// Numbas version: exam_results_page_options
{"name": "Harry's copy of Find gradient of scalar field", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Harry's copy of Find gradient of scalar field", "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "statement": "<p>Find $\\boldsymbol{\\nabla}f$ for the following function $f(x,y,z)$.</p>", "variable_groups": [], "variables": {"p2": {"name": "p2", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(0..4)"}, "p9": {"name": "p9", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(0,1)"}, "p8": {"name": "p8", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(0,1)"}, "p1": {"name": "p1", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(0..4)"}, "p3": {"name": "p3", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(0..4)"}, "p11": {"name": "p11", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(0,1)"}, "c1": {"name": "c1", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(1..9)*sign(random(-1,1))"}, "p7": {"name": "p7", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "if(p8=0 and p9=0,1,random(0,1))"}, "p6": {"name": "p6", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(0..4)"}, "b1": {"name": "b1", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(1..9)*sign(random(-1,1))"}, "e1": {"name": "e1", "templateType": "anything", "group": "Ungrouped variables", 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"definition": "random(1..9)*sign(random(-1,1))"}}, "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Gradient of $f(x,y,z)$.</p>"}, "tags": [], "parts": [{"type": "gapfill", "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>", "showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "gaps": [{"type": "jme", "checkvariablenames": true, "showpreview": true, "variableReplacements": [], "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst", "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})", "scripts": {}, "answersimplification": "all", "checkingaccuracy": 0.001, "vsetrangepoints": 5, "showFeedbackIcon": true, "checkingtype": "absdiff", "expectedvariablenames": ["x", "y", "z"], "showCorrectAnswer": true, "marks": 1}, {"type": "jme", "checkvariablenames": true, "showpreview": true, "variableReplacements": [], "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst", "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})", "scripts": {}, "answersimplification": "all", "checkingaccuracy": 0.001, "vsetrangepoints": 5, "showFeedbackIcon": true, "checkingtype": "absdiff", "expectedvariablenames": ["x", "y", "z"], "showCorrectAnswer": true, "marks": 1}, {"type": "jme", "checkvariablenames": true, "showpreview": true, "variableReplacements": [], "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst", "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})", "scripts": {}, "answersimplification": "all", "checkingaccuracy": 0.001, "vsetrangepoints": 5, "showFeedbackIcon": true, "checkingtype": "absdiff", "expectedvariablenames": ["x", "y", "z"], "showCorrectAnswer": true, "marks": 1}]}], "ungrouped_variables": ["p2", "p3", "p1", "p6", "p7", "p4", "p5", "p8", "p9", "a1", "p11", "p12", "t", "b1", "p10", "c1", "e1", "d1"], "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "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>The partial derivatives of $f$ are as follows:</p>\n<p>\\begin{align}<br/>\\frac{\\partial f}{\\partial x} &amp;=&nbsp;\\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})} \\\\[0.5em]<br/>\\frac{\\partial f}{\\partial y} &amp;=&nbsp;\\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})} \\\\[0.5em]<br/>\\frac{\\partial f}{\\partial z} &amp;=&nbsp;\\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})}<br/>\\end{align}</p>\n<p>Hence</p>\n<p>\\[ \\boldsymbol{\\nabla}f = \\pmatrix{<br/>\\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})}, &amp;<br/>\\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})}, &amp;<br/>\\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})}<br/>}.\\]</p>", "preamble": {"css": "", "js": ""}, "type": "question", "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}