{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Rates of Change" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import pandas as pd" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Slope and Rate of Change\n", "\n", "| t | speed |\n", "| ----- | ----- |\n", "| 1 | 4 |\n", "| 4 . | 10 |\n", "| 8 | 12 |" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "#rate of change from 1 to 4?\n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "#rate of change from 4 to 8?\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Slope \n", "\n", "$$(x_1, y_1) \\quad (x_2, y_2)$$" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# slope of line through (5, 2) and (8, 4)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Equation of Line\n", "\n" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "# equation of line through (5, 2) and (8,4)\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Secant\n", "\n" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def f(x): return x**2\n", "x = np.linspace(0, 3, 100)\n", "\n", "plt.figure(figsize = (15, 5))\n", "\n", "plt.plot(x, f(x))\n", "\n", "plt.plot([1, 3], [f(1), f(3)], '-o', label = '(0, 3)')\n", "\n", "plt.plot([1, 2], [f(1), f(2)], '-o', label = '(0, 2)')\n", "\n", "plt.plot([1, 1.1], [f(1), f(1.1)], '-o', label = '(0, 1)')\n", "\n", "plt.legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Definition of Derivative\n", "\n", "$$f'(x) = \\lim_{h \\to 0} \\frac{f(x + h) - f(x)}{h}$$\n", "\n", "Use this definition (and show work) to determine the derivative of:\n", "\n", "- $f(x) = 7x + 6$\n", "- $f(x) = 4x^2 - 3x + 8$" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Writing a Derivative Function\n", "\n", "The function below will compute the derivative of a function `f` at some point `x`. We are really approximating by letting `h` be a very small value." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "def ddf(f, x, h = 0.0000001):\n", " derivative = (f(x + h) - f(x))/h\n", " return np.round(derivative, 4)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "#example functions\n", "def f(x):\n", " return x**2" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "9" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#value of f at x=3\n", "f(3)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "6.0" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#derivative of f at x=3\n", "ddf(f, 3)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The slope of the tangent line to f at x = 3 is 6.0\n" ] } ], "source": [ "a = 3\n", "print(f'The slope of the tangent line to f at x = {a} is {ddf(f, a)}')" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "import numpy as np" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#plot the function and derivative together\n", "x = np.linspace(-4, 4, 1000)\n", "plt.plot(x, f(x), label = '$f$')\n", "plt.plot(3, f(3), 'o',label = \"$f(3)$\")\n", "plt.legend(fontsize = 20)\n", "plt.title('Write an equation for the line through the point (3, 3)\\ntangent to $f$.');" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#problem 2\n", "def f(x): return -x**2 + 1\n", "x = np.linspace(-4, 4, 1000)\n", "plt.plot(x, f(x), label = '$f$')\n", "plt.plot(-1, f(-1), 'o',label = \"$f(3)$\")\n", "plt.legend(fontsize = 20)\n", "plt.title('Write an equation for the line through the point (-1, -1)\\ntangent to $f$.');" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Describe a general procedure for writing an equation for the line tangent to the graph of $f$ at some point $x = a$. This should culminate in an expression for the line in terms of $f$ and $f'$." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Power Rule\n", "\n", "In general, we can compute the derivative of a given polynomial function using the *power rule* that states:\n", "\n", "$$f(x) = x^n \\quad \\rightarrow \\quad f'(x) = nx^{n-1}$$\n", "\n", "Use the power rule to evaluate the derivative of the following functions:\n", "\n", "- $f(t) = 2t^5 - 7t^3 + 4t^2 - 6t + 1$\n", "- $g(t) = 3t^4 - t^3 + 7t^2 - t + 17$" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Problem: Sequences\n", "\n", "In derivatives, we find the following two patterns:\n", "\n", "- $f(x) = mx + b \\rightarrow \\quad f'(x) = m$\n", "- $f(x) = ax^2 + bx + c \\rightarrow \\quad f'(x) = 2ax + b$\n", "\n", "Rephrase these symbolic statements in your own words. Should we have expected these results? What do they have to do with the sequences:\n", "\n", "- 1, 3, 5, 7, 9, 11\n", "- 1, 4, 9, 16, 25, 36" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.4" }, "widgets": { "application/vnd.jupyter.widget-state+json": { "state": {}, "version_major": 2, "version_minor": 0 } } }, "nbformat": 4, "nbformat_minor": 4 }