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 "cells": [
  {
   "cell_type": "markdown",
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   "source": [
    "## Calculus and Matlab"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Defining Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### inline()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using inline(), we can define custom functions that perform operations we define. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "func1 =\n",
      "     Inline function:\n",
      "     func1(x) = sqrt(x)\n"
     ]
    }
   ],
   "source": [
    "% lets create a function that takes x as an argument and computes the square root\n",
    "format compact\n",
    "func1 = inline('sqrt(x)','x') % the 1st parameter is the function we want and the 2nd is the argument "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "     2\n",
      "ans =\n",
      "    2.8284\n"
     ]
    }
   ],
   "source": [
    "% now we can use it\n",
    "func1(4)\n",
    "func1(8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c =\n",
      "     Inline function:\n",
      "     c(a,b) = sqrt(a^2+b^2)\n"
     ]
    }
   ],
   "source": [
    "% we can do more complex functions with more arguments\n",
    "c = inline('sqrt(a^2+b^2)','a','b')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "     5\n"
     ]
    }
   ],
   "source": [
    "% now we can find the hypotenuse\n",
    "c(4,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### diff()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f3 =\n",
      "     Inline function:\n",
      "     f3(x) = x^2\n"
     ]
    }
   ],
   "source": [
    "% first we'll define a custom function\n",
    "f3 = inline('x^2','x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "% x must be a symbolic\n",
    "x = sym('x');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "2*x\n"
     ]
    }
   ],
   "source": [
    "% next, we pass the function to diff()\n",
    "diff(f3(x),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "derivative =\n",
      "     Inline function:\n",
      "     derivative(x) = x.*2.0\n"
     ]
    }
   ],
   "source": [
    "% if we are using a pre-defined function, we dont use single-quotes\n",
    "derivative = inline(diff(f3(x),x),'x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "4*x\n"
     ]
    }
   ],
   "source": [
    "derivative(2*x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Symbolic Differentiation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "func =\n",
      "x^3\n"
     ]
    }
   ],
   "source": [
    "% prep, declare symbolic var, declare a function\n",
    "clear all\n",
    "x = sym('x');\n",
    "func = x^3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "3*x^2\n"
     ]
    }
   ],
   "source": [
    "% we can esily take the derivative\n",
    "diff(func,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### int()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f1 =\n",
      "     Inline function:\n",
      "     f1(x) = x\n"
     ]
    }
   ],
   "source": [
    "clear all\n",
    "f1 = inline('x','x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "     4\n"
     ]
    }
   ],
   "source": [
    "% f1 is simply a linear function\n",
    "f1(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "integral =\n",
      "x^2/2\n"
     ]
    }
   ],
   "source": [
    "x = sym('x');\n",
    "integral = int(f1(x),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n",
      "x\n",
      "--\n",
      " 2\n"
     ]
    }
   ],
   "source": [
    "pretty(integral)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Limits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### limit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f1 =\n",
      "     Inline function:\n",
      "     f1(x) = sin(x)/x\n",
      "ans =\n",
      "1\n"
     ]
    }
   ],
   "source": [
    "% the limit of x as x approaches 0\n",
    "clear all\n",
    "x = sym('x');\n",
    "f1 = inline('sin(x)/x','x')\n",
    "limit(f1(x),x,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Partial Derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f1 =\n",
      "     Inline function:\n",
      "     f1(x,y) = x^2+2*y^3\n"
     ]
    }
   ],
   "source": [
    "clear all\n",
    "x = sym('x');y = sym('y');\n",
    "f1 = inline('x^2+2*y^3','x','y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "d1 =\n",
      "2*x\n"
     ]
    }
   ],
   "source": [
    "% partial derivative wrt x\n",
    "d1 = diff(f1(x,y),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "6*y^2\n"
     ]
    }
   ],
   "source": [
    "% partial wrt y\n",
    "diff(f1(x,y),y)"
   ]
  }
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