// Numbas version: exam_results_page_options
{"name": "Section 2 (practice set)", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-shuffled", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], [], [], []], "questions": [{"name": "28_02_01a Gradient of a function", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Given the function:</p>\n<p>$f=\\simplify{{x_coeff}x+{y_coeff}y+{z_coeff}z+{const}}$</p>\n<p>Determine the gradient ${\\nabla}f$ in the form</p>\n<p>${\\nabla}f=(a,&nbsp;b, c)$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>The gradient of the function</p>\n<p>$f=\\simplify{{x_coeff}x+{y_coeff}y+{z_coeff}z+{const}}$</p>\n<p>is given by</p>\n<p>${\\nabla}f=\\left(\\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial z}\\right)$</p>\n<p>${\\nabla}f=(\\var{ans_i}, \\var{ans_j}, \\var{ans_k})$</p>\n<p>The same at all values of $(x,y,z)$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {}, "variables": {"z_coeff": {"name": "z_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "ans_k": {"name": "ans_k", "group": "Ungrouped variables", "definition": "z_coeff", "description": "", "templateType": "anything"}, "x_coeff": {"name": "x_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "y_coeff": {"name": "y_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "ans_j": {"name": "ans_j", "group": "Ungrouped variables", "definition": "y_coeff", "description": "", "templateType": "anything"}, "const": {"name": "const", "group": "Ungrouped variables", "definition": "random(-50..50)", "description": "", "templateType": "anything"}, "ans_i": {"name": "ans_i", "group": "Ungrouped variables", "definition": "x_coeff", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x_coeff", "y_coeff", "z_coeff", "const", "ans_i", "ans_j", "ans_k"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Enter the values of a, b and c in the boxes provided.</p>\n<p>$a=$[[0]], $b=$[[1]],&nbsp;$c=$[[2]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1/3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans_i", "maxValue": "ans_i", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1/3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans_j", "maxValue": "ans_j", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1/3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans_k", "maxValue": "ans_k", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "28_02_01b Gradient of a function 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Given the function:</p>\n<p>$z=\\simplify{{x_coeff}x^{xpow}+{y_coeff}y^{ypow}+{coeff}*x*y}$</p>\n<p>Determine the gradient ${\\nabla}z$ at the point $(x,y)=(\\var{xval},\\var{yval})$ in the form</p>\n<p>${\\nabla}z=(a,&nbsp;b)$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>The gradient of the function:</p>\n<p>$z=\\simplify{{x_coeff}x^{xpow}+{y_coeff}y^{ypow} + {coeff}{x}*{y}}$</p>\n<p>is given by</p>\n<p>${\\nabla}z=\\left(\\frac{\\partial z}{\\partial x}, \\frac{\\partial z}{\\partial y}\\right)$</p>\n<p>${\\nabla}z={(\\simplify{{x_coeff}*{xpow}*x^{xpow-1} + {coeff}*y},\\quad \\simplify{{y_coeff}*{ypow}*y^{ypow-1}+{coeff}*x})}$</p>\n<p>at $(x,y)=(\\var{xval},\\var{yval})$</p>\n<p>${\\nabla}z=(\\var{ans_i}, \\var{ans_j})$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {}, "variables": {"x_coeff": {"name": "x_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "y_coeff": {"name": "y_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "ans_j": {"name": "ans_j", "group": "Ungrouped variables", "definition": "y_coeff*ypow*yval^(ypow-1) + coeff*xval", "description": "", "templateType": "anything"}, "ans_i": {"name": "ans_i", "group": "Ungrouped variables", "definition": "x_coeff*xpow*xval^(xpow-1) + coeff*yval", "description": "", "templateType": "anything"}, "xpow": {"name": "xpow", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "ypow": {"name": "ypow", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "xval": {"name": "xval", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "yval": {"name": "yval", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "coeff": {"name": "coeff", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x_coeff", "y_coeff", "coeff", "xpow", "ypow", "xval", "yval", "ans_i", "ans_j"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Enter the values of $a$ and $b$ in the boxes provided.</p>\n<p>$a=$[[0]], $b=$[[1]]</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans_i", "maxValue": "ans_i", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans_j", "maxValue": "ans_j", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "28_02_01c Magnitude of the Gradient", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Given the function:</p>\n<p>$f=\\simplify{{x_coeff}x+{y_coeff}y+{z_coeff}z+{const}}$</p>\n<p>Determine the magnitude of the largest gradient $|{\\nabla}f|$&nbsp;</p>\n<p>at the point $(x,y,z)=(\\var{x},\\var{y},\\var{z})$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>The gradient of the function</p>\n<p>$f=\\simplify{{x_coeff}x+{y_coeff}y+{z_coeff}z+{const}}$</p>\n<p>is given by</p>\n<p>${\\nabla}f=\\left(\\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial z}\\right)$</p>\n<p>${\\nabla}f=(\\var{ans_i}, \\var{ans_j}, \\var{ans_k})$</p>\n<p>$|{\\nabla}f|=\\sqrt{(\\var{ans_i})^2 + (\\var{ans_j})^2 + (\\var{ans_k})^2} = \\var{answer}$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {}, "variables": {"z_coeff": {"name": "z_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "ans_k": {"name": "ans_k", "group": "Ungrouped variables", "definition": "z_coeff", "description": "", "templateType": "anything"}, "x_coeff": {"name": "x_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "y_coeff": {"name": "y_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "ans_j": {"name": "ans_j", "group": "Ungrouped variables", "definition": "y_coeff", "description": "", "templateType": "anything"}, "const": {"name": "const", "group": "Ungrouped variables", "definition": "random(-50..50)", "description": "", "templateType": "anything"}, "ans_i": {"name": "ans_i", "group": "Ungrouped variables", "definition": "x_coeff", "description": "", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "z": {"name": "z", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "sqrt((ans_i^2)+ (ans_j^2) + (ans_k^2))", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x_coeff", "y_coeff", "z_coeff", "const", "x", "y", "z", "ans_i", "ans_j", "ans_k", "answer"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Enter the&nbsp;answer in the box provided.</p>\n<p>$|\\nabla f|=$[[0]]</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "gap", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "answer", "maxValue": "answer", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "28_02_01d Magnitude of the Gradient 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Given the function:</p>\n<p>$f=\\simplify{{x_coeff}x^{xpow}+{y_coeff}y^{ypow}+{z_coeff}z^{zpow}+{const}}$</p>\n<p>Determine the magnitude of the largest gradient $|{\\nabla}f|$&nbsp;</p>\n<p>at the point $(x,y,z)=(\\var{xval},\\var{yval},\\var{zval})$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>The gradient of the function</p>\n<p>$f=\\simplify{{x_coeff}x^{xpow}+{y_coeff}y^{ypow}+{z_coeff}z^{zpow}+{const}}$</p>\n<p>is given by</p>\n<p>${\\nabla}f=\\left(\\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial z}\\right)$</p>\n<p>${\\nabla}f={(\\simplify{{x_coeff}*{xpow}*x^{xpow-1}}, \\simplify{{y_coeff}*{ypow}*y^{ypow-1}}, \\simplify{{z_coeff}*{zpow}*z^{zpow-1}})}$</p>\n<p>at $(x,y,z)=(\\var{xval},\\var{yval},\\var{zval})$</p>\n<p>${\\nabla}f=(\\var{ans_i}, \\var{ans_j}, \\var{ans_k})$</p>\n<p>$|{\\nabla}f|=\\sqrt{(\\var{ans_i})^2 + (\\var{ans_j})^2 + (\\var{ans_k})^2} = \\var{answer}$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {}, "variables": {"z_coeff": {"name": "z_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "ans_k": {"name": "ans_k", "group": "Ungrouped variables", "definition": "z_coeff*zpow*zval^(zpow-1)", "description": "", "templateType": "anything"}, "x_coeff": {"name": "x_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "y_coeff": {"name": "y_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "ans_j": {"name": "ans_j", "group": "Ungrouped variables", "definition": "y_coeff*ypow*yval^(ypow-1)", "description": "", "templateType": "anything"}, "const": {"name": "const", "group": "Ungrouped variables", "definition": "random(-50..50)", "description": "", "templateType": "anything"}, "ans_i": {"name": "ans_i", "group": "Ungrouped variables", "definition": "x_coeff*xpow*xval^(xpow-1)", "description": "", "templateType": "anything"}, "xval": {"name": "xval", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "yval": {"name": "yval", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "zval": {"name": "zval", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "sqrt((ans_i^2)+ (ans_j^2) + (ans_k^2))", "description": "", "templateType": "anything"}, "xpow": {"name": "xpow", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "ypow": {"name": "ypow", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "zpow": {"name": "zpow", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x_coeff", "y_coeff", "z_coeff", "const", "xval", "yval", "zval", "xpow", "ypow", "zpow", "ans_i", "ans_j", "ans_k", "answer"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Enter the&nbsp;answer in the box provided.</p>\n<p>$|\\nabla f|=$[[0]]</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "gap", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "answer", "maxValue": "answer", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "28_02_01e Directional Derivative", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Given the function:</p>\n<p>$z=\\simplify{{x_coeff}x^{xpow}+{y_coeff}y^{ypow}}$</p>\n<p>Determine the&nbsp;magnitude of the gradient of $z(x,y)$ at the point $(x,y) = (\\var{xval}, \\var{yval})$, in the direction of the vector $U = (\\var{a}, \\var{b})$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>The directional derivative is given by: $\\nabla z \\cdot \\hat u$;</p>\n<p>where $\\hat u = \\frac{U}{|U|}$ is the unit vector of $U$.</p>\n<p>The gradient of the function:</p>\n<p>$z=\\simplify{{x_coeff}x^{xpow}+{y_coeff}y^{ypow}}$</p>\n<p>is given by</p>\n<p>${\\nabla}z=\\left(\\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}\\right)$</p>\n<p>${\\nabla}z={(\\simplify{{x_coeff}*{xpow}*x^{xpow-1}}, \\simplify{{y_coeff}*{ypow}*y^{ypow-1}})}$</p>\n<p>at $(x,y)=(\\var{xval},\\var{yval})$</p>\n<p>${\\nabla}z=(\\var{ans_i}, \\var{ans_j})$</p>\n<p>For the direction of the vector $U = (\\var{a}, \\var{b})$, the unit vector is&nbsp;$\\hat u = \\frac{U}{|U|} = \\dfrac{(\\var{a},\\var{b})}{\\sqrt{(\\var{a})^2&nbsp;+ (\\var{b})^2}} =\\left(\\dfrac{\\var{a}}{\\sqrt{\\var{d^2}}}&nbsp; ,\\dfrac{\\var{b}}{\\sqrt{\\var{d^2}}}\\right)$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\">&nbsp;So the directional derivative is: $\\nabla z \\cdot \\hat u =(\\var{ans_i}, \\var{ans_j})\\cdot\\left(\\dfrac{\\var{a}}{\\sqrt{\\var{d^2}}}&nbsp; ,\\dfrac{\\var{b}}{\\sqrt{\\var{d^2}}}\\right) = \\left( (\\var{ans_i}) \\dfrac{\\var{a}}{\\sqrt{\\var{d^2}}} + (\\var{ans_j})\\dfrac{\\var{b}}{\\sqrt{\\var{d^2}}}\\right) = \\var{answer}$</div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {}, "variables": {"x_coeff": {"name": "x_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "y_coeff": {"name": "y_coeff", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything"}, "ans_j": {"name": "ans_j", "group": "Ungrouped variables", "definition": "y_coeff*ypow*yval^(ypow-1)", "description": "", "templateType": "anything"}, "ans_i": {"name": "ans_i", "group": "Ungrouped variables", "definition": "x_coeff*xpow*xval^(xpow-1)", "description": "", "templateType": "anything"}, "xpow": {"name": "xpow", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "ypow": {"name": "ypow", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "xval": {"name": "xval", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "yval": {"name": "yval", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "sqrt(a^2 + b^2)", "description": "", "templateType": "anything"}, "u1": {"name": "u1", "group": "Ungrouped variables", "definition": "a/d", "description": "", "templateType": "anything"}, "u2": {"name": "u2", "group": "Ungrouped variables", "definition": "b/d", "description": "", "templateType": "anything"}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "u1*ans_i + u2*ans_j", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x_coeff", "y_coeff", "xpow", "ypow", "xval", "yval", "a", "b", "d", "ans_i", "ans_j", "u1", "u2", "answer"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Enter your answer, correct to&nbsp;<strong>2 d.p.</strong>, in the box provided.</p>\n<p><span class=\"student-answer\" data-bind=\"css: {answered: scoreFeedback.answered, 'has-warnings': hasWarnings}, attr: {&quot;feedback-state&quot;: scoreFeedback.state}\" feedback-state=\"none\"></span><br/>[[0]]</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "answer", "maxValue": "answer", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "28_02_02a Calculating divergence 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Given the function:</p>\n<p>$\\underline{F}=\\left(\\simplify{ {x_exps[0]}x + {y_exps[0]}y + {z_exps[0]}z +{coeffs[0]}},\\quad \\simplify{&nbsp;{x_exps[1]}x + {y_exps[1]}y + {z_exps[1]}z +{coeffs[1]} },\\quad \\simplify{&nbsp;{x_exps[2]}x + {y_exps[2]}y + {z_exps[2]}z +{coeffs[2]} }\\right)$</p>\n<p>Evaluate the divergence $\\underline{\\nabla}.\\underline{F}$</p>\n<p>at the point $x=\\var{values[0]}$,&nbsp;$y=\\var{values[1]}$ and&nbsp;$z=\\var{values[2]}$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>The divergence of a function</p>\n<p>$\\underline{F}=\\left(\\simplify{ {x_exps[0]}x + {y_exps[0]}y + {z_exps[0]}z +{coeffs[0]} },\\quad \\simplify{&nbsp;{x_exps[1]}x + {y_exps[1]}y + {z_exps[1]}z +{coeffs[1]} },\\quad \\simplify{&nbsp;{x_exps[2]}x + {y_exps[2]}y + {z_exps[2]}z +{coeffs[2]} }\\right)$</p>\n<p>is</p>\n<p>$\\underline{\\nabla}.\\underline{F}=\\dfrac{\\partial (\\simplify{&nbsp;{x_exps[0]}x + {y_exps[0]}y + {z_exps[0]}z +{coeffs[0]} })}{\\partial x}+\\dfrac{\\partial (\\simplify{&nbsp;{x_exps[1]}x + {y_exps[1]}y + {z_exps[1]}z +{coeffs[1]}&nbsp;})}{\\partial y}+\\dfrac{\\partial (\\simplify{&nbsp;{x_exps[2]}x + {y_exps[2]}y + {z_exps[2]}z +{coeffs[2]} })}{\\partial z}$</p>\n<p>$\\phantom{\\underline{\\nabla}.\\underline{F}}=(\\var{x_exps[0]}) +(\\var{y_exps[1]}) +(\\var{z_exps[2]}) = \\var{answer}$</p>\n<p>Which is the same at every point!</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {}, "variables": {"z_exps": {"name": "z_exps", "group": "Ungrouped variables", "definition": "[random(-5..5), random(-5..5), random(-5..5)]", "description": "", "templateType": "anything"}, "x_exps": {"name": "x_exps", "group": "Ungrouped variables", "definition": "[random(-5..5), random(-5..5), random(-5..5)]", "description": "", "templateType": "anything"}, "values": {"name": "values", "group": "Ungrouped variables", "definition": "[random(-200..200)/100, random(-200..200)/100, random(-200..200)/100]", "description": "<p>Values of [x, y, z]</p>", "templateType": "anything"}, "y_exps": {"name": "y_exps", "group": "Ungrouped variables", "definition": "[random(-5..5), random(-5..5), random(-5..5)]", "description": "", "templateType": "anything"}, "coeffs": {"name": "coeffs", "group": "Ungrouped variables", "definition": "[random(-10..10), random(-10..10), random(-10..10)]", "description": "", "templateType": "anything"}, "part1": {"name": "part1", "group": "Ungrouped variables", "definition": "x_exps[0]", "description": "", "templateType": "anything"}, "part2": {"name": "part2", "group": "Ungrouped variables", "definition": "y_exps[1]", "description": "", "templateType": "anything"}, "part3": {"name": "part3", "group": "Ungrouped variables", "definition": "z_exps[2]", "description": "", "templateType": "anything"}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "part1+part2+part3", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x_exps", "y_exps", "z_exps", "coeffs", "values", "part1", "part2", "part3", "answer"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Enter your answer, correct to&nbsp;<strong>2 d.p.</strong>, in the box provided.</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "minValue": "answer", "maxValue": "answer", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "28_02_02b Calculating divergence 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Given the function:</p>\n<p>$\\underline{F}=\\left(\\simplify{{coeffs[0]}x^{x_exps[0]} +{z_exps[0]}*y^{y_exps[0]}},\\quad \\simplify{{coeffs[1]}x^{x_exps[1]} +{z_exps[1]}y^{y_exps[1]}},\\quad \\simplify{{coeffs[2]}x^{x_exps[2]} +{z_exps[2]}y^{y_exps[2]}}\\right)$</p>\n<p>Evaluate the divergence $\\underline{\\nabla}.\\underline{F}$</p>\n<p>at the point where $x=\\var{values[0]}$,&nbsp;$y=\\var{values[1]}$ and&nbsp;$z=\\var{values[2]}$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>The divergence of a function</p>\n<p>$\\underline{F}=\\left(\\simplify{ {coeffs[0]}x^{x_exps[0]} +{z_exps[0]}*y^{y_exps[0]} },\\quad \\simplify{ {coeffs[1]}x^{x_exps[1]} +{z_exps[1]}y^{y_exps[1]} },\\quad \\simplify{ {coeffs[2]}x^{x_exps[2]} +{z_exps[2]}y^{y_exps[2]} }\\right)$</p>\n<p>is</p>\n<p>$\\underline{\\nabla}.\\underline{F}=\\dfrac{\\partial (\\simplify{&nbsp;{coeffs[0]}x^{x_exps[0]} +{z_exps[0]}*y^{y_exps[0]} })}{\\partial x}+\\dfrac{\\partial (\\simplify{&nbsp;{coeffs[1]}x^{x_exps[1]} +{z_exps[1]}y^{y_exps[1]}&nbsp; }) }{\\partial y}+\\dfrac{\\partial (\\simplify{&nbsp;{coeffs[2]}x^{x_exps[2]} +{z_exps[2]}y^{y_exps[2]}&nbsp; }) }{\\partial z}$</p>\n<p>$\\phantom{\\underline{\\nabla}.\\underline{F}}=(\\simplify{&nbsp;{coeffs[0]}{x_exps[0]}x^{x_exps[0]-1}&nbsp; }) +(\\simplify{ {z_exps[1]}{y_exps[1]}&nbsp;y^{y_exps[1]-1}&nbsp; }) + (0)$</p>\n<p>At the point $x=\\var{values[0]}$,&nbsp;$y=\\var{values[1]}$ and&nbsp;$z=\\var{values[2]}$ the values can be substituted into the above to give:</p>\n<p>$\\underline{\\nabla}.\\underline{F}=\\left((\\var{part1}) + (\\var{part2}) + (\\var{part3})\\right) = \\var{answer}$</p>\n<p></p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {}, "variables": {"z_exps": {"name": "z_exps", "group": "Ungrouped variables", "definition": "[random(0..5), random(0..5), random(0..5)]", "description": "", "templateType": "anything"}, "x_exps": {"name": "x_exps", "group": "Ungrouped variables", "definition": "[random(0..5), random(0..5), random(0..5)]", "description": "", "templateType": "anything"}, "values": {"name": "values", "group": "Ungrouped variables", "definition": "[random(-200..200)/100, random(-200..200)/100, random(-200..200)/100]", "description": "<p>Values of [x, y, z]</p>", "templateType": "anything"}, "y_exps": {"name": "y_exps", "group": "Ungrouped variables", "definition": "[random(0..5), random(0..5), random(0..5)]", "description": "", "templateType": "anything"}, "coeffs": {"name": "coeffs", "group": "Ungrouped variables", "definition": "[random(0..10), random(-10..10), random(0..10)]", "description": "", "templateType": "anything"}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "part1+part2+part3", "description": "", "templateType": "anything"}, "part1": {"name": "part1", "group": "Ungrouped variables", "definition": "(coeffs[0]*x_exps[0]*(values[0]^(x_exps[0]-1)))", "description": "", "templateType": "anything"}, "part2": {"name": "part2", "group": "Ungrouped variables", "definition": "z_exps[1]*y_exps[1]*values[1]^(y_exps[1]-1)", "description": "", "templateType": "anything"}, "part3": {"name": "part3", "group": "Ungrouped variables", "definition": "0", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x_exps", "y_exps", "z_exps", "coeffs", "values", "part1", "part2", "part3", "answer"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Enter your answer, correct to&nbsp;<strong>2 d.p.</strong>, in the box provided.</p>", "minValue": "answer", "maxValue": "answer", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "28_02_02c Calculating divergence 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Given the function:</p>\n<p>$\\underline{F}=\\left(\\simplify{{coeffs[0]}x^{x_exps[0]}y^{y_exps[0]}z^{z_exps[0]}},\\quad \\simplify{{coeffs[1]}x^{x_exps[1]}y^{y_exps[1]}z^{z_exps[1]}},\\quad \\simplify{{coeffs[2]}x^{x_exps[2]}y^{y_exps[2]}z^{z_exps[2]}}\\right)$</p>\n<p>Evaluate the divergence $\\underline{\\nabla}.\\underline{F}$</p>\n<p>at the point where $x=\\var{values[0]}$,&nbsp;$y=\\var{values[1]}$ and&nbsp;$z=\\var{values[2]}$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>The divergence of a function</p>\n<p>$\\underline{F}=\\left(\\simplify{{coeffs[0]}x^{x_exps[0]}y^{y_exps[0]}z^{z_exps[0]}},&nbsp;\\simplify{{coeffs[1]}x^{x_exps[1]}y^{y_exps[1]}z^{z_exps[1]}}, \\simplify{{coeffs[2]}x^{x_exps[2]}y^{y_exps[2]}z^{z_exps[2]}}\\right)$</p>\n<p>is</p>\n<p>$\\underline{\\nabla}.\\underline{F}=\\dfrac{\\partial (\\simplify{{coeffs[0]}x^{x_exps[0]}y^{y_exps[0]}z^{z_exps[0]}})}{\\partial x}+\\dfrac{\\partial (\\simplify{{coeffs[1]}x^{x_exps[1]}y^{y_exps[1]}z^{z_exps[1]}})}{\\partial y}+\\dfrac{\\partial (\\simplify{{coeffs[2]}x^{x_exps[2]}y^{y_exps[2]}z^{z_exps[2]}})}{\\partial z}$</p>\n<p>$\\phantom{\\underline{\\nabla}.\\underline{F}}=(\\simplify{{coeffs[0]}{x_exps[0]}x^{x_exps[0]-1}y^{y_exps[0]}z^{z_exps[0]}}) + (\\simplify{{coeffs[1]}x^{x_exps[1]}{y_exps[1]}y^{y_exps[1]-1}z^{z_exps[1]}}) + (\\simplify{{coeffs[2]}x^{x_exps[2]}y^{y_exps[2]}{z_exps[2]}z^{z_exps[2]-1}})$</p>\n<p>At the point $x=\\var{values[0]}$,&nbsp;$y=\\var{values[1]}$ and&nbsp;$z=\\var{values[2]}$ the values can be substituted into the above to give:</p>\n<p>$\\underline{\\nabla}.\\underline{F}=\\left((\\var{part1}) + (\\var{part2}) + (\\var{part3})\\right) = \\var{answer}$</p>\n<p></p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {}, "variables": {"z_exps": {"name": "z_exps", "group": "Ungrouped variables", "definition": "[random(0..5), random(0..5), random(0..5)]", "description": "", "templateType": "anything"}, "x_exps": {"name": "x_exps", "group": "Ungrouped variables", "definition": "[random(0..5), random(0..5), random(0..5)]", "description": "", "templateType": "anything"}, "values": {"name": "values", "group": "Ungrouped variables", "definition": "[random(-200..200)/100, random(-200..200)/100, random(-200..200)/100]", "description": "<p>Values of [x, y, z]</p>", "templateType": "anything"}, "y_exps": {"name": "y_exps", "group": "Ungrouped variables", "definition": "[random(0..5), random(0..5), random(0..5)]", "description": "", "templateType": "anything"}, "coeffs": {"name": "coeffs", "group": "Ungrouped variables", "definition": "[random(0..10), random(-10..10), random(0..10)]", "description": "", "templateType": "anything"}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "(coeffs[0]*x_exps[0]*(values[0]^(x_exps[0]-1))*(values[1]^y_exps[0])*(values[2]^z_exps[0])) +\n(coeffs[1]*y_exps[1]*(values[1]^(y_exps[1]-1))*(values[0]^x_exps[1])*(values[2]^z_exps[1])) +\n(coeffs[2]*z_exps[2]*(values[2]^(z_exps[2]-1))*(values[0]^x_exps[2])*(values[1]^y_exps[2]))", "description": "", "templateType": "anything"}, "part1": {"name": "part1", "group": "Ungrouped variables", "definition": "(coeffs[0]*x_exps[0]*(values[0]^(x_exps[0]-1))*(values[1]^y_exps[0])*(values[2]^z_exps[0]))", "description": "", "templateType": "anything"}, "part2": {"name": "part2", "group": "Ungrouped variables", "definition": "(coeffs[1]*y_exps[1]*(values[1]^(y_exps[1]-1))*(values[0]^x_exps[1])*(values[2]^z_exps[1]))", "description": "", "templateType": "anything"}, "part3": {"name": "part3", "group": "Ungrouped variables", "definition": "(coeffs[2]*z_exps[2]*(values[2]^(z_exps[2]-1))*(values[0]^x_exps[2])*(values[1]^y_exps[2]))", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x_exps", "y_exps", "z_exps", "coeffs", "values", "answer", "part1", "part2", "part3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Enter your answer, correct to&nbsp;<strong>2 d.p.</strong>, in the box provided.</p>", "minValue": "answer", "maxValue": "answer", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "28_02_01a Curl of a function 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Given the vector function:</p>\n<p>$\\underline{F}=\\left( (\\simplify{{x_coeffs[0]}x+{y_coeffs[0]}y+{z_coeffs[0]}z+{consts[0]}}),\\quad (\\simplify{{x_coeffs[1]}x+{y_coeffs[1]}y+{z_coeffs[1]}z+{consts[1]}}),\\quad (\\simplify{{x_coeffs[2]}x+{y_coeffs[2]}y+{z_coeffs[2]}z+{consts[2]}}) \\right)$</p>\n<p>Determine the curl $\\underline{\\nabla}\\times \\underline{F}$ in the form</p>\n<p>$\\underline{\\nabla}\\times \\underline{F}=(a, b,&nbsp;c)$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>Given the function</p>\n<p>$\\underline{F}=\\left( (\\simplify{{x_coeffs[0]}x+{y_coeffs[0]}y+{z_coeffs[0]}z+{consts[0]}}),\\quad (\\simplify{{x_coeffs[1]}x+{y_coeffs[1]}y+{z_coeffs[1]}z+{consts[1]}}),\\quad (\\simplify{{x_coeffs[2]}x+{y_coeffs[2]}y+{z_coeffs[2]}z+{consts[2]}}) \\right)$</p>\n<p>the curl can be defined as</p>\n<p>$\\displaystyle\\underline{\\nabla}\\times \\underline{F} = \\left(\\frac{\\partial (\\simplify{{x_coeffs[2]}x+{y_coeffs[2]}y+{z_coeffs[2]}z+{consts[2]}})}{\\partial y}-\\frac{\\partial (\\simplify{{x_coeffs[1]}x+{y_coeffs[1]}y+{z_coeffs[1]}z+{consts[1]}})}{\\partial z}\\right),\\quad<br/>\\left(\\frac{\\partial (\\simplify{{x_coeffs[0]}x+{y_coeffs[0]}y+{z_coeffs[0]}z+{consts[0]}})}{\\partial z}-\\frac{\\partial (\\simplify{{x_coeffs[2]}x+{y_coeffs[2]}y+{z_coeffs[2]}z+{consts[2]}})}{\\partial x}\\right),\\quad<br/>\\left(\\frac{\\partial (\\simplify{{x_coeffs[1]}x+{y_coeffs[1]}y+{z_coeffs[1]}z+{consts[1]}})}{\\partial x}-\\frac{\\partial (\\simplify{{x_coeffs[0]}x+{y_coeffs[0]}y+{z_coeffs[0]}z+{consts[0]}})}{\\partial y}\\right)$</p>\n<p>$\\underline{\\nabla}\\times \\underline{F}=(\\var{a_ans}, \\var{b_ans}, \\var{c_ans})$</p>\n<p></p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {}, "variables": {"c_ans": {"name": "c_ans", "group": "Ungrouped variables", "definition": "x_coeffs[1] - y_coeffs[0]", "description": "", "templateType": "anything"}, "z_coeffs": {"name": "z_coeffs", "group": "Ungrouped variables", "definition": "[random(-10..10), random(-10..10), random(-10..10)]", "description": "<p>The coefficients of the z part of the [i, j, k] parts of the equation.</p>", "templateType": "anything"}, "a_ans": {"name": "a_ans", "group": "Ungrouped variables", "definition": "y_coeffs[2] - z_coeffs[1]", "description": "", "templateType": "anything"}, "consts": {"name": "consts", "group": "Ungrouped variables", "definition": "[random(-10..10), random(-10..10), random(-10..10)]", "description": "<p>The coefficients of the constants part of the [i, j, k] parts of the equation.</p>", "templateType": "anything"}, "b_ans": {"name": "b_ans", "group": "Ungrouped variables", "definition": "z_coeffs[0] - x_coeffs[2]", "description": "", "templateType": "anything"}, "x_coeffs": {"name": "x_coeffs", "group": "Ungrouped variables", "definition": "[random(-10..10), random(-10..10), random(-10..10)]", "description": "<p>The coefficients of the x part of the [i, j, k] parts of the equation.</p>", "templateType": "anything"}, "y_coeffs": {"name": "y_coeffs", "group": "Ungrouped variables", "definition": "[random(-10..10), random(-10..10), random(-10..10)]", "description": "<p>The coefficients of the y part of the [i, j, k] parts of the equation.</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": "250"}, "ungrouped_variables": ["x_coeffs", "y_coeffs", "z_coeffs", "consts", "a_ans", "b_ans", "c_ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Enter the values of a, b, and c in the boxes provided:</p>\n<p>$a = $ [[0]]</p>\n<p>$b = $ [[1]]</p>\n<p>$c = $ [[2]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1/3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a_ans", "maxValue": "a_ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1/3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b_ans", "maxValue": "b_ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1/3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c_ans", "maxValue": "c_ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "28_02_01b Curl of a function 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Given the vector function:</p>\n<p>$\\underline{F}=\\left( \\simplify{ {consts[0]}*x^{x_coeffs[0]}y^{y_coeffs[0]}z^{z_coeffs[0]} },\\quad \\simplify{ {consts[1]}*x^{x_coeffs[1]}y^{y_coeffs[1]}z^{z_coeffs[1]} },\\quad \\simplify{{consts[2]}*x^{x_coeffs[2]}y^{y_coeffs[2]}z^{z_coeffs[2]} } \\right)$</p>\n<p>Determine the curl $\\underline{\\nabla}\\times \\underline{F}$</p>\n<p>at the point $(x,y,z) = (\\var{x},\\var{y},\\var{z})$</p>\n<p>giving your answer in the form</p>\n<p>$\\underline{\\nabla}\\times \\underline{F}=(a, b,&nbsp;c)$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>Given the function</p>\n<p>$\\underline{F}=\\left( \\simplify{ {consts[0]}*x^{x_coeffs[0]}y^{y_coeffs[0]}z^{z_coeffs[0]} },\\quad \\simplify{ {consts[1]}*x^{x_coeffs[1]}y^{y_coeffs[1]}z^{z_coeffs[1]} },\\quad \\simplify{{consts[2]}*x^{x_coeffs[2]}y^{y_coeffs[2]}z^{z_coeffs[2]} } \\right)$</p>\n<p>the curl can be defined as</p>\n<p>$\\displaystyle\\underline{\\nabla}\\times \\underline{F} = \\left(\\frac{\\partial (\\simplify{ {consts[2]}*x^{x_coeffs[2]}y^{y_coeffs[2]}z^{z_coeffs[2]} })}{\\partial y}-\\frac{\\partial (\\simplify{ {consts[1]}*x^{x_coeffs[1]}y^{y_coeffs[1]}z^{z_coeffs[1]} })}{\\partial z}\\right),\\quad<br/>\\left(\\frac{\\partial (\\simplify{ {consts[0]}*x^{x_coeffs[0]}y^{y_coeffs[0]}z^{z_coeffs[0]} })}{\\partial z}-\\frac{\\partial (\\simplify{ {consts[2]}*x^{x_coeffs[2]}y^{y_coeffs[2]}z^{z_coeffs[2]} })}{\\partial x}\\right),\\quad<br/>\\left(\\frac{\\partial (\\simplify{ {consts[1]}*x^{x_coeffs[1]}y^{y_coeffs[1]}z^{z_coeffs[1]} })}{\\partial x}-\\frac{\\partial (\\simplify{ {consts[0]}*x^{x_coeffs[0]}y^{y_coeffs[0]}z^{z_coeffs[0]} })}{\\partial y}\\right)$</p>\n<p>$\\displaystyle\\phantom{\\underline{\\nabla}\\times \\underline{F}} = \\left({(\\simplify{ {consts[2]}*x^{x_coeffs[2]}{y_coeffs[2]}y^{y_coeffs[2]-1}z^{z_coeffs[2]} })} - {(\\simplify{ {consts[1]}*x^{x_coeffs[1]}y^{y_coeffs[1]}{z_coeffs[1]}z^{z_coeffs[1]-1} })}\\right),\\quad<br/>\\left({(\\simplify{ {consts[0]}*x^{x_coeffs[0]}y^{y_coeffs[0]}{z_coeffs[0]}z^{z_coeffs[0]-1} })} - {(\\simplify{ {consts[2]}*{x_coeffs[2]}x^{x_coeffs[2]-1}y^{y_coeffs[2]}z^{z_coeffs[2]} })}\\right),\\quad<br/>\\left({(\\simplify{ {consts[1]}*{x_coeffs[1]}x^{x_coeffs[1]-1}y^{y_coeffs[1]}z^{z_coeffs[1]} })} - {(\\simplify{ {consts[0]}*x^{x_coeffs[0]}{y_coeffs[0]}y^{y_coeffs[0]-1}z^{z_coeffs[0]} })}\\right)$</p>\n<p>Substituting values of $(x,y,z) = (\\var{x}, \\var{y}, \\var{z})$</p>\n<p>$\\displaystyle{\\underline{\\nabla}\\times \\underline{F}} =&nbsp; \\big( (\\var{term1})-(\\var{term2}), \\quad (\\var{term3})-(\\var{term4}), \\quad (\\var{term5})-(\\var{term6}) \\big)$</p>\n<p>$\\phantom{\\underline{\\nabla}\\times \\underline{F}}=(\\var{a_ans}, \\var{b_ans}, \\var{c_ans})$</p>\n<p></p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {}, "variables": {"c_ans": {"name": "c_ans", "group": "Ungrouped variables", "definition": "term5-term6", "description": "", "templateType": "anything"}, "z_coeffs": {"name": "z_coeffs", "group": "Ungrouped variables", "definition": "[random(0..2), random(0..2), random(0..2)]", "description": "<p>The coefficients of the z part of the [i, j, k] parts of the equation.</p>", "templateType": "anything"}, "a_ans": {"name": "a_ans", "group": "Ungrouped variables", "definition": "term1-term2", "description": "", "templateType": "anything"}, "consts": {"name": "consts", "group": "Ungrouped variables", "definition": "[random(-5..5), random(-5..5), random(-5..5)]", "description": "<p>The coefficients of the constants part of the [i, j, k] parts of the equation.</p>", "templateType": "anything"}, "b_ans": {"name": "b_ans", "group": "Ungrouped variables", "definition": "term3-term4", "description": "", "templateType": "anything"}, "x_coeffs": {"name": "x_coeffs", "group": "Ungrouped variables", "definition": "[random(0..2), random(0..2), random(0..2)]", "description": "<p>The coefficients of the x part of the [i, j, k] parts of the equation.</p>", "templateType": "anything"}, "y_coeffs": {"name": "y_coeffs", "group": "Ungrouped variables", "definition": "[random(0..2), random(0..2), random(0..2)]", "description": "<p>The coefficients of the y part of the [i, j, k] parts of the equation.</p>", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", 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