In the following question find the $(x,y)$ coordinates of the single stationary point $(a,b) \\in D$ of the continuous function $f: D \\rightarrow \\mathbb{R}$:
\n\\[f(x,y) = \\simplify[std]{{a} + {b}*e^(-(x-{c})^2-(y-{d})^2)}\\]
\nwhere \\[D = \\{(x,y): \\simplify[std]{(x-{c})^2+(y-{d})^2}\\} \\le \\var{r}\\]
\nThat is, $D$ is a disk of radius $\\simplify[std]{sqrt({r})}$ and centre $(\\var{c},\\var{d})$.
\nInput both cooordinates as fractions or integers and not decimals.
", "parts": [{"marks": 0, "stepsPenalty": 0, "prompt": "$x$–coordinate, $a=$ [[0]]
\n$y$–coordinate, $b=$ [[1]]
\nInput the value of $f(x,y)$ at $(a,b)$:
\n$f(a,b)=$ [[2]]
\nIf you want some help, click on Show steps. You will not lose any marks if you do so.
", "type": "gapfill", "gaps": [{"marks": 2, "scripts": {}, "allowFractions": false, "minValue": "{c}", "maxValue": "{c}", "showCorrectAnswer": true, "showPrecisionHint": false, "type": "numberentry", "correctAnswerFraction": false}, {"marks": 2, "scripts": {}, "allowFractions": false, "minValue": "{d}", "maxValue": "{d}", "showCorrectAnswer": true, "showPrecisionHint": false, "type": "numberentry", "correctAnswerFraction": false}, {"marks": 1, "scripts": {}, "allowFractions": false, "minValue": "{a+b}", "maxValue": "{a+b}", "showCorrectAnswer": true, "showPrecisionHint": false, "type": "numberentry", "correctAnswerFraction": false}], "showCorrectAnswer": true, "steps": [{"marks": 0, "showCorrectAnswer": true, "type": "information", "scripts": {}, "prompt": "\n \n \nThe $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving
the following 2 equations for $x$ and $y$
\\[\\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 \\]
\n \n \n \nIn this case you get two equations to solve for $x$ and $y$
\n \n \n "}], "scripts": {}}], "name": "Simon's copy of Functions of two variables: Stationary points 3", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "variable_groups": [], "functions": {}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"s3": {"group": "Ungrouped variables", "templateType": "anything", "name": "s3", "definition": "random(1,-1)", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "name": "b", "definition": "s2*random(1..9)", "description": ""}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "name": "s5", "definition": "random(1,-1)", "description": ""}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "name": "s1", "definition": "random(1,-1)", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "name": "d", "definition": "s4*random(1..9)", "description": ""}, "r": {"group": "Ungrouped variables", "templateType": "anything", "name": "r", "definition": "random(2,3,5,6,7)", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "name": "c", "definition": "s3*random(1..9)", "description": ""}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "name": "s4", "definition": "random(1,-1)", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "name": "a", "definition": "s1*random(1..9)", "description": ""}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "name": "s2", "definition": "random(1,-1)", "description": ""}}, "type": "question", "ungrouped_variables": ["a", "c", "b", "d", "s3", "s2", "s1", "s5", "s4", "r"], "tags": ["Calculus", "Differentiation", "Simultaneous equations", "calculus", "differentiate", "differentiation", "functions of two variables", "partial derivative", "partial differentiation", "simultaneous equations", "stationary points"], "metadata": {"notes": "\n \t\t10/07/2012:
\n \t\tAdded tags.
Question appears to be working correctly.
\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find the coordinates of the stationary point for $f: D \\rightarrow \\mathbb{R}$: $f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$, $D$ is a disk centre $(c,d)$.
"}, "advice": "\nThe $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving
the following 2 equations for $x$ and $y$
\\[\\begin{eqnarray*} \\partial f \\over \\partial x &=&0\\\\ \\\\ \\partial f \\over \\partial y &=&0 \\end{eqnarray*} \\]
\nIn this case you get two equations to solve for $x$ and $y$
\n\\[\\begin{eqnarray*} \\simplify[std]{{-2*b}*(x-{c})*e^(-(x-{c})^2-(y-{d})^2)}&=&0\\\\ \\\\ \\simplify[std]{{-2*b}*(y-{d})*e^(-(x-{c})^2-(y-{d})^2)}&=&0 \\end{eqnarray*} \\]
We can cancel off the term $\\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)}$ in both equations as $\\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)} \\neq 0,\\;\\forall x,\\;y$.
On solving these we get \\[ x = \\var{c},\\;\\;\\;y=\\var{d}\\]
\nSo the stationary point is $(\\var{c},\\var{d}) \\in D$.
\nOn substituting these values into $f(x,y)$ we get:
\n\\[f(\\var{c},\\var{d})=\\simplify[std]{{a}+{b}*e^0={a+b}}\\]
\n ", "showQuestionGroupNames": false, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}