Presentacion.ipynb

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   "source": [
    "from IPython.display import IFrame"
   ]
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   "source": [
    "# Resolviendo problemas de álgebra y cálculo usando programación"
   ]
  },
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   "source": [
    "## Presentación"
   ]
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   "source": [
    "El tema de la presentación de hoy es el uso de la librería Sympy (https://www.sympy.org/) en combinación \n",
    "con los diccionarios (un tipo de estructura de datos de Python) para resolver problemas de álgebra y\n",
    "cálculo."
   ]
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   "source": [
    "## Temario\n",
    "\n",
    "- Sympy - la libería de álgebra simbólica de Python.\n",
    "- ¿$x=x$? Variables, expresiones y simbolos.\n",
    "- Pongamos todo junto: usando diccionarios.\n",
    "- ¿Porqué resolver un problema, si podemos resolver 10?"
   ]
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   "source": [
    "## Sympy - la libería de álgebra simbólica de Python"
   ]
  },
  {
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   "id": "516c3d63-c0df-4d0a-a8d7-14e4cd8841bf",
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   "source": [
    "> SymPy es una biblioteca de Python para matemáticas simbólicas. \n",
    "> Su propósito es llegar a ser un sistema de álgebra por computadora (CAS) \n",
    "> completo manteniendo el código tan simple como sea posible para poder ser \n",
    "> legible y extensible de manera fácil. SymPy está escrito en Python enteramente. \n",
    ">\n",
    "> -- <cite>https://www.sympy.org/en/index.html</cite>"
   ]
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       "\n",
       "        <iframe\n",
       "            width=\"1000\"\n",
       "            height=\"500\"\n",
       "            src=\"http://www.sympy.org/es/\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
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   "source": [
    "IFrame('http://www.sympy.org/es/', width=1000, height=500)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "726b30d1-da9f-4cd2-bd11-c9f7ff11bf53",
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   },
   "source": [
    "¿Cómo podemos acceder a esta magia desde Python?..."
   ]
  },
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   "source": [
    "# Aquí vamos!!\n",
    "import sympy as sp"
   ]
  },
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   "source": [
    "__Nota:__ El `sp` es simplemente un _alias_ o modo alternativo para llamar a la librería, evitando escribir `sympy` cada vez. "
   ]
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   "source": [
    "Pero esperen... hay algo que es importante decir:\n",
    "__SymPy__ puede utilizar diferentes herramientas para dar la mejor presentación."
   ]
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   "source": [
    "![Latex](Imagenes/Latex.png)"
   ]
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   "source": [
    "Si __miketex__ (disponible en anaconda) está instalado permite una representación basada en __LaTeX__ (https://www.latex-project.org/). Esta representación está disponible automáticamente en Jupyter, pero para poder aprovechar esta representación en otros sitemas (por ejemplo Spyder) es necesario importar:"
   ]
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   "source": [
    "from IPython.display import display as pp # pretty print"
   ]
  },
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   "source": [
    "![Spyder](Imagenes/Spyder.png)"
   ]
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   "source": [
    "Bueno, continuemos.\n",
    "El primer paso es declarar los simbolos que utilizaremos. SymPy reconoce automáticamente la escritura de simbolos de LaTeX:"
   ]
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     "data": {
      "text/latex": [
       "$\\displaystyle x$"
      ],
      "text/plain": [
       "x"
      ]
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     "data": {
      "text/latex": [
       "$\\displaystyle P_{c}$"
      ],
      "text/plain": [
       "P_{c}"
      ]
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    }
   ],
   "source": [
    "x = sp.Symbol('x')\n",
    "Pc = sp.Symbol('P_{c}')\n",
    "pp( x, Pc )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31e8230c-880b-4e4b-ae87-6ca89806a02c",
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   "source": [
    "Estos símbolos son ahora símbolos algebraicos con los que podemos trabajar junto con numeros y armar expresiones."
   ]
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle P_{c} x + 5 x^{2} + 6$"
      ],
      "text/plain": [
       "P_{c}*x + 5*x**2 + 6"
      ]
     },
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   "source": [
    "f = 5 * x**2 + Pc * x + 6\n",
    "pp( f )"
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   "source": [
    "## ¿$x=x$? Variables, expresiones y simbolos."
   ]
  },
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   "source": [
    "¿Cómo es eso de que $x = x$?\n",
    "\n",
    "`x = sp.Symbol('x')`\n",
    "\n",
    "Lo que realmente es importante entender que lo que está a la __derecha__ es un _símbolo_ algebraico, \n",
    "mientras que lo que está a la __izquierda__ es una _variable_ de Python."
   ]
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   "source": [
    "En Python __todo__ es un __objeto__ y aunque no me quiero meter a este tema \n",
    "(y no porque no sea interesante, sino porque nos podría tomar mucho más tiempo)\n",
    "podemos afirmar que _la variable_ `x` contiene _el símbolo_ $x$."
   ]
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   "source": [
    "Así, podemos hacer, por ejemplo:"
   ]
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3}$"
      ],
      "text/plain": [
       "x**3"
      ]
     },
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   ],
   "source": [
    "y = x * x * x\n",
    "pp( y )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b9644e3-7b84-4994-b1a6-48f7227e7825",
   "metadata": {
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   "source": [
    "Aunque aquí, en realidad, `y` contiene una _expresión algebraica_ (que, al igual que los _símbolo_ `x` es un _objeto_ de __sympy__)."
   ]
  },
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   "source": [
    "Es decir:"
   ]
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   "source": [
    "![Fig3](Imagenes/TutSympy.png)"
   ]
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   },
   "source": [
    "Nada nos impide hacer cosas que pudieran no tener mucho sentido:"
   ]
  },
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle b$"
      ],
      "text/plain": [
       "b"
      ]
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     "data": {
      "text/latex": [
       "$\\displaystyle a$"
      ],
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       "a"
      ]
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   "source": [
    "a = sp.Symbol('b')\n",
    "b = sp.Symbol('a')\n",
    "pp( a,b )"
   ]
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  {
   "cell_type": "markdown",
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   "source": [
    "Pero, cómo sabemos, _con un gran poder viene una grán responsabilidad_ por lo que estoy seguro que no harán este tipo de cosas ;)."
   ]
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle a$"
      ],
      "text/plain": [
       "a"
      ]
     },
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     "data": {
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       "$\\displaystyle b$"
      ],
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       "b"
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   "source": [
    "a = sp.Symbol('a')\n",
    "b = sp.Symbol('b')\n",
    "pp( a,b )"
   ]
  },
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   "source": [
    "### Álgebra"
   ]
  },
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   "source": [
    "Una vez que hemos declarado algunos símbolos podemos manipularlos con diferentes __expresiones algebráicas__:\n",
    "- Sumas, restas, multiplicaciones y divisiones con otros símbolos o con números enteros y reales:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "b0c8474e-d9fd-410e-b39a-298de3da2729",
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.26432 x + \\frac{4 x}{2 x - 1}$"
      ],
      "text/plain": [
       "1.26432*x + 4*x/(2*x - 1)"
      ]
     },
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    }
   ],
   "source": [
    "pp( 4*x/(2*x-1) + 1.26432*x )"
   ]
  },
  {
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   "id": "41f38e81-197c-4f65-b574-edfcc0486f0d",
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   },
   "source": [
    "- Potencias y racies:"
   ]
  },
  {
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   "id": "3c3602ef-6239-4939-9ff2-9b6d1e14bb55",
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\sqrt{P_{c}} + \\sqrt[3]{x} + x^{2}$"
      ],
      "text/plain": [
       "-sqrt(P_{c}) + x**(1/3) + x**2"
      ]
     },
     "metadata": {},
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    }
   ],
   "source": [
    "pp( x**2 - sp.sqrt(Pc) + sp.root(x,3) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "afad6a3b-33fd-4bb1-835c-76e1671060f2",
   "metadata": {
    "slideshow": {
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    "tags": []
   },
   "source": [
    "- Funciones trigonométricas:"
   ]
  },
  {
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   "execution_count": 12,
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle P_{c} \\cos{\\left(x \\right)} + \\operatorname{atan}{\\left(x - 1 \\right)}$"
      ],
      "text/plain": [
       "P_{c}*cos(x) + atan(x - 1)"
      ]
     },
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    }
   ],
   "source": [
    "pp( Pc * sp.cos(x) - sp.atan(1-x) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "48fa3bbc-6a73-46d8-ae2d-eb8eef1aa1be",
   "metadata": {
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   },
   "source": [
    "- Fracciones"
   ]
  },
  {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{3}{4}$"
      ],
      "text/plain": [
       "3/4"
      ]
     },
     "metadata": {},
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    }
   ],
   "source": [
    "pp( sp.Rational(1,2) + sp.Rational(1,4) )"
   ]
  },
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   "source": [
    "### Substituciones de valores o variables:"
   ]
  },
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   },
   "source": [
    "Para substituir se puede usar el método `subs(variable,valor)`"
   ]
  },
  {
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   "execution_count": 14,
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 30$"
      ],
      "text/plain": [
       "30"
      ]
     },
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    }
   ],
   "source": [
    "pp( (x+x**2).subs(x,5) ) # 5+5x5 = 30"
   ]
  },
  {
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   },
   "source": [
    "(Los paréntesis nos permiten que la expresión sea formada antes de levar a cabo substitución)."
   ]
  },
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   },
   "source": [
    "Las substituciones pueden ocurrir una después de la otra, o usando un diccionario:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "be9d4d93-1134-4c49-bfb1-87a320a64a99",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 16$"
      ],
      "text/plain": [
       "16"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( (x**a).subs(x,4).subs(a,2) ) # 4²"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "8d02d78c-c195-473a-99ef-851a8fa015fc",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 16$"
      ],
      "text/plain": [
       "16"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x**a).subs({x:4, a:2}) # 4²"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7b995b6c-6ea9-454c-b90a-e39d759c26b5",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "### Simplificaciones, factorizaciones y expansiones"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "773d0bed-264d-4f54-80a9-24dc090a24ab",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Declaramos una función polinomial para los siguientes comandos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "f88c8f09-fd92-492d-a2e6-cf4812ab1a02",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 3 x^{3} + 4 x^{2} + x$"
      ],
      "text/plain": [
       "3*x**3 + 4*x**2 + x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z = x**3 + 4*x**2 + 2*x**3 + x\n",
    "pp(z)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4805ec57-9304-4719-a7c5-38d8b3828909",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "El método `simplify` intentará obtener una expresión más sencilla:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "d2b68431-7041-4ad9-88fe-19b3e28c723e",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x \\left(3 x^{2} + 4 x + 1\\right)$"
      ],
      "text/plain": [
       "x*(3*x**2 + 4*x + 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( sp.simplify(z) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da677d46-3931-436e-8068-986ebd33b2fd",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "__Nota__: estas funciones se pueden utilizar llamándolas como métodos de las __expresiones__ algebraicas:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "a2142516-6dff-4fab-8240-9a83f67cecb2",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x \\left(3 x^{2} + 4 x + 1\\right)$"
      ],
      "text/plain": [
       "x*(3*x**2 + 4*x + 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( z.simplify() )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "3bb16987-8ae8-405e-8d2b-6f39591447e5",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( (sp.sin(x)**2 + sp.cos(x)**2).simplify() ) # (sen(x))² + (cos(x))² = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0dc779c7-2aeb-4b3b-a709-467d6a03b83d",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "El método `factor` intentará factorizar la expresión:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "57cd681c-b0a4-4fce-ac49-7af3fb440ebc",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x \\left(x + 1\\right) \\left(3 x + 1\\right)$"
      ],
      "text/plain": [
       "x*(x + 1)*(3*x + 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( z.factor() )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "17071ebe-3126-4fc0-b079-016989d7f5dc",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Por otro lado, el método `expand` aplicará la propiedad distributiva, expandiendo la expresión:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "baa558ac-3a64-4d09-9298-bfeb1c5e5934",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 3 x^{3} + 4 x^{2} + x$"
      ],
      "text/plain": [
       "3*x**3 + 4*x**2 + x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( z.expand() )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4713fc29-18f0-48bb-a0ab-da8a58d285d7",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "### Derivación e integración"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31d6d343-2119-4c01-8417-05d9c228884d",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Se pueden derivar expresiones usando el método `diff`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "1e8f2506-ef65-4f2e-ba44-23c766da108c",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 9 x^{2} + 8 x + 1$"
      ],
      "text/plain": [
       "9*x**2 + 8*x + 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( z.diff(x) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "7ec1b0b7-9c12-4b98-bfec-ee25bff8326b",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\pi \\cos{\\left(\\pi x \\right)}$"
      ],
      "text/plain": [
       "pi*cos(pi*x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( sp.sin(sp.pi*x).diff(x) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ef93ce74-d8a8-4e9f-9f7f-db26682f414c",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Para integrar se utiliza el método `integrate`. Si se usa simplemente la variable la integral será indefinida:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "6beb709a-ae8a-4941-aab0-5b9d61f75c72",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{3 x^{4}}{4} + \\frac{4 x^{3}}{3} + \\frac{x^{2}}{2}$"
      ],
      "text/plain": [
       "3*x**4/4 + 4*x**3/3 + x**2/2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( z.integrate(x) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "1031df06-24c3-4fb0-abc8-f8dbbc52e78f",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{\\cos{\\left(\\pi x \\right)}}{\\pi}$"
      ],
      "text/plain": [
       "-cos(pi*x)/pi"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( sp.sin(sp.pi*x).integrate(x) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a0d0e1c-28e4-4dab-9469-29d80aeea0d3",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Para integrales definidas se incluye la variable y los límites en una tupla `(variable, inferior, superior)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "cd79a145-a702-4b09-a2df-fc03dc0546b6",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{31}{12}$"
      ],
      "text/plain": [
       "31/12"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( z.integrate((x,0,1)) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "b5faa014-6c83-430c-9be6-54ff4fcdb298",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2}{\\pi}$"
      ],
      "text/plain": [
       "2/pi"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( sp.sin(sp.pi*x).integrate((x,0,1)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "25869e7f-b77f-4a80-85a9-89ba1c39fcb1",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "### Vectores y matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66c8aa60-c64e-43a2-9969-b0ebbce2ed56",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Para declarar un vector o una matriz se utiliza la función `matrix` y una lista de listas:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "7bb377bd-f9ef-4783-80a6-122308085024",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2\\\\3 & 4\\\\5 & 6\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1, 2],\n",
       "[3, 4],\n",
       "[5, 6]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "m1 = sp.Matrix([[1,2],\n",
    "                [3,4],\n",
    "                [5,6]])\n",
    "pp( m1 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "8c5e2038-6069-4919-b97f-076902f4dfb7",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}6\\\\5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[6],\n",
       "[5]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "v1 = sp.Matrix([6,5])\n",
    "pp( v1 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "75d7ff65-de6d-4f8e-8361-46e3bee1e187",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3\\\\2\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[3],\n",
       "[2],\n",
       "[1]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "v2 = sp.Matrix([3,2,1])\n",
    "pp( v2 )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "662cda0e-ac23-4ae7-b438-69d44dd3c113",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Las multiplicaciones de matrices se obtienen utilizando simplemente el operador `*` (siempre y cuando se cumplan las reglas de dichas operaciones):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "45749ae4-92db-4fe5-ba2c-83ff4d81335d",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}16\\\\38\\\\60\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[16],\n",
       "[38],\n",
       "[60]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( m1 * v1 )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "726e3497-0db0-43f1-a9de-54b95b72288f",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Para transponer cualquier vector o matriz se utiliza `.T`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "cc84c6d7-1ad7-4bb2-95cd-259f98052992",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3 & 2 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([[3, 2, 1]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( v2.T )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "6de0e847-9ad7-4018-b649-cbfedee3e386",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}14 & 20\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([[14, 20]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( v2.T * m1 )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1b1effca-005c-4d7c-a5f7-557a3412f81e",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Para accesar los elementos, renglones o columnas de una matriz se utilizan los corchetes después de la expresión `[renglon, columna]`, con los dos puntos `:` para una línea o renglón completo (recordar que en __Python__ los subíndices siempre inician de cero):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "151ca00e-3fef-4644-9307-9a27242232db",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( m1[0,1] ) # Elemento 0 de la columna 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "035df63d-ef39-433b-87c7-78d2ee3d119e",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\3\\\\5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1],\n",
       "[3],\n",
       "[5]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( m1[:,0] ) # Toda la columna 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "086273b2-939b-4482-9a29-ab7b250dac48",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}5 & 6\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([[5, 6]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( m1[2,:]  )# Todo el renglón 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0b0133d6-f2db-4b42-94eb-b2545cbfd36e",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Para un vector podemos usar un sólo índice (`[número]`):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "8527935a-ddd9-417d-81d5-72842f5d63d4",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 6$"
      ],
      "text/plain": [
       "6"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( v1[0] )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "34749511-d401-4150-a94a-f6ca4d54b4ff",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Es importante hacer notar que para __Sympy__ no es igual una matriz 1x1 que un número, de tal manera que:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "28a113cc-858e-46cc-b8a2-a91f5d147c22",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}14\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([[14]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( v2.T * v2 )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "027cb2eb-0960-42d4-a8d2-9282e48b82c1",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Si queremos utilizar este valor como un número debemos extraerlo, utilizando su posición (`[0]` o `[0,0]`):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "2f581da7-d50e-4ba3-bfba-c8967d9405f8",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 19$"
      ],
      "text/plain": [
       "19"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( (v2.T * v2)[0,0] + 5  )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "0852ab47-12c9-482f-8c36-74002ba227d3",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}84\\\\70\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[84],\n",
       "[70]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( (v2.T * v2)[0,0] * v1  )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "29e8ccdd-4eb8-43fc-a091-404b939b5273",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "En el caso de dos vectores podemos usar la operación punto:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "3165d3af-616f-4717-b117-56a2ec01de61",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 14$"
      ],
      "text/plain": [
       "14"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( v2.dot(v2) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5aa39244-a4f2-4fd8-a09c-2640e7d3dc02",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Puede ser útil también el cálculo del determinante."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "0617bb1a-42fe-4ff3-91a3-5531e4a933fc",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -9$"
      ],
      "text/plain": [
       "-9"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "m2 = sp.Matrix([[1,2,3],\n",
    "                [2,3,1],\n",
    "                [4,4,1]])\n",
    "\n",
    "pp( m2.det() )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e9006dc5-5b34-406a-93a9-03f692387056",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "### Combinando lo anterior"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8742c8e6-fdc1-4ff4-9d24-f5581119f2b3",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Todo lo anterior se puede usar en conjunto:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "2a59f07c-8ed2-4f0b-8ff6-9049a1c5637b",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & \\frac{2}{3}\\\\\\frac{1}{2} & \\infty\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  1, 2/3],\n",
       "[1/2,  oo]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pp( (a * sp.Matrix([[x,    x**2 ],\n",
    "                [x**3, x**-1]])).integrate((x,0,1)).subs(a,2) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "39c0eddd-f554-4796-ae64-be697531680e",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "### Gráficas de funciones"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dfaeeedd-e4ec-4156-a1f8-f28c6b3c1c48",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Para graficar podemos usar `sp.plotting.plot`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "6d061d62-7404-43fc-9d0e-0a116912b6f0",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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94gR2Hq/Es0lRmBwyXO44AwrLiojIAj7ML8P+4lrcM3kU7p4UJHecAYdlRUTUz3adqMRTm4/AxdEef5wbCUnigIorxbIiIupHh8rqsfK97xHu5470uyfAwZ5vu1eDW42IqJ+U1jRj2bv7MdzNEf+8dyLcOPLvqrGsiIj6QXWTHkveyYNJCGQui4ePu7PckQY01jwRUR9raTdg+bv7UdHQhvfvm8xr/vUB7lkREfWhDqMJj354CEe0DXhz0XWYMGqY3JFsAsuKiKiPGIwmPPzvgzima8RL86MxK4oXp+0rLCsioj5gMgk8vukwth7W4a5JQVgYz3Op+hLLiojoGplMAr//+Ag2H9Di0YSxSJ06Ru5INodlRUR0DYQQeO7TQmTtL8Nvp6nw4IxQuSPZJJYVEdFVEkJg9WfHsH5vKVKnhuDRWWPljmSzWFZERFdBCIFXvziBt3cXY+n1o/FUYjgvo9SPeJ4VEdEVMpkEVm87hrwfL0z7LK/31++4Z0U2Z9myZfDx8cG4cePM02pra5GQkIDQ0FAkJCSgrq7OPC8tLQ0qlQphYWHIzc01Ty8oKEB0dDRUKhVWrVoFIYRF14Osk8FowuMfHca63cWYMGoYnk+KYlFZAMuKbM7SpUuRk5Nz0bQ1a9ZgxowZ0Gg0mDFjBtasWQMAKCoqQlZWFgoLC5GTk4OVK1fCaDQCAFasWIGMjAxoNBpoNJouP5MGn7YOIx54/3tsKijH72aG4tm5kbwlvYWwrMjmTJ06FV5eXhdNy87ORkpKCgAgJSUFW7ZsMU9fuHAhnJycEBwcDJVKhby8POh0OjQ2NmLKlCmQJAlLliwxL0ODU5PegGXv7kdu4Tk8OzcSv5s5lntUFsTPrGhQOHfuHPz9/QEA/v7+qKysBABotVpMnjzZ/DilUgmtVgsHBwcolcou07uTkZGBjIwMAEBVVVV/rQLJqK65HUv/mYejZxvxl+TxuD1O2fNC1Ke4Z0WDWnefQ0mSdMnp3UlNTUV+fj7y8/Ph7e3d5xlJXhUNbUj+v704VnEef797AotKJiwrGhR8fX2h0+kAADqdDj4+PgA695jKysrMjysvL0dAQACUSiXKy8u7TKfBpbiqCXekfwtdQxsy741HQqSv3JEGLZYVDQpJSUnIzMwEAGRmZmLevHnm6VlZWdDr9SguLoZGo0F8fDz8/f3h7u6Offv2QQiB9evXm5ehweG7MzV48IMDsLeT8P59kzBlzHC5Iw1q/MyKbM6iRYvw1Vdfobq6GkqlEs8//zyefPJJJCcnY926dQgKCsLGjRsBAFFRUUhOTkZkZCQUCgXWrl0Le3t7AEB6ejqWLl2K1tZWJCYmIjExUc7VIgv69/4f8MyWowgc5orMZRMRPIL3o5Kb1MO5IzyxhOgKqNVq5Ofnyx2DrpLBaMJL247jnT3FuCl0BN5aFAdPVwe5Y9m6Xg2p5J4VERGAhtYOPPjBAfznZBXuvWE0nr41Agp7flJiLVhWRDToFVc3Y3nmfvxQ04K026OxiPeisjosKyIa1L7RVOKBf3UOpHjv15MwOYQDKawRy4qIBiWD0YTXt2vw1q5TuC3GH0/MDkegl6vcsegSWFZENOjoGlrx0AcHkVdSiwXqQDyXFAUXR3u5Y9FlsKyIaFDZefwcHv3wEPQGE15fEItfXjdS7kjUCywrIhoU9B1GvPblSWT85wwi/D2w9q7rEOLN86cGCpYVEdm8Y7pGPPrhQSjs7XD35CA8MycSzg487DeQsKyIyGYZjCb8/evTeGOHBp4uDki7PRoJkX5yx6KrwLIiIpukOXcej248hMPlDbgtxh8vzBsHryGOcseiq8SyIiKb0m4w4V/flSJt23G4OSuw9q44zInxlzsWXSOWFRHZjG9PVeOPnxTibF0L5sUG4InEcIxwc5I7FvUBlhURDXgVDW1Yve0YPj10FoFeLnjzrjjMiOC9p2wJy4qIBqwOowmZ35bgr1+eRIdJ4KEZoVjxizEc6WeDWFZENOAIIfDVySps2FuCncerMC3MG88lRWHU8CFyR6N+wrIiogHl+x/q8PLnx/FdcS1uGDMcGfdMQEKkLySpV7dFogGKZUVEA8Kpyia8knscuYXnMMLNES/Mi8LCiUFwVPCeU4MBy4qIrNrpqias+6YYH31fDgd7OzySMBbLbwzGECe+fQ0m/G0TkVUqPNuAv+06jW1HdXBS2OHBGSosmhiE4RyKPiixrIjIagghUFBah799dRo7j1fC3UmBFTePwbIbg3m+1CDHsiIi2bV1GLH1sA7vfluMU+ea4OvpjEcTxmLJ9aPh6eIgdzyyAiwrIpKNtr4VG/eXYcO+UtQ0tyPUxw1/mBuJ+bEj4crPpOgCfDUQkUXpDUZ8WXQOH+aX4xtNFdSjhuG6oKFYen0wblAN5xB06hbLioj6nRACh8obsO2wDh8WlKG+pQMBns54cHoo7pygRJCXq9wRycqxrIioXwghUKRrxNbDOmw9fBZlta24fsxw3KgagWR1IG5QjYC9HfeiqHdYVkTUZwxGEwpK67DjeCVOVpzHVyerYG8n4QbVCKyaHopZUX4cMEFXhWVFRNek8nwbvj1Vg++Ka/D50QrUt3TAwb6zoNLmR+OWcX686SFdM5YVEV2RhpYOHPihDv/RVGPPqWqcOHceAHDzWG9MD/dBQoQvbhrrDTeO5qM+xFcTEV2SyWRCSU0LCkrrzP9pKpsQP3oYDpU3ID7YC/PjRuJG1QhE+nvAjp9BUT9hWRERgM57Q52qbELh2UYUnW1E4dkGHK9oRLtBoLXDCA9nBSaMGoZ5sQGYONoL4wOH8r5RZDEsK6JBpr6lHaU1LThd1YQzVc04XdWE01VN8HRxwP6SOgCAs4Mdwv08MCcmAOpRwxA90hNjvN2450SyYVkRXUZOTg4eeughGI1G/PrXv8aTTz4pd6RLEkKgSW9A5Xk9Khv1qDzfhsZWA85UN6G8rvXH/1rQYTChzWACANjbSRjl5YoQbzeoRw3D3ZNHISrAA8Ej3DisnKwKy4roEoxGIx544AF8+eWXUCqVmDhxIpKSkhAZGdm3z2MSaDeYoDcYoTeYzF+3tBvRrDeiWW9Ac7vB/HVLuwHVTe2ob+1AfUs7Glo74DXEEd+dqUVrh/Ginx3iPQQVDW0IHOYK5TAXxI8eBuUwV4wa7ooQ7yEI8hrC+0HRgCAJIS43/7IzifqT0SRQ39KOxtYONOmNOK/vQFNb5xt3U5sBAkB1Uzv0HZ1v8m0dRrT9+DUANLcbYTCaMHKoC05XNcFoEgga7opTlc0YPsQRbs4KQAAGkwlDHO1hJ9lBIQHRgUPx0fflcEMLDh86jJtunobj586jubYSemEPr+HDMS7AE0KYcEx3HsPdnKDyGQJ9hwkbnrwLd7ywAR3Gzjylta0Y6uoAXw9nHClvgFEIhIwYgiJdIwxGAVdHe9Q0t3e7/v6eztA1tHWZHujlgvNtBgxzdYSniwOGujogePgQ2NtJ8PFwgre7E3zcneHj3vm1p4sDL2FE1qxXL87LltXs2bNFdXV1n6SpqqqCt7d3n/wsSxqouQHrzW40CXQYTegwChiMJnT8+L0EoLXD2DnfYILpx8fbSRJM3bxOXR3s0dJhhPTjYyTpv/9X2EkwCcBOAhwV9uaf76Cwg8EoYG8n4cKjXHY/vZlLgIuDParP62EvDNC36+Hh4dmZt12PDoMRrq6ucLC3AyDQ1GaEyWiAobUJQgI6ztfCb/RYAAJCdK6rnZ0Ee0mCwSTM2YxCQEJnBgmdmS/ML0kS7M3/l2BnJ8FOkmBvd0HWPmatr5eeDNTcwMDN3pe5CwoKcoUQs3t6nMX2rNRqNfLz8/vqx1nMQM0NyJddCIGKxjacrmxCaW0LfqhpQWlNC0prW1BW24LgEUNwRNtw0TJeQxwxJWQ46lvbMczVEds/24LUpXfDy9UBQ10dMcRJATcnBdydFeavhzjaw8XRvt/2GjZu3Ijc3Fy8/fbbAIANGzYgLy8Pb7755iWXGTJkCJqbm/slT38bqK/1gZobGLjZ+zh3r/6A+ZkVXZOG1g6cPHcexyvO40RFI05UnMeJivNobDPA280JVU16ONrbIdDLBUFerpgU7AWVjxt+c/MY+Ho4wdfDGT4eTnBSXDwEWv2XVDyS8IJMa9VJqVSirKzM/H15eTkCAgJkTEQ0eLGsqNc6jCYc0zXi+9I6fP9DPU5UnMfJyvP4aefc3UmBMD93zB0fgHA/d4T6uCFo+BD4eTgPyCHPEydOhEajQXFxMUaOHImsrCy8//77csciGpQsVlapqamWeqo+NVBzA9eevaXdgO+Ka3Hwh3p8e7oah8sbzIMXfD2cEBc0DLfHjUSorxvC/DwQ4OncZ4fkrGG7KxQKvPXWW7jllltgNBqxbNkyREVFXXaZESNGWChd37OGbX41BmpuYOBmlyM3RwOSmckkUHi2Ef/RVGG3phoFpXVoN5owYdQwGE0CcUHDEDdqKOKChsG/D4vJlgzUzyCIZMTPrKhnTXoDdh2vxOHyemwqKEddSwcAINzPHUtvGI0bVSMwcfQwuDjypUJE8uE70CDU0NKB7cfO4fOjFfiPpgrtBhOmhAzHtHAf3BQ6AjeoRsDH3VnumEREZn166vrGjRsRFRUFOzu7LodC0tLSoFKpEBYWhtzc3G6Xr62tRUJCAkJDQ5GQkIC6urq+jNcrCxYsQGxsLGJjYzF69GjExsZ2+7jRo0cjOjoasbGxUKvVlg15Cc899xxGjhxpzr9t2zbzvNZ2Iz4+UI4nPjqM6/70BR7deAjbC44jzK4SH94/Be/9ehL+khyL+dcpLV5Ujz32GMLDwxETE4P58+ejvr6+28dZ0zbPyclBWFgYVCoV1qxZ02W+EAKrVq2CSqVCTEwMvv/+exlSXqysrAzTpk1DREQEoqKi8MYbb3R5zFdffQVPT0/za+iFF+QdkXmhnn7/1rjNAeDEiRPm7RkbGwsPDw+8/vrrFz3GWrb7smXL4OPjg3Hjxpmn9fZ9uae/iWsmhLjcf1ekqKhIHD9+XNx8881i//795umFhYUiJiZGtLW1iTNnzoiQkBBhMBi6LP/YY4+JtLQ0IYQQaWlp4vHHH7/SCH3qkUceEc8//3y380aNGiWqqqosnOjynn32WfHKK6+YvzeZTGJ/cY14YtMhEfXHHDHqia0iZd13YvS834mte4+KtrY2ERMTIwoLC2VMLURubq7o6OgQQgjx+OOPX/L3bi3b3GAwiJCQEHH69Gmh1+sv2oYTJkwQQgjx2WefidmzZwuTyST27t0r4uPj5YwshBDi7NmzoqCgQAghRGNjowgNDe3yu9+1a5eYM2eOHPF61NPv3xq3+c8ZDAbh6+srSkpKLppuLdv966+/FgUFBSIqKso8rTfvy5f7m+iFnnoIQoi+3bOKiIhAWFhYl+nZ2dlYuHAhnJycEBwcDJVKhby8vG4fl5KSAgBISUnBli1b+jLeFRFC4MMPP8SiRYtky3C1apr02LCvFNNf+xp3/n0vPjl0FrPH+SErdTLuCzdgbGsR5kyOgpOTExYuXIjs7GxZ886aNQsKRecR6cmTJ6O8vFzWPD3Jy8uDSqVCSEgIHB0du92G2dnZWLJkCSRJwuTJk1FfXw+dTidT4k7+/v6Ii4sDALi7uyMiIgJarVbWTH3JGrf5z+3YsQNjxozBqFGj5I7SralTp8LLy+uiab15X+7N38S1ssgVLLVaLQIDA83fK5XKbv9Izp07B39/fwCdf1iVlZWWiNetb775Br6+vggNDe12viRJmDVrFiZMmICMjAwLp+tejckFb+XVY8Lzn+PFTd9hmLMdXrkzBvufnolXfzUek0OGQ3f2bK9+F3J55513kJiY2O08a9nmvXk99/Y1L5eSkhIcOHAAkyZN6jJv7969GD9+PBITE1FYWChDuu719Pu39m0OAFlZWZf8B7C1bvfevC9bYttf8QCLmTNnoqKiosv01atXY968ed0uI7oZHi/nsOferMMHH3xw2b2qPXv2ICAgAJWVlUhISEB4eDimTp3ab5l/8vPsQrKD0TcSI6fdjdPt4+ASZo/b40aiYX82Wr7Pwa9++85Fy8v1u+jNNl+9ejUUCgUWL17c7c+Qa5v/XG+2obW95i/U1NSEO+64A6+//jo8PDwumhcXF4fS0lK4ublh27Zt+OUvfwmNRiNT0ov19Pu35m0OAO3t7fjkk0+QlpbWZZ41b/fesMS2v+Ky2r59+xU/SW8vW+Pr6wudTgd/f3/odDr4+Phc8XP1Rk/rYDAYsHnzZhQUFFzyMT/l9/Hxwfz585GXl2eRN86fsrcbTPj4QDkyvy1FSU0z9ApHPDNnNH41IRCerg4ouc4dt912W5fl5bqEUE/bPDMzE1u3bsWOHTsu+SKXa5v/XG+2obVeqqmjowN33HEHFi9ejNtvv73L/AvL69Zbb8XKlStRXV1tFSc79/T7t9Zt/pPPP/8ccXFx8PX17TLPmrd7b96XLbHtLXIYMCkpCVlZWdDr9SguLoZGo0F8fHy3j8vMzATQ+eZ1qT21/rZ9+3aEh4dDqVR2O7+5uRnnz583f/3FF19cNHqmP+kNRmzYV4ppr36FJz46AoW9hNcXxOLrx6ZhjsoFnq4OAICPP/6420wXXkKovb0dWVlZSEpKskj2S8nJycHLL7+MTz75BK6urt0+Rs5t/nO92YZJSUlYv349hBDYt28fPD09zYdS5CKEwPLlyxEREYFHHnmk28dUVFSY/5Wcl5cHk8mE4cOHWzJmt3rz+7fGbX6hyx2tsdbtDvTufdki7ys9jMC4Ips3bxYjR44Ujo6OwsfHR8yaNcs878UXXxQhISFi7NixYtu2bebpy5cvN48crK6uFtOnTxcqlUpMnz5d1NTUXGmEPpGSkiLS09MvmqbVakViYqIQQojTp0+LmJgYERMTIyIjI8WLL77Y75la2w3ind1nxKTV28WoJ7aK+Wt3i13HzwmTyWR+zN133y3GjRsnoqOjxdy5c8XZs2e7ZBeic9RUaGioCAkJsUj2nowZM0YolUoxfvx4MX78eHH//fcLIeTf5pfT3TZMT08XQUFBQojOkZgrV64UISEhYty4cReNjpXLN998IwCI6Oho87b+7LPPRHp6uvn1/uabb4rIyEgRExMjJk2aJPbs2SNz6k6X+v1fmN0at/lPmpubhZeXl6ivrzdPs8btvnDhQuHn5ycUCoUYOXKkePvtty/5vtyH7yu9Gg3Iyy1ZOZNJ4OMDWrz7bQmOVzTiuqBheGhGKK4fM9yqjsdTJ15uieiK8XJLA93e0zVYva0IR7WNGK/0xIZlkzB5jHUcGiAisiSWlRU6XdWEtG3Hsf3YOYwc6oI3FsZibkzAgLzNBhFRX2BZWZH65nb8704N1u8thbODPR6fHYZlNwTD2cG+54WJiGwYy8oKCCHwyaGz+NPWIoR4u2HBxEA8nDAWI9yc5I5GRGQVWFYyK6ttwTNbjuLrk1UYr/TEs7dFImqkp9yxiIisCstKJgajCe/sKcZfvjwJe0nCs3MjsWTKaNjzcykioi5YVjIoOtuA5z4tQl5xLWZG+OCFeeMQMNRF7lhERFbLIlewoE5Gk8Dfvz6NeWv3wFlhh/TFcfjHEjWLiogGlf379yMmJgZtbW2QJGmIJEmFkiRd9pI03LOykPK6Fjzy4SHkFddidpQfXro9Gl5DHOWORURkcRMnTkRSUhKeeeYZAPgzgPeEEEcvtwzLqp8J0XkFimezC2ESAq/cGYM7Jyh59QkiGtT++Mc/YuLEiQCgBrCqp8ezrPpRS7sBf9xyFIfKGxDm546/JMciaHj3F2olIhpMamtr0dTUBADuAJwBNF/u8fzMqp+crmrC/LXf4qMDWvwyNgD/vn8Ki4qI6Eepqan405/+BAD/AvByT49nWfWDzw7rMO+tPag834bMe+PxwPRQDkknIvrR+vXroVAocNdddwHAGgATJUmafrlleNX1PtRuMCHt82P4554SXBc0FGvviuNIv0GGV10numK86rolVZ5vw5ptx7H5gBb33jAaTyVGwFHBHVcior7AsuoDxysasfzdfLS0G/Dmouswd7z13EqbiMgWsKyu0a7jlfjt+9/DzVmB9csmIVrJ6/oREfU1ltVVEkLg3W9L8KetRYjw98C6lInw83SWOxYRkU1iWV0Fg9GE5z4txHv7fsCsSF+8vjAWro7clERE/YXvsFeoSW/Anz4twr/zy3D/zSF44pZw3sGXiKifcbjaFahtbsfif+zDtiNn8ec7Y/BUYgSLyops3LgRUVFRsLOz6zJ8PC0tDSqVCmFhYcjNzTVPLygoQHR0NFQqFVatWoWfTuXQ6/VYsGABVCoVJk2ahJKSEkuuChH9DMuql87Wt+JXf/8WxyvO468LrkOyOlDuSPQz48aNw+bNmzF16tSLphcVFSErKwuFhYXIycnBypUrYTQaAQArVqxARkYGNBoNNBoNcnJyAADr1q3DsGHDcOrUKTz88MN44oknLL4+RPRfLKteOFXZhDvTv0Vlox7rl8VjZqSv3JGoGxEREQgLC+syPTs7GwsXLoSTkxOCg4OhUqmQl5cHnU6HxsZGTJkyBZIkYcmSJdiyZYt5mZSUFADAnXfeiR07dqCHE+iJqB+xrHpwuLweyf+3F+1GE7Lun4xJIcPljkRXSKvVIjDwv3vCSqUSWq0WWq0WSqWyy/SfL6NQKODp6Ymamppuf35GRgbUajXUajWqqqr6cU2IBi8OsLiMgtI6vLStCENdHfBOykSMHjFE7kiD3syZM1FRUdFl+urVqzFv3rxul+luj0iSpEtOv9wy3UlNTUVqaiqAzsstEVHfY1ldQkFpHVLeycMIN0e8f99kXuPPSmzfvv2Kl1EqlSgrKzN/X15ejoCAACiVSpSXl3eZfuEySqUSBoMBDQ0N8PLyuvYVIKKrwsOA3SgorTUXVVbqFBbVAJeUlISsrCzo9XoUFxdDo9EgPj4e/v7+cHd3x759+yCEwPr16817Z0lJScjMzAQAbNq0CdOnT+cNM4lkxD2rn+ksqv3mouJVKQaOjz/+GA8++CCqqqowZ84cxMbGIjc3F1FRUUhOTkZkZCQUCgXWrl0Le3t7AEB6ejqWLl2K1tZWJCYmIjExEQCwfPly3HPPPVCpVPDy8kJWVpacq0Y06PEWIRf4qai83Z3wwX2TWVR0xXiLEKIr1qtDFjwM+KODP9Tjd1kHWVRERFaIZQXgRMV5pPwzD36eziwqIiIrNOjL6oeaFtyz7js4O9jhL8mxLCoiIis0qAdYVDa24e5136HdaMKH909BoJer3JGIiKgbg3bPqr6lHfesy0N1kx7v3huPsb7uckciIqJLGJRl1dpuxB+yj6K4uhn/WKJGbOBQuSMREdFlDLrDgEaTwO/+fQBfnajE2sVxuEE1Qu5IRETUg0G3Z/XStmPILTyHJxMjkBDpJ3ccIiLqhUFVVu/uKca63cW494bRuPeGYLnjEBFRLw2asvqy6Bxe2FqEhEhfPDMnUu44RER0BQZFWR0ur8eqDw4geqQn3lgYC3veip6IaECx+bI6W9+K3/37IIa7OeLtlIlwdRx0Y0qIiAY8my6r1nYjUjfkw04C3kmZCG93J7kjERHRVbDZshJC4KnNh1F4thFPJUZgrB9P+iUiGqhstqzW7S7GloNn8WjCWMyI8JU7DhERXQObLKtvNFV4adsxJI7zwwPTVHLHISKia2RzZfVDTQt++/4BhPq449VfjeetyImIbIBNlVWz3oD71nfepTVjyQQMceLIPyIiW2AzZSWEwF++PImyuha8ddd1GDV8iNyRiIioj9hMWb23rxTrdhfj4ZljcVOot9xxiIioD9lEWR0pb8Cfth7DtDBvLL+R1/wjIrI1A76sGts68MD732O4myNeS46FHS+lRERkcwb0CAQhBJ786DC09a348P7J8BriKHckIiLqBwN6z2r93lJsO1KBx28Jw4RRXnLHISKifjJgy+pweT1Wf3YMM8J9cN9NIXLHISKifjQgy6pJb8CruScQ7td54i8/pyIism0Dsqxe+LQQu09V4+k5ERjGz6mIiGzegCurnKM6fJhfjhW/GINJIcPljkNERBYwoMrqXGMbntx8BDFKT/xu5li54xARkYUMmLIymQT+Z+MhtHUY8dcFsXCwHzDRiYjoGg2Yd/zMvSX4RlONZ+ZEYoy3m9xxiIjIggZEWZ08dx5pnx/H9HAfLJ4UJHccslKPPfYYwsPDERMTg/nz56O+vt48Ly0tDSqVCmFhYcjNzTVPLygoQHR0NFQqFVatWgUhBABAr9djwYIFUKlUmDRpEkpKSiy8NkR0Iasvq3aDCa/lnoC7kwIv3xHD+1PRJSUkJODo0aM4fPgwxo4di7S0NABAUVERsrKyUFhYiJycHKxcuRJGoxEAsGLFCmRkZECj0UCj0SAnJwcAsG7dOgwbNgynTp3Cww8/jCeeeEK29SKiAVBWf/vqFL4oOodXfhUDb3cnueOQFZs1axYUis4riE2ePBnl5eUAgOzsbCxcuBBOTk4IDg6GSqVCXl4edDodGhsbMWXKFEiShCVLlmDLli3mZVJSUgAAd955J3bs2GHe6yIiy7Pqsjqma8RbO08hKTYA08N95Y5DA8g777yDxMREAIBWq0VgYKB5nlKphFarhVarhVKp7DL958soFAp4enqipqam2+fKyMiAWq2GWq1GVVVVf60S0aBmtReyNRhNeHzTYXi6OODZuVFyxyErMXPmTFRUVHSZvnr1asybN8/8tUKhwOLFiwGg2z0iSZIuOf1yy3QnNTUVqampAAC1Wt3LNSGiK2G1ZfWPb4pxRNuAtXfF8WrqZLZ9+/bLzs/MzMTWrVuxY8cOc7kolUqUlZWZH1NeXo6AgAAolUrzocILp1+4jFKphMFgQENDA7y8eLFkIrlY5WHA01VN+Ov2k5gd5Ydbo/3kjkMDRE5ODl5++WV88skncHV1NU9PSkpCVlYW9Ho9iouLodFoEB8fD39/f7i7u2Pfvn0QQmD9+vXmvbOkpCRkZmYCADZt2oTp06dzcA+RjKxuz8poEnh802G4ONjjhV9G8Q2Ceu23v/0t9Ho9EhISAHQOsvj73/+OqKgoJCcnIzIyEgqFAmvXroW9vT0AID09HUuXLkVraysSExPNn3MtX74c99xzD1QqFby8vJCVlSXbehERIPUwwsniw5/+uacYz39ahL8kj8ftccqeFyCyImq1Gvn5+XLHIBpIerVHYlWHAX+oacGfc05gWpg35l83Uu44RERkJaymrIQQeHLzYSjsJLx0ezQP/xERkZnVlNXG/DK0dRjx9JwI+Hu6yB2HiIisiFWUVV1zO9I+Pw47SUKyOrDnBYiIaFCxirJ6Oec4GtsMeHH+ON6inoiIupC9rApKa5G1vwzLbwxGuJ+H3HGIiMgKyVpWHUYTnv74KAI8nfHQjFA5oxARkRWTtaze3VOC4xXn8ce5URjiZHXnJxMRkZWQrazO1rfir9tPYnq4D26J4hXViYjo0mQrqxc+LYJJCDyfxEsqERHR5clSVl+fqEJtczsenK5CoJdrzwsQEdGgZvEPijqMJjy/tRBCAPfdNMbST09ERAOQxfesMr8twZmqZvzhtgg4KmQfOU9ERAOARduiukmPN7ZrcPNYb0wL87HkUxMR0QBm0bJ67YsTaO0w4g+3RXJQBRER9ZrFyuqotgFZ+8uQcv1oqHzcLPW0RERkAyxSVkIIPP9pIbxcHbGKV6ogIqIrZJGy+vSwDvtL6vA/t4TB08XBEk9JREQ2xCJlVdnYhrigobz9BxERXRVJCHG5+ZedeSWMJgF73v6DbJxarUZ+fr7cMYgGkl4Vg8UGWLCoiIjoavGsXCIisnosKyIisnosKyIisnosKyIisnosKyIisnosKyIisnosKyIisnosKyIisnosKyIisnosKyIisno9XRuQiK6AJEk5QojZcucgsjUsKyIisno8DEhERFaPZUVERFaPZUVERFaPZUVERFaPZUVERFbv/wGsr7cjbsu56AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "sp.plotting.plot(z);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1d89c0a9-6e82-4c64-9d8f-4acd38d3d61b",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Si queremos definir límites de la función se utiliza la misma notación que la integración. Por otro lado, si queremos definir los límites de la gráfica, se usan las palbras `xlim=` y `ylim=`, seguidos de los valores `[inferior, superior]`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "48a714a0-7772-4d63-b1da-1b9326853b78",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "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": [
    "sp.plotting.plot(sp.sin(sp.pi*x), (x,-1,1), xlim=[-1.5, 1.5]);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8046813-fd43-4777-8213-5141e78471ce",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "Esta función tiene otras opciones, pero no se explorarán aqui."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ce15a19-ba5e-407a-86de-b59510e68ec2",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "### Solución de ecuaciones"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5049282d-5dcb-4488-b130-91f925bd3686",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Suponiendo que tengamos una ecuación del tipo:\n",
    "\n",
    "$$x^2 + 2*x = 5$$\n",
    "\n",
    "Sympy buscará algebraicamente las soluciones usando el comando `sp.solve()`. Es necesario _igualar la expresión a cero_:\n",
    "\n",
    "$$x^2 + 2*x - 5 = 0$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "e0962dd7-a3ae-4a70-a53a-3412fc4d9f55",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-3, 1]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr = x**2 + 2*x - 3\n",
    "\n",
    "pp( sp.solve(expr) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b796a0ad-a4bc-4a9c-8dac-424e15f36bc1",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Sistemas de ecuaciones pueden ser contenidos en una matriz (ya sea directamente o por una multiplicación) y resueltos con el mismo comando. Por ejemplo, un sistema $Mx=b$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "c8f7d8bb-f410-47cf-b796-992d53ac4d30",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}x_{1} + 2 x_{2} - 8\\\\3 x_{1} - 2 x_{2} - 16\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[   x_1 + 2*x_2 - 8],\n",
       "[3*x_1 - 2*x_2 - 16]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x1, x2 = sp.symbols('x_1, x_2')\n",
    "\n",
    "x = sp.Matrix([x1, x2])\n",
    "M = sp.Matrix([[1,2],[3,-2]])\n",
    "b = sp.Matrix([8,16])\n",
    "\n",
    "expr = M * x - b\n",
    "pp( expr )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "03adf832-70fd-4b40-b27d-6f4a62780361",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{x_1: 6, x_2: 1}"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.solve(expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8ce25430-b0f4-430d-ae9b-354578177ed0",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "## Pongamos todo junto: usando diccionarios"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bae3767a-157b-4df9-b63b-54901867b304",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Los _diccionarios_ en __Python__ son estructuras de datos propias del lenguaje en cierta forma similares a las _tuplas_ y a los _listas_ con una diferencia: en las _tuplas_ y _listas_ los elementos se encuentran asociados cada uno a un entero que forma una secuencia, mientras que en los diccionarios es posible asociar a los elementos a cualquier objeto."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ecdeb5eb-a111-42dc-90f9-3b029583f0d7",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Así, en una _lista_ (por ejemplo) tenemos:\n",
    "\n",
    "    dirección -> objeto\n",
    "       0      ->   'a'\n",
    "       1      ->   'b'\n",
    "       2      ->   'c'\n",
    "           ........"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed1da354-be61-4ab6-821a-745169259fb2",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "En un _diccionario_ podemos tener, por ejemplo:\n",
    "\n",
    "       key          ->   value\n",
    "       'pais'       ->   'Argentina'\n",
    "       'provincia'  ->   'Salta'\n",
    "       'longitud'   ->   -65.410556\n",
    "       'latitud'    ->   -24.788333"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "63724182-f5b4-434c-81ce-e90c159392d1",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Así el diccionario nos permite guardar informaciones __complejas__ o __relacionadas__."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b098c44-ad0e-4ff5-b03b-2ec269af912b",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "El ejemplo anterior se declara como diccionario usando las `{}` y separando cada _value_ de su _key_ utilizando `:`. Es decir:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "2d68ad55-874c-4c48-b2f6-2ca84800d2be",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "Salta = {'pais':'Argentina',\n",
    "         'provincia':'Salta',\n",
    "         'longitud':-65.410556,\n",
    "         'latitud':-24.788333}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c0a6f1ba-b495-4ac7-809a-29977a8318ad",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Siempre y cuando estemos dentro de las llaves (`{}`) y fuera de los apostrofes (`'`) podemos incluir nuevas líneas para mejorar la legibilidad."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c9b83b99-a350-42f6-8919-7e50785db280",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "__Resolvamos un problema__:\n",
    "\n",
    "Para explorar como se puede aplicar un diccionario a la solución de un problema, tomemos por ejemplo el siguiente problema:\n",
    "\n",
    "Un automóvil parte de Salta hacia Asunción a las 12:00 horas. A las 14:00, otro sale de Asunción hacia Salta. Los automóviles llevan velocidades de 100 km/h y 50 km/h respectivamente. Sabiendo que la distancia que separa Salta de Asunción es de 1100 km, calcular:\n",
    "\n",
    "1. ¿A qué distancia de Salta se cruzan?\n",
    "2. ¿A qué distancia de Asunción se cruzan?\n",
    "3. ¿A qué hora se cruzan?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4532771f-f31d-45ea-9a39-91737da79e14",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Declaramos los _símbolos_ que utilizaremos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "b31ba11d-4b28-4cec-a7bc-b456ef628652",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "t = sp.Symbol('t') # tiempo que tardan en encontrarse\n",
    "d = sp.Symbol('d') # distancia de encuentro\n",
    "hsal1 = sp.Symbol('hsal_1') # hora de salida 1\n",
    "hsal2 = sp.Symbol('hsal_2') # hora de salida 2                  \n",
    "v1 = sp.Symbol('v_1') # velocidad 1                  \n",
    "v2 = sp.Symbol('v_2') # velocidad 2\n",
    "henc = sp.Symbol('henc') # hora a la que se encuentran"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6b153eac-3488-452a-9817-3d0c0616498a",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Sabemos que la distancia recorrida es igual a la integral de la velocidad con respecto al tiempo, usando como límites la hora de inicio y la hora de final del movimiento:\n",
    "\n",
    "$$distancia = \\int_{hsal_1}^{henc} v_1 dt$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7aa22c0-33da-4590-b1b2-83238574e323",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Para obtener el tiempo que tardan en encontrarse podemos usar la ecuación:\n",
    "\n",
    "$$\\int_{hsal_1}^{henc} v_1 dt + \\int_{hsal_2}^{henc} v_2 dt  = d$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "3259a0f9-8ea4-4fc1-b904-de45eac2d613",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - d + henc v_{1} + henc v_{2} - hsal_{1} v_{1} - hsal_{2} v_{2}$"
      ],
      "text/plain": [
       "-d + henc*v_1 + henc*v_2 - hsal_1*v_1 - hsal_2*v_2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eq = sp.integrate(v1, (t, hsal1, henc)) + sp.integrate(v2, (t, hsal2, henc)) - d\n",
    "pp(eq)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "425e8956-a7ff-4567-bed5-c67fb8e6b10a",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Los datos del problema se pueden poner en un diccionario:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "be10bcd2-acea-4a33-9571-3cfbd7372184",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[20.0000000000000]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P1 =  {d:1100, hsal1:12.0, hsal2:14.0, v1:100, v2:50}\n",
    "pp( sp.solve(eq.subs(P1)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5c63e927-fc93-407a-ac34-551144b3f431",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "__Nota__: El comando `solve` automáticamente detecta que nuestra única incognita es la `henc` y nos regresa una lista, ya que podría haber varias soluciones."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cef3b704-bf36-4443-9e61-46186e56c18b",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Las preguntas 1 y 2 se pueden contestar usando este valor en ambas integrales."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "2b616111-ffe3-4af6-8722-41ad8ae8eaa3",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 800.0$"
      ],
      "text/plain": [
       "800.000000000000"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.integrate(v1, (t, hsal1, henc)).subs(P1).subs(henc,20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "71e96fb5-c480-4d3e-9b2b-4a7d90b00177",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 300.0$"
      ],
      "text/plain": [
       "300.000000000000"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.integrate(v2, (t, hsal2, henc)).subs(P1).subs(henc,20)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68859a1f-998f-4e41-9260-ad40659cd72e",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Comandante Fontana\n",
    "\n",
    "![Comandante Fontana](Imagenes/ComandanteFontana.png)\n",
    "\n",
    "__Fuente__: Google Maps "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f198833-3e47-434b-a7c2-3df50bb1fc98",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## ¿Porqué resolver un problema, si podemos resolver 10?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "816b4b6a-58c9-4c97-9665-5abb970110bd",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Una vez que hemos hecho lo anterior es fácil resolver el problema si los datos cambian."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7e511d8-8831-409c-ab63-29e2c403f5fe",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Si ahora el segundo automóviles no salió de Asunción, sino de Buenos Aires:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "68bd5379-57ef-4c8a-b424-2fc62a043f0e",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hora de encuentro:\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[22.6666666666667]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P2 =  {d:1500, hsal1:12.0, hsal2:14.0, v1:100, v2:50}\n",
    "print('Hora de encuentro:')\n",
    "pp( sp.solve(eq.subs(P2)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa4b7955-0ce3-41aa-bde2-ffad5fd747d0",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Si la velocidad del primer automóvil baja a 90 km/h."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "5d70104b-4c57-402d-b176-0a98e0b84d31",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hora de encuentro:\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[20.5714285714286]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P3 =  {d:1100, hsal1:12.0, hsal2:14.0, v1:90, v2:50}\n",
    "print('Hora de encuentro:')\n",
    "pp( sp.solve(eq.subs(P3)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "043fdbb7-4e27-4252-8456-bcd89b8eb514",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Si el primer automóvil se retrasó y salió hasta las 5pm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "a362796c-b622-4384-91e9-11982afbd5ba",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hora de encuentro:\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[23.3333333333333]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P4 =  {d:1100, hsal1:17.0, hsal2:14.0, v1:100, v2:50}\n",
    "print('Hora de encuentro:')\n",
    "pp( sp.solve(eq.subs(P4)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6016266e-779e-4cf8-b8f1-43c1aefb3c0c",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Si el que sale de Asunción no es un automóvil, sino un avión:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "7347bbc8-ef4c-43b7-b50f-c9d5392c66ba",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hora de encuentro:\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[16.5714285714286]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P5 =  {d:1100, hsal1:12.0, hsal2:14.0, v1:100, v2:250}\n",
    "print('Hora de encuentro:')\n",
    "pp( sp.solve(eq.subs(P5)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e9244597-405a-48fb-b718-6bd25d52ebf1",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Pero bueno, al final ahí estamos más o menos con el mismo problema, imaginen que ahora ignoramos la velocidad del primer automóvil, pero sabemos que se encontraron a las 21:30."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "12a950ee-982d-42f1-8cb3-50db542c27d6",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Velocidad del automóvil que salió de Salta:\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[76.3157894736842]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P6 =  {d:1100, hsal1:12.0, hsal2:14.0, v2:50, henc:21.5}\n",
    "print('Velocidad del automóvil que salió de Salta:')\n",
    "pp( sp.solve(eq.subs(P6)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ed4dda4-e33a-4983-a0ed-28b6668a928a",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "A las 22:15."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "5b7321d8-866d-489f-9a99-5b9083eeaa27",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Velocidad del automóvil que salió de Salta:\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[67.0731707317073]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P7 =  {d:1100, hsal1:12.0, hsal2:14.0, v2:50, henc:22.25}\n",
    "print('Velocidad del automóvil que salió de Salta:')\n",
    "pp( sp.solve(eq.subs(P7)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7606c517-2ca6-4e44-8a2e-162da247c3ab",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Si conocemos la velocidad y la hora a la que se encontraron, pero no la hora a la que salió de Salta."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "fe50490a-cf87-40e5-9079-e2eafbeb7df9",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hora de salida del automóvil que salió de Salta:\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[11.2500000000000]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P8 =  {d:1100, hsal2:14.0, v1:100, v2:50, henc:19.5}\n",
    "print('Hora de salida del automóvil que salió de Salta:')\n",
    "pp( sp.solve(eq.subs(P8)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3e172fa-e49f-462f-b067-ffe040f6a0ae",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "Incluso, podríamos saber que el automóvil de asunción salió 30 minutos después (si se encontraron a las 20hrs)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "8aabad57-33c1-41be-899f-039fe9851cc3",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hora de salida del automóvil que salió de Salta:\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[12.5000000000000]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P9 =  {d:1100, hsal2:hsal1+0.5, v1:100, v2:50, henc:20.0}\n",
    "print('Hora de salida del automóvil que salió de Salta:')\n",
    "pp( sp.solve(eq.subs(P9)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4615ae11-7863-4980-9072-2c7e2f1f3a67",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "O que el automóvil de Asunción viajaba 20 km/h más rápido que el de Salta (nuevamente encontrandose a las 20hrs)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "804e54ff-524d-4cc8-86ff-8ae857684e53",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Velocidad automóvil que salió de Salta:\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[70.0000000000000]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P10 =  {d:1100, hsal1:12.0, hsal2:14.0, v2:v1+20, henc:20}\n",
    "print('Velocidad automóvil que salió de Salta:')\n",
    "pp( sp.solve(eq.subs(P10)) )"
   ]
  }
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