// 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": [{"variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..8)", "name": "c1", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "name": "d", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a*d=b*c1,c1+1,c1)", "name": "c", "description": ""}, "det": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*d-b*c", "name": "det", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "name": "b", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s2", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "det", "c1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "name": "Quotient rule - differentiate quotient of linear terms", "showQuestionGroupNames": false, "variable_groups": [], "functions": {}, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "{a}/{c}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{-det}/{c}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "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", "steps": [{"type": "information", "prompt": "<p>$\\simplify[std]{{a}x+{b}=a*({c}x+{d})+b}$ for suitable numbers $a$ and $b$.</p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{-c}/({c}x+{d})^2", "showCorrectAnswer": true, "vsetrange": [10, 11], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}, {"answer": "{det}/({c}x+{d})^2", "showCorrectAnswer": true, "vsetrange": [10, 11], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "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", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Let \\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x+{d})}\\]</p>", "tags": ["algebraic manipulation", "Calculus", "checked2015", "derivatives", "derivatives ", "deriving a quotient", "differentiate a quotient", "differentiation", "dividing linear polynomials", "MAS1601"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t<p><strong>1/08/2012:</strong></p>\n\t\t<p>Added tags.</p>\n\t\t<p>Checked calculation. OK.</p>\n\t\t<p>Added description.</p>\n\t\t<p>All round improvement in display.</p>\n\t\t<p>Added &nbsp;forbidden instructions on using decimals.</p>\n\t\t<p>Added information on losing 1 mark if use Show steps in part a).</p>\n\t\t", "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>"}, "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", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}