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    "# Tutorial A8"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__1__ Write a functions that takes two numbers as input and returns the sum."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__2__ Write a function that computes the cumulative sum of values in a sequence and returns the result as a list. Demonstrate that it works. Hint: You can use a list to pass a sequence to a function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__3__ Write a function `factorial(n)` that returns the factorial of an integer $n$. What happens if you pass a float?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Advanced 1__ Write a function `exp(x)` that takes a float `x` \n",
    "as an argument and returns $\\mathrm{e}^x$ approximated as the Taylor series up to order $n$ around the expansion point $a = 0$. Make the order $n$ an optional argument that defaults to $n = 5$. You can call `factorial(n)` from within `exp(x)` to calculate the factorials."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$T(x; a) = \\sum_{n = 0}^\\infty \\frac{f^{(n)}(a)}{n!} (x - a)^n$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For $f(x) = \\mathrm{e}^x$, $a = 0$:\n",
    "\n",
    "$$T(x; 0) = \\sum_{n = 0}^\\infty \\frac{x^{n}}{n!}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Advanced 2__ What is wrong with these two functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def broken():\n",
    "    print(x)\n",
    "    x = \"NEW\"\n",
    "    \n",
    "    return(x)\n",
    "\n",
    "x = \"STRING\""
   ]
  },
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   "execution_count": 12,
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    "def broken_counter():\n",
    "    counter += 1\n",
    "\n",
    "counter = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Advanced 3__ (Warning: the solution to this problem touches a concept we have not discussed in the lesson. This is a really advanced exercise.) Ever solved a \"Jumble\"-puzzle? Write a function that expects a string\n",
    "and returns every possible re-combination of the letters to help you find\n",
    "the terms hidden behind \"rta\", \"atob\" and \"rbani\". You can test your function with the built-in function\n",
    "`itertools.permutations` from the `itertools` module. How many possible combinations does this return?\n",
    "Does the solver help you with this one: \"ibaptlienxraoc\"? Hint: Functions can call themselves. Try to pick letters from the string one by one to find a possible re-combination. "
   ]
  }
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