02_Differentiation_Integration.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Differentation & Integration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some elementary functions and their derivatives:\n",
    "\n",
    "$$\n",
    "    \\cos (x)                           \\qquad \\frac{d}{dx} \\left( \\cos (x) \\right) = - \\sin(x)\n",
    "    \\\\\n",
    "    \\sin (x)                           \\qquad \\frac{d}{dx} \\left( \\sin (x) \\right) = \\cos (x)\n",
    "    \\\\\n",
    "    \\tan (x)                           \\qquad \\frac{d}{dx} \\left( \\tan (x)  \\right) = \\sec^2 (x)\n",
    "    \\\\\n",
    "    \\ln (x)                            \\qquad \\frac{d}{dx} \\left( \\ln (x)  \\right) = \\frac{1}{x}\n",
    "    \\\\\n",
    "    x^n + x^{n-1} + \\ldots + x + c     \n",
    "    \\qquad \\frac{d}{dx} \\left( x^n + x^{n-1} + \\ldots + x + c \\right) \n",
    "    =  (n)x^{n-1} + (n-1)x^{n-2} + \\ldots + 1 + 0\n",
    "$$\n",
    "\n",
    "Let's validate these with `sympy`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Setup\n",
    "from sympy import *\n",
    "from sympy import __version__ as sympy_version\n",
    "print(\"Sympy version is: \\t\" + sympy_version)\n",
    "x = Symbol('x')\n",
    "# from sympy.abc import x     # Prepare the variable x.\n",
    "init_printing()             # Make things print nicely"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = cos(x)\n",
    "y_prime = diff(y)\n",
    "\n",
    "(y_prime)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = sin(x)\n",
    "y_prime = diff(y)\n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = tan(x)\n",
    "y_prime = diff(y)\n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "y = ln(x)\n",
    "y_prime = diff(y)\n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = x**3 + x**2 + x + 1\n",
    "y_prime = diff(y)\n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## More complex derivatives\n",
    "\n",
    "Say we have a more difficult function to differentiate, using a symbolic package we can easily obtain an accurate and useful answer to our problem in a few lines of code.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = cos(3*x)*sin(4*x)*tan(5*x)\n",
    "\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_prime = diff(y)\n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Perhaps we don't know the coefficients?\n",
    "\n",
    "Still doable!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.abc import a,b,c\n",
    "y = cos(a*x)*sin(b*x)*tan(c*x)\n",
    "\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Differentiate with respect to x. Optional second argument indicates what to differentiate with\n",
    "y_prime = diff(y, x)            \n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Arbitrary Functions\n",
    "\n",
    "We can even manipulate functions that have not been explicitly specified with `sympy`. We do this with the `Function()` function.\n",
    "\n",
    "We can even do single and multi-variable functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Symbol, Function\n",
    "\n",
    "x = Symbol('x')\n",
    "\n",
    "f = Function('f')(x)\n",
    "\n",
    "display(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Symbol, Function\n",
    "\n",
    "x = Symbol('x')\n",
    "y = Symbol('y')\n",
    "\n",
    "f = Function('f')(x,y)\n",
    "\n",
    "display(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can then manipulate them with integration and differentiation. This is useful for validating transformation rules and how they affect function (whether a transformation is linear or not, how multiplication of functions affects differentiation, etc.)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import integrate, diff\n",
    "F = integrate(f,x)\n",
    "f_prime = diff(f,x)\n",
    "\n",
    "display(F)\n",
    "display(f_prime)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Repetitive Derivatives"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Say we want to differentiate to a certain power, we can specify that with a third argument to `diff()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = cos(a*x)\n",
    "\n",
    "# Here, the third argument indicates how many times to differentiate.\n",
    "# We are taking the 100th derivative.\n",
    "y_prime = diff(y, x, 100)            \n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's take a high-order derivative of a more complicated function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = tan(x)*1/x\n",
    "y_prime = diff(y, x, 10)\n",
    "\n",
    "y_prime   # Note that large functions format poorly on github, but nicely when run on locally on your computer (hopefully)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, computer-based symbolic manipulations show their strength."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integration\n",
    "\n",
    "[Link](https://docs.sympy.org/1.3/modules/integrals/integrals.html?highlight=integrate#sympy.integrals.integrate) \n",
    "to the documentation for `integrate` for `sympy` version 1.3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "from sympy.abc import x\n",
    "from sympy import __version__ as sympy_version\n",
    "print(\"Sympy version is: \\t\" + str(sympy_version))\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simple example\n",
    "Indefinite integral."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = cos(x)\n",
    "Y = integrate(y)\n",
    "\n",
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definite integral.\n",
    "\n",
    "See how we pass a `tuple` as the second argument to indicate to the `integrate()` function what to integrate with respect to and with what bounds. \n",
    "\n",
    "It's formatted like: `integrate(function, (variable, lower_limit, upper_limit))`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = sin(x)\n",
    "attributes = (x, 0, pi)\n",
    "Y = integrate(y, attributes)     # Integrate sin(x) from 0 to pi\n",
    "\n",
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A more complicated example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = cos(x)*exp(x)\n",
    "Y = integrate(y)\n",
    "\n",
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Perhaps with some coefficients?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.abc import a\n",
    "y = a*cos(3*x)*exp(4*x)\n",
    "Y = integrate(y, x)\n",
    "\n",
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symbolic limits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Y = integrate(log(x), (x, 1, a))\n",
    "\n",
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# -------- Differentation Questions --------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 1:\n",
    "\n",
    "Calculate the second derivative of:\n",
    "\n",
    "$$\n",
    "\\frac\n",
    "{\\tan (x) + \\sin(x)}\n",
    "{x}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 1:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 2: (Requires the knowledge of `plotting`)\n",
    "\n",
    "Consider the function:\n",
    "\n",
    "$$\n",
    "    \\sqrt{x} \\cdot \\sin\\left(x\\right) \n",
    "$$\n",
    "\n",
    "Symbolically plot it in blue(`'b'`)  and its derivative in red (`'r'`) on the same plot over the domain $[0,2\\pi]$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 3:\n",
    "\n",
    "\n",
    "If $n$ is the 3rd number in your Student ID number, find the $n$-th derivative of $y = e^x \\cdot \\sin(x)$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 4:\n",
    "\n",
    "If $f$ is a function of $x$ and $g$ is a function of $x$, find:\n",
    "\n",
    "$$\n",
    "    \\frac{d}{dx} \\left( f\\cdot g \\right) \n",
    "$$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 4:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 5:\n",
    "\n",
    "### (i)\n",
    "If $f$ is a function of $x$ and $g$ is a function of $x$, find:\n",
    "\n",
    "$$\n",
    "    \\frac{d}{dx} \\left( \\frac{f}{g} \\right) \n",
    "$$\n",
    "\n",
    "### (ii)\n",
    "How does this stack up with your knowledge of the quotient rule? Is the answer equivalent?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 5:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 6:\n",
    "\n",
    "Differentiate the following function four times.\n",
    "\n",
    "$$ f(x) = \\frac{1}{(x^2+x+1)^2} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 6:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# -------- Integration Questions --------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 1:\n",
    "\n",
    "Evaluate:\n",
    "\n",
    "$$\n",
    "\\int^{\\pi / 2}_0 \\sin^2(x) \\cos^3(x) dx\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 1:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 2:\n",
    "\n",
    "In the cell below (after you evaluate it with `Shift+Enter`) you will have a function $f(r) = 2\\pi r$\n",
    "\n",
    "### (i)\n",
    "What is the formula $2\\pi r$ used to calculate?\n",
    "\n",
    "### (ii)\n",
    "Integrate this function, print and interpret the result. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 3:\n",
    "\n",
    "Evaluate the following repeated integral (all integrations are done with respect to $x$).\n",
    "\n",
    "$$\n",
    "\\iiint \\frac{dx^3 }{(x+2)^2 (x+3)^3}\n",
    "$$\n",
    "\n",
    "A note on repeated integrals [here](http://mathworld.wolfram.com/RepeatedIntegral.html), which discusses better (and more rigorous ways) of representing them."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 4:\n",
    "\n",
    "Make a python function that takes in a mathematical function to be integrated $n$ times with respect to some variable, $x$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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