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{"metadata": {"notes": "", "description": "<p>Implicit differentiation including finding tangents</p>", "licence": "Creative Commons Attribution 4.0 International"}, "navigation": {"showfrontpage": false, "reverse": true, "onleave": {"action": "none", "message": ""}, "browse": true, "preventleave": false, "allowregen": true, "showresultspage": "never"}, "name": "Maria's copy of mathcentre: Implicit differentiation", "showQuestionGroupNames": false, "pickQuestions": 0, "question_groups": [{"name": "", "questions": [{"name": "SFY0004 Implicit 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "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}], "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": []}]}, {"name": "SFY0004 Implicit 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d"], "tags": ["Calculus", "Differentiation", "calculus", "derivative", "deriving an implicit relation", "differentiate", "differentiate implicitly", "differentiation", "equation of tangent", "first derivative using implicit differentiation", "gradient", "implicit differentiation", "implicit relation", "tangent at a point"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\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>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<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>", "rulesets": {"std": ["all", "fractionNumbers"]}, "parts": [{"stepsPenalty": 0, "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      <p>Input all numbers as integers not as decimals.</p>\n      <p>If you want more help click on Show steps - you will not lose any marks.</p>\n      ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Input all numbers as integers or as fractions, not as decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "all,!collectNumbers", "scripts": {}, "answer": "(({( - a)} + ( - (2 * x))-{d}y) / ({b} + (2 * y)+{d}x))", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"type": "information", "showCorrectAnswer": true, "scripts": {}, "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          ", "marks": 0}], "type": "gapfill"}, {"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      ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "-c", "minValue": "-c", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "1", "minValue": "1", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"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      ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Input as an integer or as a fraction, not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a-d*c}/{c+1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"notallowed": {"message": "<p>Input all numbers as integers or as fractions, not as decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a-d*c}/{c+1}*x-{c}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"notallowed": {"message": "<p>Input as an integer or as a fraction, not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{-a-d}/{c+1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"notallowed": {"message": "<p>Input all numbers as integers or as fractions, not as decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{-a-d}/{c+1}*x+{1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "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>", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "-random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(2..9 except -a+1)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "c-1", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-3..3 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"notes": "\n    \t\t<p><strong>30/04/2013</strong></p>\n    \t\t        <p>Created new question for SFY0004 out of 1041 CBA2_5.</p>\n    \t\t        <p>Added more tags.</p>\n    \t\t", "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", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}], "pickQuestions": 0, "pickingStrategy": "all-ordered"}], "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "allQuestions": true, "percentPass": 0, "duration": 0, "questions": [], "type": "exam", "feedback": {"showtotalmark": true, "showactualmark": true, "advicethreshold": 0, "showanswerstate": true, "allowrevealanswer": true}, "shuffleQuestions": false, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}], "extensions": [], "custom_part_types": [], "resources": []}