// Numbas version: finer_feedback_settings
{"name": "Simon's copy of SFY0004 Implicit 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "sortAnswers": false, "stepsPenalty": 0, "marks": 0, "unitTests": [], "prompt": "<p>Using implicit differentiation find $\\displaystyle \\frac{dy}{dx}$&nbsp;in terms of $x$ and $y$.</p>\n<p>Input your answer here:</p>\n<p>$\\displaystyle \\frac{dy}{dx}= $ [[0]]</p>\n<p><strong>Input all numbers as integers not as decimals.</strong></p>\n<p>If you want more help click on Show steps - you will not lose any marks.</p>", "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "\n          <p>Note that we regard $y$ as a function of $x$.</p>\n          <p>Hence we have (using the chain rule):</p>\n          <p>$\\displaystyle \\frac{d(y^2)}{dx} = 2y\\frac{dy}{dx}$</p>\n          <p>And , using the product rule:</p>\n          <p>$\\displaystyle \\frac{d(xy)}{dx} = y+x\\frac{dy}{dx}$.</p>\n          <p>&nbsp;Now differentiate both sides of the relation with respect to $x$.</p>\n          ", "type": "information", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "marks": 0, "extendBaseMarkingAlgorithm": true}], "gaps": [{"notallowed": {"strings": ["."], "message": "<p>Input all numbers as integers or as fractions, not as decimals.</p>", "partialCredit": 0, "showStrings": false}, "checkingType": "absdiff", "showPreview": true, "failureRate": 1, "answerSimplification": "all,!collectNumbers", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "vsetRangePoints": 5, "vsetRange": [0, 1], "marks": 2, "unitTests": [], "checkingAccuracy": 0.001, "checkVariableNames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answer": "(({( - a)} + ( - (2 * x))-{d}y) / ({b} + (2 * y)+{d}x))", "variableReplacements": [], "expectedVariableNames": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true}], "variableReplacements": [], "customMarkingAlgorithm": "", "type": "gapfill"}, {"extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "sortAnswers": false, "marks": 0, "unitTests": [], "prompt": "\n      <p>Find the two points $(0,a),\\;\\;(0,b),\\;\\;a \\lt b$ which lie on the curve given by the relation.</p>\n        <p>$a=\\;$[[0]]</p>\n        <p>$b=\\;$[[1]]</p>\n        <p>(remember that $a \\lt b$).</p>\n      ", "variableReplacementStrategy": "originalfirst", "gaps": [{"minValue": "-c", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "marks": 1, "unitTests": [], "mustBeReduced": false, "mustBeReducedPC": 0, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "maxValue": "-c", "correctAnswerFraction": false, "variableReplacements": [], "customMarkingAlgorithm": "", "allowFractions": false, "extendBaseMarkingAlgorithm": true}, {"minValue": "1", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "marks": 1, "unitTests": [], "mustBeReduced": false, "mustBeReducedPC": 0, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "maxValue": "1", "correctAnswerFraction": false, "variableReplacements": [], "customMarkingAlgorithm": "", "allowFractions": false, "extendBaseMarkingAlgorithm": true}], "variableReplacements": [], "customMarkingAlgorithm": "", "type": "gapfill"}, {"extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "sortAnswers": false, "marks": 0, "unitTests": [], "prompt": "\n      <p>Hence find the equations of the tangents at the points $(0,a)$ and $(0,b)$.</p>\n      <p>Input all numbers as integers or as fractions, not as decimals.</p>\n      <p><strong>Equation of tangent at $(0,a)$:</strong></p>\n      <p>Find the gradient of the tangent at $(0,a)$.</p>\n      <p>Gradient=[[0]].</p>\n      <p>Hence the equation of the tangent at $(0,a)$ is:</p>\n      <p>$y = \\;$[[1]]</p>\n      <p><strong>Equation of tangent at $(0,b)$:</strong></p>\n      <p>Find the gradient of the tangent at $(0,b)$.</p>\n      <p>Gradient=[[2]].</p>\n      <p>Hence the equation of the tangent at $(0,b)$ is:</p>\n      <p>$y = \\;$[[3]]</p>\n      ", "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"strings": ["."], "message": "<p>Input as an integer or as a fraction, not as a decimal.</p>", "partialCredit": 0, "showStrings": false}, "checkingType": "absdiff", "showPreview": true, "failureRate": 1, "answerSimplification": "std", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "vsetRangePoints": 5, "vsetRange": [0, 1], "marks": 1, "unitTests": [], "checkingAccuracy": 0.001, "checkVariableNames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answer": "{a-d*c}/{c+1}", "variableReplacements": [], "expectedVariableNames": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true}, {"notallowed": {"strings": ["."], "message": "<p>Input all numbers as integers or as fractions, not as decimals.</p>", "partialCredit": 0, "showStrings": false}, "checkingType": "absdiff", "showPreview": true, "failureRate": 1, "answerSimplification": "std", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "vsetRangePoints": 5, "vsetRange": [0, 1], "marks": 1, "unitTests": [], "checkingAccuracy": 0.001, "checkVariableNames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answer": "{a-d*c}/{c+1}*x-{c}", "variableReplacements": [], "expectedVariableNames": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true}, {"notallowed": {"strings": ["."], "message": "<p>Input as an integer or as a fraction, not as a decimal.</p>", "partialCredit": 0, "showStrings": false}, "checkingType": "absdiff", "showPreview": true, "failureRate": 1, "answerSimplification": "std", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "vsetRangePoints": 5, "vsetRange": [0, 1], "marks": 1, "unitTests": [], "checkingAccuracy": 0.001, "checkVariableNames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answer": "{-a-d}/{c+1}", "variableReplacements": [], "expectedVariableNames": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true}, {"notallowed": {"strings": ["."], "message": "<p>Input all numbers as integers or as fractions, not as decimals.</p>", "partialCredit": 0, "showStrings": false}, "checkingType": "absdiff", "showPreview": true, "failureRate": 1, "answerSimplification": "std", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "vsetRangePoints": 5, "vsetRange": [0, 1], "marks": 1, "unitTests": [], "checkingAccuracy": 0.001, "checkVariableNames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answer": "{-a-d}/{c+1}*x+{1}", "variableReplacements": [], "expectedVariableNames": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true}], "variableReplacements": [], "customMarkingAlgorithm": "", "type": "gapfill"}], "extensions": [], "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "\n    \t\t<p>Implicit differentiation.</p>\n    \t\t            <p>Given $x^2+y^2+dxy +ax+by=c$ find $\\displaystyle \\frac{dy}{dx}$ in terms of $x$ and $y$.</p>\n    \t\t            <p>Also find two points on the curve where $x=0$ and find the equation of the tangent at those points.</p>\n    \t\t            <p>&nbsp;</p>\n    \t\t"}, "name": "Simon's copy of SFY0004 Implicit 2", "statement": "<p>Given the following relation between $x$ and $y$<br/>\\[\\simplify[all,!collectNumbers]{x^2+y^2+{d}x y+{a}x+{b}y}=\\var{c}\\]<br/>answer the following questions.</p>", "variable_groups": [], "advice": "<p>Note from the chain rule that $\\frac{d}{dx}(xy) = x\\frac{dy}{dx}+y\\frac{d}{dx}(x)=&nbsp;x\\frac{dy}{dx}+y$</p>\n<p><strong></strong></p>\n<p><strong>(a)</strong></p>\n<p>On differentiating both sides of the equation implicitly we get<br/>\\[2x + \\simplify[all,!collectNumbers]{2y*Diff(y,x,1) +{d}(y+x*Diff(y,x,1))+ {a} + {b} *Diff(y,x,1)} = 0\\]<br/>Collecting terms in $\\displaystyle\\frac{dy}{dx}$ and rearranging the equation we get <br/>\\[( \\simplify[all,!collectNumbers]{({b} + 2y+{d}x)} )\\frac{dy}{dx} = \\simplify[all,!collectNumbers]{{ -a} -2x-{d}y}\\] and hence on further rearranging:</p>\n<p>\\[\\frac{dy}{dx} = \\simplify[all,!collectNumbers]{({ - a} - 2 * x-{d}y) / ({b} + (2 * y)+{d}x)}\\]</p>\n<h4></h4>\n<h4>(b)</h4>\n<p>On putting $x=0$ in the relation we get:</p>\n<p>\\[\\simplify{y^2+{b}y={c}} \\Rightarrow \\simplify{y^2+{b}y-{c}=0 }\\Rightarrow (y+\\var{c})(y-1)=0\\]</p>\n<p>Hence $a=-\\var{c}$ and $b=1$.</p>\n<p></p>\n<h4>(c)</h4>\n<p>First we find the tangent at the point $(0,-\\var{c})$.</p>\n<p>We find using the formula we found for $\\frac{dy}{dx}$ in part (a) that the gradient at &nbsp;$(0,-\\var{c})$ is:</p>\n<p>\\[\\frac{dy}{dx}=\\frac{\\simplify[all,!collectnumbers]{{-a}+{d*c}}}{\\var{b}-\\var{2*c}}=\\simplify[all,fractionNumbers]{{a-d*c}/{c+1}}\\]</p>\n<p>As the tangent goes through the point $(0,\\var{-c})$ i.e. at $x=0,\\;\\;y=-\\var{c}$ we see that the equation of the tangent is:</p>\n<p>\\[y=\\simplify[all,fractionNumbers]{{a-d*c}/{c+1}}x-\\var{c}\\]</p>\n<p>Next we find that the gradient at &nbsp;$(0,1)$ is:</p>\n<p>\\[\\frac{dy}{dx}=\\frac{\\simplify[all,!collectnumbers]{{-a}-{d}}}{\\var{b}+2}=\\simplify[all,fractionNumbers]{-{a+d}/{c+1}}\\]</p>\n<p>As the tangent goes through the point $(0,1)$ i.e. at $x=0,\\;\\;y=1$ we see that the equation of the tangent is:</p>\n<p>\\[y=\\simplify[all,fractionNumbers]{-{a+d}/{c+1}}x+1\\]</p>", "ungrouped_variables": ["a", "c", "b", "d"], "variables": {"d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "templateType": "anything", "description": ""}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..9 except -a+1)", "templateType": "anything", "description": ""}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "c-1", "templateType": "anything", "description": ""}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "-random(1..9)", "templateType": "anything", "description": ""}}, "preamble": {"css": "", "js": ""}, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {"std": ["all", "fractionNumbers"]}, "type": "question", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}