// Numbas version: finer_feedback_settings
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"functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>$f(x,y,z)=\\simplify[std]{{a1}*x^{p1}*y^{p2}*z^{p3}+{b1}*x^{p4}*y^{p5}*z^{p6}+{c1}*({1-t}*sin({d1}*x^{p7}*y^{p8}*z^{p9})+{t}*cos({e1}*x^{p10}*y^{p11}*z^{p12}))}$.</p>\n<p>$\\boldsymbol{\\nabla}f=($[[0]]$,$[[1]]$,$[[2]]$)$.</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}+{c1*d1*p7*(1-t)}*x^{p7-1}*y^{p8}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p10*t}*x^{p10-1}*y^{p11}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "answerSimplification": "all", "extendBaseMarkingAlgorithm": true, "showPreview": true, "checkVariableNames": true, "checkingType": "absdiff", "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "failureRate": 1, "scripts": {}, "vsetRangePoints": 5, "type": "jme", "unitTests": [], "checkingAccuracy": 0.001, "expectedVariableNames": ["x", "y", "z"], "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}, {"answer": "{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}+{c1*d1*p8*(1-t)}*x^{p7}*y^{p8-1}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p11*t}*x^{p10}*y^{p11-1}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "answerSimplification": "all", "extendBaseMarkingAlgorithm": true, "showPreview": true, "checkVariableNames": true, "checkingType": "absdiff", "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "failureRate": 1, "scripts": {}, "vsetRangePoints": 5, "type": "jme", "unitTests": [], "checkingAccuracy": 0.001, "expectedVariableNames": ["x", "y", "z"], "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}, {"answer": "{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}+{c1*d1*p9*(1-t)}*x^{p7}*y^{p8}*z^{p9-1}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p12*t}*x^{p10}*y^{p11}*z^{p12-1}*sin({e1}*x^{p10}*y^{p11}*z^{p12})", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "answerSimplification": "all", "extendBaseMarkingAlgorithm": true, "showPreview": true, "checkVariableNames": true, "checkingType": "absdiff", "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "failureRate": 1, "scripts": {}, "vsetRangePoints": 5, "type": "jme", "unitTests": [], "checkingAccuracy": 0.001, "expectedVariableNames": ["x", "y", "z"], "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Find $\\boldsymbol{\\nabla}f$ for the following function $f(x,y,z)$.</p>", "tags": ["checked2015", "gradient", "nabla", "partial derivatives", "scalar field"], "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Gradient of $f(x,y,z)$.</p>\n<p>Should warn that multiplied terms need * to denote multiplication.</p>"}, "advice": "<p>This question is simply an exercise in partial differentiation, using the fact that</p>\n<p>\\[\\boldsymbol{\\nabla}f=\\pmatrix{\\frac{\\partial f}{\\partial x},\\frac{\\partial f}{\\partial y},\\frac{\\partial f}{\\partial z}}.\\]</p>\n<p>Hence</p>\n<p>\\[\\boldsymbol{\\nabla}f=\\pmatrix{\\simplify{{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}+{c1*d1*p7*(1-t)}*x^{p7-1}*y^{p8}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p10*t}*x^{p10-1}*y^{p11}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})},\\simplify{{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}+{c1*d1*p8*(1-t)}*x^{p7}*y^{p8-1}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p11*t}*x^{p10}*y^{p11-1}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})},\\simplify{{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}+{c1*d1*p9*(1-t)}*x^{p7}*y^{p8}*z^{p9-1}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p12*t}*x^{p10}*y^{p11}*z^{p12-1}*sin({e1}*x^{p10}*y^{p11}*z^{p12})}}.\\]</p>", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}