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    "# Homework: Derivatives Intro\n",
    "\n",
    "**GOALS**:\n",
    "\n",
    "- Explain the definition of the derivative\n",
    "- Use the definition of the derivative to approximate slopes of tangent lines\n",
    "- Use the power rule to evaluate derivatives of polynomials\n",
    "- Use a functions derivative to explain where original function is increasing, decreasing, is maximized, or is minimized.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
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   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import sympy as sy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### PROBLEM I: Definition\n",
    "\n",
    "Below are two definitions of the derivative.  In your own words and with pictures, explain what these mean.\n",
    "\n",
    "$$\\lim_{h \\to 0} \\frac{f(x + h) - f(x)}{h} \\quad \\text{and} \\quad \\lim_{x_2 \\to x_1} \\frac{f(x_2) - f(x_1)}{x_2 - x_1}$$"
   ]
  },
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  {
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   "source": [
    "##### PROBLEM II: Tables\n",
    "\n",
    "Using the definition of the derivative below, fill in the tables for appropriate functions at $a = 0$.\n",
    "\n",
    "- $f(x) = .5x^3 - 1$\n",
    "- $g(x) = \\sin(x)$\n",
    "- $h(x) = \\frac{1}{x}$\n",
    "\n",
    "$$\\lim_{h \\to 0} \\frac{f(a + h) - f(a)}{h}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src = 'images/table_ddf.png' width = 50%>"
   ]
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  {
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   "source": [
    "##### PROBLEM III: Equations of Tangent Lines\n",
    "\n",
    "For each of the following functions, use the **power rule** to find the derivative $f'(x)$ and write equation for the line tangent to the graph at $x = a$.  Draw a picture of $f$ and the tangent line to verify you are correct.\n",
    "\n",
    "1.  $f(x) = 2x^3 + 4x^2 - 5x - 3$ at $x = -1$.\n",
    "\n",
    "2. $f(x) = x^2 + \\frac{4}{x^3} - 10$ at $x = 8$\n",
    "\n",
    "\n"
   ]
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  {
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   "source": [
    "##### PROBLEM IV: Shapes of curves\n",
    "\n",
    "Given the plots of the two derivatives below, identify the following:\n",
    "\n",
    "- Intervals where $f$ is increasing\n",
    "- Intervals where $f$ is decreasing\n",
    "- Any local maximum or minimum values in $f$"
   ]
  },
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   "metadata": {},
   "source": [
    "<img src = 'images/15_p1.png' width = 50%/><img src = 'images/15_p2.png' width = 50%/>"
   ]
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   "execution_count": null,
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  {
   "cell_type": "code",
   "execution_count": 105,
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   "source": [
    "# x = np.linspace(-3, 3, 1000)\n",
    "# def f(x): return (x + 2)*(x + 1)**2 * x * (x - 1)\n",
    "# # # set the x-spine (see below for more info on `set_position`)\n",
    "# fig, ax = plt.subplots()\n",
    "# ax.spines['left'].set_position('zero')\n",
    "\n",
    "# # turn off the right spine/ticks\n",
    "# ax.spines['right'].set_color('none')\n",
    "# ax.yaxis.tick_left()\n",
    "\n",
    "# # set the y-spine\n",
    "# ax.spines['bottom'].set_position('zero')\n",
    "\n",
    "# # turn off the top spine/ticks\n",
    "# ax.spines['top'].set_color('none')\n",
    "# ax.xaxis.tick_bottom()\n",
    "# ax.plot(x, f(x), label = \"f'(x)\")\n",
    "# plt.ylim(-5, 5)\n",
    "# plt.axvline(color = 'black')\n",
    "# plt.axhline(color = 'black')\n",
    "# plt.legend(fontsize = 20, frameon = False)\n",
    "# plt.savefig('images/15_p1.png')"
   ]
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  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [],
   "source": [
    "# def f(x): return (x - 2)*(x + 2)*(x + 1)*(x - 1)*x**2\n",
    "# # # set the x-spine (see below for more info on `set_position`)\n",
    "# fig, ax = plt.subplots()\n",
    "# ax.spines['left'].set_position('zero')\n",
    "\n",
    "# # turn off the right spine/ticks\n",
    "# ax.spines['right'].set_color('none')\n",
    "# ax.yaxis.tick_left()\n",
    "\n",
    "# # set the y-spine\n",
    "# ax.spines['bottom'].set_position('zero')\n",
    "\n",
    "# # turn off the top spine/ticks\n",
    "# ax.spines['top'].set_color('none')\n",
    "# ax.xaxis.tick_bottom()\n",
    "# def f(x): return (x - 2)*(x + 2)*(x + 1)*(x - 1)*x**2\n",
    "# ax.plot(x, f(x), label = \"f'(x)\")\n",
    "# plt.ylim(-8, 5)\n",
    "# plt.axvline(color = 'black')\n",
    "# plt.axhline(color = 'black')\n",
    "# plt.legend(fontsize = 20, frameon = False)\n",
    "# plt.savefig('images/15_p2.png')"
   ]
  }
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