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{"name": "SFY0004 Implicit 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "SFY0004 Implicit 1", "tags": ["Calculus", "calculus", "derivative", "deriving an implicit relation", "differentiate", "differentiate implicitly", "differentiation", "first derivative using implicit differentiation", "implicit differentiation", "implicit relation"], "advice": "<p>On differentiating both sides of the equation implicitly we get<br/>\\[2x + \\simplify[all,!collectNumbers]{2y*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/>\\[(\\var{b} + 2y) \\frac{dy}{dx} = \\simplify[all,!collectNumbers]{{ -a} -2x}\\] and hence on further rearranging:<br/>\\[\\frac{dy}{dx} = \\simplify[all,!collectNumbers]{({ - a} - 2 * x) / ({b} + (2 * y))}\\]</p>", "rulesets": {}, "parts": [{"prompt": "\n        <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        \n      \n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "all,!collectNumbers", "marks": 2.0, "answer": "(({( - a)} + ( - (2 * x))) / ({b} + (2 * y)))", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "<p>Given the following relation between $x$ and $y$<br/>\\[\\simplify[all,!collectNumbers]{x^2+y^2+{a}x+{b}y}=\\var{c}\\]<br/>answer the following question.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "-random(1..9)", "name": "a"}, "c": {"definition": "random(1..9)", "name": "c"}, "b": {"definition": "random(1..9)", "name": "b"}}, "metadata": {"notes": "\n    \t\t        \t\t                    \t\t<p><strong>20/06/2012:</strong></p>\n    \t\t        \t\t                    \t\t<p>Added tags.</p>\n    \t\t        \t\t                    \t\t<p>Improved display using \\displaystyle where appropriate.</p>\n    \t\t        \t\t                    \t\t<p>Changed marks to 2.</p>\n    \t\t        \t\t                    \t\t<p>&nbsp;</p>\n    \t\t        \t\t                    \t\t<p><strong>3/07/2012:<br /></strong></p>\n    \t\t        \t\t                    \t\t<p>Added tags.</p>\n    \t\t        \t\t                    \n    \t\t        \n    \t\t", "description": "\n    \t\t        \t\t                    \t\t<p>Implicit differentiation.</p>\n    \t\t        \t\t                    \t\t<p>Given $x^2+y^2+ax+by=c$ find $\\displaystyle \\frac{dy}{dx}$ in terms of $x$ and $y$.</p>\n    \t\t        \t\t                    \t\t<p>&nbsp;</p>\n    \t\t        \t\t                    \n    \t\t        \n    \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}