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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Homework 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Your Name: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Big-$O$ Notation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 1 (3 points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is $3^{n+1}=O(3^n)$? Justify your answer using [Definition 3.1](https://6001-majhi.datasci.land/notes/big-o.html#def-o)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 2 (3 points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is $5^{2n}=O(5^n)$? Justify your answer using [Definition 3.1](https://6001-majhi.datasci.land/notes/big-o.html#def-o)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 3 (3 points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using [Definition 3.1](https://6001-majhi.datasci.land/notes/big-o.html#def-o), show that the function $700n\\lg{n}$ is $O(n^2-1)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 4 (1+1 points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Describe the difference between an upper bound and a tight upper bound? Explain using an example.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 5 (1+1 points)\n",
    "Provide an example of an $O(1)$ and $O(n)$ algorithm.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 6 (🏅 Bonus)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If $f(n)=O(g(n))$, then $2^{f(n)} = O(2^{g(n)})$  True or false? Justify your answer.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Maximum difference"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 7 (5 points)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given an integer array nums, return the maximum difference between two successive elements in its sorted form. If the array contains less than two elements, return 0.  \n",
    "Note : Do not use built-in sorting functions in Python.\n",
    "\n",
    "Eg:  \n",
    "Input: x = [3,6,9,1]   \n",
    "Output: 3  \n",
    "Explanation : The sorted form of the array is [1,3,6,9], either (3,6) or (6,9) has the maximum difference 3."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Implement your algorithm in Python."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def max_diff(A):\n",
    "    # implement me\n",
    "    pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 8 (2 points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the input list has $n$ elements, describe the time complexity of your code. A thorough explanation is required.\n"
   ]
  }
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