práctica}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{VI}$. Con las funciones definidas en ($\\textbf{IV}$.) calcula la $n-esima$ derivada para los valores de $n$ correspondientes. Compara los resultados obtenidos con los que se obtienen usando el módulo 6.\n",
    "\n",
    "a) $n=2$\n",
    "\n",
    "b) $n=3$\n",
    "\n",
    "c) $n=4$\n",
    "\n",
    "d) $n=3$\n",
    "\n",
    "e) $n=2$\n",
    "\n",
    "f) $n=3$\n",
    "\n",
    "g) $n=3$\n",
    "\n",
    "h) $n=1$, $n=2$\n",
    "\n",
    "i) $n=3$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Parte IV: APLICACIONES DE LA DERIVADA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## MÁXIMOS Y MÍNIMOS RELATIVOS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Definición.}$\n",
    "\n",
    "Una función $f(x)$ tiene un máximo relativo en $x=x_0$ si existe un intervalo abierto $(a,b)$ tal que $x_0\\in (a,b)$ y $f(x) \\geq f(x_0)$ para todo $x\\in (a,b)$; $x \\neq x_0$\n",
    "\n",
    "Análogamente, una función $f(x)$ tiene un mínimo relativo en $x=x_0$ si existe un intervalo abierto $(a,b)$ tal que $x_0\\in (a,b)$ y $f(x) \\leq f(x_0)$ para todo $x\\in (a,b)$; $x \\neq x_0$\n",
    "\n",
    "A un máximo o mínimo relativo se le llama extremo relativo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Caracterización de los máximos y mínimos por la primera derivada"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si el punto $(x_0,f(x_0))$ es un punto máximo o mínimo relativo de de $f(x)$, entonces $f'(x_0)=0$, o $f(x)$ no es derivable en $x_0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pasos para hallar los extremos relativos y clasificarlos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Calcular la primera derivada, $f'(x)$\n",
    "\n",
    "2. Resolver la ecuación $f'(x)=0$\n",
    "\n",
    "3. Hallar la segunda derivada, $f''(x)$\n",
    "\n",
    "4. Sustituir en $f''(x)$ las raíces $x$ obtenidas en el paso 2\n",
    "\n",
    "$\\qquad$ * Si $f''(x)<0$, entonces $f$ tiene un máximo relativo en este punto.\n",
    "\n",
    "$\\qquad$ * Si $f''(x)>0$, entonces $f$ tiene un mínimo relativo en este punto."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Ejemplo}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hallar los extremos relativos para la funcion $f(x)=2x^3-5x^2-1$.\n",
    "\n",
    "Paso 1. Hallamos la primera derivada $f'(x)=6x^2-10x$\n",
    "\n",
    "Paso 2. Resolvemos la ecuación $f'(x)=6x^2-10x=0$, que da dos soluciones $\\{0, \\frac{5}{3}  \\} $\n",
    "\n",
    "Paso 3. Hallamos la segunda derivada $f''(x)=12x-10$\n",
    "\n",
    "Paso 4. Sustituimos las raíces obtenidas en el Paso 2 en la segunda derivada $f''(x)$\n",
    "\n",
    "* $f''(0)=-10$, al ser menor que cero estamos ante un máximo relativo\n",
    "\n",
    "* $f''(5/3)=10$, al ser mayor que cero estamos ante un mínimo relativo\n",
    "\n",
    "Por lo tanto la función $f$ alcanza un máximo relativo en $x=0$, donde $f(0)=-1$ y alcanza un mínimo relativo en $x=\\frac{5}{3}$, donde $f(\\frac{5}{3})=-\\frac{152}{27}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el módulo 7, para una función $f$ dada (no periódica), se calculan los extremos relativos, caracterizándolos mediante el criterio de la segunda derivada, y se grafica la función dada."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Módulo 7. Calcula y caracteriza los extremos relativos para una función $f$ dada y muestra su gráfica"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[1mSOLUCIÓN: CÁLCULO DE EXTREMOS RELATIVOS\n",
      "\u001b[0m\n",
      "Función inicial f(x)= 2*x**3 - 5*x**2 - 1\n",
      "\n",
      "Ecuación f´(x)=0 ===> 6*x**2 - 10*x=0\n",
      "\n",
      "Raíces de la ecuación:[0, 5/3]\n",
      "\n",
      "Función segunda derivada: f´´(x)=2*(6*x - 5)\n",
      "\n",
      "En x=0, f´´(0)=-10, la función alcanza un máximo relativo, y f(0)=-1\n",
      "\n",
      "En x=5/3, f´´(5/3)=10, la función alcanza un mínimo relativo, y f(5/3)=-152/27\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ***************** PARA LA DEFINICIÓN DE FUNCIONES*************************************************\n",
    "# El producto hay que indicarlo con asterisco (*).Las potencias se indican con doble asterisco (**).\n",
    "# La función seno(x) se define como sin(x),# la funcion arcoseno(x) SE DEFINE COMO asin(x),\n",
    "# La función arcoscosno(x) como acos(x), la funíon arcotangente(x) como atan(x)\n",
    "# La función logaritmo neperiano como log(x). La función exponencial como exp(x).\n",
    "# La función valor absoluto como abs(x).\n",
    "# Definir la función f y asignar el valor para n después de la línea \"AQUÍ SE DEFINE LA FUNCIÓN Y SE\n",
    "# ASIGNA EL VALOR DE n\"\n",
    "# **************************************************************************************************\n",
    "#Llamada de funciones de sympy\n",
    "from sympy import sin, cos, tan, asin, acos, atan, log, exp\n",
    "import sympy as sp\n",
    "x=sp.Symbol('x')\n",
    "\n",
    "# ****************************************************************************\n",
    "#            AQUÍ SE DEFINE LA FUNCIÓN Y SE ASIGNA EL VALOR DE n\n",
    "f=2*x**3-5*x**2-1\n",
    "\n",
    "# ****************************************************************************\n",
    "\n",
    "%matplotlib inline\n",
    "from sympy.plotting import plot\n",
    "\n",
    "\n",
    "fp=sp.diff(f,x,1) #Cálculo de la derivada como función\n",
    "\n",
    "sol=sp.solveset(fp, x, domain=sp.S.Reals) # Halla las raíces de f'(x)=0\n",
    "\n",
    "sol=list(sol) # Convierte las soluciones de formato FiniteSet a formato Lista\n",
    "sol=sorted(sol)\n",
    "\n",
    "print('\\033[1m'+'SOLUCIÓN: CÁLCULO DE EXTREMOS RELATIVOS\\n'+'\\033[0m')\n",
    "print('Función inicial f(x)= '+str(f)+'\\n')\n",
    "print('Ecuación f´(x)=0 ===> '+ str(fp)+'=0\\n')\n",
    "print('Raíces de la ecuación:'+str(sol)+'\\n')\n",
    "\n",
    "if len(sol)>0:\n",
    "    fpp=sp.diff(f,x,2) #Cálculo de la derivada como función    \n",
    "    print('Función segunda derivada: f´´(x)='+str(fpp)+'\\n')\n",
    "    for i in range(0,len(sol)):\n",
    "        fppx=fpp.subs(x,sol[i]) #Evaluar la segunda derivada en cada solución        \n",
    "        if fppx==0:\n",
    "            print('En x='+str(sol[i])+', no se puede decir nada, pues en este valor la segunda derivada es cero')\n",
    "        if fppx>0:\n",
    "            fx=f.subs(x,sol[i]) #Evaluar f en la raiz\n",
    "            print('En x='+str(sol[i])+', f´´('+str(sol[i])+')='+str(fppx) +', la función alcanza un mínimo relativo, y f('+str(sol[i])+')='+str(fx)+'\\n') \n",
    "        if fppx<0:\n",
    "            fx=f.subs(x,sol[i]) #Evaluar f en la raiz            \n",
    "            print('En x='+str(sol[i])+', f´´('+str(sol[i])+')='+str(fppx) +', la función alcanza un máximo relativo, y f('+str(sol[i])+')='+str(fx)+'\\n') \n",
    "    p=plot(f,(x,min(sol)-2,max(sol)+2),title='Grafica de la función f',line_color='blue',show=False)\n",
    "else:\n",
    "    print('No existen extremos relativos')\n",
    "    p=plot(f,title='Grafica de la función f',line_color='blue',show=False)\n",
    "     \n",
    "p.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Práctica}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{VII}$. Halla y caracteriza los extremos relativos de las siguientes funciones. Compara el resultado obtenido con el resultado que se obtiene usando el módulo 7.\n",
    "\n",
    "a) $f(x)=(3x^2+5)^2$\n",
    "\n",
    "b) $f(x)=x^5-x^3+2$\n",
    "\n",
    "c) $f(x)=(x^5-x^3+3)^4$\n",
    "\n",
    "d) $f(x)=-\\frac{3}{5}x^3+\\frac{2}{3}x^2-4$\n",
    "\n",
    "e) $f(x)-\\frac{1}{x^3}+\\frac{1}{x^2}-4x$\n",
    "\n",
    "f) $f(x)=\\frac{4x^4-2x^3}{x^5}$\n",
    "\n",
    "g) $f(x)=\\sqrt{2^x}$\n",
    "\n",
    "h) $f(x)=-\\frac{x}{\\sqrt{2^x}}$\n",
    "\n",
    "i) $f(x)=\\frac{7^{x^2}}{x^3}$\n",
    "\n",
    "j) $f(x)=3^{x-5}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## MONOTONÍA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Definición.}$\n",
    "\n",
    "Una función $f$ es creciente en un intervalo $(a,b)$ si para cualquier par de números $x_1,x_2 \\in (a,b)$, $x_1 < x_2$ implica que $f(x_1)<f(x_2)$.\n",
    "\n",
    "Análogamente una función $f$ es decreciente en un intervalo $(a,b)$ si para cualquier par de números $x_1,x_2 \\in (a,b)$, $x_1 < x_2$ implica que $f(x_1)>f(x_2)$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Criterios para funciones crecientes o decrecientes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sea $f(x)$ una función continua en $[a,b]$ y derivable en $(a,b)$\n",
    "\n",
    "1. Si $f'(x)>0$  para todo $x \\in (a,b)$, $f(x)$ es creciente en $[a,b]$\n",
    "\n",
    "2. Si $f'(x)<0$  para todo $x \\in (a,b)$, $f(x)$ es decreciente en $[a,b]$\n",
    "\n",
    "3. Si $f'(x)=0$  para todo $x \\in (a,b)$, $f(x)$ es constante en $[a,b]$\n",
    "\n",
    "Es decir estudiar la monotonía de una función es estudiar los intervalos en los que la función es creciente o decreciente. Estos intervalos están separados por los extremos relativos y las discontinuidades de la primera derivada.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pasos para identificar los intervalos de crecimiento o decrecimiento"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Se calculan los máximos y mínimos.\n",
    "\n",
    "2. Se hallan las discontinuidades de la primera derivada de la función $f'(x)$\n",
    "\n",
    "3. Con los puntos de los pasos 1 y 2 se determinan los intervalos a analizar.\n",
    "\n",
    "3. Se prueba un punto de cada uno de los intervalos determinados en el paso 3 y se clasifican los intervalos\n",
    "\n",
    "$\\qquad$ * Es intervalo de crecimiento si $f'(x)>0$\n",
    "\n",
    "$\\qquad$ * Es intervalo de decrecimiento si $f'(x)<0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Ejemplo}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Determina los interevalos de crecimiento y decrecimiento de la función $f(x)=2x^3-5x^2-1$\n",
    "\n",
    "Paso 1. Se calculan los valores donde $f$ alcanza máximos y mínimos. Para la función dada, estos son $\\{0, \\frac{5}{3}  \\} $\n",
    "\n",
    "Paso 2. Se hallan discontinuidades de $f'(x)=6x^2-10x$. No existen\n",
    "\n",
    "Paso 3. Con los puntos encontrados en los pasos 1 y 2, se determinan los intervalos a analizar:\n",
    "\n",
    "* Intervalo $I_1=(-\\infty,0)$, $f'(-1)=16>0$, por lo que es un intervalo de crecimiento\n",
    "\n",
    "* Intervalo $I_2=(0,\\frac{5}{3})$, $f'(1)=-4 < 0$, por lo que es un intervalo de decrecimiento\n",
    "\n",
    "* Intervalo $I_3=(\\frac{5}{3},\\infty)$, $f'(2)=4>0$, por lo que es un intervlao de crecimiento"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el módulo 8, para una función $f$  dada (no periódica), se determina los intervalos de crecimiento y decrecimiento y se grafica la función $f$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Módulo 8. Identifica los intervalos de crecimiento y decrecimiento para una función $f$ dada."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[1mSOLUCIÓN: ANÁLISIS DE CRECIMIENTO Y DECRECIMIENTO\n",
      "\u001b[0m\n",
      "Función dada: f(x)=2*x**3 - 5*x**2 - 1\n",
      "\n",
      "En el intervalo (-∞,0) la función es creciente\n",
      "\n",
      "En el intervalo (0,5/3) la función es decreciente\n",
      "\n",
      "En el intervalo (5/3,+∞) la función es creciente\n",
      "\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ***************** PARA LA DEFINICIÓN DE FUNCIONES*************************************************\n",
    "# El producto hay que indicarlo con asterisco (*).Las potencias se indican con doble asterisco (**).\n",
    "# La función seno(x) se define como sin(x),# la funcion arcoseno(x) SE DEFINE COMO asin(x),\n",
    "# La función arcoscosno(x) como acos(x), la funíon arcotangente(x) como atan(x)\n",
    "# La función logaritmo neperiano como log(x). La función exponencial como exp(x).\n",
    "# La función valor absoluto como abs(x).\n",
    "# Definir la función f y asignar el valor para n después de la línea \"AQUÍ SE DEFINE LA FUNCIÓN Y\n",
    "# SE ASIGNA EL VALOR DE n\"\n",
    "# *************************************************************************************************\n",
    "#Llamada de funciones de sympy\n",
    "from sympy import sin, cos, tan, asin, acos, atan, log, exp\n",
    "import sympy as sp\n",
    "x=sp.Symbol('x')\n",
    "\n",
    "# ****************************************************************************\n",
    "#            AQUÍ SE DEFINE LA FUNCIÓN Y SE ASIGNA EL VALOR DE n\n",
    "f=2*x**3-5*x**2-1\n",
    "\n",
    "# ****************************************************************************\n",
    "\n",
    "\n",
    "%matplotlib inline\n",
    "from sympy.plotting import plot\n",
    "\n",
    "fp=sp.diff(f,x,1) #Cálculo de la derivada como función\n",
    "\n",
    "sol=sp.solveset(fp, x, domain=sp.S.Reals) # Halla las raíces de f'(x)=0\n",
    "\n",
    "sol=list(sol) # Convierte las soluciones de formato FiniteSet a formato Lista\n",
    "\n",
    "if len(sol)==0:    # Se definen los intervalos\n",
    "    sol[len(sol):]=[0,1]\n",
    "else:\n",
    "    sol[len(sol):]=[min(sol)-1,max(sol)+1]\n",
    "\n",
    "sol=sorted(sol)\n",
    "\n",
    "#for i in range(0,len(sol)): # Se define la lista linter con los extremos de los intervalos\n",
    "#    linter[i]=(str(sol[i]))\n",
    "#linter[0]=('-∞'); linter[len(linter)-1]=('+∞')\n",
    "\n",
    "linter=list()\n",
    "for i in range(0,len(sol)): # Se define la lista linter con los extremos de los intervalos\n",
    "    linter.append(str(sol[i]))\n",
    "\n",
    "linter[0]=('-∞'); linter[len(linter)-1]=('+∞')\n",
    "\n",
    "\n",
    "print('\\033[1m'+'SOLUCIÓN: ANÁLISIS DE CRECIMIENTO Y DECRECIMIENTO\\n'+'\\033[0m')\n",
    "print('Función dada: f(x)='+str(f)+'\\n')\n",
    "#Análisis de intervalos de crecimiento y decrecimiento\n",
    "for i in range(0,len(sol)-1):\n",
    "    xm=(sol[i+1]+sol[i])/2\n",
    "    fpval=fp.subs(x,xm)\n",
    "    if fpval==0:\n",
    "        fmono='constante'\n",
    "    else:\n",
    "        if fpval>0:\n",
    "            fmono='creciente'\n",
    "        else:\n",
    "            fmono='decreciente'\n",
    "    print('En el intervalo ('+linter[i]+','+linter[i+1]+') la función es '+fmono+'\\n')\n",
    "\n",
    "\n",
    "p=plot(f,(x,min(sol)-1,max(sol)+1),title='Grafica de la función f',line_color='blue',show=False)\n",
    "p.show()              \n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Práctica}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{VIII}$. Con las funciones definidas en (\\textbf{VII}.) estudia la monotonía de tales funciones. Compara el resultado obtenido con los que se obtienen usando el módulo 8."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## CONCAVIDAD, CONVEXIDAD Y PUNTOS DE INFLEXIÓN"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{Definición.}$\n",
    "\n",
    "Sea $f(x)$ una función derivable en $(a,b)$. La gráfica de $f(x)$ es convexa $(\\cup)$ en $(a,b)$ si $f'(x)$ es creciente en $(a,b)$, y es cóncava $(\\cap)$ si $f'(x)$ es decreciente en $(a,b)$.\n",
    "\n",
    "Un punto de inflexión de una función es un punto en el que la función cambia de cóncava a convexa, o viceversa, y la tangente atraviesa la gráfica.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pasos para identificar los puntos de inflexión"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Se calcula la segunda derivada, $f''(x)$\n",
    "\n",
    "2. Se resuelve la ecuación, $f''(x)=0$\n",
    "\n",
    "3. Se halla la tercera derivada, $f'''(x)$\n",
    "\n",
    "4. Se sustituyen las raíces halladas en el Paso 2 en la tercera derivada. Si $f'''(x) \\neq 0$, son puntos de inflexión."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Ejemplo}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dada la función $f(x)=2x^3-5x^2-1$, hallar los puntos de inflexión\n",
    "\n",
    "Paso 1. Calculamos la segunda derivada, $f''(x)=12x-5$\n",
    "\n",
    "Paso 2. Resolvemos la ecuación $f''(x)=12x-5=0$, que tiene como solución $x=\\frac{5}{6}$\n",
    "\n",
    "Paso 3. Hallamos la tercera derivada, $f'''(x)=12$\n",
    "\n",
    "Paso 4. Sustituimos en la tercera derivada la raíz obtenida en el Paso 2. $f'''(\\frac{5}{6})=12$, que es distinto de cero, por lo que el punto $x=\\frac{5}{6}$ es un punto de inflexión."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el módulo 9, para una función $f$ dada (no periódica), se calculan sus puntos de inflexión."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Módulo 9. Calcula los puntos de inflexión de una función $f$ dada"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[1mSOLUCIÓN: PUNTOS DE INFLEXIÓN\n",
      "\u001b[0m\n",
      "(5/6,-179/54) es punto de inflexión de la función f(x)=2*x**3 - 5*x**2 - 1\n",
      "\n"
     ]
    },
    {
     "data": {
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DGE7uduzYQfXq1b0Oo8D8EPfu3bvZu3cvtWvXBmDnzp0cOHCACy+88LjjduzYwdH/6/T0dGJiYoIdaqH54XyfDsUdXOEU96FDsG4d1KwJNWp4FFg+csa9bNmyL621cQE/2Fqb1x9faNGihdchnBY/xD1x4kR77733Hvt6zJgx9pFHHsnzMeXLly/usIqFH8736VDcwRVOcd9+u7WtWlm7Y4cHAQXohLjzyznH/dEUX5irVasW27ZtO/Z1UlISNWvW9DAiESkKGzbAp5/CNddAtWpeR1M8lKDC3GWXXcbGjRvZvHkzhw8fZsKECXTt2tXrsESkkAYMgHLlwmth7olCYh1U3759vQ7htPgh7tKlS/POO+/QsWNHjhw5Qp8+fYiOjs7zMdVC9OOYH8736VDcwRUOcW/cCB9/DI8/7gok/Kww5zvPIgkgzzslPMXGxpKYmOh1GCJyCr17w8SJsHkznHuu19EUiCnIwZriExEJIZs2wbhx8MADIZecCkwJSkQkhAwY4HbKfeopryMpfr5MUJ999hnR0dGUKlUqz6kmv7Xw2bVrFx06dKB+/fp06NCB3bt353pcnTp1aNKkCTExMcR62Nkxv/NnreXRRx+lXr16NG3alOXLl3sQ5cnyi3vBggVUrlyZmJgYYmJiePnllz2I8nh9+vQhMjKSxo0b53q/X891fnH78VwDbNu2jXbt2hEVFUV0dDTDhg076Rg/nvP84t68GUaPtmRljaBTJ/+c87S0NFq2bEmzZs2Ijo7mhRdeOOkYay3GmLeMMT8ZY1YbY5rn+8T51KF7Yt26dXb9+vW2TZs2dunSpbkek5mZaevWrWs3bdpk09PTbdOmTe3atWuDHOnxnnrqKTtw4EBrrbUDBw60/fr1y/W42rVr2x0eL1zI6/wdXbcwc+ZMGxcXZ7OysuyiRYtsy5YtvQzZWhvY//v8+fNtly5dPIowd998841dtmyZjY6OzvV+P55ra/OP24/n2lprt2/fbpctW2attXbv3r22fv36J/2c+PGc5xf3ffdZW6bMEXvddXd5FWKusrKy7L59+6y11h4+fNi2bNnSLlq06LhjZs6caYFZuOtQrYH/2FBcBxUVFUWDBg3yPMaPLXymTZtGr169AOjVqxdTp071NJ68BHL+pk2bxt13340xhtatW7Nnzx5+/fVXjyJ2/Pj/HohrrrmGqlWrnvJ+P55ryD9uv6pRowbNm7sP6JUqVSIqKork5OTjjvHjOc8r7i1bYNQouOGG7ZQrt8u7IHNhjKFixYoAZGRkkJGRgTHH10Nk/56Oyc5Xi4Eqxpg8+1/4MkEFIjk5mQsuuODY17Vq1TrpBzDYfvvtN2pk9xupUaMGv//+e67HGWO4/vrradGiBfHx8cEM8ZhAzp8fz3GgMS1atIhmzZrRqVMn1q5dG8wQT4sfz3Wg/H6ut2zZwooVK2jVqtVxt/v9nJ8Y9wcfwDnnwO23b/PlOT9y5AgxMTFERkbSoUOHXM83sC3HTUnA+Xk9p2froNq3b09KSspJtw8YMICbbrop38fbXMrjT8zYxSGvuAOVkJBAzZo1+f333+nQoQMNGzbkmmuuKcow8xXI+fPqHOclkJiaN2/O1q1bqVixIl988QXdunVj48aNwQrxtPjxXAfC7+d6//79dO/enaFDh3LWWWcdd5+fz/mJca9fD2+8Af/7v9CxY7Qvz/kZZ5zBypUr2bNnDzfffDNr1qw57tplbuebfJYyeZag5s6dW6jHe9XCJ6+4zz33XH799Vdq1KjBr7/+SuQpVtAdjTMyMpKbb76ZJUuWFHuCOnQI1q+Hn3+GpCRYsaI18+aV5c9/hl9+gaSktpQpcyVxcW4RYKdOsHbtQB577Gxq14by5WHFip5MmVKfH36ACy+EBg3cIsFg/k4H8v+e842oc+fOPPTQQ6Smpvp6AXKotqTy87nOyMige/fu3HHHHdxyyy0n3e/Xc55b3M88434Hn3nG3+ccoEqVKrRt25bZs2cfl6Bq1aoFcEGOQ2sB2/N6rpCd4vNjC5+uXbsyevRoAEaPHp3rSPDAgQPs27fv2L+/+uqrU1ZInS5rXZL56CP3A3399VCxIjz2GDz7rJvHTkk5n127APZyySVH2L9/Hh07VuCaa9yxV1wBl19+JqmpiVSqZElO3klWVhTffHMWH3zgnve88+Dqq6FdO3jpJZgwwc2T5732u3AC+X9PSUk59mltyZIlZGVlcc455xRfUEWga9eujBkzBmstixcvpnLlysemi/3Mr+faWsu9995LVFQUTzzxRK7H+PGc5xb3okUwZQr06wfVq/vznO/YsYM9e/YAcOjQIebOnUvDhg2POyb79/Ru47QG/rDW5n3RL58qCk9MnjzZnn/++TYiIsJGRkba66+/3lprbXJysu3UqdOx42bOnGnr169v69ata1999VWvwj0mNTXVXnvttbZevXr22muvtTt37rTWHh/3pk2bbNOmTW3Tpk1to0aNiizuw4et/fprax94wNpu3awFa6Ojrb3uOmtvvNHav/3N2ilTrF292tqDB91jcjt/I0aMsBdeeKG11lXmPPTQQ7Zu3bq2cePGdunSpfbwYWs3bbJ27lxrhw2z9plnrG3Z0tp27dxrtmljba1a1j78sLUjR1q7fXuRfHvHOVXcI0aMsNZa+/bbb9tGjRrZpk2b2latWtmEhISiD6KAevbsac877zxbunRpe/7559sPP/zwuJhzO9d+kF/cfjzX1lr77bffWsA2adLENmvWzDZr1szOnDnT9+f8xLibNm1mo6N32rPOOmCHDv3AWuvPc75q1SobExNjmzRpYqOjo+1LL71krbUnnW/gXWAT8D0Qa/Op4lOroxBmLSQkwCefwGefwe7d7hNW165w6aVuZHPJJQWfgjudVkcZGfD997B8OcyZA/v2waxZ7r5mzdxmau3aQfPmwZ0SFAll//qX+30eMcJ1jggDBfrtV4IKQX/8AaNHw/Dhrs3+8uVw441w++0QF+c6HBdGUfTisxZWr4bZs92flBR3DaxuXbjtNujRwyVRJSuR3B05Ak2bQmYmrFnjukeEASWocJWcDG++CV984a4xtWrlqnpuusldNyoqxdEsdtcumDrVNbj8+mv3S9eggRtZ3X23K7oQkf8aORL69IFJk6B7d6+jKTJKUOEmORlefdUVPRw54oodbr8dWrQontcr7m7mO3e6ZPXvf8PYsW4UFRcHfftCly5QOiQ2gREpPocOuen5mjVh8eKwmmlQN/NwkZ7uGkNedpn7FNW7txs5DR5cfMkpGM45B+69101T/vyzqyxcuRK6dYPateG552DrVq+jFPHO++9DRAQMGhRWyanANILyqTlz4JFH4Mcf3fD+zTfhoouC89pe7AeVmemmLuPjXXFF3bouCT/1VGgnY5GCSkpy099xcfD5515HU+Q0ggplycnwpz+5tUtZWa7AYNKk4CUnr5Qu7aqVZsxw+93cfLNLVLGxcN117jwU5/oqEb948kn3uz94sNeReE8JyicyMmDIEGjYEKZPh5dfdmXbHTt6HVnw1anjpjZ++cX9vX6962zRrBmMGePOlUg4WrAAPv0U+vcP/w+lgVCC8oF169z1pSefhLZtYe1a+NvfCl8uHuoqV3ZTfJs3u+4XWVnu+lTjxu7rzEyvIxQpOhkZriq3Th3XNSJcLF26lKZNm5KWloYxpoIxZq0xJqD2OUpQHhs3zhVBJCS4+eZ//ctdfykKTz31FA0bNqRp06bcfPPNx1qRhJqICOjVy40o338fKlWCe+6B6GgYP94lLpFQN3y4W+80dCiceabX0RSdyy67jK5du/Lcc88BDALGWWvXBPJYJSiPpKXB/ffDXXe56yyLFkEu/SwLpUOHDqxZs4bVq1dzySWXMHDgwKJ9gSAzxk31LV3qepOVLQt//rNbzPj550pUErp++w2ef95N6XvcUrRYPP/888yZMwcgFpekAqIE5YGffoLLL3cVa08/7RauFkePyuuvv57S2YuKWrduTVJSUtG/iAeMcSXpK1e6BrWZmXDrra7a76uvVEwhoad/f7f2adiw8Cwr37VrF/v37weoBAR88UIJKsjmzHFvpL/8AjNnwmuvBWdh6kcffUSnTp1OeX98fDyxsbHExsayY8eO4g+oCJQq5Soe16xxa6r273ej0vbtXfISCQWLF7trqk884crLw1Hfvn155ZVXAD4G3gj0cVoHFUTvvQcPP+w+7b/5ZtG09wlk48cBAwaQmJjI5MmTA9qQzYt1UEXh8GE3Kn3xRdda6e67XQcOtw2NiP9kZrqp/WXLYMOGom1Z5hdjxoxh6tSpR99/SgPfAU9ba+fl++B82p1LEcjKsvbFF912FF26WHvgQPBee9SoUbZ169b2QAFetEWLFsUYUfHbvdvafv2sjYiw9swzrX3uOWv37vU6KpGTDRxorTHWTp/udSRBk1/OOe6PpviK2ZEjriPEiy+6SrQpU9zOmMEwe/Zs3njjDaZPn075YL2oD1Sp4rbH3rDBNdJ99VXXjWPkSBVSiH+sWwcvvOBGUDfc4HU0/qQEVYzS06FnT1c+2q+fe4MMZsv8Rx55hH379tGhQwdiYmJ4IEw2lAlUnTquDH3xYlfx16ePK05ZssTryKSky8x0SyUqVYJ33w3PwoiioL7RxWTvXldpNn++6xBxil2ni9VPP/0U/Bf1oVatXHeOjz92HxRatXJvDgMHwrnneh2dlER//7v7oDR+vH4G86IRVDHYs8cVQ3z7rdtOwovkJMczxu09tWED/N//uQXSV17pFkWqI4UE0w8/uDVPt9ziqlDl1JSgitiBA9C5s9uCfdo096Yo/lGpErz+uitNv/ZaePxxV/afkOB1ZFISHDniRu8VKripf03t5U0Jqgilp7su3P/5j5tO6tzZ64jkVC65xLVNmjwZdu+Gq65ybxwhsgRMQtTf/+7eH955R1N7gVCCKiKZmW6X2zlz3M63YbRFc9gyxn2g+OGH/077NWjgOqar2k+K2vr1rgl0t26ueErypwRVBLKyXIXYlCnw1luunFxCR4UKbtpv9Wq44go37Xflle5rkaJwtGqvQgUYMUJTe4FSgioka+HRR10xxCuvuHb5Epqiolw3+aFDXb/E5s1d1d+BA15HJqHu5Zfde8Xw4XDeeV5HEzqUoApp0CA3p/zkk/Dss15HI4VljOswv2GD+8T75ptuW48ZM7yOTELVV1+5xeING0KPHl5HE1qUoAph8mTXhfiSS1yi0rA9fFStCh984JYKVKgAN97oSoKTk72OTEJJUhLccYf7kKOqvYJTgjpNK1a4T9otW8KHH+oHL1xddZX7v37tNdi5000DvvuuKxcWyUtGhvtQk5bmlp2UoG5jRUYJ6jSkpLhNxapWhalTw2v3SzlZRITbt+v9910XikcecYnr+++9jkz87Jln4LvvXIf9hg29jiY0KUEVUFqaKxPdtcu1zymOjQbFny6+2F1PGDv2v0UUzzzjNpoTyWnaNBg8GB580C0/kdOjBFUA1rpPz//5j3uTuvRSryOSYDvaMmn9ejfFO3Cgu8Ywd67XkYlfbN4MvXu7DiX/+IfX0YQ2JagCiI+HBQvcm9Itt3gdjXjpnHPcgux581yy6tDBbZCoThQlW3o63Hab+zA7caLroi+nTwkqQN9/D489BvXqubUxIgDt2sHy5fDcc64zdVSU60SR90bVEo6sdU1gf/gBRo+GunW9jij0KUEF4OBBV41TubL7wSulsyY5lCvnFmmvWOGWHPTqBf/zP/Djj15HJsH0xhtuuclzz7mNMqXw9FYbgMcec5+Kxo5Vg0c5tcaNYeFC18rmq6+gaVOXuNLTvY5Mitv48a7Ss2dP19dRioYSVD4+/dQt2Ozf311nEMlLqVLwwAOukKZbNzflExMD//6315FJcfn3v11RxNVXw6hRmmEpSjqVedi8Gfr2hdatXS8tkUDVqAETJsAXX7ilCW3awL33usW+Ej7Wr3cfRC66yK2JVFFE0VKCOoWMDDdcN8YN38uU8Tqi0zd48GCMMaSmpnodSonTqROsXeumfcaMcUUUH3+sIopw8Ntvbs+30qXdB5GqVb2OKPwoQZ3Cc8/BkiVueq9OHa+jOX3btm1jzpw5XHjhhV6HUmKVL++281i+3CWsO++Etm1d4pLQtG+fGxGnpLhGwqrYKx5KULmYM8dV49x/v1vTEMoef/xxBg0ahFGzQM81aQIjR7oPPWvWuGtT/frB/v1eRyYFsX+/GzktXQqTJrl+nFI8lKBOsHcvPPGEKxMN9VXg06dP5/zzz6dZs2ZehyLZSpWC++5z23n06uW284iKcp3xNe3nf0eT06JFbtv2zp29jii8lfY6AL95+mlYtw7++c/QaALbvn17UlJSTrp9wIABvPbaa3z11VcBPU98fDzx8fEA7FA7hGJXrZrrgt+nj+vX1r27K8j561/dWirxnwMH4IYbICEBPvkk9GdXQoGxeX9sK1Gf6RISXKnoX/4S+qOn77//nuuuu47y2T3+k5KSqFmzJkuWLOG8fLb0jI2NJTExMRhhCm478HffhSFD3DWNxx9310ArVfI6Mjnq4EHo0sWVlI8bpwawhVCgaw1KUNnS013z14MH3fWBihW9jqho1alTh8TERKpVq5bvsUpQ3khJcSP4UaNcmfqgQa4RrS4femvvXre27dNP3WL9P//Z64hCWoF+mnUNKtvrr7tuEe+9F37JSULDeee5IorFi6FWLdct/e67XTWpeCM5Ga65BmbPduvalJyCSwkKd81pwAD3wxcX53U0xWPLli0BjZ7Ee61auST14YewcqX7+o474JdfvI6sZFmzxi3S37TJrYXUNafgK/EJKivLNfY86ywYOtTraEScUqXcOpvvvnObIk6eDA0awLPPujU4UrzmzXO7Jh85At9+Cx07eh1RyVTiE9R777k3gb//HapX9zoakeNVquRG9xs2uEq/115zW76MHOmKK6TojR3rZlJq1XIj2ZgYryMquUp0kURSEjRq5IbxX36pi9FHqUjCv5YscaXoqanu0/0rr7ipJzUoLbz0dNeSavly175o8mSoUsXrqMKOiiQCYS089JD7FPree0pOEhpatnSlzm+84RqT9uzpthafNUsLfQtj0ya48koYNgyaN3cfWJWcvFdiE9SsWTBzpvsEqj5aEkqMga5dXQHF2LHwxx+uo8Fdd7lrJ0pUBfPZZy4pbdoEU6a4a9Gh3Bw6nJTIBJWW5kZPHTu6RbkioeiMM1zj2fXr3ULfLVvguuvcYvOvvlKiys/R94EePdxU/8qVbusM8Y8SmaCGDYOtW+HJJ91cs0goi4hwb7Rz57r+cFu3ug9frVu7bSCysryO0H/mz4drr3Uti5580k2b1q7tdVRyohKXoH77zVVFde3qfkBFwkW5cvDww/DTT/D+++5n/fnnXRf1jz7S1vMAv/7q1pRde607P1Onuoa9mtLzpxKXoF54AQ4dcj+UIuGobFnXeHbjRnjsMTfCuvdeN0IYMABKYi/gzEw3c9KwIXz+uXsfWLPG7csl/lWiEtT337u9eB5+WB2jJfyVKeOuUS1f7qb/Lr0U3nrLFQXddRcsXBj+16msdYUjsbEuWV9+uUtML74YGrsVlHQlJkFZ69aPVK7spj1ESgpjXPHErFmwYIHbh2r6dFdM0aQJvP027NnjdZRFy1q30+1VV8H110OFCm5zwVmz3EJnCQ0lJkHNnu12yn3hBaha1etoRLwRFeUKKbZvd73+ypeHRx+FW25x1WxTp4b2tarMTNc3LyYGbrzRLcYfOtT97nfvrvWOoaZEdJLIyIBmzdwP75o1bk5eTk2dJEqW5cth4kS3SWdqqlugeuut7k/btu6alt/t3OnWhMXHu10JoqKgf3+3b5MKIHxF+0GdaPhwd91p6lS3lbvkTQmqZMrIcNeqPvnEJa1169zWM9df7zbr69zZbQniFwcOuDL6BQvcaPDwYTd92a2bq9JV+ydfUoLKac8eN+fctCl8/bWG+IFQgpJDh1xxwYwZ7k9Skrv9lltcNWCbNq41ULB3cNm82SXRVatc6fyhQ9C+vbuWds897m/xNSWonPr1g8GDYdkyV8Uk+VOCkpyshdWrXaJauNAtck1Ph3POcdOBnTvDhRdCdLQr465du2hGL6mpsHYtrFjhfn8XLnQl8mlprsCjcWN3Xenqq11XDQkJBUpQYd1HITnZ9dl68kklJ5HTZYy7htusmfs6LQ0SEyEhAZYuhW3b3Ghr7VpXJdu8uStEqloVzj7b3VapkttzrWxZ173FGJf40tPd8+3b57ZWT02F3bvd7+6+fW6EBK7fYFycmwlp394lQs2GhL+wTlCvv+6mJh54wOtIRMJHuXKufPuqq/57265dLkGtXw+//+5GOqmp7t8HD7pEFBHhRlaHDrm/y5RxW4aUKePuL1fOJbGzz3bThxdc4NZsRUdDjRrefb/inbBNUMnJrqKnVy91KxcpblWruqm2q6/O/f6DB90I6eBBV01r7X+TVIUKbqpQ1XZyorBNUK+/7ppkPvus15F46+233+add96hdOnSdOnShUGDBnkdkpRA5cu7PyIFEZYJ6ujoqXdvuOgir6Pxzvz585k2bRqrV6+mbNmy/P77716HJCISsLBcKaDRkzNixAj69+9P2eyVlpGRkR5HJCISuLBLUElJbvR0zz1Qp47X0Xjrxx9/5Ntvv6VVq1a0adOGpUuXeh2SiEjAwm6K7+jo6ZlnvI4kONq3b09KSspJtw8YMIDMzEx2797N4sWLWbp0KT169ODnn3/G5FKfGx8fT3x8PAA7SuJ+DCLiO2G1UDcpCS6+2FXuZb/XlmhxcXH079+fttmb3lx88cUsXryY6tWr5/k4LdQVkWJSoNVrYTXFp2tPx+vWrRvz5s0D3HTf4cOHqRbs3jQiIqcpbKb4tm1zmxH26eNarQj06dOHPn360LhxYyIiIhg9enSu03siIn4UNgnq9dfd4r+Scu0pEBEREYwbN87rMERETktYTPFt2+ba7Wv0JCISPsIiQR0dPT39tNeRiIhIUQn5BLV9O2zY4BrCavQkIhI+Qv4a1Ntvu/1pVFYuIhJeQnoEtX8/vPee2+VTHctFRMJLSCeojz5yW7r/9a9eRyIiIkUtZBNUZiYMHeo2Nmvd2utoRESkqIVsgpoyBTZv1uhJRCRchWSCshYGD4Z69aBrV6+jERGR4hCSVXwJCbBkCbz7LpxxhtfRiIhIcQjJEdSQIXDOOW7HXBERCU8hl6A2boRp0+DBB6F8ea+jERGR4hJyCeof/4AyZeCRR7yOREREilNIJajUVBg5Eu66C8491+toRESkOIVUghoxAtLS4IknvI5ERESKW8gkqIwM+OQTuP9+aNTI62hERKS4hUyCmjoV1q/XuicRkZIiZBLU8OFQpw507Oh1JCIiEgwhkaB++AEWLHDTe1qYKyJSMoREghoxAiIi3JbuEriVK1fSunVrYmJiiI2NZcmSJV6HJCISMN8nqAMHYPRouPVWiIz0OprQ0q9fP1544QVWrlzJyy+/TL9+/bwOSUQkYL5PUJ98Anv3wkMPeR1J6DHGsHfvXgD++OMPatas6XFEIiKBM9bavO7P887iZi20aOH2flq1CozxMprQ88MPP9CxY0estWRlZfHdd99Ru3btXI+Nj48nPj4egB07drB169ZghioiJUOB3sV9naD+8x+3GeHw4a73npysffv2pKSknHT7gAED+Prrr2nTpg3du3dn4sSJxMfHM3fu3HyfMzY2lsTExOIIV0RKtvBJUL16weTJsH07VKrkZSShqXLlyuzZswdjDNZaKleufGzKLy9KUCJSTAqUoHx7DWrnTvj0U9d3T8np9NSsWZNvvvkGgHnz5lG/fn2PIxIRCZxvNywcORLS0zW1VxgffPABf/nLX8jMzKRcuXLHrjGJiIQCX07xZWXBJZdAjRrw7bdeRFCyaYpPRIpJ6E/xzZkDmzaptFxEpCTzZYIaMwaqV4dbbvE6EhER8YrvElRKCkyaBI8+CmXLeh2NiIh4xXcJatw4OHwYbrvN60hERMRLvkpQ1rrqvcsvhwYNvI5GRES85KsEtXQprFsH99zjdSQiIuI1XyWokSPhzDPhT3/yOhIREfGabxJUWhqMHw/du8NZZ3kdjYiIeM03CWrqVPjjD+jd2+tIRETED3yToEaOhNq1oV07ryMRERE/8EWC2rbNdY/o1QtK+SIiERHxmi/SwZgxrsRc03siInKU5wnKWhg1Ctq2hYsu8joaERHxC88TVEIC/PST1j6JiMjxPE9QI0dCxYquvFxEROQoTxPUgQMwcSL06AEVKngZiYiI+I2nCWraNNi/X9N7IiJyMk8T1EcfueKIK6/0MgoREfEjzxJUSgrMn++SkynQJsAiIlISeJagPvsMsrLg9tu9iiB8fPbZZ0RHR1OqVCkSExOPu2/gwIHUq1ePBg0a8OWXX3oUoYhIwZX26oUnTIAmTSA62qsIwkfjxo2ZPHky999//3G3r1u3jgkTJrB27Vq2b99O+/bt+fHHHznjjDM8ilREJHCejKC2boXvvtPoqahERUXRIJcdHqdNm0bPnj0pW7YsF110EfXq1WPJkiUeRCgiUnCeJKgJE9zfPXt68eolR3JyMhdccMGxr2vVqkVycnKux8bHxxMbG0tsbCw7duwIVogiIqfkyRTf+PHQurVaGxVE+/btSUlJOen2AQMGcNNNN+X6GGvtSbeZU1Sk9O3bl759+wIQGxtbiEhFRIpG0BPUDz/AqlUwdGiwXzm0zZ07t8CPqVWrFtu2bTv2dVJSEjVr1izKsEREik3Qp/jGj3dbavToEexXLnm6du3KhAkTSE9PZ/PmzWzcuJGWLVt6HZaISECCmqCsdQmqbVuoUSOYrxzepkyZQq1atVi0aBFdunShY8eOAERHR9OjRw8aNWpEXFwc7777rir4RCRkmNyuU+SQ550FlZgIl10GH3wA991XlM8sRSk2Nvak9VQiIkWgQG0ZgjqCGj8eypRR53IREclf0BJUVhZ8+inExcHZZwfrVUVEJFQFLUF9+y0kJ2txroiIBCZoCWr8eChfHrp2DdYriohIKAtKgsrIgEmTXHLSxoQiIhKIoCSouXNh505N74mISOCCkqDGj4cqVSB7eY6IiEi+ij1BZWTAjBmutLxs2eJ+NRERCRfF3ouvTBnYsAHS0or7lUREJJwEpVls9erBeBUREQknnm35LiIikhclKBER8SUlKBER8SUlKBER8SUlKBER8SUlKBER8SUlKBER8SUlKBER8SUlKBER8SVjrfU6BvEZY8xsa22c13GISMmmBCUiIr6kKT4REfElJSgREfElJSgREfElJSgREfElJSgREfGl/weiaH/EB+qi0AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ***************** PARA LA DEFINICIÓN DE FUNCIONES*************************************************\n",
    "# El producto hay que indicarlo con asterisco (*).Las potencias se indican con doble asterisco (**).\n",
    "# La función seno(x) se define como sin(x),# la funcion arcoseno(x) SE DEFINE COMO asin(x),\n",
    "# La función arcoscosno(x) como acos(x), la funíon arcotangente(x) como atan(x)\n",
    "# La función logaritmo neperiano como log(x). La función exponencial como exp(x).\n",
    "# La función valor absoluto como abs(x).\n",
    "# Definir la función f y asignar el valor para n después de la línea \"AQUÍ SE DEFINE LA FUNCIÓN Y\n",
    "# SE ASIGNA EL VALOR DE n\"\n",
    "# ************************************************************************************************************************\n",
    "#Llamada de funciones de sympy\n",
    "from sympy import sin, cos, tan, asin, acos, atan, log, exp\n",
    "import sympy as sp\n",
    "x=sp.Symbol('x')\n",
    "\n",
    "# ****************************************************************************\n",
    "#            AQUÍ SE DEFINE LA FUNCIÓN Y SE ASIGNA EL VALOR DE n\n",
    "f=2*x**3-5*x**2-1\n",
    "\n",
    "# ****************************************************************************\n",
    "\n",
    "\n",
    "%matplotlib inline\n",
    "from sympy.plotting import plot\n",
    "\n",
    "fpp=sp.diff(f,x,2) #Cálculo de la derivada como función\n",
    "\n",
    "sol=sp.solveset(fpp, x, domain=sp.S.Reals) # Halla las raíces de f''(x)=0\n",
    "\n",
    "sol=list(sol) # Convierte las soluciones de formato FiniteSet a formato Lista\n",
    "\n",
    "sol=sorted(sol)\n",
    "\n",
    "\n",
    "fppp=sp.diff(f,x,3) #Cálculo de la derivada como función\n",
    "\n",
    "print('\\033[1m'+'SOLUCIÓN: PUNTOS DE INFLEXIÓN\\n'+'\\033[0m')\n",
    "if len(sol)==0:\n",
    "    print('La función f(x)='+str(f)+' no tiene puntos de inflexión\\n')\n",
    "else:\n",
    "    for i in range(0,len(sol)):\n",
    "        if fppp.subs(x,sol[i])!=0:\n",
    "            print('('+str(sol[i])+','+str(f.subs(x,sol[i]))+')'+' es punto de inflexión de la función f(x)='+str(f)+'\\n')\n",
    "        else:\n",
    "            print('La función f(x)='+str(f)+' no tiene puntos de inflexión\\n')\n",
    "            \n",
    "if len(sol)!=0:\n",
    "    p=plot(f,(x,min(sol)-2,max(sol)+2),title='Grafica de la función f\\n',line_color='blue',show=False)\n",
    "else:\n",
    "    p=plot(f,title='Grafica de la función f',line_color='blue',show=False)    \n",
    "p.show()              \n",
    "                "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Criterio para el estudio de la curvatura"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sea $f(x)$ una función cuya segunda derivada existe en $(a,b)$:\n",
    "    \n",
    "a) Si $f''(x)>0$ para todo $x \\in (a,b)$, la gráfica de $f(x)$ es convexa en $(a,b)$\n",
    "\n",
    "b) Si $f''(x)<0$ para todo $x \\in (a,b)$, la gráfica de $f(x)$ es cóncaba en $(a,b)$\n",
    "\n",
    "Analizar la curvatura de una función consiste en estudiar en qué intervalos es cóncava o convexa. Cabe notar que los intervalos de curvatura está separados por los puntos de inflexión y las discontinuidades de $f''(x)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pasos para analizar la curvatura de una función"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Se calculan los puntos de inflexión\n",
    "\n",
    "2. Se hallan las discontinuidades de la tercera derivada de la función, $f'''(x)$\n",
    "\n",
    "3. Con los puntos de los pasos 1 y 2 se determinan los intervalos a analizar.\n",
    "\n",
    "4. Para cada intervalo determinado en el paso 3 se toma un punto del intervalo y se evalúa la segunda derivada $f''(x)$\n",
    "\n",
    "$\\qquad$ * Si $f''(x)>0$ la función es convexa\n",
    "\n",
    "$\\qquad$ * Si $f''(x)<0$ la función es cóncava\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Ejemplo}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dada la función $f(x)=2x^3-5x^2-1$, analizar la curvatura\n",
    "\n",
    "Paso 1. Calculamos los puntos de inflexión, tiene un punto de inflexión en $x=\\frac{5}{6}$\n",
    "\n",
    "Paso 2. Se hallan las discontinuidades de la tercera derivada de la función, $f'''(x)=12$. No tiene\n",
    "\n",
    "Paso 3. Se determinan los intervalos a analizar $(-\\infty,\\frac{5}{6})$ y $(\\frac{5}{6},+\\infty)$\n",
    "\n",
    "Paso 4. Identificamos la curvatura en cada intervalo\n",
    "* Calculamos la segunda derivada, $f''(x)=12x-10$\n",
    "\n",
    "* En el intervalo $I_1=(-\\infty,\\frac{5}{6})$, $f''(0)=-10$ que es menor que cero, por lo que es cóncava.\n",
    "\n",
    "* En el intervalo $I_2=(-\\infty,\\frac{5}{6})$, $f''(1)=2$ que es mayor que cero, por lo que es convexa.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el módulo 10, para una función  𝑓  dada (no periódica), se identifican los intervalos de curvatura."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Módulo 10. Identifica los intervalos de curvatura de una función $f$ dada"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[1mSOLUCIÓN: ANÁLISIS DE CURVATURA\n",
      "\u001b[0m\n",
      "Función dada: f(x)=2*x**3 - 5*x**2 - 1\n",
      "\n",
      "Segunda derivada: f´´(x)=2*(6*x - 5)\n",
      "\n",
      "(5/6,-179/54) es punto de inflexión\n",
      "\n",
      "En el intervalo (-∞,5/6) la función es cóncava\n",
      "\n",
      "En el intervalo (5/6,+∞) la función es convexa\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
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     "output_type": "display_data"
    }
   ],
   "source": [
    "# ***************** PARA LA DEFINICIÓN DE FUNCIONES************************************************\n",
    "# El producto hay que indicarlo con asterisco (*).Las potencias se indican con doble asterisco (**).\n",
    "# La función seno(x) se define como sin(x),# la funcion arcoseno(x) SE DEFINE COMO asin(x),\n",
    "# La función arcoscosno(x) como acos(x), la funíon arcotangente(x) como atan(x)\n",
    "# La función logaritmo neperiano como log(x). La función exponencial como exp(x).\n",
    "# La función valor absoluto como abs(x).\n",
    "# Definir la función f y asignar el valor para n después de la línea \"AQUÍ SE DEFINE LA FUNCIÓN Y\n",
    "# SE ASIGNA EL VALOR DE n\"\n",
    "# ***********************************************************************************************\n",
    "#Llamada de funciones de sympy\n",
    "from sympy import sin, cos, tan, asin, acos, atan, log, exp\n",
    "import sympy as sp\n",
    "x=sp.Symbol('x')\n",
    "\n",
    "# ****************************************************************************\n",
    "#            AQUÍ SE DEFINE LA FUNCIÓN Y SE ASIGNA EL VALOR DE n\n",
    "f=2*x**3-5*x**2-1\n",
    "#f=5+(x**2-2)\n",
    "# ****************************************************************************\n",
    "\n",
    "\n",
    "%matplotlib inline\n",
    "from sympy.plotting import plot\n",
    "\n",
    "\n",
    "fpp=sp.diff(f,x,2) #Cálculo de la segunda derivada como función\n",
    "\n",
    "# Halla los puntos de inflexión\n",
    "\n",
    "sol=sp.solveset(fpp, x, domain=sp.S.Reals) # Halla las raíces de f''(x)=0\n",
    "\n",
    "sol=list(sol) # Convierte las soluciones de formato FiniteSet a formato Lista\n",
    "\n",
    "sol=sorted(sol) #Ordenamos la lista de menor a mayor\n",
    "\n",
    "fppp=sp.diff(f,x,3) #Cálculo de la derivada como función\n",
    "\n",
    "print('\\033[1m'+'SOLUCIÓN: ANÁLISIS DE CURVATURA\\n'+'\\033[0m')\n",
    "print('Función dada: f(x)='+str(f)+'\\n')\n",
    "print('Segunda derivada: f´´(x)='+str(fpp)+'\\n')\n",
    "\n",
    "\n",
    "solinflex=list()\n",
    "if len(sol)==0:\n",
    "    print('La función f(x)='+str(f)+' no tiene puntos de inflexión\\n')\n",
    "else:\n",
    "    for i in range(0,len(sol)):\n",
    "        if fppp.subs(x,sol[i])!=0:           \n",
    "            solinflex.append(sol[i])            \n",
    "            print('('+str(sol[i])+','+str(f.subs(x,sol[i]))+')'+' es punto de inflexión\\n')\n",
    "        #else:\n",
    "            #print('La función f(x)='+str(f)+' no tiene puntos de inflexión\\n')\n",
    "\n",
    "#Puntos de inflexión en solinflex[]\n",
    "# Se identifica la curvatura\n",
    "\n",
    "if len(solinflex)==0:    # Se definen los intervalos\n",
    "    solinflex[len(solinflex):]=[0,1]\n",
    "else:\n",
    "    solinflex[len(solinflex):]=[min(solinflex)-1,max(solinflex)+1]\n",
    "\n",
    "solinflex=sorted(solinflex)    \n",
    "    \n",
    "linter=list()\n",
    "for i in range(0,len(solinflex)): # Se define la lista linter con los extremos de los intervalos\n",
    "    linter.append(str(solinflex[i]))\n",
    "\n",
    "linter[0]=('-∞'); linter[len(linter)-1]=('+∞')\n",
    "\n",
    "\n",
    "#Análisis de intervalos de curvatura\n",
    "\n",
    "for i in range(0,len(solinflex)-1):\n",
    "    xm=(solinflex[i+1]+solinflex[i])/2\n",
    "    fppval=fpp.subs(x,xm)\n",
    "    if fppval==0:\n",
    "        fcurva='constante'\n",
    "    else:\n",
    "        if fppval>0:\n",
    "            fcurva='convexa'\n",
    "        else:\n",
    "            fcurva='cóncava'\n",
    "    print('En el intervalo ('+linter[i]+','+linter[i+1]+') la función es '+fcurva+'\\n')\n",
    "\n",
    "\n",
    "p=plot(f,(x,min(solinflex)-1,max(solinflex)+1),title='Grafica de la función f',line_color='blue',show=False)\n",
    "p.show()    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large{\\textbf{Práctica}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{IX}$. Con las funciones definidas en ($\\textbf{VII}$.) calcula los puntos de inflexión de tales funciones. Compara el resultado obtenido con los que se obtienen usando el módulo 9.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textbf{X}$. Con las funciones definidas en ($\\textbf{VII}$.) estudia la curvatura de tales funciones. Compara el resultado obtenido con los que se obtienen usando el módulo 10."
   ]
  }
 ],
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