{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "b55a0ff2",
   "metadata": {},
   "source": [
    "# Homework Assignment 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ea8e741",
   "metadata": {},
   "source": [
    "*Deadline: March 23, 2025. Your answer to the assignment must be written by hand and uploaded as a pdf to \"Assignments\" on Learn. Read the rules here: [](afsnit:regler-hjemafl). You may use a calculator or similar tools for simple calculations, e.g., calculating $P_4(1/2)$.*\n",
    "\n",
    "> **Remember, all answers must be justified and well-reasoned.**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "331a14db",
   "metadata": {},
   "source": [
    "## Exercise 1: Computation of Cosine Values"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4f8e8f38",
   "metadata": {},
   "source": [
    "In Python the value of e.g. $\\cos(\\sqrt{17})$ is computed approximatively with the call:\n",
    " \n",
    "```python\n",
    "import math\n",
    "\n",
    "print(math.cos(math.sqrt(17)))\n",
    "```\n",
    "\n",
    "But how does a computer actually find this approximation? A computer either uses a *precomputed lookup table* or approximating polynomials or a mix. In this exercise we will study how the computer would carry this out using Taylor polynomials. \n",
    "\n",
    "We consider a function $f:\\mathbb{R} \\rightarrow \\mathbb{R}$ given by the expression:\n",
    "\n",
    "\\begin{equation*}\n",
    "    f(x)=\\cos (x).\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69bd672b",
   "metadata": {},
   "source": [
    "### Question a"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b895c166",
   "metadata": {},
   "source": [
    "Determine the approximating polynomial $P_4 = P_{4,f,x_0}$ of degree (no higher than) 4 with expansion point $x_0 = 0$ for the cosinus function."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51eee2b5",
   "metadata": {},
   "source": [
    "### Question b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5b9df301",
   "metadata": {},
   "source": [
    "Use the polynomial $P_4$ you have found in the previous question to give an approximated value of $\\cos(1/2)$, and estimate how far the found value is from the exact value.\n",
    "% How large is the error at $x=\\pi/2$? (By \"error\" we mean the difference between $P_4(\\pi/2)$ and $\\cos(\\pi/2)$)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ef64f692",
   "metadata": {},
   "source": [
    "### Question c"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bf33c687",
   "metadata": {},
   "source": [
    "Let $I = [-\\pi/2, \\pi/2]$. Use <a href=\"../_assets/textbook_mat1b_en.pdf#ta2\">Theorem 4.3.3</a> to show that \n",
    "\n",
    "\\begin{equation*}\n",
    "   \\lvert \\cos(x) - P_4(x) \\rvert \\le \\frac{(\\pi/2)^5}{5!} < \\frac{2^5}{5!} \n",
    "\\end{equation*}\n",
    "\n",
    "for all $x \\in [-\\pi/2, \\pi/2]$. Explain in your own words what the number\n",
    "\n",
    "\\begin{equation*}\n",
    "   \\max_{x \\in [-\\pi/2, \\pi/2]} \\lvert \\cos(x) - P_4(x) \\rvert\n",
    "\\end{equation*}\n",
    "\n",
    "describes."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "794f40c4",
   "metadata": {},
   "source": [
    "### Question d"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dcc8d55c",
   "metadata": {},
   "source": [
    "Show that the $K$'th-degree Taylor polynomial $P_K$ with the expansion point $x_0 = 0$ for the cosinus function fulfills\n",
    "\n",
    "\\begin{equation*}\n",
    "   \\lvert \\cos(x) - P_K(x) \\rvert < \\frac{2^{K+1}}{(K+1)!} \n",
    "\\end{equation*}\n",
    "\n",
    "for all $x \\in [-\\pi/2, \\pi/2]$. Determine $K$ such that the error is smaller than $10^{-10}$. *By \"error\" we mean the numerical difference between $P_K(x)$ and $\\cos(x)$ for $x \\in [-\\pi/2, \\pi/2]$*.  \n",
    "\n",
    "```{hint}\n",
    ":class: dropdown\n",
    "You do not need to calculate $P_K(x)$. \n",
    "\n",
    "When you are trying to determine $K$, it is enough to look at the right-hand side $\\frac{2^{K+1}}{(K+1)!}$. You might need a calculator or similar tool. \n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb2dbfa8",
   "metadata": {},
   "source": [
    "### Question e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe64de58",
   "metadata": {},
   "source": [
    "We can now with fairly good accuracy compute $\\cos(x)$ for any $x \\in [-\\pi/2, \\pi/2]$ by use of the Taylor polynomial $P_K$ with expansion point $x_0 = 0$ (for a fixed, but sufficiently large chosen $K$). But how do we calculate an approximated value of $\\cos$ on the rest of the real line, meaning when $x$ is outside the interval $[-\\pi/2, \\pi/2]$?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4cbe7781",
   "metadata": {},
   "source": [
    "In order to answer this question we let $x \\in [-\\pi/2, \\pi/2]$ be given and assume that the value of $\\cos(x)$ is known (i.e. calculated approximately as above). Explain how you for an arbitrarily given $y \\in \\mathbb{R}$ now can find the value of $\\cos(y)$.\n",
    "\n",
    "```{hint}\n",
    ":class: dropdown\n",
    "You only need some properties of the cosine function. Draw the graph of $\\cos$ over $[-2\\pi,2\\pi]$. Use the fact that $\\cos$ is periodic and \"odd\" (not even) around $x = \\pi/2$. \n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "065dd561",
   "metadata": {},
   "source": [
    "## Exercise 2: L'Hôpital's Rule and Taylor's Limit Formula"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f64e5cfc",
   "metadata": {},
   "source": [
    "L'Hôpital's rule is used for determination of limits of the type $\\lim_{x \\to a} \\frac{f(x)}{g(x)}$ where $f(a)=0$ and $g(a)=0$. The limit value is in this case said to be of the type $\\frac{0}{0}$. The rule is stated as follows:\n",
    "\n",
    "\n",
    "```{admonition} Theorem (L'Hôpital's Rule)\n",
    "Let $I \\subseteq \\mathbb{R}$ be an open interval, let $a \\in I$, and let $f$ and $g$ be differentiable on $I\\setminus \\{a\\}$. Assume the following:\n",
    "\n",
    "1. $\\lim_{x \\to a} f(x) = \\lim_{x \\to a} g(x) = 0$\n",
    "2. $g'(x) \\neq 0$ for $x \\in I \\setminus \\{a\\}$\n",
    "3. The limit $\\lim_{x \\to a} \\frac{f'(x)}{g'(x)}$ exists\n",
    "\n",
    "Then it holds that:\n",
    "\n",
    "\\begin{equation*}\n",
    "     \\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\lim_{x \\to a} \\frac{f'(x)}{g'(x)}.\n",
    "\\end{equation*}\n",
    "```\n",
    "\n",
    "Consider the limit:\n",
    "\n",
    "\\begin{equation*}\n",
    "   \\lim_{x \\to 0} \\frac{\\ln(1 + x)}{1 - \\sqrt{1 + x}}.\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "610db9fd",
   "metadata": {},
   "source": [
    "### Question a"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05c5f46b",
   "metadata": {},
   "source": [
    "Show that this limit is of the type $\\frac{0}{0}$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b02820b5",
   "metadata": {},
   "source": [
    "### Question b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "431bdabf",
   "metadata": {},
   "source": [
    "Check that $g'(x) \\neq 0$ for $x$ close to $0$, so that we can use the rule."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f83fca1e",
   "metadata": {},
   "source": [
    "### Question c"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e3526f33",
   "metadata": {},
   "source": [
    "Use L'Hôpital's rule to determine the limit value."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cae52a7b",
   "metadata": {},
   "source": [
    "### Question d"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "531d8148",
   "metadata": {},
   "source": [
    "Determine this same limit limit using Taylor's limit formula for $\\ln(1 + x)$ and $\\sqrt{1 + x}$ up to a fitting order."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dcc3f056",
   "metadata": {},
   "source": [
    "### Question e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c269581a",
   "metadata": {},
   "source": [
    "Prove L'Hôpital's rule using Taylor's limit formula for general functions $f(x)$ and $g(x)$ that fulfill the assumptions in the theorem.\n",
    "\n",
    "\n",
    "```{hint}\n",
    ":class: dropdown\n",
    "Use a Taylor expansion of $f(x)$ and $g(x)$ around $ x = a $ up to the first order with *epsilon* functions:\n",
    "\\begin{equation*}\n",
    "    f(x) = f(a) + f'(a)(x-a) + \\epsilon_f(x-a)(x-a),\n",
    "\\end{equation*}\n",
    "and\n",
    "\\begin{equation*}\n",
    "    g(x) = g(a) + g'(a)(x-a) + \\epsilon_g(x-a)(x-a).\n",
    "\\end{equation*}\n",
    "\n",
    "```\n",
    "\n",
    "```{admonition} Update March 12\n",
    "You may assume that $g'(a) \\neq 0$. This is often the case (for example, in our application), and it simplifies the proof.\n",
    "```\n",
    "\n",
    "```{hint}\n",
    ":class: dropdown\n",
    "Use what you know about $f(a)$ and $g(a)$.\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b53a7d4e",
   "metadata": {},
   "source": [
    "## Exercise 3: Image Set of a Continuous, Differentiable Function"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "93768786",
   "metadata": {},
   "source": [
    "Let $B$ be the set $B=\\left\\{ (x_1 , x_2) \\in \\mathbb{R}^2 \\mid x_1^2 + x_2^2 \\leq 2 \\wedge x_1 \\leq 0 \\right\\}$.\n",
    "\n",
    "A function $f:B \\rightarrow \\mathbb{R}$ is given by: \n",
    "\n",
    "\\begin{equation*}\n",
    "    f(x_1 ,x_2) =  x_1^2​ + x_2^2 + x_1​ + 1.\n",
    "\\end{equation*}\n",
    "\n",
    "State the image set of $f$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a75f44f",
   "metadata": {},
   "source": [
    "## Exercise 4: Local Extrema"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4481f23d",
   "metadata": {},
   "source": [
    "A function $f:\\mathbb{R}^2 \\rightarrow \\mathbb{R}$ is given by: \n",
    "\n",
    "\\begin{equation*}\n",
    "f(x_1 ,x_2 )=x_1^2 - 2x_1 +3x_2^5 - 5x_2^3.\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66198c70",
   "metadata": {},
   "source": [
    "### Question a"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87c0b986",
   "metadata": {},
   "source": [
    "Find all stationary points for $f$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "952d8eaa",
   "metadata": {},
   "source": [
    "### Question b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d72ccc32",
   "metadata": {},
   "source": [
    "State for each stationary point whether it is a local maximum, a local minimum or a saddle point."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2724d2ee",
   "metadata": {},
   "source": [
    "### Question c"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc6f1b03",
   "metadata": {},
   "source": [
    "For each stationary point, plot the function in Python along with the approximating polynomial $P_{2}$ of degree (no high than) 2 expanded from the stationary point."
   ]
  }
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