// Numbas version: exam_results_page_options
{"name": "Productiefunctie 1 variabele", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "parts": [{"sortAnswers": false, "unitTests": [], "showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "gaps": [{"checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "scripts": {}, "vsetRangePoints": 5, "variableReplacements": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "showPreview": true, "valuegenerators": [{"value": "", "name": "x"}], "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "customName": "Cx", "adaptiveMarkingPenalty": 0, "marks": "0.5", "checkVariableNames": false, "useCustomName": true, "mustmatchpattern": {"pattern": "$n * x + $n", "nameToCompare": "", "message": "", "partialCredit": 0}, "unitTests": [], "answer": "{VC}*x+{FC}"}], "marks": 0, "useCustomName": false, "prompt": "<p>Bepaal de totale kosten in termen van het aantal uur arbeid \\( x \\):</p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp; &nbsp;&nbsp; [[0]]</p>", "customName": "", "adaptiveMarkingPenalty": 0, "variableReplacements": [], "customMarkingAlgorithm": ""}, {"sortAnswers": false, "unitTests": [], "showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "gaps": [{"checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all, fractionNumbers", "scripts": {}, "vsetRangePoints": 5, "variableReplacements": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "showPreview": true, "valuegenerators": [{"value": "", "name": "q"}], "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "customName": "xifvq", "adaptiveMarkingPenalty": 0, "marks": "0.5", "checkVariableNames": false, "useCustomName": true, "unitTests": [], "answer": "{b} + q^{4/3}"}], "marks": 0, "useCustomName": false, "prompt": "<p>Bereken \\( x \\) in termen van \\( q \\) uit de productiefunctie:</p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp; \\( x = \\)&nbsp; [[0]]</p>", "customName": "", "adaptiveMarkingPenalty": 0, "variableReplacements": [], "customMarkingAlgorithm": ""}, {"sortAnswers": false, "unitTests": [], "showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "gaps": [{"checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all,fractionNumbers", "scripts": {}, "vsetRangePoints": 5, "variableReplacements": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "showPreview": true, "valuegenerators": [{"value": "", "name": "q"}], "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "customName": "Cifvq", "adaptiveMarkingPenalty": 0, "marks": "0.5", "checkVariableNames": false, "useCustomName": true, "mustmatchpattern": {"pattern": "$n * q^(4/3) + rational:$n", "nameToCompare": "", "message": "", "partialCredit": 0}, "unitTests": [], "answer": "{VC}*q^{4/3}+{VC*b+FC}"}], "marks": 0, "useCustomName": false, "prompt": "<p>Bepaal nu de totale kosten in functie van \\( q \\):</p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp; \\( C(q) = \\) [[0]]</p>", "customName": "", "adaptiveMarkingPenalty": 0, "variableReplacements": [], "customMarkingAlgorithm": ""}, {"sortAnswers": false, "unitTests": [], "showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "gaps": [{"checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all,FractionNumbers", "scripts": {}, "vsetRangePoints": 5, "variableReplacements": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "showPreview": true, "valuegenerators": [{"value": "", "name": "q"}], "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "customName": "ACq", "adaptiveMarkingPenalty": 0, "marks": "0.5", "checkVariableNames": false, "useCustomName": true, "mustmatchpattern": {"pattern": "$n * q^(1/3) + rational:$n / q `| $n * q^(1/3) + rational:$n * q^(-1)", "nameToCompare": "", "message": "", "partialCredit": 0}, "unitTests": [], "answer": "{VC}*q^{1/3}+{VC*b+FC}/q"}], "marks": 0, "useCustomName": false, "prompt": "<p>De gemiddelde kostenfunctie wordt gegeven door</p>\n<p>&nbsp; &nbsp; &nbsp; [[0]]</p>", "customName": "", "adaptiveMarkingPenalty": 0, "variableReplacements": [], "customMarkingAlgorithm": ""}, {"sortAnswers": false, "unitTests": [], "showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "gaps": [{"checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all,fractionNumbers", "scripts": {}, "vsetRangePoints": 5, "variableReplacements": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "showPreview": true, "valuegenerators": [{"value": "", "name": "q"}], "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "customName": "afgAC", "adaptiveMarkingPenalty": 0, "marks": 1, "checkVariableNames": false, "useCustomName": true, "mustmatchpattern": {"pattern": "rational:$n * q^(-2/3) - rational:$n / q^2 `| rational:$n * q^(-2/3) - rational:$n * q^(-2) `| rational:$n / (q^(2/3)) - rational:$n / q^2 `| rational:$n / (q^(2/3)) - rational:$n * q^(-2)", "nameToCompare": "", "message": "", "partialCredit": 0}, "unitTests": [], "answer": "{1/3*VC}*q^{-2/3}-{VC*b+FC}/q^2"}], "marks": 0, "useCustomName": false, "prompt": "<p>Bepaal nu de eerste orde afgeleide van de gemiddelde kostenfunctie naar \\( q \\):</p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp; \\( \\frac {dAC} {dq} (q) = \\) [[0]]</p>", "customName": "", "adaptiveMarkingPenalty": 0, "variableReplacements": [], "customMarkingAlgorithm": ""}, {"sortAnswers": false, "unitTests": [], "showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "gaps": [{"checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "scripts": {}, "vsetRangePoints": 5, "variableReplacements": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "showPreview": true, "valuegenerators": [], "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "customName": "nulpafgAC", "adaptiveMarkingPenalty": 0, "marks": 1, "checkVariableNames": false, "useCustomName": true, "mustmatchpattern": {"pattern": "$n", "nameToCompare": "", "message": "", "partialCredit": 0}, "unitTests": [], "answer": "27*{mu}^3"}], "marks": 0, "useCustomName": false, "prompt": "<p>Deze afgeleide van \\( AC \\) is 0 voor</p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp; \\( q = \\)&nbsp; [[0]]</p>", "customName": "", "adaptiveMarkingPenalty": 0, "variableReplacements": [], "customMarkingAlgorithm": ""}, {"sortAnswers": false, "unitTests": [], "showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "gaps": [{"checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "scripts": {}, "vsetRangePoints": 5, "variableReplacements": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "showPreview": true, "valuegenerators": [], "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "customName": "pcomp", "adaptiveMarkingPenalty": 0, "marks": 1, "checkVariableNames": false, "useCustomName": true, "mustmatchpattern": {"pattern": "$n", "nameToCompare": "", "message": "", "partialCredit": 0}, "unitTests": [], "answer": "{8*mu*z}"}], "marks": 0, "useCustomName": false, "prompt": "<p>Controleer of je hierdoor inderdaad een minimum van de gemiddelde kosten bekomt.</p>\n<p>De competitieve prijs is gelijk aan [[0]].</p>", "customName": "", "adaptiveMarkingPenalty": 0, "variableReplacements": [], "customMarkingAlgorithm": ""}], "advice": "<p>De gevraagde competitieve prijs wordt gevonden als het minimum van de gemiddelde kosten: \\( p_{comp}&nbsp; = min AC \\).</p>\n<p>De totale kostprijs, uitgedrukt in termen van \\( x \\) , wordt gegeven door \\[&nbsp; \\var{VC} \\cdot x+\\var{FC}\\]</p>\n<p>Uit \\( q= (x- \\var{b} )^{3/4} \\) volgt dat \\[&nbsp; x= \\var{b} +q^{4/3} . \\]</p>\n<p>Bijgevolg is de totale kostprijs, uitgedrukt in termen van \\( q \\) , gelijk aan \\[ C(q) = \\var{VC} \\cdot (\\var{b} +q^{4/3})+\\var{FC} ,\\]<br/>d.w.z. \\[ C(q) = \\var{VC} \\cdot q^{4/3}+\\var{b*VC+FC} .\\]</p>\n<p>Hieruit bereken je de gemiddelde kostprijs:<br/> \\[ AC(q) = \\frac{C(q)} {q} = \\var{VC} \\cdot q^{1/3} + \\frac{\\var{b*VC+FC}}{q} ,\\]<br/>met als afgeleide<br/>\\[&nbsp; \\frac{dAC} {dq} (q) = \\simplify[all,fractionNumbers]{{1/3*VC}} \\cdot q^{-2/3} - \\frac{\\var{b*VC+FC}}{q^2} .\\]</p>\n<p>Deze laatste is gelijk aan 0 als \\[&nbsp;&nbsp; \\simplify[all,fractionNumbers]{{1/3*VC}} \\cdot q^{-2/3} = \\frac{\\var{b*VC+FC}}{q^2} ,\\]<br/>d.w.z. als<br/>\\[&nbsp;&nbsp; q^{4/3} = \\frac{\\var{b*VC+FC}}{\\simplify[all,fractionNumbers]{{1/3*VC}}} =&nbsp;\\simplify[all,fractionNumbers]{{(b*VC+FC)/(1/3*VC)}} ,\\] <br/>wat als oplossing <br/>\\[&nbsp;&nbsp; q_{comp} = \\var{27*mu^3} \\] <br/>geeft.</p>\n<p>Voor deze q-waarde geldt dat<br/>\\[&nbsp; \\frac{d^2{AC}} {d^2 q} (\\var{27*mu^3}) &gt; 0,\\]<br/>zodat er inderdaad een minimum van \\( AC \\) wordt bereikt.</p>\n<p>De competitieve prijs wordt dan ook gegeven door<br/>\\[&nbsp; p_{comp}&nbsp; = AC(\\var{27*mu^3}) = \\var{8*mu*z} .\\]</p>", "ungrouped_variables": ["mu", "w", "b", "z", "FC", "VC"], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "Calculate the competitive price as the minimum of the average cost, given a production function in one variable for a situation of perfect competition.<br>"}, "variables": {"b": {"description": "", "definition": "22*w", "group": "Ungrouped variables", "name": "b", "templateType": "anything"}, "VC": {"description": "", "definition": "2*z", "group": "Ungrouped variables", "name": "VC", "templateType": "anything"}, "mu": {"description": "<p>mu</p>", "definition": "random(1..6)", "group": "Ungrouped variables", "name": "mu", "templateType": "anything"}, "FC": {"description": "", "definition": "10*z", "group": "Ungrouped variables", "name": "FC", "templateType": "anything"}, "w": {"description": "", "definition": "1/22*(27*mu^4-5)", "group": "Ungrouped variables", "name": "w", "templateType": "anything"}, "z": {"description": "", "definition": "random(1..5)", "group": "Ungrouped variables", "name": "z", "templateType": "anything"}}, "preamble": {"js": "", "css": ""}, "name": "Productiefunctie 1 variabele", "statement": "<p>Gebruik steeds gehele getallen of breuken in jouw antwoorden, maar geen decimalen!</p>\n<h3>Een onderneming opereert op een markt van volledige mededinging en heeft als productiefunctie \\[&nbsp; q= (x- \\var{b} )^{3/4} \\] waarbij \\( x \\) het aantal uren arbeid voorstelt.<br/>De kosten per uur arbeid bedragen &euro; {VC}, en er zijn &euro; {FC} vaste kosten.<br/>Bepaal de competitieve prijs voor deze onderneming.</h3>", "functions": {}, "tags": [], "rulesets": {}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "contributors": [{"name": "Paul Verheyen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3610/"}, {"name": "Dirk Janssens", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3705/"}]}]}], "contributors": [{"name": "Paul Verheyen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3610/"}, {"name": "Dirk Janssens", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3705/"}]}