{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "# Include these two lines of code at the beginning of every notebook you open.\n", "# It will allow you to receive more than one output from a single code chunk/cell.\n", "from IPython.core.interactiveshell import InteractiveShell\n", "InteractiveShell.ast_node_interactivity = \"all\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# MATH1110 Lab 3: Limits\n", "\n", "Remember to run the chunk of code above as you get started!" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 3 graphics primitives" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = var('x')\n", "y = var('y')\n", "func = tan(y)==y/sin(x/2) # remember this guy?! You made them in an exercise set last week!\n", "Aqua = implicit_plot(func, (x, -20, 20), (y, -10, 10), color = 'hotpink', \\\n", " fill = True, fillcolor = 'aqua', axes = False)\n", "Barbie = points( [[14*sin(pi*x/7)-3, 6*(-1)^x] for x in range(-10, 10)], color = 'hotpink', size = 40)\n", "Barbie+Aqua" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "How'd Assignment 1 go? Terribly? Not too shabby?\n", "If you're still feeling clueless, please come talk to Maya or Stefan and we'll get you up to speed!\n", "\n", "**Objectives for today:**\n", "1. Limits\n", "2. The Solve function" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1. Limits\n", "\n", "Looking at a plot, we can get a good sense of potential areas of discontinuity. Limits are a more precise way we can test the behaviour of a function, such as around asymptotes, for example.\n", "\n", "Let's use Sage to calculate some limits to find the exact values. Let's first look at $\\arctan{(x)}^2$. Do you think this function has any asymptotes? How could we use limits to find out?\n", "\n", "*Hint:* Infinite limits can be inputted with *infinity* or *oo* (two lowercase o's)." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Limits can also be one-sided with the *dir=* argument. Take the regular and one-sided limits as $x\\rightarrow1$, where the function is discontinuous." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 3 graphics primitives" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y2 = 1/(x^4-1)\n", "plot(y2, x, 0, 5, ymin = -10, ymax = 10, detect_poles = 'show', color = 'chocolate')\n", "# remember that the detect_poles argument is how we show asymptotes in Sage\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercise questions\n", "\n", "1. Is the function $\\sqrt{t}-t^2+\\cos{(4t)}$ continuous? Make its plot, and find the limit as $t \\rightarrow \\infty$.\n", "\n", "\n", "2. What is $\\lim_{x\\rightarrow 0}f(x)$ for $f(x) = \\large{ \\frac{x^3-1}{x^2-x} }$? Do the one-sided limits equal each other?" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "clear_vars()\n", "#1\n", "var('t')\n", "func = sqrt(t)-t^2+cos(4*t)\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "clear_vars()\n", "#2\n", "x = var('x')\n", "f = (x^3-1)/(x^2-x)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. The Solve function\n", "\n", "The solve command has many different applications - it's a good multipurpose tool for isolating variables. As you can find in the *Glossary of Commands*, there are two arguments needed for *solve()*: the expression you're given, and what you're solving for.\n", "\n", "Solve the quadratic $x^2+x-5=0$ for $x$. Remember to use the double *==* when you're denoting an actual mathematical equality, as in an equation." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we have a *system of equations* (any linear algebra students in the establishment?!) we can solve for multiple variables, assuming we have enough information about those variables.\n", "Find $x, y$, and $z$ in the following system:\n", "$$x-y=2 \\\\ y-3z=1 \\\\ x+y+z=4$$" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One last thing about solving. In a situation where we *don't* have enough information to find what a variable is equal to, we can still use Sage to isolate the variable, rewriting it in terms of everything else. This can be especially useful when we want to find inverse equations." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[x == arccos(y)]" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# a simple example: what is the inverse of cos(x)?\n", "solve( y==cos(x), x)\n", "# arccos(y)\n", "# Note that we can write this by hand as cos^-1(y), but Sage will not understand this notation." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercise questions\n", "\n", "3. Solve for $z$ in the equation $\\frac{y}{z}+z^2=x$.\n", " * Once you see your output, you'll be glad you didn't attempt to do this by hand!\n", " * You'll notice in two of the solutions, there is a little *i*. That means there is an imaginary number. Not a real number. For this course, you can disregard imaginary solutions.\n", "\n", "\n", "4. Solve for $x$ and $y$ given that $9x+2y=1$ and $x-y^2=0$. Then plot the two curves together.\n", " * Wherever the curves intersect, there is a solution for the system of equations." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That's it! That's all." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.0", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.6" } }, "nbformat": 4, "nbformat_minor": 2 }