{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Glossary of commands\n",
    "\n",
    "This file holds a list of all the commands used in MATH1110 Calculus 1.\n",
    "\n",
    "\n",
    "Remember that if you are unsure of how to use a command, you can type its name with a ? and Sage will recall information regarding the command."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### *Contents:*\n",
    "\n",
    "\n",
    "* abs\n",
    "\n",
    "* clear_vars\n",
    "\n",
    "* derivative\n",
    "* desolve\n",
    "* diff\n",
    "\n",
    "* expand\n",
    "\n",
    "* factor\n",
    "* factorial\n",
    "* find_local_minimum\n",
    "* find_root\n",
    "* function\n",
    "\n",
    "* implicit_plot\n",
    "* infinity\n",
    "* integral / integrate\n",
    "\n",
    "* lim / limit\n",
    "* line\n",
    "* ln / log\n",
    "\n",
    "* N\n",
    "\n",
    "* Operators (various)\n",
    "\n",
    "* parametric_plot\n",
    "* plot\n",
    "* point\n",
    "* polar_plot\n",
    "* print\n",
    "\n",
    "* reset\n",
    "* round\n",
    "\n",
    "* show\n",
    "* solve\n",
    "* sqrt\n",
    "* sum\n",
    "\n",
    "* Trigonometry functions (various)\n",
    "\n",
    "* var"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "abs ~ absolute value of a number or expression.\n",
    "\n",
    "    sage: abs( -7 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### C"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "clear_vars ~ used to reset all previously ran and stored variables. \n",
    "\n",
    "$\\hspace{0.6cm}$ No arguments needed. If variables specified, only those within the parentheses will be cleared.\n",
    "\n",
    "    sage: clear_vars()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "derivative ~ compute the derivative of a function.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required argument: (1) function that is being differentiated.\n",
    "\n",
    "$\\hspace{0.6cm}$ Second and third arguments: (2) the variable that is being differentiated with respect two, (3) the order of the derivative.\n",
    "\n",
    "    sage: derivative(f, x, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "desolve ~ solve a differential equation for the function being derived.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required arguments: (1) differential equation, (2) function that is being solved for\n",
    "\n",
    "$\\hspace{0.6cm}$ Additional argument: (3) initial conditions given for the variable and the function\n",
    "\n",
    "    sage: desolve( eqn, y, ics = [1, 2] )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "diff ~ alternative for derivative (see above).\n",
    "\n",
    "    sage: diff(f, x, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### E"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "expand ~ expand a factored expression.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required argument: expression that is being expanded.\n",
    "\n",
    "    sage: expand( (x-1)*(x+1) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### F"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "factor ~ factor an expression.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required argument: expression that is being factored.\n",
    "\n",
    "    sage: factor( x^2 - 1 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "factorial ~ takes the factorial (!) of a number or expression.\n",
    "\n",
    "    sage: factorial(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "find_local_minimum ~ find the local minimum value of a function in a given range.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required arguments: (1) expression/function, (2) lower bound, and (3) upper bound within which the minimum is being located.\n",
    "\n",
    "    sage: find_local_minimum( f, 0, 10 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "find_root ~ find the roots of an expression (i.e. where the function is equal to zero).\n",
    "\n",
    "$\\hspace{0.6cm}$ Required arguments: (1) expression for which the roots are being located, (2) lower bound, and (3) upper bound within which a root is being located.\n",
    "\n",
    "    sage: find_root( f, -5, 0 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "function ~ define a symbol as a function.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required argument: symbol that is being defined as a function.\n",
    "\n",
    "$\\hspace{0.6cm}$ In the example below, f is defined as a function of a (previously defined) variable x.\n",
    "\n",
    "    sage: f = function('f')(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### I"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "infinity ~ positive infinity, also denoted 'oo'.\n",
    "\n",
    "    sage: infinity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implicit_plot ~ plot an implicitly defined curve.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required arguments: (1) the equation that is being plotted, (2) the first variable and its domain, (3) the second variable and its domain.\n",
    "\n",
    "$\\hspace{0.6cm}$ Additional arguments: see options listed for the 'plot' funciton.\n",
    "\n",
    "    sage: implicit_plot( x*y^3 == tan(y) , (x, -10, 10), (y, -10, 10) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "integral ~ compute the integral of a function.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required arguments: (1) the function that is being integrated, (2) the variable that is being integrated with respect to.\n",
    "\n",
    "$\\hspace{0.6cm}$ Additional arguments: (3) the lower bound of the definite integral, (4) the upper bound of the definite integral.\n",
    "\n",
    "    sage: integral(f, x)\n",
    "    \n",
    "\n",
    "integrate ~ alternative for integral (see above).\n",
    "\n",
    "    sage: integrate(f, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### L"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "lim ~ alterative for limit (see below).\n",
    "\n",
    "    sage: lim(f, x = oo)\n",
    "    \n",
    "limit ~ take the limit of an expression.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required arguments: (1) the function for which the limit is being evaluated, (2) the independent variable assigned to the value which it is approaching.\n",
    "\n",
    "    sage: limit(f, x = oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "line ~ create a line between two points.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required argument: the location of the start and end of the line, represented as two ordered pairs in a larger set of square brackets.\n",
    "\n",
    "$\\hspace{0.6cm}$ Additional arguments: linestyle, color, alpha, and soforth.\n",
    "\n",
    "    sage: line( [(-pi, 5), (pi, 5)], color = 'green' )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ln ~ the natural logarithm function.\n",
    "\n",
    "    sage: ln(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "log ~ the logarithmic function.\n",
    "\n",
    "$\\hspace{0.6cm}$ The default base is $e$, **not** base 10, making it equivalent to the natural logarithm.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required argument: number or expression within the logarithm.\n",
    "\n",
    "$\\hspace{0.6cm}$ Second argument: the base.\n",
    "\n",
    "    sage: log(x, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "N ~ express a number as a decimal.\n",
    "\n",
    "$\\hspace{0.6cm}$ Additional argument: number of digits desired.\n",
    "\n",
    "    sage: N( pi, digits = 4 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### O"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Operators, assorted\n",
    "\n",
    " * \\+ is the addition operator\n",
    " * \\- is the subtraction operator\n",
    " * \\* is the multiplication operator\n",
    " * / is the division operator\n",
    " * ^ (or \\*\\*) is the exponent operator\n",
    " * sqrt(\\_) is the square root operator\n",
    " \n",
    " \n",
    " * (), [], {} are parentheses, square brackets, and curly brackets\n",
    " \n",
    " \n",
    " * = is the assignment operator\n",
    " * == is an equality\n",
    " * \\> is greater than\n",
    " * \\< is less than\n",
    " * \\>= is greater than or equal to\n",
    " * \\<= is less than or equal to"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### P"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "parametric_plot ~ plot a parametrically defined curve.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required arguments: (1) parametric functions, (2) parameter (variable) and its domain.\n",
    "\n",
    "$\\hspace{0,6cm}$ Additional arguments: see options listed for the 'plot' function.\n",
    "\n",
    "    sage: parametric_plot( (sin(t), cos(t)), (t, -pi, pi) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "plot ~ create a graph of an expression.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required argument: (1) function that is being plotted.\n",
    "\n",
    "$\\hspace{0.6cm}$ Additional arguments: (2) independent variable, (3) the lower bound of domain, (4) the upper bound of domain, and the scaling on the y-axis (ymin / ymax).\n",
    "\n",
    "$\\hspace{1cm}$ Graphs can be customized with titles, legends, text, and lines of different colours, styles, thicknesses, etc.\n",
    "\n",
    "$\\hspace{1cm}$ See https://doc.sagemath.org/html/en/reference/plotting/sage/plot/plot.html for more options.\n",
    "\n",
    "    sage: plot(f, x, -10, 10, ymin = -10, ymax = 10, color = 'violet', title = 'Plot of a function')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "point ~ create a point or multiple points on a plot.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required argument: location of point(s), inputted as an ordered pair.\n",
    "\n",
    "$\\hspace{0.6cm}$ Additional arguments: color, size, and shape of the point.\n",
    "\n",
    "$\\hspace{0.6cm}$ In the first example, only one point is made at (1, 1). The second example demonstrates correct syntax when creating multiple points.\n",
    "\n",
    "    sage: point( [1, 1], size = 30, color = 'red')\n",
    "    sage: point( [ [1, 1], [2, 2], [3, 3] ], size = 40, color = 'black')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "polar_plot ~ plot a curve in polar coordinates.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required arguments: (1) the expression being plotted, (2) the independent variables with its domain.\n",
    "\n",
    "$\\hspace{0.6cm}$ Additional arguments: see options listed for the 'plot' function.\n",
    "\n",
    "    sage: ploar_plot( sin(theta), (theta, -2*pi, 2*pi) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "print ~ print desired numbers, symbols, or text in the code output.\n",
    "\n",
    "    sage: print('This is a piece of text.')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "reset ~ wipe any stored meaning of a previously defined symbol(s).\n",
    "\n",
    "$\\hspace{0.6cm}$ No required arguments. If none are specified, all previously defined symbols will be cleared. In the example below, only the variable $x$ will be wiped.\n",
    "\n",
    "    sage: reset('x')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "round ~ round a number to a specified decimal place.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required argument: the number being rounded.\n",
    "\n",
    "$\\hspace{0.6cm}$ Second argument: the number of decimal places desired.\n",
    "\n",
    "    sage: round( sqrt(2), 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### S"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "show ~ display input in nicer script.\n",
    "\n",
    "    sage: show( f(x) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "solve ~ solve an equation for one variable.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required arguments: (1) equation, (2) variable to be isolated.\n",
    "\n",
    "    sage: solve( x^2 - 1 == 0, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "sqrt ~ take the square root of a number and/or variable.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required argument: item being rooted.\n",
    "\n",
    "    sage: sqrt( 81 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "sum ~ add a sequence of terms together.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required arguments: (1) the expression to be summed, (2) the variable, (3) starting value, (4) ending value.\n",
    "\n",
    "    sage: sum( x^2, x, 0, 10 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Trigonometry functions, assorted\n",
    "\n",
    "    sin(x), cos(x), tan(x)\n",
    "    \n",
    "    csc(x), sec(x), cot(x)\n",
    "    \n",
    "    arcsin(x), arccos(x), arctan(x)\n",
    "    \n",
    "    arccsc(x), arcsec(x), arccot(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "var ~ define a symbol as a variable.\n",
    "\n",
    "$\\hspace{0.6cm}$ Required argument: symbol that is being defined as a variable\n",
    "\n",
    "    sage: x = var('x')"
   ]
  }
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