// Numbas version: finer_feedback_settings
{"name": "Binome & Verwandtes", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Binome & Verwandtes", "tags": [], "metadata": {"description": "<p>Ein paar Beispiele zum Arbeiten mit mathematischen Ausdr&uuml;cken</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Binome und Verwandtes</strong></p>\n<p>Verwenden Sie <code>*</code> als Multiplikationszeichen und <code>^</code> f&uuml;r Exponenten.</p>", "advice": "<p><strong>L&ouml;sung zu a):</strong></p>\n<p>$\\left(\\var{a}\\cdot x+\\var{b}\\right)^2=\\left(\\var{a}\\cdot x+\\var{b}\\right)\\cdot\\left(\\var{a}\\cdot x+\\var{b}\\right)$</p>\n<p>$\\quad=\\var{a}\\cdot x\\cdot\\var{a}\\cdot x+\\var{a}\\cdot x\\cdot\\var{b}+\\var{b}\\cdot\\var{a}\\cdot x+\\var{b}\\cdot\\var{b}=\\var{a^2}x^2+\\var{2*a*b}\\cdot x+\\var{b^2}$</p>\n<p><strong>L&ouml;sung zu b):</strong></p>\n<p>$\\left(\\simplify{{a2}*y+{b2}}\\right)^2=\\left((\\var{a2})\\cdot y \\var{if(b2&gt;0,latex('+{b2}'),b2)}\\right)\\cdot\\left((\\var{a2})\\cdot y\\var{if(b2&gt;0,latex('+{b2}'),b2)}\\right)$</p>\n<p>$\\quad=(\\var{a2})\\cdot y\\cdot(\\var{a2})\\cdot y+(\\var{a2})\\cdot y\\cdot\\var{if(b2&gt;0,b2,latex('({b2})'))}+\\var{if(b2&gt;0,b2,latex('({b2})'))}\\cdot(\\var{a2})\\cdot y+\\var{if(b2&gt;0,b2,latex('({b2})'))}\\cdot\\var{if(b2&gt;0,b2,latex('({b2})'))}$</p>\n<p><strong>L&ouml;sung zu c):</strong></p>\n<p>$\\var{d^2}-\\var{c^2}\\cdot z^2=\\var{d}^2-\\left(\\var{c}\\cdot z\\right)^2$ wof&uuml;r wir die dritte binomische Formel $a^2-b^2=(a+b)\\cdot(a-b)$ mit $a=\\var{d},b=\\var{c}\\cdot z$ anwenden k&ouml;nnen und somit erhalten:</p>\n<p>$\\var{d}^2-\\left(\\var{c}\\cdot z\\right)^2=(\\var{d}+\\var{c}\\cdot z)\\cdot(\\var{d}-\\var{c}\\cdot z)$</p>\n<p><strong>L&ouml;sung zu d):</strong></p>\n<p>Wir ben&ouml;tigen die Summenregel $f(x)=g(x)+h(x) \\rightarrow f'(x)=g'(x)+h'(x)$ und die Potenzregel $f(x)=a\\cdot x^n\\rightarrow f'(x)=n\\cdot a\\cdot x^{n-1}$ und erhalten:</p>\n<p>$f'(x)=3\\cdot\\var[fractionNumbers]{b^2/3}x^2+2\\cdot\\var{a*b}x^2+\\var{a^2}=\\var{if(b^2=1,\"\",b^2)}x^2+\\var{2*a*b}x^2+\\var{a^2}=(\\var{if(b^2=1,\"\",b^2)} x+\\var{a})^2$</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 30px; width: 0px; top: 62px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 15px; width: 50px; top: 47px; height: 20px; font-size: 10px; color: black;\"></span></p>", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2,4,6)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1,3,5)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2,3,4)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1,5,6)", "description": "", "templateType": "anything"}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "-a", "description": "", "templateType": "anything"}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(-1,-3,-5,2,4,6)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "a2", "b2", "c", "d"], "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>Multiplizieren Sie aus!</p>\n<p>$\\left(\\var{a}\\cdot x+\\var{b}\\right)^2=$[[0]]</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 17px; width: 0px; top: 68px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 2px; width: 50px; top: 53px; height: 20px; font-size: 10px; color: black;\"></span></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "binom1", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}^2x^2+2*{a}*{b}*x+{b}^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "mustmatchpattern": {"pattern": "$n*$v^2+$n*$v+$n", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Multiplizieren Sie aus!</p>\n<p>$\\left(\\simplify{{a2}* y+{b2}}\\right)^2=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "binom2", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a2}^2*y^2+2*{a2}*{b2}*y+{b2}^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "mustmatchpattern": {"pattern": "$n* $v^2+`+-$n*$v+$n", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "y", "value": ""}]}], "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>Faktorisieren&nbsp;Sie!</p>\n<p>$\\var{d^2}-\\var{c^2}\\cdot z^2=$[[0]]</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 53px; width: 0px; top: 65px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 38px; width: 50px; top: 50px; height: 20px; font-size: 10px; color: black;\"></span></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "binom3", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({d}-{c}*z)*({d}+{c}*z)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "mustmatchpattern": {"pattern": "($n-$n*$v)*($n+$n*$v)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "z", "value": ""}]}], "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>Bestimmen Sie die erste Abletung von $f(x)=\\var[fractionNumbers]{b^2/3}x^3+\\var{a*b}x^2+\\var{a^2}\\cdot x$!</p>\n<p>$f'(x)=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "binom4", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({b}x+{a})^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "resources": []}]}], "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}