Practice_Final_Exam.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Practice Final Exam, ChE 263\n",
    "\n",
    "This exam covers the following topics:\n",
    "\n",
    "* Problem 1: Variable types and formats\n",
    "* Problem 2: Debugging\n",
    "* Problem 3: Solve Linear Equations\n",
    "* Problem 4: Differentiate and Integrate\n",
    "* Problem 5: Interpolation\n",
    "* Problem 6: General Nonlinear Regression and Plotting\n",
    "* Problem 7: Equation Solution and Plotting\n",
    "* Problem 8: Parameter Estimation\n",
    "* Problem 9: Implicit Equation Solve\n",
    "\n",
    "A template for each problem is provided with instructions on what is needed. In some cases, the problem is to debug a section of code to produce a specific result. In other cases, the problem is to import a module and produce a result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import packages"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 1: Variable types and formats\n",
    "\n",
    "#### Part A\n",
    "Add **x=2.0** (float) and **y='3'** (string) together to produce a number 5 (integer). Show the type of variable with **type()** to confirm that the result is an integer.\n",
    "\n",
    "Hint: A string is converted to a number with **int()** or **float()**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Part B\n",
    "\n",
    "Show $\\pi$ to 30 decimal places."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2: Debugging\n",
    "\n",
    "You are working on the launch sequence of a rocket. The following code is intended to count backwards from 5 to 0 with steps of -1 in 1 second intervals. Print `blast off` after reaching zero.\n",
    "\n",
    "```python\n",
    "import time\n",
    "for i in range(5,1)\n",
    "    time.wait(1.0)\n",
    "    print(i)\n",
    "print('Blast off\")\n",
    "```\n",
    "\n",
    "The code has a few bugs (errors) that prevent it from running or producing the correct result. Find the errors in the code to produce:\n",
    "\n",
    "```\n",
    "5\n",
    "4\n",
    "3\n",
    "2\n",
    "1\n",
    "0\n",
    "Blast off\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 3: Solve Linear Equations\n",
    "\n",
    "Find the values of $x_0$, $x_1$ that satisfy the following equations:\n",
    "\n",
    "$5 x_0 + x_1 = 2$\n",
    "\n",
    "$3 x_0 + 12 x_1 = 1$\n",
    "\n",
    "Put equations into matrix form with $A \\;x = b$ and solve $x = A^{-1}\\; b$. The $A$ matrix is given below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 4: Differentiate and Integrate\n",
    "\n",
    "#### Part A\n",
    "\n",
    "Compute the derivative $\\frac{d f(x)}{dx}$ for the following function:\n",
    "\n",
    "$f(x) = \\frac{1}{1+e^{-x}}$\n",
    "\n",
    "Find a solution with scipy (numerically) and sympy (symbolically). For the numeric solution (scipy), use value of $x=0.2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Part B\n",
    "\n",
    "Compute the integral for the function:\n",
    "\n",
    "$f(x) = \\frac{1}{1+e^{-x}}$\n",
    "\n",
    "Find a solution with scipy (numerically) and sympy (symbolically). For the numeric solution, use limits of integration for $x$ between $-1$ and $1$:\n",
    "\n",
    "$\\int_{-1}^{1} \\left(\\frac{1}{1+e^{-x}}\\right) dx$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 5: Interpolation\n",
    "\n",
    "Data for `x` and `y` is shown below:\n",
    "\n",
    "```python\n",
    "x = [1,3,7,10]\n",
    "y = [2,9,20,35]\n",
    "```\n",
    "\n",
    "Create a cubic spline interpolation to approximate the value of `y` at `x=5`. Show a cubic spline interpolation on a plot together with the data points. Label the plot with appropriate $x$ and $y$ axis labels, and a legend. Save the plot as `plot.png`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 6: Nonlinear Regression and Plotting\n",
    "\n",
    "Pressure in a tank is recorded as $P = [1.5,2.6,3.5,10.2,20.3,30.2]$ at respective times of $[0,1,2,3,4,5]$ min. Create a nonlinear approximation of the pressure trend as:\n",
    "\n",
    "$P = A \\, e^{n\\,t}$\n",
    "\n",
    "where $t$ is time, $A$ is an unknown pre-exponential factor, and $n$ is also an unknown parameter. Show the optimized parameter values as well as a plot with the data points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 7: Equation Solution and Plotting\n",
    "\n",
    "#### Part A\n",
    "\n",
    "The ideal gas law is shown below:\n",
    "\n",
    "$P V_m=R_g T$\n",
    "\n",
    "where $P$ is the pressure, $V_m$ is the molar volume, $T$ is the temperature, and $R_g$ is the universal gas constant. Use $R_g=8.314 \\frac{J}{mol\\,K}$.\n",
    "\n",
    "Create the function $P(V_m,T)$ such that the pressure ($P$) is a function of molar volume $\\left(V_m\\right)$ and temperature ($T$). Convert the pressure value from $Pa$ to $bar$.\n",
    "\n",
    "Use the function and range variables to create a $P$ vs $T$ plot where $T$ ranges from $100K$ to $1200K$ and $V_m$ = 2.24 $\\frac{L}{mol}$ = 0.00224 $\\frac{m^3}{mol}$ . Include $x$ and $y$ labels on the plot and a legend."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Part B\n",
    "\n",
    "Repeat part A but use the non-ideal gas Van der Waals Equation of State:\n",
    "\n",
    "$P+\\frac{a}{V_m^2}=\\frac{R_g T}{V_m-b}$\n",
    "\n",
    "with constants $a$ and $b$ for ethanol with critical properties $T_c=514 K$ and $P_c=6.137\\mathrm{x}10^6 Pa$:\n",
    "\n",
    "$a = 25 \\frac{\\left(R_g T_c\\right)^2}{64 P_c} = 25 \\frac{\\left(8.314 \\frac{J}{mol\\,K} 514 K\\right)^2}{64 \\,\\mathrm{x}\\,6.137\\mathrm{x}10^6 Pa}$ = $1.1623 \\frac{Pa\\,mol^2}{m^6}$\n",
    "\n",
    "$b = \\frac{R_g T_c}{8 P_c} = \\frac{8.314 \\frac{J}{mol\\,K} 514 K}{8 \\,\\mathrm{x}\\,6.137\\mathrm{x}10^6 Pa}$ = $8.70416\\mathrm{x}10^{-5} \\frac{m^3}{mol}$\n",
    "\n",
    "Compare the ideal gas and non-ideal gas results on the same plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 8: Parameter Estimation and Statistics\n",
    "\n",
    "The yield (concentration) of green fluorescent protein produced in a reaction over time has been determined and is shown below.\n",
    "\n",
    "```\n",
    "Time (minutes)  Concentration (mg/mL)\n",
    "15              107.32\n",
    "30              203.05\n",
    "45              341.26\n",
    "60              401.24\n",
    "180             844.01\n",
    "480             1135.12\n",
    "720             1374.70\n",
    "1200            1651.26\n",
    "```\n",
    "\n",
    "Fit the data to the equation \n",
    "\n",
    "$G = A t^2 + B\\ln{t}+D$\n",
    "\n",
    "Where $G$ is the concentration of GFP in mg/mL at a given time ($t$) in minutes and $A$, $B$, and $D$ are adjustable parameters to minimize the difference between predicted and measured $G$.\n",
    "\n",
    "Solve for $A$, $B$, and $D$ to fit the data. Create a plot of the measured and predicted values and determine the $R^2$ value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9. Implicit Equation Solve\n",
    "\n",
    "Determine the two roots of the polynomial using `fsolve` from `scipy.optimize`.\n",
    "\n",
    "$x^2+3x+2=0$\n",
    "\n",
    "Verify the solutions from the quadratic formula:\n",
    "\n",
    "$\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}$\n",
    "\n",
    "with solutions:\n",
    "\n",
    "$x_0=\\frac{-3+\\sqrt{9-8}}{2} = \\frac{-2}{2} = -1$\n",
    "\n",
    "$x_1=\\frac{-3-\\sqrt{9-8}}{2} = \\frac{-4}{2} = -2$\n",
    "\n",
    "A better way to find polynomial roots is with `numpy` such as\n",
    "\n",
    "```python\n",
    "import numpy as np\n",
    "np.polynomial.polynomial.polyroots([2,3,1])\n",
    "```\n",
    "\n",
    "but use this exercise to practice `fsolve` and how to use a different initial guess to find alternate solutions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}