// Numbas version: exam_results_page_options
{"name": "Quotient rule - differentiate quotient of linear terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quotient rule - differentiate quotient of linear terms", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Other method. Find $p,\\;q$ such that&nbsp;$\\displaystyle \\frac{ax+b}{cx+d}= p+ \\frac{q}{cx+d}$. Find the derivative of $\\displaystyle \\frac{ax+b}{cx+d}$.</p>"}, "extensions": [], "preamble": {"js": "", "css": ""}, "statement": "<p>Let \\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x+{d})}\\]</p>", "variable_groups": [], "functions": {}, "variables": {"b": {"name": "b", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "s1*random(1..9)"}, "a": {"name": "a", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(2..9)"}, "s1": {"name": "s1", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(1,-1)"}, "s2": {"name": "s2", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(1,-1)"}, "c1": {"name": "c1", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(1..8)"}, "c": {"name": "c", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "if(a*d=b*c1,c1+1,c1)"}, "det": {"name": "det", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "a*d-b*c"}, "d": {"name": "d", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "s2*random(1..9)"}}, "tags": [], "parts": [{"variableReplacements": [], "stepsPenalty": 1, "type": "gapfill", "useCustomName": false, "steps": [{"variableReplacements": [], "type": "information", "useCustomName": false, "prompt": "<p>$\\simplify[std]{{a}x+{b}=a*({c}x+{d})+b}$ for suitable numbers $a$ and $b$.</p>", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}], "prompt": "\n\t\t\t<p>Find numbers $a$ and $b$ such that<br/>\\[\\simplify[std]{f(x) = a + b/({c}x+{d})}\\]<br/>Enter a and b as integers or fractions, but not as decimals.</p>\n\t\t\t        <p>$a=\\;$[[0]]</p>\n\t\t\t        <p>$b=\\;$[[1]]</p>\n\t\t\t        <p>You can click on Show steps to get some help, but you will lose 1 mark if you do so.</p>\n\t\t\t", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "gaps": [{"variableReplacements": [], "notallowed": {"showStrings": false, "strings": ["."], "partialCredit": 0, "message": "<p>Input numbers as fractions or integers and not as decimals.</p>"}, "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "showCorrectAnswer": true, "showPreview": true, "answer": "{a}/{c}", "unitTests": [], "scripts": {}, "vsetRange": [0, 1], "marks": 1, "vsetRangePoints": 5, "valuegenerators": [], "type": "jme", "useCustomName": false, "customName": "", "failureRate": 1, "checkingAccuracy": 0.001, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "customMarkingAlgorithm": ""}, {"variableReplacements": [], "notallowed": {"showStrings": false, "strings": ["."], "partialCredit": 0, "message": "<p>Input numbers as fractions or integers and not as decimals.</p>"}, "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "showCorrectAnswer": true, "showPreview": true, "answer": "{-det}/{c}", "unitTests": [], "scripts": {}, "vsetRange": [0, 1], "marks": 1, "vsetRangePoints": 5, "valuegenerators": [], "type": "jme", "useCustomName": false, "customName": "", "failureRate": 1, "checkingAccuracy": 0.001, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "customMarkingAlgorithm": ""}], "showCorrectAnswer": true, "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}, {"variableReplacements": [], "gaps": [{"variableReplacements": [], "notallowed": {"showStrings": false, "strings": ["."], "partialCredit": 0, "message": "<p>Input numbers as fractions or integers and not as decimals.</p>"}, "answerSimplification": "std", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "showCorrectAnswer": true, "showPreview": true, "answer": "{-c}/({c}x+{d})^2", "unitTests": [], "scripts": {}, "vsetRange": [10, 11], "marks": 1, "vsetRangePoints": 5, "valuegenerators": [{"name": "x", "value": ""}], "type": "jme", "useCustomName": false, "customName": "", "failureRate": 1, "checkingAccuracy": 0.001, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "customMarkingAlgorithm": ""}, {"variableReplacements": [], "notallowed": {"showStrings": false, "strings": ["."], "partialCredit": 0, "message": "<p>Input numbers as fractions or integers and not as decimals.</p>"}, "answerSimplification": "std", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "showCorrectAnswer": true, "showPreview": true, "answer": "{det}/({c}x+{d})^2", "unitTests": [], "scripts": {}, "vsetRange": [10, 11], "marks": 1, "vsetRangePoints": 5, "valuegenerators": [{"name": "x", "value": ""}], "type": "jme", "useCustomName": false, "customName": "", "failureRate": 1, "checkingAccuracy": 0.001, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "customMarkingAlgorithm": ""}], "type": "gapfill", "useCustomName": false, "prompt": "\n\t\t\t<p>Differentiate<br/>\\[\\simplify[std]{g(x) = 1/({c}x+{d})}\\]</p>\n\t\t\t        <p>$\\displaystyle \\frac{dg}{dx}=\\;$[[0]]</p>\n\t\t\t        <p>Hence using the first part of the question differentiate \\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x+{d})}\\]</p>\n\t\t\t        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[1]]</p>\n\t\t\t        <p>Input numbers as fractions or integers and not as decimals.</p>\n\t\t\t", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t<p><strong>a)</strong></p>\n\t    <p>We have $\\displaystyle \\simplify[std]{{a}x+{b}={a}/{c}*({c}x+{d})+{b}-{a}*{d}/{c}={a}/{c}*({c}x+{d})+{-det}/{c}}$<br/>Hence \\[\\begin{eqnarray*} \\simplify[std]{({a} * x+{b})/({c}x+{d})}&amp;=&amp;\\simplify[std]{({a}/{c}*({c}x+{d})+{-det}/{c})/({c}x+{d})}\\\\ &amp;=&amp;\\simplify[std]{{a}/{c}+({-det}/{c})/({c}x+{d})} \\end{eqnarray*}\\]<br/>Where we have divided out by $\\simplify[std]{{c}x+{d}}$ at the last step.</p>\n\t    <p><strong>b)</strong></p>\n\t    <p>We have \\[\\frac{dg}{dx} = \\simplify[std]{{-c}/({c}x+{d})^2}\\]<br/>using standard rules of differentiation.<br/>Since from <strong>a)</strong>,&nbsp;\\[f(x) = \\simplify[std]{{a}/{c}+({-det}/{c})/({c}x+{d})}\\]<br/>&nbsp;we see that<br/>\\[\\begin{eqnarray*}\\frac{df}{dx} &amp;=&amp;\\simplify[std,!unitPower,!unitDenominator,!zeroFactor,!zeroTerm,!zeroPower]{(-{c})*(({-det}/{c})/({c}x+{d})^2)}\\\\ &amp;=&amp;\\simplify[std]{{det}/({c}x+{d})^2} \\end{eqnarray*}\\]</p>\n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "det", "c1"], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}