// Numbas version: exam_results_page_options
{"name": "Functions of two variables: Stationary points 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "s3", "s2", "s1", "s5", "s4", "r"], "name": "Functions of two variables: Stationary points 3", "tags": ["Calculus", "Differentiation", "Simultaneous equations", "calculus", "differentiate", "differentiation", "functions of two variables", "partial derivative", "partial differentiation", "simultaneous equations", "stationary points"], "preamble": {"css": "", "js": ""}, "advice": "\n    <p>The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving<br/>the following 2 equations for $x$ and $y$</p>\n    <p>\\[\\begin{eqnarray*} \\partial f \\over \\partial x &amp;=&amp;0\\\\ \\\\ \\partial f \\over \\partial y &amp;=&amp;0 \\end{eqnarray*} \\]</p>\n    <p>In this case you get two equations to solve for $x$ and $y$</p>\n    <p>\\[\\begin{eqnarray*} \\simplify[std]{{-2*b}*(x-{c})*e^(-(x-{c})^2-(y-{d})^2)}&amp;=&amp;0\\\\ \\\\ \\simplify[std]{{-2*b}*(y-{d})*e^(-(x-{c})^2-(y-{d})^2)}&amp;=&amp;0 \\end{eqnarray*} \\]<br/>We can cancel off the term $\\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)}$ in both equations as&nbsp; &nbsp;$\\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)} \\neq 0,\\;\\forall x,\\;y$.&nbsp;&nbsp;</p>\n    <p>On solving these we get \\[ x = \\var{c},\\;\\;\\;y=\\var{d}\\]</p>\n    <p>So the stationary point is $(\\var{c},\\var{d}) \\in D$.</p>\n    <p>On substituting these values into $f(x,y)$ we get:</p>\n    <p>\\[f(\\var{c},\\var{d})=\\simplify[std]{{a}+{b}*e^0={a+b}}\\]</p>\n    ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 0, "prompt": "<p>$x$&ndash;coordinate, $a=$&nbsp;[[0]]</p>\n<p>$y$&ndash;coordinate, $b=$&nbsp;[[1]]</p>\n<p>Input the value of $f(x,y)$ at $(a,b)$:</p>\n<p>$f(a,b)=$&nbsp;[[2]]</p>\n<p>If you want some help, click on <em>Show steps.</em>&nbsp;You will not lose any marks if you do so.</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 2, "maxValue": "{c}", "minValue": "{c}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 2, "maxValue": "{d}", "minValue": "{d}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "{a+b}", "minValue": "{a+b}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"type": "information", "showCorrectAnswer": true, "scripts": {}, "prompt": "\n          \n          \n          <p>The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving<br/>the following 2 equations for $x$ and $y$</p>\n          \n          \n          \n          <p>\\[\\begin{eqnarray*}\n          \n          \\partial f \\over \\partial x &=&0\\\\\n          \n          \\\\\n          \n          \\partial f \\over \\partial y &=&0\n          \n          \\end{eqnarray*}\n          \n          \\]</p>\n          \n          \n          \n          <p>In this case you get two equations to solve for $x$ and $y$</p>\n          \n          \n            ", "marks": 0}], "type": "gapfill"}], "statement": "<p>In the following question find the $(x,y)$ coordinates of the single stationary point $(a,b) \\in D$ of the continuous function $f: D \\rightarrow \\mathbb{R}$:</p>\n<p>\\[f(x,y) = \\simplify[std]{{a} + {b}*e^(-(x-{c})^2-(y-{d})^2)}\\]</p>\n<p>where \\[D = \\{(x,y): \\simplify[std]{(x-{c})^2+(y-{d})^2}\\} \\le \\var{r}\\]</p>\n<p>That is, $D$ is a disk of radius $\\simplify[std]{sqrt({r})}$ and centre $(\\var{c},\\var{d})$.</p>\n<p>Input both cooordinates as fractions or integers and not decimals.</p>", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "s3*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "s2*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "s4*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "s5": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "s4": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s4", "description": ""}, "r": {"definition": "random(2,3,5,6,7)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>10/07/2012:</strong></p>\n        \t\t<p>Added tags.<strong><br /></strong></p>\n        \t\t<p>Question appears to be working correctly.</p>\n        \t\t<p><strong>&nbsp;</strong></p>\n        \t\t", "description": "<p>Find the coordinates of the stationary point for $f: D \\rightarrow \\mathbb{R}$:&nbsp;$f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$,&nbsp;$D$ is a disk centre $(c,d)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}