// Numbas version: exam_results_page_options
{"name": "Find the stationary point of a function on a disk", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2,3,5,6,7)", "description": "", "name": "r"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s5"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "name": "d"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "name": "c"}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s4"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "b"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "s3", "s2", "s1", "s5", "s4", "r"], "name": "Find the stationary point of a function on a disk", "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "showCorrectAnswer": true, "minValue": "{c}", "maxValue": "{c}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 2, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "showCorrectAnswer": true, "minValue": "{d}", "maxValue": "{d}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 2, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "showCorrectAnswer": true, "minValue": "{a+b}", "maxValue": "{a+b}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "information", "showCorrectAnswer": true, "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            ", "variableReplacements": [], "marks": 0}], "prompt": "<p>$x$ &ndash; coordinate, $a=$&nbsp;[[0]]</p>\n<p>$y$ &ndash; coordinate, $b=$&nbsp;[[1]]</p>\n<p>Input 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></p>", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "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</p>\n<p>\\[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><em>Input both cooordinates as fractions or integers and not decimals.</em></p>", "tags": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "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>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<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{align} <br/>\\partial f \\over \\partial x &amp;= 0 \\\\[1em]<br/> \\partial f \\over \\partial y &amp;= 0 <br/>\\end{align}</p>\n<p>In this case you get two equations to solve for $x$ and $y$</p>\n<p>\\begin{align} <br/>\\simplify[std]{{-2*b}*(x-{c})*e^(-(x-{c})^2-(y-{d})^2)} &amp;= 0 \\\\[1em]<br/>\\simplify[std]{{-2*b}*(y-{d})*e^(-(x-{c})^2-(y-{d})^2)} &amp;= 0 <br/>\\end{align}</p>\n<p>We can cancel off the term $\\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)}$ in both equations as $\\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)} \\neq 0,\\;\\forall x,\\;y$.&nbsp;&nbsp;</p>\n<p>On solving these, we get</p>\n<p>\\[ x = \\var{c}, \\quad \\;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,!zeropower,!othernumbers]{{a}+{b}*e^0={a+b}} \\]</p>", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}