// Numbas version: exam_results_page_options
{"name": "Find the fixed points of a 1D dynamical system", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "r1"}, "ans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(0.5*(c1+sqrt(c1^2-4*d1)),3)", "description": "", "name": "ans3"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "2*sqrt(d1)+random(1..9)", "description": "", "name": "c1"}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "ans1"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(map(x^2,x,1..6))", "description": "", "name": "d1"}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(0.5*(c1-sqrt(c1^2-4*d1)),3)", "description": "", "name": "ans2"}}, "ungrouped_variables": ["r1", "ans1", "ans2", "ans3", "tol", "c1", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find the fixed points of a 1D dynamical system", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans1+tol", "minValue": "ans1-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans3+tol", "minValue": "ans3-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>\\[\\dot{x}=\\simplify{{-r1}*(x^3-{c1}*x^2+{d1}*x)}\\]</p>\n        <p>There are&nbsp;<strong>three</strong> fixed points; enter them in ascending numerical order.</p>\n        <p>Fixed point 1: $x_0=$ [[0]] (Enter your answer to 3d.p.)</p>\n        <p>Fixed point 2: $x_0=$ [[1]]&nbsp;(Enter your answer to 3d.p.)</p>\n        <p>Fixed point 3: $x_0=$ [[2]]&nbsp;(Enter your answer to 3d.p.)</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Find the fixed points $x_0$ of the following one-dimensional dynamical system $\\dot{x}=f(x)$.</p>", "tags": ["checked2015", "MAS2106"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Fixed points of a 1D dynamical system.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The fixed points are given by $\\dot{x}=0$.</p>\n    <p>Rewrite the system as</p>\n    <p>\\[\\dot{x}=\\simplify{{-r1}*x*(x^2-{c1}*x+{d1})},\\]</p>\n    <p>then $\\dot{x}=0\\implies x=0$ (which is the first fixed point) or $\\simplify{x^2-{c1}*x+{d1}=0}$.</p>\n    <p>Then solve the quadratic equation for $x$ to determine the other two fixed points.</p>\n    <p>The three fixed points are therefore $x_0=\\var{ans1}$, $x_0=\\var{ans2}$, and $x_0=\\var{ans3}$ to 3d.p.</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}