Finding_changes_in_functions_Solutions.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Finding changes in functions - Solutions.ipynb",
      "provenance": [],
      "authorship_tag": "ABX9TyOGzHWcgfme2Ws30ij061DU",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/awangberg/calculus_playground/blob/main/Finding_changes_in_functions_Solutions.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#To Start:  Make your own copy of this worksheet.\n",
        "\n",
        "![image.png](data:image/png;base64,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)"
      ],
      "metadata": {
        "id": "tlaGzP8neHqQ"
      }
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qQcuOUtvYdma"
      },
      "source": [
        "# Changes in Functions\n",
        "\n",
        "In earlier projects, we found formulas for the change in area \n",
        "\n",
        "$$dA = 2s ds$$\n",
        "\n",
        "of a square $A = s^2$ when the side length was increased various amounts $ds$.  We did several mathematical experiments for $ds = 0.20, ds = 0.10, ds = 0.05, $ and $ds = 0.10$.\n",
        "\n",
        "We extended this work to include estimates for the changes $V = s^3, Q = s^7,$ and $p = \\sqrt{s} = s^{0.5}$, finding expressions for $dV, dQ,$ and $dp$.\n",
        "\n",
        "In this activity, you'll work with a partner to handle three functions from the following list:\n",
        "\n",
        "| List A          |  List B           | List C              | List D          |\n",
        "|-----------------|-------------------|---------------------|-----------------|\n",
        "| $$E=e^x$$       |$$L=\\ln(x)$$      |$$S=\\sin(x) $$       |$$C=\\cos(x)$$    |\n",
        "| $$F=e^{4x}$$    |$$M=-2\\ln(3x) $$  |$$T=2\\sin(3x + 5) $$ |$$D=3\\cos(2x-4)$$|\n",
        "| $$G=3e^{-2x-3}$$|$$N=3\\ln(0.5x) $$ |$$U=4\\sin(-x + 6) $$ |$$B=-2\\cos(4x)$$ |\n",
        "\n",
        "Like before, we will choose several different values of $dx$ to account for the change in $x$ for each function, arriving at en expression like\n",
        "\n",
        "$$  dE = ( \\hspace{1cm} ) dx $$\n",
        "\n",
        "We're going to be a bit more efficient this time:  Instead of copying and pasting tables into Desmos four times, one for each small change, we'll do this all at once with a separate column in our spreadsheet for each small change $dx$."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DVeRfuIXZGgu"
      },
      "source": [
        "# Spreadsheet set-up\n",
        "\n",
        "\n",
        "In the spreadsheeet, click on 'File' and 'Make a Copy'.  This will make a copy of your spreadsheet in a separate tab.  This is ok - it will be easier to work in that tab.  If it doesn't open in a new tab, copy the url and paste it into a new browser tab.\n",
        "\n",
        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)"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "In the spreadsheet, you'll notice that we have four columns (one for each different $dx$), four columns (one each for the change in the function), and   a separate column for each change in x, a separate column "
      ],
      "metadata": {
        "id": "i_x-CZT2eZr5"
      }
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 395
        },
        "id": "bVnNOc5SZDa7",
        "outputId": "5f92b523-d340-40f0-e9c0-bf08db3e4582"
      },
      "source": [
        "from IPython.display import IFrame\n",
        "IFrame(src=\"https://docs.google.com/spreadsheets/d/15GW-krAdm4HvjpSfjmGdyufhb4_vV-2lOdeRvYcJpGQ/edit#gid=0\", width=1200, height=375)"
      ],
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/html": [
              "\n",
              "        <iframe\n",
              "            width=\"1200\"\n",
              "            height=\"375\"\n",
              "            src=\"https://docs.google.com/spreadsheets/d/15GW-krAdm4HvjpSfjmGdyufhb4_vV-2lOdeRvYcJpGQ/edit#gid=0\"\n",
              "            frameborder=\"0\"\n",
              "            allowfullscreen\n",
              "        ></iframe>\n",
              "        "
            ],
            "text/plain": [
              "<IPython.lib.display.IFrame at 0x7f4dec8be390>"
            ]
          },
          "metadata": {},
          "execution_count": 1
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 896
        },
        "id": "olAAeWDYYXan",
        "outputId": "321c3f22-fd8a-4c23-cf23-ad710153e79d"
      },
      "source": [
        "from IPython.display import IFrame\n",
        "IFrame(src=\"https://www.desmos.com/calculator/acfpj6uls0\", width=1200, height=875)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/html": [
              "\n",
              "        <iframe\n",
              "            width=\"1200\"\n",
              "            height=\"875\"\n",
              "            src=\"https://www.desmos.com/calculator/acfpj6uls0\"\n",
              "            frameborder=\"0\"\n",
              "            allowfullscreen\n",
              "        ></iframe>\n",
              "        "
            ],
            "text/plain": [
              "<IPython.lib.display.IFrame at 0x7fb8f27772d0>"
            ]
          },
          "metadata": {},
          "execution_count": 1
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Now, repeat this again for the other functions in your list."
      ],
      "metadata": {
        "id": "oMmiSvWtjaqg"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#Record your data and Save to Github:\n",
        "\n",
        "We'll make a record of our data in the following table.  \n",
        "\n",
        "To add your data, double click your mouse in the cell, and then either modify the existing table row, or add a new table row.  Use | to separate columns, a new line for a new row, and --- to put a horizontal line in the table.\n",
        "\n",
        "Then, save this file out to your github account."
      ],
      "metadata": {
        "id": "Fj45BjD34gMw"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# HW List A:  Relationship for exponential functions\n",
        "\n",
        "Part 1:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d(e^x)$ $      |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(e^x) = (  e^{1.03021 x}   ) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(e^x) = (  e^{1.01499}   ) ( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(e^x) = ( e^{1.00747}   ) ( 0.05)$$  |\n",
        "|  Experiment  | 0.01  | $$d(e^x) = ( e^{1.00149}   ) ( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "$$d(e^x) = (  e^{1 x}  ) ( dx )$$"
      ],
      "metadata": {
        "id": "fzn5stKc8UIl"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Part 2:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d(e^{4x})$      |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(e^{4x}) = ( 6.1277e^{4x}    ) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(e^{4x}) = ( 4.91825e^{4x}    ) ( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(e^{4x}) = ( 4.428e^{4x}    ) ( 0.05)$$   |\n",
        "|  Experiment  | 0.01  | $$d(e^{4x}) = ( 4.081e^{4x}  ) ( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "$$d(e^{4x}) = (  4e^{4x}   ) ( dx)$$"
      ],
      "metadata": {
        "id": "FaeelQWvkLC3"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Part 3:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d(3e^{-2x-3})$ $      |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(3e^{-2x-3}) = ( -4.945e^{-2x-3}   ) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(3e^{-2x-3}) = ( -5.438e^{-2x-3}   ) ( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(3e^{-2x-3}) = ( -5.709e^{-2x-3}  ) ( 0.05)$$   |\n",
        "|  Experiment  | 0.01  | $$d(3e^{-2x-3}) = ( -5.94e^{-2x-3}   ) ( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "$$d(3e^{-2x-3}) = (  -6 e^{-2x-3}   ) ( dx)$$"
      ],
      "metadata": {
        "id": "OkMnYlyhkL7S"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# HW List B:  Relationship for exponential functions\n",
        "\n",
        "Part 1:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d(\\ln(x))$ $      |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(\\ln(x)) = \\left(  \\frac{0.5672}{x} \\right) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(\\ln(x)) = \\left(  \\frac{0.7063}{x}   \\right) ( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(\\ln(x)) = \\left(  \\frac{0.8194}{x}  \\right) ( 0.05)$$  |\n",
        "|  Experiment  | 0.01  | $$d(\\ln(x)) =\\left(  \\frac{0.955}{x}  \\right) ( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "$$d(\\ln(x)) = \\left(  \\frac{1}{x}  \\right) ( dx )$$"
      ],
      "metadata": {
        "id": "YSFNZnUu9IGe"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Part 2:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d(-2\\ln(3x))$ $      |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(-2\\ln(3x)) = \\left( \\frac{-1.13}{x}  \\right) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(-2\\ln(3x)) = \\left( \\frac{-1.41}{x}  \\right) ( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(-2\\ln(3x)) = \\left( \\frac{-1.6388}{x}  \\right) ( 0.05)$$  |\n",
        "|  Experiment  | 0.01  | $$d(-2\\ln(3x)) = \\left( \\frac{-1.9105}{x}  \\right) ( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "$$d(-2\\ln(3x)) = \\left( \\frac{-2}{x}  \\right) ( dx )$$"
      ],
      "metadata": {
        "id": "rBJyzt1olAas"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Part 3:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d(3\\ln(0.5x))$ $      |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(3\\ln(0.5x)) = \\left( \\frac{.2859}{x}  \\right) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(3\\ln(0.5x)) = \\left( \\frac{2.11849}{x}  \\right) ( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(3\\ln(0.5x)) = \\left( \\frac{2.458}{x}  \\right) ( 0.05)$$  |\n",
        "|  Experiment  | 0.01  | $$d(3\\ln(0.5x)) = \\left( \\frac{2.866}{x}  \\right) ( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "$$d(3\\ln(0.5x)) = \\left( \\frac{3}{x}  \\right) ( dx )$$"
      ],
      "metadata": {
        "id": "n5Lebox2lLT3"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# HW List c:  Relationship for sine functions\n",
        "\n",
        "Part 1:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d(\\sin(x))$ $      |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(\\sin(x)) = \\left(  0.986 \\cos(1.002x)   \\right) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(\\sin(x)) = \\left(  0.9949 \\cos(1.001x)   \\right) ( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(\\sin(x)) = \\left(  0.9979 \\cos(1.0006x)   \\right) ( 0.05)$$  |\n",
        "|  Experiment  | 0.01  | $$d(\\sin(x)) = \\left(  0.9996 \\cos(1.0001x)   \\right) ( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "\n",
        "*   List item\n",
        "*   List item\n",
        "\n",
        "\n",
        "$$d(\\sin(x)) = \\left(  \\cos(x) \\right) ( dx )$$"
      ],
      "metadata": {
        "id": "B2r0Tnsr9gQR"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Part 2:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d(2\\sin(3x+5)$     |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(2\\sin(3x+5)) = \\left( 5.8247\\cos(2.86x+5) \\right) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(2\\sin(3x+5)) = \\left( 5.997\\cos(2.98x+5) \\right) ( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(2\\sin(3x+5)) = \\left( 6.0203\\cos(2.99x+5) \\right)( 0.05)$$  |\n",
        "|  Experiment  | 0.01  | $$d(2\\sin(3x+5)) = \\left( 6.0075\\cos(2.998x+5) \\right) ( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "$$d(2\\sin(3x+5)) = \\left( 6 \\cos(3x+5)  \\right) ( dx )$$"
      ],
      "metadata": {
        "id": "3uQApyR6lb0C"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Part 3:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d\\left(4\\sin(-x+6)\\right)$      |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(4\\sin(-x+6)) = \\left(-3.93395\\cos(-x+6) \\right) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(4\\sin(-x+6)) = \\left(-3.97357\\cos(-x+6) \\right)( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(4\\sin(-x+6)) = \\left(-3.98844\\cos(-x+6) \\right) ( 0.05)$$  |\n",
        "|  Experiment  | 0.01  | $$d(4\\sin(-x+6)) = \\left(-3.99796\\cos(-x+6) \\right)( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "$$d(4\\sin(-x+6)) = \\left(-4\\cos(-x+6) \\right) ( dx )$$"
      ],
      "metadata": {
        "id": "-KVAONyull8k"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# HW List D:  Relationship for exponential functions\n",
        "\n",
        "Part 1:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d(\\cos(x))$      |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(\\cos(x)) = \\left( -1.00064 \\sin(x)    \\right) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(\\cos(x)) = \\left( -1.00199 \\sin(x)    \\right) ( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(\\cos(x)) = \\left( -1.00141 \\sin(x)    \\right) ( 0.05)$$  |\n",
        "|  Experiment  | 0.01  | $$d(\\cos(x)) = \\left( -1.00035 \\sin(x)    \\right) ( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "$$d(\\cos(x)) = \\left( -sin(x) \\right) ( dx )$$"
      ],
      "metadata": {
        "id": "icLUTf34lw0I"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Part 2:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d\\left(3\\cos(2x-4)\\right)$      |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(3\\cos(2x-4)) = \\left(-5.75\\sin(2.008x - 4)\\right) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(3\\cos(2x-4)) = \\left(-5.916\\sin(2.004x - 4)\\right) ( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(3\\cos(2x-4)) = \\left(-5.968\\sin(2.002x - 4)\\right) ( 0.05)$$  |\n",
        "|  Experiment  | 0.01  | $$d(3\\cos(2x-4)) = \\left(-5.995\\sin(2.0004x - 4)\\right) ( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "$$d(3\\cos(2x-4)) = \\left(-6\\sin(2x - 4)\\right) ( dx )$$"
      ],
      "metadata": {
        "id": "cbO9fZOol2vv"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Part 3:\n",
        "\n",
        "| Type of Data | $dx$ |    Equation for $d(-2\\cos(4x))$      |\n",
        "|--------------|------|-----------------------------|\n",
        "|  Experiment  | 0.2  | $$d(-2\\cos(4x)) = \\left( 7.788\\sin(4x+.4)\\right) ( 0.2)$$   |\n",
        "|  Experiment  | 0.1  | $$d(-2\\cos(4x)) = \\left( 7.946\\sin(4x+.2)\\right) ( 0.1)$$   |\n",
        "|  Experiment  | 0.05  | $$d(-2\\cos(4x)) = \\left( 7.9867\\sin(4x+.01)\\right) ( 0.05)$$  |\n",
        "|  Experiment  | 0.01  | $$d(-2\\cos(4x)) = \\left( 7.9994\\sin(4x+.02)\\right) ( 0.01)$$   |\n",
        "|--------------|-------|-----------------------------|\n",
        "\n",
        "So, as $dx$ shrinks toward $0$, our logical guess is: \n",
        "$$d(-2\\cos(4x)) = \\left( 8\\sin(4x )\\right) ( dx )$$"
      ],
      "metadata": {
        "id": "aXDOrQ5yl-vB"
      }
    }
  ]
}