MAT120_LAB_Lec03.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 39
        },
        "id": "pd-2WoiLmHgK",
        "outputId": "00a09ef2-ce12-4ccf-98c3-8b9f971f90f6"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "pi"
            ],
            "text/latex": "$\\displaystyle \\pi$"
          },
          "metadata": {},
          "execution_count": 1
        }
      ],
      "source": [
        "# Importing sympy\n",
        "import sympy as sp\n",
        "sp.pi\n"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Maxima Minima of Function\n",
        "Change the interval and play with different expressions\n",
        "\n"
      ],
      "metadata": {
        "id": "51X2e5ZX7Fo4"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Declaring symbols\n",
        "x=sp.symbols('x')\n",
        "\n",
        "# Writing the expression\n",
        "y=x**4-2/3*x**3-2*x**2+2*x\n",
        "\n",
        "# first derivative\n",
        "diff1=sp.diff(y,x,1)\n",
        "# Second derivative\n",
        "diff2=sp.diff(y,x,2)\n",
        "\n",
        "# solution of first derivative\n",
        "soln=sp.solve(diff1)\n",
        "\n",
        "# Finding real solution within interval\n",
        "soln1=[]\n",
        "for i in range(len(soln)):\n",
        "  # Checking imaginary solutions\n",
        "  if sp.im(soln[i])!=0:\n",
        "    continue\n",
        "  # Checking solutions within range 0 to 1\n",
        "  elif soln[i].evalf()>=0 and soln[i].evalf()<=1:\n",
        "    soln1.append(soln[i])\n",
        "  else:\n",
        "    continue\n",
        "\n",
        "\n",
        "# Finding the maximum and minimum value\n",
        "\n",
        "if len(soln1)>0:\n",
        "  for k in range(len(soln1)):\n",
        "    # Checking if soln1[k] corresponds to maxima\n",
        "    if diff2.subs(x,soln1[k])<0:\n",
        "      max_pt=soln1[k]\n",
        "      # Evaluating the maximum value\n",
        "      max_val=y.subs(x,soln1[k]).evalf()\n",
        "      print(f'The maximum value of the function at {max_pt} is {max_val}')\n",
        "    # Checking if soln1[k] corresponds to manima\n",
        "    elif diff2.subs(x,soln1[k])>0:\n",
        "      min_pt=soln1[k]\n",
        "      # Evaluating the minimum value\n",
        "      min_val=y.subs(x,soln1[k]).evalf()\n",
        "      print(f'The mimimum value of the function at {min_pt} is {min_val}')\n",
        "else:\n",
        "  print(\"No maxima or minima of the function within the given range.\")\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "4X-DcKpamv9K",
        "outputId": "aed18737-3111-44a8-f741-287c4ceb8137"
      },
      "execution_count": 64,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The maximum value of the function at 0.500000000000000 is 0.479166666666667\n",
            "The mimimum value of the function at 1.00000000000000 is 0.333333333333333\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Plotting markers corresponding to maximum and minimum value\n",
        "p1=sp.plot(y,(x,0,1.5),markers=[{'args': [max_pt,max_val,'r*'], 'ms': 10},\n",
        "                             {'args': [min_pt,min_val,'r*'], 'ms': 10}],show=False)\n",
        "p2=sp.plot(max_val,(x,0.2,0.8),line_color='red',show=False)\n",
        "p3=sp.plot(min_val,(x,0.8,1.2),line_color='red',show=False)\n",
        "p1.extend(p2)\n",
        "p1.extend(p3)\n",
        "p1.show()"
      ],
      "metadata": {
        "id": "lcMtlmklnUj4",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 484
        },
        "outputId": "c7efd214-a5bb-4810-8642-e650c4c8505d"
      },
      "execution_count": 57,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Solving using sympy solveset function\n",
        "z=sp.sin(x)\n",
        "sp.solveset(z)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 39
        },
        "id": "efiBv9QoInmj",
        "outputId": "811ac4cf-fd92-4d20-9572-5be5098f6ab6"
      },
      "execution_count": 59,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "{2⋅n⋅π │ n ∊ ℤ} ∪ {2⋅n⋅π + π │ n ∊ ℤ}"
            ],
            "text/latex": "$\\displaystyle \\left\\{2 n \\pi\\; \\middle|\\; n \\in \\mathbb{Z}\\right\\} \\cup \\left\\{2 n \\pi + \\pi\\; \\middle|\\; n \\in \\mathbb{Z}\\right\\}$"
          },
          "metadata": {},
          "execution_count": 59
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Setting interval in the solveset function\n",
        "from sympy import Interval\n",
        "# Solveset by default gives solutions within the range as a python set\n",
        "sp.solveset(z,x,Interval(0,2*sp.pi))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 39
        },
        "id": "b-pvA-4kIw-u",
        "outputId": "cecb8129-40d7-44a3-ef5e-34cb444d1637"
      },
      "execution_count": 61,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "{0, π, 2⋅π}"
            ],
            "text/latex": "$\\displaystyle \\left\\{0, \\pi, 2 \\pi\\right\\}$"
          },
          "metadata": {},
          "execution_count": 61
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#Determining maxima-minima of a function within a given interval using solveset() Function"
      ],
      "metadata": {
        "id": "wXuVn6VpJy1W"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Declaring symbols\n",
        "x=sp.symbols('x')\n",
        "\n",
        "# Writing the expression\n",
        "y=x**4-2/3*x**3-2*x**2+2*x\n",
        "\n",
        "# first derivative\n",
        "diff1=sp.diff(y,x,1)\n",
        "# Second derivative\n",
        "diff2=sp.diff(y,x,2)\n",
        "\n",
        "# solution of first derivative within interval\n",
        "soln=sp.solveset(diff1,x,Interval(0,1))\n",
        "# Changing the solution set into a list\n",
        "soln1=list(soln)\n",
        "\n",
        "# Finding the maximum and minimum value\n",
        "if len(soln1)>0:\n",
        "  for k in range(len(soln1)):\n",
        "    # Checking imaginary solutions\n",
        "    if sp.im(soln1[k])!=0:\n",
        "       continue\n",
        "    # Checking if soln1[k] corresponds to maxima\n",
        "    elif diff2.subs(x,soln1[k])<0:\n",
        "      max_pt=soln1[k]\n",
        "      # Evaluating the maximum value\n",
        "      max_val=y.subs(x,soln1[k]).evalf()\n",
        "      print(f'The maximum value of the function at {max_pt} is {max_val}')\n",
        "    # Checking if soln1[k] corresponds to manima\n",
        "    elif diff2.subs(x,soln1[k])>0:\n",
        "      min_pt=soln1[k]\n",
        "      # Evaluating the minimum value\n",
        "      min_val=y.subs(x,soln1[k]).evalf()\n",
        "      print(f'The mimimum value of the function at {min_pt} is {min_val}')\n",
        "else:\n",
        "  print(\"No maxima or minima of the function within the given range.\")\n",
        "\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ZXK2Ouxl_VxU",
        "outputId": "5475a300-f401-428d-a5d5-63cabf022d53"
      },
      "execution_count": 72,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The maximum value of the function at 0.500000000000000 is 0.479166666666667\n",
            "The mimimum value of the function at 1.00000000000000 is 0.333333333333333\n"
          ]
        }
      ]
    }
  ]
}