// Numbas version: exam_results_page_options
{"variable_groups": [], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Function approximation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "n"}, "est": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(a^(m/n)+h*m*a^(m/n-1)/n,5)", "description": "", "name": "est"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(a^(1/n),0)", "description": "", "name": "c"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(n=2,random(4,9,16,25,36),if(n=3,random(8,27,64),if(n=4,random(16,81),random(32,243))))", "description": "", "name": "a"}, "tr": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((a+h)^(m/n),5)", "description": "", "name": "tr"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "m-n", "description": "", "name": "p"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(n=2 or n=4,random(1,3,5), if(n=3,random(1,2,4,5),random(1,2,3,4,6)))", "description": "", "name": "m"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(random(1,-1)*random(0.001..0.1#0.001),3)", "description": "", "name": "h"}}, "ungrouped_variables": ["a", "c", "est", "h", "s1", "tr", "m", "n", "p"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "x^({m}/{n})", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 0.5, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a}", "minValue": "{a}", "correctAnswerFraction": false, "marks": 0.25, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{h}", "minValue": "{h}", "correctAnswerFraction": false, "marks": 0.25, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{est}", "minValue": "{est}", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{tr}", "minValue": "{tr}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n        <p>In this case $f(x) =\\;$ [[0]].</p>\n      <p>$a= \\;$[[1]] and $h=\\; $[[2]]<br/>Input your estimation to $5$ decimal places: [[3]]</p>\n      <p>True value is: [[4]] (input to 5 decimal places).</p>\n        ", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Use the approximation $f(a+h) \\approx f(a)+hf^{\\prime}(a)$ to estimate \\[\\var{a+h}^{\\frac{\\var{m}}{\\var{n}}} \\]for a suitable function $f(x)$.</p>", "tags": ["application of differentiation", "approximation", "approximation of the value of a function using the tangent", "approximations", "calculus", "Calculus", "checked2015", "equation of tangent", "first order approximation", "functions", "maclaurin series", "MacLaurin series", "mas1601", "MAS1601", "tangent equation"], "rulesets": {"std": ["all", "!collectNumbers"], "dpoly": ["std", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n                        \t\t<p><strong>3/07/2012<br /></strong></p>\n                        \t\t<p>Added tags</p>\n                        \t\t<p><strong>20/06/2012:</strong></p>\n                        \t\t<p><strong><br /></strong>Added tags.</p>\n                        \t\t<p>Got rid of request for 5dps for the function!</p>\n                        \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Approximate $f(x)=(a+h)^{m/n}$ by $f(a)+hf^{\\prime}(a)$ to 5 decimal places and compare with true value.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>We have $f(x)=x^\\frac{\\var{m}}{\\var{n}}$ and $a= \\var{a}$, $h=\\var{h}$.<br/>Note that $\\var{a}^\\frac{1}{\\var{n}}=\\var{c}$ and so using the approximation :<br/>$f(a+h)\\approx f(a)+hf^{\\prime}(a)$ and $f^{\\prime}(x) = \\frac{\\var{m}}{\\var{n}}\\simplify[std]{x^({m-n}/{n})}$ <br/>we have:<br/>\\[\\simplify[std]{{a+h}^({m}/{n})}\\approx \\simplify[simplifyFractions]{{a}^({m}/{n})+{h}*({m}/{n})*{a}^({p}/{n})}=\\simplify[std,!sqrtSquare]{{c}^{m}+ {h}*({m}/{n})*{c}^{m-n}}=\\var{est}\\]<br/>to 5 decimal places.<br/>The true value to 5 decimal places is {tr}.</p>"}, {"name": "Limiting speed given acceleration function, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "b"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..7)", "description": "", "name": "n"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "b", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{a} / {(b * (n -1))}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Input as a fraction or an integer and not as a decimal.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>Maximum possible speed = [[0]]$m/s$</p>\n      <p>Input as a fraction or an integer and not as a decimal.</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>An object moves in a straight line with an acceleration given by:<br/>\\[f(t)=\\frac{\\var{a}}{(1+\\var{b}t)^{\\var{n}}}\\]</p>\n  <p>where $t$ is time.</p>\n  <p>Given that the object starts from rest, find its maximum possible speed (i.e. its limiting speed).</p>", "tags": ["1st order differential equation", "acceleration and speed", "applied mathematics", "Calculus", "checked2015", "differential equation", "differential equation ", "first order differential equation", "initial conditions", "integration", "limiting value", "MAS1603", "MAS1902", "modelling", "ode"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>29/06/2012:</strong></p>\n    <p>Added, edited tags</p>\n    <p>Slight changes in display in prompt.</p>\n    <p>Added a line to explain the limit in advice (last line).</p>\n    <p>Checked calculation. No problem with checking range as variables $\\gt 0$.</p>\n    <p><strong>10/07/2012:</strong></p>\n    <p>Added requirement in prompt that numbers entered as fractions or integers. Added decimal point as forbidden string.</p>\n    <p><strong>18/07/2012:</strong></p>\n    <p>Added description.</p>\n    <p><strong>23/07/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Question appears to be working correctly.</p>\n    <p><strong>04/11/2012:</strong></p>\n    <p>Added m/s units where needed.</p>\n    <p>&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>An object moves in a straight line, acceleration given by:</p>\n<p>$\\displaystyle f(t)=\\frac{a}{(1+bt)^n}$. The object starts from rest. Find its maximum speed.&nbsp;</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>If $v(t)$ is the velocity at time $t$ then the acceleration at time $t$ is $\\displaystyle{\\frac{dv}{dt}}$.</p>\n  <p>Hence we have the differential equation for the velocity:</p>\n  <p>\\[\\frac{dv}{dt}=\\frac{\\var{a}}{(1+\\var{b}t)^{\\var{n}}}=\\simplify[std]{{a}(1+{b}t)^{-n}}\\] where $v(0)=0$ as the object starts from rest.<br/>Integrating this gives: $\\displaystyle{v(t) = \\simplify[std]{-{a}/{b*(n-1)}*(1+{b}*t)^{-n+1}+A}}$.</p>\n  <p>Since $v(0)=0$ this gives $\\displaystyle{A=\\simplify[std]{{a}/{b*(n-1)}}}$ and so the solution is:</p>\n  <p>$\\displaystyle{v(t) = \\simplify[std]{{a}/{b*(n-1)}(1-(1+{b}*t)^{-n+1})}}$</p>\n  <p>It follows that the limiting speed is:</p>\n  <p>\\[\\lim_{t \\to \\infty} v(t) = \\simplify[std]{{a}/{b*(n-1)}}m/s\\] as</p>\n  <p>\\[(1+\\var{b}t)^{\\var{-n+1}}=\\simplify[std]{1/(1+{b}t)^{n-1}} \\rightarrow 0,\\;\\;t \\rightarrow \\infty\\]</p>"}, {"name": "Second order ODE with constant coefficients and boundary conditions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s*random(1..5)", "description": "", "name": "a"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..7)", "description": "", "name": "b"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "description": "", "name": "c"}, "s": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s"}}, "ungrouped_variables": ["a", "s", "b", "c"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"answer": "exp({- a} * x) * (cos({b} * x) + ({c+a} / {b}) * sin({b} * x))", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 1e-05, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input all numbers as integers or fractions.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Solution is:</p>\n      <p>$y=\\;\\;$[[0]]</p>\n      <p>Input all numbers as integers or fractions &ndash; not as decimals.</p>", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Solve:<br/>\\[\\simplify[std]{(d^2y/dx^2)+{2*a}*(dy/dx)+{a^2+b^2}y}=0\\]<br/>which satisfies $y(0)=1$ and $y'(0)=\\var{c}$ (where prime denotes the derivative).</p>", "tags": ["2nd order differential equation", "auxiliary equation", "boundary conditions on differential equation", "Calculus", "checked2015", "complex roots of auxillary equation", "constant coefficients", "differential equation", "differential equation ", "exponential function", "finding the auxillary equation", "linear differential equation", "MAS1603", "ode", "quadratic equation", "second order differential equation", "solving differential equations", "solving quadratic equation", "trigonometric functions"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>29/06/2012:</strong></p>\n    <p><strong><br /></strong>Added tags. Edited tags.</p>\n    <p>Improved display.</p>\n    <p>Checked answer.</p>\n    <p><strong>23/07/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Question appears to be working correctly.</p>\n    <p><strong>04/11/2012:</strong></p>\n    <p><strong><br /></strong>Corrected mistake in solution.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve:&nbsp;$\\displaystyle \\frac{d^2y}{dx^2}+2a\\frac{dy}{dx}+(a^2+b^2)y=0,\\;y(0)=1$ and $y'(0)=c$.&nbsp;</p>"}, "advice": "<p>The auxillary equation is $\\simplify[std]{lambda^2+{2*a}lambda+{a^2+b^2}}=0$.</p>\n  <p>On solving this equation we get $\\lambda=\\simplify[std]{{-a}+{b}i}$ and $\\lambda=\\simplify[std]{{-a}-{b}i}$.</p>\n  <p>Hence the general solution is:<br/>\\[y = \\simplify[std]{e^({-a}x)(A*sin({b}x)+B*cos({b}x))}\\]<br/>Note that <br/>\\[y'(x)=\\simplify[std]{-{a}e^({-a}x)(A*sin({b}x)+B*cos({b}x))+e^({-a}x)({b}*A*cos({b}x)-{b}*B*sin({b}x))}\\]<br/>Using the conditions $y(0)=1$ and $y'(0)=\\var{c}$ gives:<br/>\\[\\begin{eqnarray*} B &amp;=&amp; 1\\\\ \\simplify[std]{{b}A+{-a}B}&amp;=&amp; \\var{c} \\end{eqnarray*} \\]<br/>This gives $\\displaystyle{A = \\simplify[std]{{c+a}/{b}}}$.</p>\n  <p>Hence the solution is:</p>\n  <p>\\[y=\\simplify[std]{exp({- a} * x) * (cos({b} * x) + &nbsp;({c+a} / {b}) * sin({b} * x))}\\]</p>"}, {"name": "Separable first order ODE with boundary condition", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "b"}, "n": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "description": "", "name": "n"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "b", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "({(b ^ (n + 1))} + ({(a * (n + 1))} * t)) ^ (1 / {(n + 1)}) - {b}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"message": "<p>Input all numbers as integers or fractions.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>The thickness at time $t$ is given by:</p>\n        <p>$x(t)=\\;\\;$[[0]]</p>\n        <p>Input all numbers as integers or fractions &ndash; not as decimals.</p>", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>The thickness of ice on water, $x(t)$, grows according to the equation:<br/>\\[\\frac{dx}{dt}=\\simplify[std]{{a}/(x+{b})^{n}}\\]<br/>Given that $x(0)=0$ find $x(t)$.</p>", "tags": ["1st order differential equation", "Calculus", "checked2015", "differential equation", "differential equation ", "first order differential equation", "growth", "initial conditions", "MAS1603", "modelling", "ode", "separable variables", "separation of variables"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>29/06/2012:</strong></p>\n    <p>Added and edited tags.</p>\n    <p>Checked answer. Checking range OK as we are taking roots of positive numbers, given the choice of ranges for the variables.</p>\n    <p><strong>18/07/2012:</strong></p>\n    <p>Added description.</p>\n    <p><strong>23/07/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>The arbitrary constant A should be relabelled as A_1 in the Advice section part way though the solution.</p>\n    <p>Question appears to be working correctly.</p>\n    <p><strong>&nbsp;</strong></p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve for $x(t)$, $\\displaystyle\\frac{dx}{dt}=\\frac{a}{(x+b)^n},\\;x(0)=0$</p>"}, "advice": "<p>On rearranging the equation we get $\\displaystyle{\\simplify[std]{(x+{b})^{n}*(dx/dt) = {a}}}$ and on integrating we obtain:<br/>$\\displaystyle{\\simplify[std]{(x+{b})^{n+1}/{n+1}={a}t +A} \\Rightarrow x+\\var{b}=(A+\\var{a*(n+1)}t)^{1/\\var{n+1}}}$</p>\n    <p>Using the condition $x(0)=0$ gives $\\displaystyle{A^{1/\\var{n+1}}=\\var{b} \\Rightarrow A=\\var{b^(n+1)}}$</p>\n    <p>Hence the solution is:<br/>\\[x(t) = \\simplify[std]{({(b ^ (n + 1))} + ({(a * (n + 1))} * t)) ^ (1 / {(n + 1)}) - {b}}\\]</p>"}, {"name": "Solve a second order ODE with repeated roots, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..7)", "description": "", "name": "b"}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(f,3)", "description": "", "name": "f1"}, "s": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6)", "description": "", "name": "d"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6)", "description": "", "name": "c"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s*random(1..7)", "description": "", "name": "a"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "d*exp(a)-c", "description": "", "name": "f"}}, "ungrouped_variables": ["a", "f1", "c", "b", "d", "f", "s"], "functions": {}, "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "<p>Solution is:</p>\n      <p>$y=\\;\\;$[[0]]</p>\n      <p>Input all numbers correct to 3 decimal places.</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{c} * Exp({ - a} * x) + {f1} * x * Exp({- a} * x)", "showCorrectAnswer": true, "checkingAccuracy": 1e-05, "customMarkingAlgorithm": "", "answerSimplification": "std,!fractionNumbers", "expectedVariableNames": [], "showPreview": true, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "vsetRangePoints": 5, "showFeedbackIcon": true, "scripts": {}, "marks": 3, "type": "jme", "unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "failureRate": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "<p>Find the solution of:<br/>\\[\\simplify[std]{(d^2y/(d*x^2))+{2*a}*((d*y/(d*x)))+{a^2}y}=0\\]<br/>which satisfies $y(0)=\\var{c}$ and $y(1)=\\var{d}$.</p>", "tags": ["2nd order differential equation", "auxillary equation", "auxillary equation with equal roots", "boundary conditions", "calculus", "Calculus", "checked2015", "constant coefficients", "differential equations", "Differential equations", "equal roots", "exponential function", "general solution", "linear differential equations", "linear differential equations with constant coefficients", "ODE", "ode", "quadratic function", "repeated roots for auxillary equation", "second order differential equations", "solving differential equations", "solving quadratic function", "trigonometric functions"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "advice": "<p>The auxillary equation is $\\simplify[std]{lambda^2+{2*a}lambda+{a^2}}=0$.</p>\n  <p>On solving this equation we get $\\lambda=\\var{-a}$ twice.</p>\n  <p>Hence the general solution is:<br/>\\[y = \\simplify[std]{A*e^({-a}x)+B*x*e^({-a}x)}\\]<br/>The boundary conditions give:</p>\n  <p>$y(0)=\\var{c} \\Rightarrow A=\\var{c}$</p>\n  <p>$y(1)=\\var{d} \\Rightarrow \\simplify{Ae^{-a}+Be^{-a}={d}}\\Rightarrow A+B = \\simplify{{d}e^{a}}$</p>\n  <p>So $B=\\simplify{{d}e^{a}-{c}}=\\var{f1}$ to 3 decimal places.</p>\n  <p>Hence the solution is:<br/>\\[y=\\simplify{(({c} * Exp(({( - a)} * x))) + ({f1} * x * Exp(({( - a)} * x))))}\\]</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve:&nbsp;$\\displaystyle \\frac{d^2y}{dx^2}+2a\\frac{dy}{dx}+a^2y=0,\\;y(0)=c$ and $y(1)=d$. &nbsp;(Equal roots example).</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Taylor series (three terms)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"answer": "{tm0}+{tm1}/{a*n}*(x-{c})+{tm2}/{2*a^2*n^2}*(x-{c})^2", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "prompt": "\n        <p>Input the first three terms in the Taylor series in the form $a+b(x-\\var{c})+c(x-\\var{c})^2$ for suitable coefficients $a,\\;b$ and $c$.</p>\n        <p><br/>Input coefficients as fractions, not as decimals. Also do not use factorials in your answer. For example, input 6 rather than 3!.</p>\n        ", "notallowed": {"message": "<p>Do not input factorials or decimals in the Taylor series.</p>", "showStrings": false, "partialCredit": 0, "strings": ["!", "."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 1e-06, "type": "jme", "answersimplification": "all,fractionNumbers,!collectNumbers", "marks": 4, "vsetrangepoints": 5}], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "tm1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "tm0*b", "name": "tm1", "description": ""}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a=4 or a=9 or a=25 or a=36 or a=49,2,if(a=8 or a=27,3,if(a=32,5,if(a=16,random(2,4),random(2..5)))))", "name": "n", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,4,8,9,16,27,32,25,36,49)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "c", "description": ""}, "tm0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a^(1/n)", "name": "tm0", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*switch(a=1,random(2..9),a=4,random(3,5,7,9),a=8,random(1,3,5,7,9),a=9,random(1,2,4,5,7,8),a=16,random(1,3,5,7,9),a=32,random(1,3,5,7,9),a=25,random(1,2,4,6,7,9),a=27,random(1,2,4,5,7,8),a=36,random(1,5,7,9),random(1,2,3,4,5,8,9))", "name": "b", "description": ""}, "tm2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-(n-1)*tm1*b", "name": "tm2", "description": ""}}, "ungrouped_variables": ["a", "tm0", "tm2", "b", "tm1", "s1", "c", "n"], "rulesets": {}, "variable_groups": [], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>You are asked to find the first 3 terms in the Taylor series at $x=\\var{c}$ for $f(x)=(\\simplify[all]{{a-b*c}+{b}*x})^{1/\\var{n}}$ i.e. up to terms in $x^2$.</p>", "tags": ["3 term Taylor series", "approximation", "approximations", "Calculus", "calculus", "checked2015", "function", "functions", "mas1601", "MAS1601", "series approximation", "series expansion", "Taylor series", "taylor series", "three term Taylor series"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n                                        \t\t<p><strong>20/06/2012:</strong></p>\n                                        \t\t<p>Added tags.</p>\n                                        \t\t<p>Added !collectNumbers to some rules so that polynomials presented in standard order.</p>\n                                        \t\t<p>Added more explanation to prompt in question.</p>\n                                        \t\t<p>Also included ! and . in forbidden strings together with message.</p>\n                                        \t\t<p><strong>3/07/2012:<br /></strong></p>\n                                        \t\t<p>Added tags.</p>\n                                        \t\t<p><strong>9/07/2012:</strong></p>\n                                        \t\t<p>Improved display of first line of Advice.</p>\n                                        \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the first 3 terms in the Taylor series at $x=c$ for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.</p>"}, "functions": {}, "advice": "<p>The first three terms in the Taylor series are given by $\\simplify[all]{a+b(x-{c})+c(x-{c})^2}$ where $\\displaystyle a=f(\\var{c}),\\;\\;b=f'(\\var{c}),\\;\\;c=\\frac{f''(\\var{c})}{2}$<br/>For this example, <br/>\\[\\begin{eqnarray*} f'(x)&amp;=&amp;\\simplify[all,fractionNumbers]{{b}/{n}*({a-b*c}+{b}x)^(-{n-1}/{n})}\\\\ f''(x)&amp;=&amp;\\simplify[all,fractionNumbers]{-{b^2*(n-1)}/{n^2}*({a-b*c}+{b}x)^(-{2*n-1}/{n})} \\end{eqnarray*} \\]<br/>and so we get:<br/>\\[\\begin{eqnarray*} a&amp;=&amp;f(\\var{c})=\\simplify[all]{{a}^(1/{n})={tm0}}\\\\ b&amp;=&amp;f'(\\var{c})=\\simplify[all,fractionNumbers]{{tm1}/{a*n}}\\\\ c&amp;=&amp;\\frac{f''(\\var{c})}{2}=\\simplify[all,fractionNumbers]{{tm2}/{2*a^2*n^2}} \\end{eqnarray*}\\]<br/>Hence the first three terms of the Taylor series are:<br/>\\[\\simplify[all,fractionNumbers,!collectNumbers]{{tm0}+{tm1}/{a*n}*(x-{c})+{tm2}/{2*a^2*n^2}*(x-{c})^2} \\]</p>"}, {"name": "Determine long-term behaviour of 1D dynamical systems", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "r1"}, "incans21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (ans2=0,a2,\n      if (ans2=a2,b2,\n    if (ans2=b2,infinity,0)\n      )\n    )", "description": "", "name": "incans21"}, "x01": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..9)", "description": "", "name": "x01"}, "x02": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..15)", "description": "", "name": "x02"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "a1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a2+random(1..5)", "description": "", "name": "b2"}, "incans13": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (ans1=0,latex('$\\\\simplify{{a1}/{random(2..5)}}$'),\n      if (ans1=a1,-infinity,a1*random(2..5))\n    )", "description": "", "name": "incans13"}, "incans22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (ans2=0,infinity,\n      if (ans2=a2,infinity,\n    if (ans2=b2,a2,b2)\n      )\n    )", "description": "", "name": "incans22"}, "incans11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (ans1=0,infinity,\n      if (ans1=a1,0,a1)\n    )", "description": "", "name": "incans11"}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (r2<0,\n      if (x02=0,0,\n        if (x02=a2,a2,\n          if (x02=b2,b2,\n            if (x02<b2,a2,infinity)\n          )\n        )\n      ),\n      if (x02=0,0,\n        if (x02=a2,a2,\n          if (x02=b2,b2,\n            if (x02<a2,0,b2)\n          )\n        )\n      )\n    )", "description": "", "name": "ans2"}, "incans12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (ans1=0,a1,\n      if (ans1=a1,infinity,0)\n    )", "description": "", "name": "incans12"}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (r1<0,\n      if (x01=0,0,\n    if (x01=a1,a1,\n      if (x01<a1,0,infinity)\n    )\n      ),\n      if (x01=0,0,a1)\n    )\n      ", "description": "", "name": "ans1"}, "incans23": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (ans2=0,b2,\n      if (ans2=a2,0,\n    if (ans2=b2,0,a2)\n      )\n    )", "description": "", "name": "incans23"}, "r2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "r2"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "a2"}}, "ungrouped_variables": ["r1", "r2", "incans13", "ans1", "incans11", "incans22", "incans23", "incans21", "a1", "a2", "b2", "incans12", "x02", "x01", "ans2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"displayType": "radiogroup", "choices": ["<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{ans1}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans11}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans12}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans13}$.</p>"], "displayColumns": 1, "prompt": "<p>\\[\\dot{x}=f(x)=\\simplify{{r1}*x*(1-x/{a1})}\\]</p>\n        <p>with initial condition $x(0)=\\var{x01}$ and $x\\geqslant 0$.</p>", "distractors": ["", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0], "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{ans2}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans21}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans22}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans23}$.</p>"], "displayColumns": 1, "prompt": "<p>\\[\\dot{x}=g(x)=\\simplify{{r2}*x*({a2}-x)*(x-{b2})}\\]</p>\n        <p>with initial condition $x(0)=\\var{x02}$ and $x\\geqslant 0$.</p>", "distractors": ["", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0], "marks": 0}], "statement": "<p>For the given one-dimensional dynamical systems and initial conditions below, determine the long-term behaviour of the solution $x(t)$ as $t\\rightarrow\\infty$.</p>", "tags": ["checked2015", "MAS2106"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Determine the long-term behaviour of 1D dynamical systems.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>It is helpful to draw the one-dimensional phase flow for systems of this type.</p>\n    <p>a)</p>\n    <p>The ODE in this part can be written in the general form</p>\n    <p>\\[\\dot{x}=f(x)=rx\\left(1-\\frac{x}{a}\\right),\\]</p>\n    <p>where $r=\\var{r1}$ and $a=\\var{a1}$.</p>\n    <p>The system has fixed points at $x=0$ and $x=a$, and the function $f(x)$ is a parabola.</p>\n    <p>If $r&gt;0$, then the function (with phase flow as arrows) is as shown below.</p>\n    <p><img src=\"resources/images/parabola1.png\"/></p>\n    <p>If $r&lt;0$, then the function (with phase flow as arrows) is as shown below.</p>\n    <p><img src=\"resources/images/parabola2.png\"/></p>\n    <p>Hence:</p>\n    <ul>\n    <li>if $r&gt;0$, then $x\\rightarrow a$ as $t\\rightarrow\\infty$, irrespective of whether $x(0)&lt; a$ or $x(0)&gt;a$;</li>\n    <li>if $r&lt;0$, then $x\\rightarrow 0$ as $t\\rightarrow\\infty$ if $0\\leqslant x(0)&lt; a$, and $x\\rightarrow\\infty$ as $t\\rightarrow\\infty$ if $x(0)&gt;a$;</li>\n    <li>if $x(0)=0$ or $x(0)=a$, then $x(t)$ will remain at these points for all time, since they are the fixed points $x_0$ of the system.</li>\n    </ul>\n    <p></p>\n    <p>If $r&gt;0$, then $x_0=0$ is unstable and $x_0=a$ is stable.</p>\n    <p>If $r&lt;0$, then $x_0=0$ is stable and $x_0=a$ is unstable.</p>\n    <p>For the particular values of $r$ and $a$ in this part $x\\rightarrow\\var{ans1}$ as $t\\rightarrow\\infty$.</p>\n    <p></p>\n    <p>b)</p>\n    <p>The ODE in this part can be written in the general form</p>\n    <p>\\[\\dot{x}=g(x)=rx(a-x)(x-b),\\]</p>\n    <p>where $r=\\var{r2}$, $a=\\var{a2}$, and $b=\\var{b2}$.</p>\n    <p>The system has fixed points at $x=0$, $x=a$, and $x=b$, and the function $g(x)$ is a cubic.</p>\n    <p>If $r&gt;0$, then the function (with phase flow as arrows) is as shown below.</p>\n    <p><img src=\"resources/images/cubic1.png\"/></p>\n    <p>If $r&lt;0$, then the function (with phase flow as arrows) is as shown below.</p>\n    <p><img src=\"resources/images/cubic2.png\"/></p>\n    <p>Hence:</p>\n    <ul>\n    <li>if $r&gt;0$, then $x\\rightarrow 0$ as $t\\rightarrow\\infty$ if $0\\leqslant x(0)&lt; a$, and $x\\rightarrow b$ as $t\\rightarrow\\infty$ if $x(0)&gt;a$;</li>\n    <li>if $r&lt;0$, then $x\\rightarrow a$ as $t\\rightarrow\\infty$ if&nbsp;$0\\leqslant x(0)&lt; b$, and $x\\rightarrow\\infty$ as $t\\rightarrow\\infty$ if $x(0)&gt;b$;</li>\n    <li>if $x(0)=0$, $x(0)=a$, or $x(0)=b$, then $x(t)$ will remain at these points for all time, since they are the fixed points $x_0$ of the system.</li>\n    </ul>\n    <p></p>\n    <p>If $r&gt;0$, then $x_0=0$ and $x_0=b$ are stable, and $x_0=a$ is unstable.</p>\n    <p>If $r&lt;0$, then $x_0=0$ and $x_0=b$ are unstable, and $x_0=a$ is stable.</p>\n    <p>For the particular values of $r$, $a$, and $b$ in this part $x\\rightarrow\\var{ans2}$ as $t\\rightarrow\\infty$.</p>"}, {"name": "Find fixed points of each of two dynamical systems", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"ans3a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(if (additional,sqrt(c3/d3),0),3)", "description": "", "name": "ans3a"}, "ans3b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(if (additional,sqrt(a3/b3),0),3)", "description": "", "name": "ans3b"}, "anyxorxeq0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (disc=0,\"any $x$\",\"$x=0$\")", "description": "", "name": "anyxorxeq0"}, "disc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b2*c2-a2*d2", "description": "", "name": "disc"}, "additional": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a3/b3>0 and c3/d3>0", "description": "", "name": "additional"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b1"}, "fixedpoints": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (additional,\n      \"which are both greater than $0$, so there are five fixed points\",\n      \"both of which are not greater than $0$, so there is one fixed point\"\n    )", "description": "", "name": "fixedpoints"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b2"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b3"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c2"}, "d2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "d2"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "c1"}, "d3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "d3"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a3"}, "todp": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (additional,\" to 3d.p.\",\".\")", "description": "", "name": "todp"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "c3"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a2"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "d1"}, "w2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (disc=0,1,0)", "description": "", "name": "w2"}}, "ungrouped_variables": ["anyxorxeq0", "todp", "fixedpoints", "additional", "ans3a", "ans3b", "b2", "a1", "a3", "disc", "w2", "tol", "b3", "c3", "c2", "c1", "d1", "d2", "d3", "a2", "b1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{c1}/{d1}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}, {"answer": "{a1}/{b1}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>\\[\\begin{align}\\dot{x}&amp;=x(\\simplify[std]{{a1}-{b1}*y}),\\\\\\dot{y}&amp;=y(\\simplify[std]{{c1}-{d1}*x}).\\end{align}\\]</p>\n        <p>There are <strong>two</strong> fixed points. &nbsp;Enter the fixed point with the&nbsp;<strong>smallest $x$-component</strong> in the first set of boxes. &nbsp;<strong>Do not enter decimals in your answers</strong>.</p>\n        <p>Fixed point 1: $\\boldsymbol{x}_0=($[[0]]$,$[[1]]$)^\\mathsf{T}$.</p>\n        <p>Fixed point 2: $\\boldsymbol{x}_0=($[[2]]$,$[[3]]$)^\\mathsf{T}$.</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans3a+tol", "minValue": "ans3a-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans3b+tol", "minValue": "ans3b-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>\\[\\begin{align}\\dot{x}&amp;=x(\\simplify[std]{{a3}-{b3}*y^2}),\\\\\\dot{y}&amp;=y(\\simplify[std]{{c3}-{d3}*x^2}).\\end{align}\\]</p>\n        <p>One fixed point always exists; enter it in the first set of boxes.</p>\n        <p>Fixed point 1: $\\boldsymbol{x}_0=($[[0]]$,$[[1]]$)^\\mathsf{T}$.</p>\n        <p>Four additional fixed points can exist, depending on the coefficients in the system. &nbsp;For this particular set of coefficients, decide whether these additional fixed points exist. &nbsp;If they do, enter $x_0$ and $y_0$ in the boxes below. &nbsp;If no additional fixed points exist, enter the value $0$ for both components.</p>\n        <p>Additional fixed points: $\\boldsymbol{x}_0=(\\pm$[[2]]$,\\pm$[[3]]$)^\\mathsf{T}$. &nbsp;(Enter your answers to 3d.p.)</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Find the fixed points $\\boldsymbol{x}_0=\\pmatrix{x_0,y_0}^\\mathsf{T}$ of the following two-dimensional dynamical systems $\\dot{\\boldsymbol{x}}=\\boldsymbol{f}(\\boldsymbol{x})$, where $\\boldsymbol{x}=\\pmatrix{x,y}^\\mathsf{T}$.</p>", "tags": ["checked2015", "MAS2106"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Fixed points of 2D dynamical systems.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The fixed points of a dynamical system</p>\n  <p>\\[\\begin{align}\\dot{x}&amp;=f(x,y),\\\\\\dot{y}&amp;=g(x,y),\\end{align}\\]</p>\n  <p>are given by $\\dot{x}=0$ and $\\dot{y}=0$, i.e. when $f(x,y)=0$ and $g(x,y)=0$.</p>\n  <p>a)</p>\n  <p>We have</p>\n  <p>\\[\\begin{align}\\dot{x}&amp;=x(\\simplify[std]{{a1}-{b1}*y}),\\\\\\dot{y}&amp;=y(\\simplify[std]{{c1}-{d1}*x}).\\end{align}\\]</p>\n  <p>For $\\dot{x}=0$, either $x=0$ or $y=\\simplify{{a1}/{b1}}$.</p>\n  <p>If $x=0$ then $\\dot{y}=0$ requires that $y=0$.</p>\n  <p>If $y=\\simplify{{a1}/{b1}}$, then $\\dot{y}=0$ requires $x=\\simplify{{c1}/{d1}}$.</p>\n  <p>Hence, there are two fixed points $\\boldsymbol{x}_0=\\pmatrix{0,0}^\\mathsf{T}$ and $\\boldsymbol{x}_0=\\pmatrix{\\simplify{{c1}/{d1}},\\simplify{{a1}/{b1}}}^\\mathsf{T}$.</p>\n  <p></p>\n  <p>b)</p>\n  <p>We have</p>\n  <p>\\[\\begin{align}\\dot{x}&amp;=x(\\simplify[std]{{a3}-{b3}*y^2}),\\\\\\dot{y}&amp;=y(\\simplify[std]{{c3}-{d3}*x^2}).\\end{align}\\]</p>\n  <p>To begin with, write the system in the general form</p>\n  <p>\\[\\begin{align}\\dot{x}&amp;=x(a-by^2),\\\\\\dot{y}&amp;=y(c-dx^2).\\end{align}\\]</p>\n  <p>It is clear that $\\boldsymbol{x}_0=\\pmatrix{0,0}^\\mathsf{T}$ is always a fixed point.</p>\n  <p>For $\\dot{x}=0$, then $x\\ne 0\\implies y=\\pm\\sqrt{\\frac{a}{b}}$, which is possible only if $\\frac{a}{b}&gt;0$.</p>\n  <p>Assuming this is true, then $\\dot{y}=0$ requires $x=\\pm\\sqrt{\\frac{c}{d}}$, which is possible only if $\\frac{c}{d}&gt;0$.</p>\n  <p>If this is also true, then there are four additional fixed points $\\boldsymbol{x}_0=\\pmatrix{\\pm\\sqrt{\\frac{c}{d}},\\pm\\sqrt{\\frac{a}{b}}}^\\mathsf{T}$.</p>\n  <p>In this case $\\frac{a}{b}=\\simplify{{a3}/{b3}}$ and $\\frac{c}{d}=\\simplify{{c3}/{d3}}$, {fixedpoints}.</p>\n  <p>The answers in the boxes should therefore be $\\boldsymbol{x}_0=\\pmatrix{0,0}^\\mathsf{T}$ and $\\boldsymbol{x}_0=\\pmatrix{\\pm\\var{ans3a},\\pm\\var{ans3b}}^\\mathsf{T}${todp}</p>"}, {"name": "Find the fixed points of a 1D dynamical system", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "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}], "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>"}, {"name": "Transform second order ODE into 2D and 3D dynamical systems", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a1"}}, "ungrouped_variables": ["a1", "c1", "b1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"answer": "y", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{-b1}*x-{a1}*y-{c1}*sin(t)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>By setting $y=\\dot{x}$, transform the differential equation into a 2-dimensional non-autonomous dynamical system of the form</p>\n        <p>\\[\\begin{align}\\dot{x}&amp;=f(x,y,t),\\\\\\dot{y}&amp;=g(x,y,t).\\end{align}\\]</p>\n        <p>$f(x,y,t)=$ [[0]].</p>\n        <p>$g(x,y,t)=$ [[1]].</p>", "marks": 0}, {"scripts": {}, "gaps": [{"answer": "y", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{-b1}*x-{a1}*y-{c1}*sin(z)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "1", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>By setting $y=\\dot{x}$ and $z=t$, transform the differential equation into a 3-dimensional autonomous dynamical system of the form</p>\n        <p>\\[\\begin{align}\\dot{x}&amp;=f(x,y,z),\\\\\\dot{y}&amp;=g(x,y,z),\\\\\\dot{z}&amp;=h(x,y,z).\\end{align}\\]</p>\n        <p>$f(x,y,z)=$ [[0]].</p>\n        <p>$g(x,y,z)=$ [[1]].</p>\n        <p>$h(x,y,z)=$ [[2]].</p>", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Consider the following second order differential equation:</p>\n  <p>\\[\\simplify[std]{ddot:x+{a1}*dot:x+{b1}*x+{c1}*sin(t)}=0,\\]</p>\n  <p>where $\\dot{x}$ denotes the derivative of $x$ with respect to $t$.</p>", "tags": ["checked2015", "MAS2106"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Transform a second order ODE into 2D non-autonomous and 3D autonomous dynamical systems of ODEs.</p>"}, "advice": "<p>a)</p>\n  <p>Let $y=\\dot{x}$, then $\\ddot{x}=\\dot{y}$, and the equation becomes</p>\n  <p>\\[\\simplify[std]{dot:y+{a1}*y+{b1}*x+{c1}*sin(t)}=0,\\]</p>\n  <p>and so</p>\n  <p>\\[\\begin{align}\\dot{x}&amp;=y=f(x,y,t),\\\\\\dot{y}&amp;=\\simplify[std]{{-b1}*x-{a1}*y-{c1}*sin(t)}=g(x,y,t).\\end{align}\\]</p>\n  <p></p>\n  <p>b)</p>\n  <p>Now also let $z=t$, and so $\\dot{z}=1$, then</p>\n  <p>\\[\\begin{align}\\dot{x}&amp;=y=f(x,y,z),\\\\\\dot{y}&amp;=\\simplify[std]{{-b1}*x-{a1}*y-{c1}*sin(z)}=g(x,y,z),\\\\\\dot{z}&amp;=1=h(x,y,z).\\end{align}\\]</p>"}, {"name": "Transform second order ODE into 2D and 3D dynamical systems", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c1"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a1"}, "n1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "description": "", "name": "n1"}}, "ungrouped_variables": ["a1", "n1", "c1", "b1"], "functions": {}, "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\n        <p>By setting $y=\\dot{x}$, transform the differential equation into a 2-dimensional non-autonomous dynamical system of the form</p>\n      <p>\\[\\begin{align}\\dot{x}&amp;=f(x,y,t),\\\\\\dot{y}&amp;=g(x,y,t).\\end{align}\\]</p>\n      <p>$f(x,y,t)=$ [[0]].</p>\n      <p>$g(x,y,t)=$ [[1]].</p>\n        ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "y", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": [], "failureRate": 1, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "jme", "unitTests": [], "showPreview": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"answer": "{-b1}*sin(x)-{a1}*y-{c1}*t^{n1}", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": [], "failureRate": 1, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "jme", "unitTests": [], "showPreview": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\n        <p>By setting $y=\\dot{x}$ and $z=t$, transform the differential equation into a 3-dimensional autonomous dynamical system of the form</p>\n      <p>\\[\\begin{align}\\dot{x}&amp;=f(x,y,z),\\\\\\dot{y}&amp;=g(x,y,z),\\\\\\dot{z}&amp;=h(x,y,z).\\end{align}\\]</p>\n      <p>$f(x,y,z)=$ [[0]].</p>\n      <p>$g(x,y,z)=$ [[1]].</p>\n      <p>$h(x,y,z)=$ [[2]].</p>\n        ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "y", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": [], "failureRate": 1, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "jme", "unitTests": [], "showPreview": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"answer": "{-b1}*sin(x)-{a1}*y-{c1}*z^{n1}", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": [], "failureRate": 1, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "jme", "unitTests": [], "showPreview": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"answer": "1", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": [], "failureRate": 1, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "jme", "unitTests": [], "showPreview": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Consider the following second order differential equation:</p>\n<p>\\[\\simplify[std]{ddot:x+{a1}*dot:x+{b1}*sin(x)+{c1}*t^{n1}}=0,\\]</p>\n<p>where $\\dot{x}$ denotes the derivative of $x$ with respect to $t$.</p>", "tags": ["checked2015"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Transform a second order ODE into 2D non-autonomous and 3D autonomous dynamical systems of ODEs.</p>"}, "advice": "<p>a)</p>\n  <p>Let $y=\\dot{x}$, then $\\ddot{x}=\\dot{y}$, and the equation becomes</p>\n  <p>\\[\\simplify[std]{dot:y+{a1}*y+{b1}*sin(x)+{c1}*t^{n1}}=0,\\]</p>\n  <p>and so</p>\n  <p>\\[\\begin{align}\\dot{x}&amp;=y=f(x,y,t),\\\\\\dot{y}&amp;=\\simplify[std]{{-b1}*sin(x)-{a1}*y-{c1}*t^{n1}}=g(x,y,t).\\end{align}\\]</p>\n  <p></p>\n  <p>b)</p>\n  <p>Now also let $z=t$, and so $\\dot{z}=1$, then</p>\n  <p>\\[\\begin{align}\\dot{x}&amp;=y=f(x,y,z),\\\\\\dot{y}&amp;=\\simplify[std]{{-b1}*sin(x)-{a1}*y-{c1}*z^{n1}}=g(x,y,z),\\\\\\dot{z}&amp;=1=h(x,y,z).\\end{align}\\]</p>"}, {"name": "Classify fixed points of a 2D dynamical system", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["checked2015"], "metadata": {"description": "<p>Nature of fixed points of a 2D dynamical system.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Consider the two-dimensional dynamical system</p>\n    <p>\\[\\begin{align}\\dot{x}&amp;=\\simplify[std]{{a}*x+{b}*y},\\\\\\dot{y}&amp;=\\simplify[std]{{c}*x+{d}*y}.\\end{align}\\]</p>", "advice": "<p>a)</p>\n<p>The elements of the matrix $\\mathsf{A}$ are the coefficients of $x$ and $y$ in the system of equations, hence</p>\n<p>\\[\\mathsf{A}=\\pmatrix{\\var{a} &amp; \\var{b}\\\\\\var{c} &amp; \\var{d}}.\\]</p>\n<p>&nbsp;</p>\n<p>b)</p>\n<p>The trace $\\tau$ and determinant $\\delta$ of a matrix</p>\n<p>\\[\\mathsf{A}=\\pmatrix{a &amp; b\\\\c &amp; d}\\]</p>\n<p>are given by $\\tau=a+d$ and $\\delta=ad-bc$.</p>\n<p>A straight forward substitution for $a=\\var{a}$, $b=\\var{b}$, $c=\\var{c}$, and $d=\\var{d}$ reveals that $\\tau=\\var{tau}$ and $\\delta=\\var{delta}$.</p>\n<p>&nbsp;</p>\n<p>c)</p>\n<p>The eigenvalues $\\lambda$ of a matrix $\\mathsf{A}$ can be calculated from the expression</p>\n<p>\\[\\mathsf{A}\\boldsymbol{x}=\\lambda\\boldsymbol{x},\\]</p>\n<p>which is equivalent to</p>\n<p>\\[\\left(\\mathsf{A}-\\lambda\\mathsf{I}\\right)\\boldsymbol{x}=\\boldsymbol{0},\\]</p>\n<p>where $\\mathsf{I}$ is the identity matrix.</p>\n<p>This linear system has a solution only when</p>\n<p>\\[\\det\\left(\\mathsf{A}-\\lambda\\mathsf{I}\\right)=0,\\]</p>\n<p>i.e. when $(a-\\lambda)(d-\\lambda)-bc=0$. &nbsp;This leads to $\\lambda^2-(a+d)\\lambda+(ad-bc)=0$, which is equivalent to</p>\n<p>\\[\\lambda^2-\\tau\\lambda+\\delta=0.\\]</p>\n<p>The eigenvalues $\\lambda_1$ and $\\lambda_2$ of the matrix $\\mathsf{A}$ are therefore given by</p>\n<p>\\[\\lambda_{1,2}=\\frac{\\tau\\pm\\sqrt{\\tau^2-4\\delta}}{2}.\\]</p>\n<p>Following through this calculation for the specific values of $a$, $b$, $c$, and $d$ in this question leads to $\\lambda_1=\\var{lam1}$ and $\\lambda_2=\\var{lam2}$.</p>\n<p>&nbsp;</p>\n<p>d)</p>\n<p>The eigenvectors corresponding to the eigenvalues $\\lambda_1$ and $\\lambda_2$ can be found by solving the linear system of equations given by</p>\n<p>\\[\\mathsf{A}\\boldsymbol{x}=\\lambda\\boldsymbol{x},\\]</p>\n<p>or</p>\n<p>\\[\\begin{align}ax+by&amp;=\\lambda x,\\\\cx+dy&amp;=\\lambda y,\\end{align}\\]</p>\n<p>where $\\lambda$ is either $\\lambda_1$ or $\\lambda_2$.</p>\n<p>Because this is an eigenvalue problem, we can arbitrarily choose one of the eigenvector components to be $1$, the $x$-component in this case. &nbsp;The $y$-component can then be found from</p>\n<p>\\[y=\\frac{x(\\lambda-a)}{b}\\quad\\text{or}\\quad y=\\frac{cx}{\\lambda-d},\\]</p>\n<p>with $x=1$. &nbsp;Both expressions are equivalent, and will lead to the same value for $\\lambda$.</p>\n<p>Making the necessary substitutions reveals that the $y$-component of the eigenvector corresponding to $\\lambda_1$ is $\\var{vec1}$, and the $y$-component of the eigenvector corresponding to $\\lambda_2$ is $\\var{vec2}$.</p>\n<p>&nbsp;</p>\n<p>e)</p>\n<p>The nature of the fixed point at the origin can be determined by examining the eigenvalues. &nbsp;If the eigenvalues are real, then the sign of each eigenvalue determines the nature of the fixed point. &nbsp;If the eigenvalues are complex, then the sign of the real part of each eigenvalue determines the nature of the fixed point.</p>\n<p>In this case, {naturetext}.</p>", "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "variables": {"natures": {"name": "natures", "group": "Ungrouped variables", "definition": "shuffle(['unstable node','stable node','saddle point','unstable spiral','stable spiral','centre'] except nature)", "description": "", "templateType": "anything"}, "vec2": {"name": "vec2", "group": "Ungrouped variables", "definition": "(lam2-a)/b", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "sign(b)*((tau-d)*d-delta)", "description": "", "templateType": "anything"}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "random(2..18#2 except abs(tau))", "description": "", "templateType": "anything"}, "delta": {"name": "delta", "group": "Ungrouped variables", "definition": "0.25*(tau^2+sign(random(-1,1))*r^2)", "description": "", "templateType": "anything"}, "tau": {"name": "tau", "group": "Ungrouped variables", "definition": "random([0]+repeat(2*random(0..9)*sign(random(-1,1)),5))", "description": "", "templateType": "anything"}, "lam2": {"name": "lam2", "group": "Ungrouped variables", "definition": "(tau-sqrt(disc))/2", "description": "", "templateType": "anything"}, "nature": {"name": "nature", "group": "Ungrouped variables", "definition": "if (im(lam1)=0 and im(lam2)=0,\n      switch (\n    lam1>0 and lam2>0, 'unstable node',\n    lam1<0 and lam2<0, 'stable node',\n    lam1>0 and lam2<0, 'saddle point'\n      ),\n      switch (\n    re(lam1)>0, 'unstable spiral',\n    re(lam1)<0, 'stable spiral',\n    re(lam1)=0, 'centre'\n      )\n    )", "description": "", "templateType": "anything"}, "lam1": {"name": "lam1", "group": "Ungrouped variables", "definition": "(tau+sqrt(disc))/2", "description": "", "templateType": "anything"}, "disc": {"name": "disc", "group": "Ungrouped variables", "definition": "tau^2-4*delta", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "tau-d", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..15 except abs(tau))*sign(random(-1,1))", "description": "", "templateType": "anything"}, "vec1": {"name": "vec1", "group": "Ungrouped variables", "definition": "(lam1-a)/b", "description": "", "templateType": "anything"}, "naturetext": {"name": "naturetext", "group": "Ungrouped variables", "definition": "switch (\n      nature='unstable node', \"because $\\\\lambda_1>0$ and $\\\\lambda_2>0$, the fixed point is an unstable node\",\n      nature='stable node', \"because $\\\\lambda_1<0$ and $\\\\lambda_2<0$, the fixed point is a stable node\",\n      nature='saddle point', \"because $\\\\lambda_1>0$ and $\\\\lambda_2<0$, the fixed point is a saddle point\",\n      nature='unstable spiral', \"because $\\\\mathrm{Im}(\\\\lambda_1)\\\\ne 0$ and $\\\\mathrm{Im}(\\\\lambda_2)\\\\ne 0$, and $\\\\mathrm{Re}(\\\\lambda_1)=\\\\mathrm{Re}(\\\\lambda_2)>0$, the fixed point is an unstable spiral\",\n      nature='stable spiral', \"because $\\\\mathrm{Im}(\\\\lambda_1)\\\\ne 0$ and $\\\\mathrm{Im}(\\\\lambda_2)\\\\ne 0$, and $\\\\mathrm{Re}(\\\\lambda_1)=\\\\mathrm{Re}(\\\\lambda_2)<0$, the fixed point is a stable spiral\",\n      nature='centre', \"because $\\\\mathrm{Im}(\\\\lambda_1)\\\\ne 0$ and $\\\\mathrm{Im}(\\\\lambda_2)\\\\ne 0$, and $\\\\mathrm{Re}(\\\\lambda_1)=\\\\mathrm{Re}(\\\\lambda_2)=0$, the fixed point is a centre\",\n      false\n    )", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["a", "tau", "c", "b", "d", "nature", "vec1", "lam2", "lam1", "vec2", "naturetext", "delta", "r", "disc", "natures"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The system can be written in the form $\\dot{\\boldsymbol{x}}=\\mathsf{A}\\boldsymbol{x}$, where $\\boldsymbol{x}=\\pmatrix{x,y}^\\mathsf{T}$.</p>\n<p>Identify the components of the matrix $\\mathsf{A}$.</p>\n<p>$\\mathsf{A}=$ [[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([[a,b],[c,d]])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the trace $\\tau$, and the determinant $\\delta$ of the matrix $\\mathsf{A}$.</p>\n        <p>Trace $\\tau=$ [[0]].</p>\n        <p>Determinant $\\delta=$ [[1]].</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Trace $\\tau$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "tau", "maxValue": "tau", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Determinant $\\delta$", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "delta", "maxValue": "delta", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": "0", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the eigenvalues $\\lambda_1$ and $\\lambda_2$ of the matrix $\\mathsf{A}$.</p>\n<p>If the eigenvalues have <strong>zero imaginary part</strong>, enter the eigenvalue with the <strong>largest real part</strong> first. &nbsp;If the eigenvalues have <strong>non-zero imaginary part</strong>, enter the eigenvalue with the <strong>largest imaginary part</strong> first.</p>\n<p>$\\lambda_1=$ [[0]]</p>\n<p>$\\lambda_2=$ [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "$\\lambda_1$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{lam1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "$\\lambda_2$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{lam2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Assuming that the $x$-component of each eigenvector is $1$, find the $y$-components of the eigenvectors corresponding to the eigenvalues $\\lambda_1$ and $\\lambda_2$.</p>\n        <p>$y$-component of eigenvector corresponding to $\\lambda_1$: [[0]].</p>\n        <p>$y$-component of eigenvector corresponding to $\\lambda_2$: [[1]].</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Corresponding to $\\lambda_1$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{vec1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "Corresponding to $\\lambda_2$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{vec2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Determine the nature of the fixed point at the origin.</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 1, "showCellAnswerState": true, "choices": ["<p>{nature}</p>", "<p>{natures[0]}</p>", "<p>{natures[1]}</p>", "<p>{natures[2]}</p>", "<p>{natures[3]}</p>", "<p>{natures[4]}</p>"], "matrix": [1, 0, 0, 0, 0, 0], "distractors": ["", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}], "name": "", "pickQuestions": 0}], "name": "Chaos theory", "showQuestionGroupNames": false, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Questions used in a university course titled \"Chaos theory\""}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "extensions": [], "custom_part_types": [], "resources": []}