// Numbas version: exam_results_page_options
{"name": "Robert's copy of Functions of two variables: Stationary points 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["rawstatval", "statval", "sol", "lmin", "q1", "dvalue", "s3", "s2", "s1", "s5", "s4", "neither", "b", "c1", "a", "c", "p1", "d", "f", "lmax", "y", "x"], "name": "Robert's copy of Functions of two variables: Stationary points 1", "tags": ["Calculus", "Differentiation", "calculus", "derivative", "differentiation", "function of 2 variables", "functions of 2 variables", "functions of two variables", "partial derivatives", "partial differentiation", "stationary points", "stationary points of functions of two variables"], "advice": "<p>The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving<br/>the following 2 equations for $x$ and $y$:</p>\n<p>\\[\\begin{eqnarray*} \\partial f \\over \\partial x &amp;=&amp;0\\\\ \\\\ \\partial f \\over \\partial y &amp;=&amp;0 \\end{eqnarray*} \\]</p>\n<p>In this case you get two linear equations to solve for $x$ and $y$:</p>\n<p>\\[\\begin{eqnarray*} \\simplify[std]{{2*a}x+{b}y+{d}}&amp;=&amp;0\\\\ \\\\ \\simplify[std]{{b}x+{2*c1}y+{f}}&amp;=&amp;0 \\end{eqnarray*} \\]<br/>On solving these we get \\[ x = \\simplify[std]{{2*c1*d-b*f}/{b^2-4*a*c1}},\\;\\;\\;y=\\simplify[std]{{2*a*f-b*d}/{b^2-4*a*c1}}\\]<br/>On substituting these values into $f(x,y)$ we get:</p>\n<p>\\[f\\left(\\simplify[std]{{2*c1*d-b*f}/{b^2-4*a*c1}},\\simplify[std]{{2*a*f-b*d}/{b^2-4*a*c1}}\\right) = \\var{rawstatval}\\approx\\var{statval}\\]<br/>to 2 decimal places.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 0, "prompt": "<p>Input both cooordinates as fractions or integers and not decimals.</p>\n<p>$x$&ndash;coordinate, $a=$&nbsp;[[0]].</p>\n<p>$y$&ndash;coordinate, $b=$&nbsp;[[1]].</p>\n<p>Input value of $f(x,y)$ at $(a,b)$:</p>\n<p>$f(a,b)=\\;\\;$[[2]] (Input to 2 decimal places).</p>\n<p>If you want some help, click on <em>Show steps</em>. You will not lose any marks if you do so.</p>", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Input answer as a fraction or an integer, not a decimal</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{2*c1*d-b*f}/{b^2-4*a*c1}", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"notallowed": {"message": "<p>Input answer as a fraction or an integer, not a decimal</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{2*a*f-b*d}/{b^2-4*a*c1}", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"allowFractions": false, "scripts": {}, "maxValue": "statval+0.01", "minValue": "statval-0.01", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"type": "information", "showCorrectAnswer": true, "marks": 0, "prompt": "\n            \n            \n            <p>The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving<br/>the following 2 equations for $x$ and $y$</p>\n            \n            \n            \n            <p>\\[\\begin{eqnarray*}\n            \n            \\partial f \\over \\partial x &=&0\\\\\n            \n            \\\\\n            \n            \\partial f \\over \\partial y &=&0\n            \n            \\end{eqnarray*}\n            \n            \\]</p>\n            \n            \n            \n            <p>In this case you get two linear equations to solve for $x$ and $y$</p>\n            \n            \n            ", "scripts": {}}], "type": "gapfill"}], "statement": "<p>In the following question find the $(x,y)$ coordinates of the single stationary point $(a,b)$ of the function</p>\n<p>\\[f(x,y)=\\simplify[std]{{a}*x^2+{b}*x*y+{c1}*y^2+{d}*x+{f}*y}\\]</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"rawstatval": {"definition": "a*x^2+b*x*y+c1*y^2+d*x+f*y", "templateType": "anything", "group": "Ungrouped variables", "name": "rawstatval", "description": ""}, "statval": {"definition": "precround(rawstatval,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "statval", "description": ""}, "sol": {"definition": "switch(b=1,'by multiplying the second equation by '+2*a,b=-1,'by multiplying the second equation by '+2*a,'by multiplying the second equation by '+2*a+' and multiplying the first equation by '+b)", "templateType": "anything", "group": "Ungrouped variables", "name": "sol", "description": ""}, "lmin": {"definition": "'Local minimum'", "templateType": "anything", "group": "Ungrouped variables", "name": "lmin", "description": ""}, "q1": {"definition": "random(-3..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "q1", "description": ""}, "dvalue": {"definition": "4*a*c1-b^2", "templateType": "anything", "group": "Ungrouped variables", "name": "dvalue", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "s5": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "s4": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s4", "description": ""}, "neither": {"definition": "'Saddle point'", "templateType": "anything", "group": "Ungrouped variables", "name": "neither", "description": ""}, "p1": {"definition": "random(-3..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "p1", "description": ""}, "c1": {"definition": "if(b^2=4*a*c,c+1,c)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "a": {"definition": "s1*random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "s3*random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "s2*random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "s4*random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "f": {"definition": "s5*random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "lmax": {"definition": "'Local maximum'", "templateType": "anything", "group": "Ungrouped variables", "name": "lmax", "description": ""}, "y": {"definition": "(2*a*f-b*d)/(b^2-4*a*c1)", "templateType": "anything", "group": "Ungrouped variables", "name": "y", "description": ""}, "x": {"definition": "(2*c1*d-b*f)/(b^2-4*a*c1)", "templateType": "anything", "group": "Ungrouped variables", "name": "x", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>10/07/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p>Question appears to be working correctly.</p>\n        \t\t", "description": "<p>Find the stationary point $(p,q)$ of the function:&nbsp;$f(x,y)=ax^2+bxy+cy^2+dx+gy$. Calculate $f(p,q)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Robert Anderson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/809/"}]}]}], "contributors": [{"name": "Robert Anderson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/809/"}]}