// Numbas version: exam_results_page_options
{"name": "T6Q5 (custom feedback)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "metadata": {"description": "<p>Partial differentiation question with customised feedback to catch some common errors.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "parts": [{"scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": [], "prompt": "<p>$\\frac{\\partial z}{\\partial x} =$[[0]]</p>", "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"vsetRangePoints": 5, "type": "jme", "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "malrules:\n  [\n    [\"-1/x\", \"Don't forget the product rule!\"],\n    [\"ln(y/x)+x^2\", \"Be careful when multiplying by $\\\\frac{\\\\partial}{\\\\partial x} \\\\left( \\\\frac{y}{x} \\\\right) = \\\\frac{\\\\partial}{\\\\partial x} (yx^{-1}) = ?$\"],\n    [\"ln(y/x)+y^2\", \"The derivative of $\\\\ln \\\\left( \\\\frac{y}{x} \\\\right)$ is $\\\\frac{1}{\\\\left(\\\\frac{y}{x} \\\\right)} \\\\neq \\\\frac{y}{x}$. Also, be careful when multiplying by $\\\\frac{d}{dx} \\\\left( \\\\frac{y}{x} \\\\right) = \\\\frac{d}{dx} (yx^{-1}) = ?$\"],\n    [\"ln(y/x)+x^2/y\", \"Don't forget to multiply by the derivative of $\\\\left( \\\\frac{y}{x} \\\\right)$ (chain rule). Remember $\\\\frac{\\\\partial}{\\\\partial x} \\\\ln x = \\\\frac{1}{x}$ while $\\\\frac{\\\\partial}{\\\\partial x} \\\\left( \\\\ln \\\\left( f(x) \\\\right) \\\\right) = \\\\frac{1}{f(x)} \\\\times f'(x).$ Therefore, $\\\\frac{\\\\partial}{\\\\partial x} \\\\left( \\\\ln \\\\left(\\\\frac{y}{x} \\\\right) \\\\right) = \\\\frac{1}{\\\\frac{y}{x}} \\\\times \\\\frac{\\\\partial}{\\\\partial x} \\\\left( \\\\frac{y}{x} \\\\right) = \\\\frac{x}{y} \\\\times ?$\"],\n    [\"ln(y/x)-y^2/x^2\", \"Look very closely at how you differentiated $\\\\ln \\\\left( \\\\frac{y}{x} \\\\right)$. Hint: $\\\\frac{1}{\\\\left(\\\\frac{y}{x} \\\\right)} \\\\neq \\\\frac{y}{x}$.\"]\n    ]\n\n\nparsed_malrules: \n    map(\n        [\"expr\":parse(x[0]),\"feedback\":x[1]],\n        x,\n        malrules\n    )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n    map(\n        len(filter(not x ,x,map(\n            try(\n                resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n                message,\n                false\n            ),\n            vars,\n            vset\n        )))<settings[\"failureRate\"],\n        malrule,\n        parsed_malrules\n    )\n\nmatched_malrules:\n  filter(x[0],x,zip(agree_malrules,parsed_malrules))\n  \nmalrule_feedback:\n    assert(numericallyCorrect,\n        assert(len(matched_malrules)=0,\n            set_credit(0,matched_malrules[0][1][\"feedback\"])\n        )\n    )\n\nmark:\n    apply(studentExpr);\n    apply(unexpectedVariables);\n    apply(numericallyCorrect);\n    apply(malrule_feedback);\n    apply(failMinLength);\n    apply(failMaxLength);\n    apply(forbiddenStringsPenalty);\n    apply(requiredStringsPenalty) ", "unitTests": [], "vsetRange": [0, 1], "showCorrectAnswer": true, "checkVariableNames": false, "checkingType": "absdiff", "answer": "ln(y/x)-1", "extendBaseMarkingAlgorithm": true, "showPreview": true, "expectedVariableNames": [], "failureRate": 1, "marks": 1, "checkingAccuracy": 0.001, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "extendBaseMarkingAlgorithm": true, "marks": 0, "type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": [], "prompt": "<p>$\\frac{\\partial z}{\\partial y}=$ [[0]]</p>", "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"vsetRangePoints": 5, "type": "jme", "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "malrules:\n  [\n    [\"y/x\", \"Look closely at how you have differentiated $\\\\ln \\\\left( \\\\frac{y}{x} \\\\right)$. Hint: $\\\\frac{1}{\\\\left(\\\\frac{y}{x} \\\\right)} \\\\neq \\\\frac{y}{x}$.\"],\n    [\"y\", \"It looks like you have differentiated $\\\\ln \\\\left( \\\\frac{y}{x} \\\\right)$ incorrectly. Note that $\\\\frac{1}{\\\\left(\\\\frac{y}{x} \\\\right)} \\\\neq \\\\frac{y}{x}$. Also, don't forget to multiply by the derivative (with respect to $y$) of $ \\\\left( \\\\frac{y}{x} \\\\right) $. Remember $\\\\frac{\\\\partial}{\\\\partial y} \\\\ln y = \\\\frac{1}{y}$ while $\\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\ln \\\\left( f(y) \\\\right) \\\\right) = \\\\frac{1}{f(y)} \\\\times f'(y).$ Therefore, $\\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\ln \\\\left(\\\\frac{y}{x} \\\\right) \\\\right) = \\\\frac{1}{\\\\frac{y}{x}} \\\\times \\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\frac{y}{x} \\\\right) = \\\\frac{x}{y} \\\\times ?$\"],\n    [\"x^2/y\", \"Don't forget to multiply by the derivative (with respect to $y$) of $ \\\\frac{y}{x} $ (chain rule). Remember $\\\\frac{\\\\partial}{\\\\partial y} \\\\ln y = \\\\frac{1}{y}$ while $\\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\ln \\\\left( f(y) \\\\right) \\\\right) = \\\\frac{1}{f(y)} \\\\times f'(y).$ Therefore, $\\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\ln \\\\left(\\\\frac{y}{x} \\\\right) \\\\right) = \\\\frac{1}{\\\\frac{y}{x}} \\\\times \\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\frac{y}{x} \\\\right) = \\\\frac{x}{y} \\\\times ?$\"]\n    ]\n\n\nparsed_malrules: \n    map(\n        [\"expr\":parse(x[0]),\"feedback\":x[1]],\n        x,\n        malrules\n    )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n    map(\n        len(filter(not x ,x,map(\n            try(\n                resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n                message,\n                false\n            ),\n            vars,\n            vset\n        )))<settings[\"failureRate\"],\n        malrule,\n        parsed_malrules\n    )\n\nmatched_malrules:\n  filter(x[0],x,zip(agree_malrules,parsed_malrules))\n  \nmalrule_feedback:\n    assert(numericallyCorrect,\n        assert(len(matched_malrules)=0,\n            set_credit(0,matched_malrules[0][1][\"feedback\"])\n        )\n    )\n\nmark:\n    apply(studentExpr);\n    apply(unexpectedVariables);\n    apply(numericallyCorrect);\n    apply(malrule_feedback);\n    apply(failMinLength);\n    apply(failMaxLength);\n    apply(forbiddenStringsPenalty);\n    apply(requiredStringsPenalty) ", "unitTests": [], "vsetRange": [0, 1], "showCorrectAnswer": true, "checkVariableNames": false, "checkingType": "absdiff", "answer": "x/y", "extendBaseMarkingAlgorithm": true, "showPreview": true, "expectedVariableNames": [], "failureRate": 1, "marks": 1, "checkingAccuracy": 0.001, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "extendBaseMarkingAlgorithm": true, "marks": 0, "type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": [], "prompt": "<p>$\\frac{\\partial^2 z}{\\partial x \\partial y} =$ [[0]]</p>", "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"vsetRangePoints": 5, "type": "jme", "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": [], "vsetRange": [0, 1], "showCorrectAnswer": true, "checkVariableNames": false, "checkingType": "absdiff", "answer": "1/y", "extendBaseMarkingAlgorithm": true, "showPreview": true, "expectedVariableNames": [], "failureRate": 1, "marks": 1, "checkingAccuracy": 0.001, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "extendBaseMarkingAlgorithm": true, "marks": 0, "type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": [], "prompt": "<p></p>\n<p>$\\frac{\\partial^2 z}{\\partial y&nbsp;\\partial x} =$ [[0]]</p>", "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"vsetRangePoints": 5, "type": "jme", "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "malrules:\n  [\n    [\"y/x^2\", \"Look very closely at how you differentiated $\\\\ln \\\\left( \\\\frac{y}{x} \\\\right)$. Hint: $\\\\frac{1}{\\\\left(\\\\frac{y}{x} \\\\right)} \\\\neq \\\\frac{y}{x}$.\"],\n    [\"x/y\", \"Don't forget to multiply by the derivative (with respect to $y$) of $ \\\\frac{y}{x} $ (chain rule). Remember $\\\\frac{\\\\partial}{\\\\partial y} \\\\ln y = \\\\frac{1}{y}$ while $\\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\ln \\\\left( f(y) \\\\right) \\\\right) = \\\\frac{1}{f(y)} \\\\times f'(y).$ Therefore, $\\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\ln \\\\left(\\\\frac{y}{x} \\\\right) \\\\right) = \\\\frac{1}{\\\\frac{y}{x}} \\\\times \\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\frac{y}{x} \\\\right) = \\\\frac{x}{y} \\\\times ?$\"],\n    [\"y/x\", \"It looks like you have differentiated $\\\\ln \\\\left( \\\\frac{y}{x} \\\\right)$ incorrectly. Hint: $\\\\frac{1}{\\\\left(\\\\frac{y}{x} \\\\right)} \\\\neq \\\\frac{y}{x}$. Also,don't forget to multiply by the derivative (with respect to $y$) of $ \\\\frac{y}{x} $ (chain rule). Remember $\\\\frac{\\\\partial}{\\\\partial y} \\\\ln y = \\\\frac{1}{y}$ while $\\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\ln \\\\left( f(y) \\\\right) \\\\right) = \\\\frac{1}{f(y)} \\\\times f'(y).$ Therefore, $\\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\ln \\\\left(\\\\frac{y}{x} \\\\right) \\\\right) = \\\\frac{1}{\\\\frac{y}{x}} \\\\times \\\\frac{\\\\partial}{\\\\partial y} \\\\left( \\\\frac{y}{x} \\\\right) = \\\\frac{x}{y} \\\\times ?$\"]\n    ]\n\n\nparsed_malrules: \n    map(\n        [\"expr\":parse(x[0]),\"feedback\":x[1]],\n        x,\n        malrules\n    )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n    map(\n        len(filter(not x ,x,map(\n            try(\n                resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n                message,\n                false\n            ),\n            vars,\n            vset\n        )))<settings[\"failureRate\"],\n        malrule,\n        parsed_malrules\n    )\n\nmatched_malrules:\n  filter(x[0],x,zip(agree_malrules,parsed_malrules))\n  \nmalrule_feedback:\n    assert(numericallyCorrect,\n        assert(len(matched_malrules)=0,\n            set_credit(0,matched_malrules[0][1][\"feedback\"])\n        )\n    )\n\nmark:\n    apply(studentExpr);\n    apply(unexpectedVariables);\n    apply(numericallyCorrect);\n    apply(malrule_feedback);\n    apply(failMinLength);\n    apply(failMaxLength);\n    apply(forbiddenStringsPenalty);\n    apply(requiredStringsPenalty) ", "unitTests": [], "vsetRange": [0, 1], "showCorrectAnswer": true, "checkVariableNames": false, "checkingType": "absdiff", "answer": "1/y", "extendBaseMarkingAlgorithm": true, "showPreview": true, "expectedVariableNames": [], "failureRate": 1, "marks": 1, "checkingAccuracy": 0.001, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "extendBaseMarkingAlgorithm": true, "marks": 0, "type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "functions": {}, "statement": "<p>Let $z=x \\ln \\frac{y}{x}$. Show that $\\frac{\\partial^2 z}{\\partial x \\partial y} = \\frac{\\partial^2 z}{\\partial y \\partial x}$.</p>", "rulesets": {}, "name": "T6Q5 (custom feedback)", "variables": {}, "preamble": {"css": "", "js": ""}, "advice": "", "ungrouped_variables": [], "extensions": [], "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}]}]}], "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}]}