Committing the ipynb internal stuff (hidden dot directory and files)

.ipynb_checkpoints/Quiz_3_Review-checkpoint.ipynb

@@ -0,0 +1,129 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Quiz 3 Review\n",
+    "\n",
+    "### Problem 1 \n",
+    "3 points\n",
+    "\n",
+    "$$\\int_2^\\infty e^{-3x} dx$$\n",
+    "\n",
+    "$$= \\lim_{t\\to \\infty} \\int_2^t e^{-3x} dx$$\n",
+    "\n",
+    "$$= \\lim_{t\\to \\infty} \\left[ \\frac{e^{-3x}}{-3} \\right]_{2}^{t}$$\n",
+    "\n",
+    "$$= -\\frac{1}{3} \\lim_{t\\to \\infty} \\left( e^{-3t} - e^{-6}  \\right)$$\n",
+    "\n",
+    "$$= -\\frac{1}{3} \\left( 0 - \\frac{1}{e^6} \\right)$$\n",
+    "\n",
+    "$$\\text{Solution }= \\frac{1}{3e^6}$$\n",
+    "\n",
+    "__Converges__"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Problem 2\n",
+    "3 points\n",
+    "\n",
+    "$$\\int_0^1 \\frac{3}{x^-5} dx$$\n",
+    "\n",
+    "$$= \\lim_{t\\to 0^{+}} \\int_t^1 3x^5 dx$$\n",
+    "\n",
+    "$$= \\lim_{t\\to 0^{+}} \\left[ \\frac{3x^{-4}}{-4} \\right]_{t}^{1}$$\n",
+    "\n",
+    "$$= \\lim_{t\\to 0^{+}} \\left( \\frac{-3}{4} + \\frac{3}{4t^4} \\right)$$\n",
+    "\n",
+    "$$\\text{Solution }= \\infty$$\n",
+    "\n",
+    "__Diverges__"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x1085faad0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "%matplotlib inline\n",
+    "\n",
+    "import numpy as np\n",
+    "import sympy as sp\n",
+    "from matplotlib import pyplot as plt\n",
+    "\n",
+    "# f(x) = 1/x --> Diverges\n",
+    "f = lambda x: 3/(x**5)\n",
+    "\n",
+    "sp.mpmath.plot([f], xlim=[0,5], ylim=[0,5], points=500)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Problem 3\n",
+    "4 points\n",
+    "\n",
+    "$$\\{a_n\\} = \\frac{n^4}{5n^4+3}$$\n",
+    "\n",
+    "$$= \\lim_{n\\to \\infty} \\frac{n^4}{5n^4+3}$$\n",
+    "\n",
+    "$$= \\lim_{n\\to \\infty} \\frac{\\frac{n^4}{n^4}}{5\\left(\\frac{n^4}{n^4}\\right) + 3\\left(\\frac{1}{n^4}\\right)}$$\n",
+    "\n",
+    "$$= \\lim_{n\\to \\infty} \\frac{1}{5 + 3(0)}$$\n",
+    "\n",
+    "$$= \\frac{1}{5}$$\n",
+    "\n",
+    "__Converges__"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 2",
+   "language": "python",
+   "name": "python2"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 2
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython2",
+   "version": "2.7.10"
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+}