// Numbas version: exam_results_page_options
{"name": "Differenzieren wie Isaac Barrow", "extensions": [], "custom_part_types": [], "resources": [["question-resources/barrow01.png", "/srv/numbas/media/question-resources/barrow01.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differenzieren wie Isaac Barrow", "tags": [], "metadata": {"description": "<p>Fr&uuml;he Ableitungsregeln nutzen</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>In seinen \"Lectiones Geometricae\" geht Isaac Barrow davon aus, dass man sich Kurven (vgl. Abb. unten) aus unendlich kleinen Geradenstr&uuml;cken (\"linelets'') zusammengesetzt vorstellen kann.</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/barrow01.png\"/></p>\n<p style=\"text-align: left;\">Sei $MN$ so ein unendlich kleines Geradenst&uuml;ck ($M$ und $N$ also infinitesimal benachbarte Punkte) einer Kurve mit der Gleichung $f(x,y)=0$, so darf man nach Barrow folgern, dass gilt:<br/> <br/>$f(x+e,y+a)=f(x,y)=0$<br/> <br/> und dass alle auftretenden $e^m,a^n$ mit $m,n&gt;1$ zu Null werden.<br/> <br/> Ermitteln Sie mit diesem Ansatz die Tangentensteigung $\\frac{a}{e}$ f&uuml;r die folgende Kreisgleichung, d. h. f&uuml;r die implizite Funktion $f(x,y)=x^2+y^2-\\var{c}^2=0$</p>\n<p style=\"text-align: center;\"></p>", "advice": "<p>Bitte einfach die korrekten Antworten der L&uuml;cken ansehen.</p>", "rulesets": {}, "extensions": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c"], "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>Es folgt zun&auml;chst:</p>\n<p>[[0]]$=x^2+y^2-\\var{c}^2=0$</p>\n<p>Durch Klammeraufl&ouml;sen und Unterdr&uuml;cken aller &uuml;bereinstimmenden Terme erhalten wir daraus:</p>\n<p>[[1]]$+a^2=0$</p>\n<p>Durch Unterdr&uuml;cken aller&nbsp;$e^m,a^n$ mit $m,n&gt;1$ bleibt schlie&szlig;lich:</p>\n<p>[[2]]$=0$, was uns zur Ableitung $\\frac{a}{e}=$[[3]] f&uuml;hrt.</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Term1", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(x+e)^2+(y+a)^2-{c}^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "a", "value": ""}, {"name": "x", "value": ""}, {"name": "y", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Term2", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "2*e*x+e^2+2*a*y", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "a", "value": ""}, {"name": "x", "value": ""}, {"name": "y", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Term3", "marks": "1/2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "2*e*x+2*a*y", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "a", "value": ""}, {"name": "x", "value": ""}, {"name": "y", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Term4", "marks": "3/2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-x/y", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}]}], "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}