// Numbas version: exam_results_page_options
{"name": "Robert's copy of Functions of two variables: Stationary points 2", "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", "ch", "d", "m", "s5", "a1", "other", "b1", "c2", "c1", "b", "check", "d1"], "name": "Robert's copy of Functions of two variables: Stationary points 2", "tags": ["Calculus", "Differentiation", "calculus", "derivative", "differentiation", "functions of two variables", "partial derivative", "partial differentiation", "stationary points", "stationary points of functions of two variables"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>\\[\\begin{eqnarray*} {\\partial f \\over \\partial x} &amp;=&amp;\\simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))}\\\\ \\\\ \\partial f \\over \\partial y &amp;=&amp;\\simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})} \\end{eqnarray*}\\]</p>\n<h4>b)</h4>\n<p>$(a,b)$ is a stationary point for the function $f(x,y)$ if $f_x=0,\\;\\;f_y=0$,where the partial derivatives are evaluated at $x=a,\\;\\;y=b$. <br/>So you have to make sure that both these partial derivatives are $0$ at the stationary point.</p>\n<p>For this example we have from the above equations that:<br/>\\[\\begin{eqnarray*} \\simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))}&amp;=&amp;0,\\qquad &amp;\\mathbf{(1)}&amp;\\\\ \\\\ \\simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})}&amp;=&amp;0, \\qquad &amp;\\mathbf{(2)}&amp; \\end{eqnarray*}\\]</p>\n<p>The left hand side of equation (1) can be factorised as:</p>\n<p>\\[\\simplify[std]{({a1}x+{b1}y)*({c1}x+{d1}y)=0}\\]</p>\n<p>and so we have:<br/>\\[y=\\simplify[std]{{-a1}/{b1}*x},\\mbox{ or } y= \\simplify[std]{{-c1}/{d1}*x}\\]</p>\n<h5>First case: $y= \\simplify[std]{{-a1}/{b1}*x}$</h5>\n<p>Substituting this into equation (2) gives:<br/>\\[\\simplify[std]{{b}/2*x^2-{2c*a1}/{b1}*x^2+{d}}=0 \\Rightarrow \\simplify[std]{{-b*b1+4*c*a1}/{2*b1}*x^2={d}}\\]</p>\n<p>Hence $x=\\var{m}\\mbox{ or } x = \\var{-m}$ and the stationary points which are on the list and which you had to choose are:<br/>\\[\\left(\\var{m},\\simplify[std]{-{a1*m}/{b1}}\\right)\\mbox{ and }\\left(\\var{-m},\\simplify[std]{{a1*m}/{b1}}\\right)\\]</p>\n<h5>Second case: $y= \\simplify[std]{{-c1}/{d1}*x}$</h5>\n<p>{check}</p>\n<p>Substituting this into equation (2) gives:<br/>\\[\\simplify[std]{{b}/2*x^2-{2c*c1}/{d1}*x^2+{d}}=0 \\Rightarrow \\simplify[std]{{-b*d1+4*c*c1}/{2*d1}*x^2={d}}\\]</p>\n<p>{other}</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Enter the partial derivatives here. Note if you want to enter a product of unknowns, such as $xy$ then you input the expression in the form <code>x*y</code>.</p>\n<p>$\\displaystyle { \\partial f \\over \\partial x}=$&nbsp;[[0]]</p>\n<p>$\\displaystyle {\\partial f \\over \\partial y}=$&nbsp;[[1]]</p>", "marks": 0, "gaps": [{"expectedvariablenames": ["x", "y"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))", "marks": 2, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme"}, {"expectedvariablenames": ["x", "y"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})", "marks": 2, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"maxAnswers": 0, "displayColumns": 3, "prompt": "\n      <h4>Finding Stationary Points.</h4>\n        <p>Tick the two choices which give stationary points for $f(x,y)$.</p>\n        <p>Note that the easiest way to do this question is to substitute the values for $x$ and for $y$ into the expressions for $\\displaystyle {\\partial f \\over \\partial x}$ and $\\displaystyle{\\partial f \\over \\partial y}$ and see if you get $0$ for both.</p>\n      ", "matrix": [2, 2, 0, 0, 0, 0], "shuffleChoices": true, "minAnswers": 0, "choices": ["<p>$x=\\var{m},\\;\\;y=\\simplify[std]{-{a1*m}/{b1}}$</p>", "<p>$x=\\var{-m},\\;\\;y=\\simplify[std]{{a1*m}/{b1}}$</p>", "<p>$x=\\var{m+1},\\;\\;y=\\simplify[std]{-{c1*(m+1)}/{d1}}$</p>", "<p>$x=\\var{-m-1},\\;\\;y=\\simplify[std]{{c1*(m+1)}/{d1}}$</p>", "<p>$x=\\var{m-1},\\;\\;y=\\simplify[std]{-{a1+2*b1}/{b1}}$</p>", "<p>$x=\\var{-m+1},\\;\\;y=\\simplify[std]{{a1+2*b1}/{b1}}$</p>"], "warningType": "none", "displayType": "checkbox", "maxMarks": 0, "scripts": {}, "distractors": ["", "", "", "", "", ""], "marks": 0, "showCorrectAnswer": true, "type": "m_n_2", "minMarks": 0}], "statement": "\n  <p>Answer the following questions about the function:</p>\n  <p>\\[f(x,y)=\\simplify[std]{ ({a} / 3) * x ^ 3 + ({b} / 2) * x ^ 2 * y + {c} * y ^ 2 * x + {d} * y}\\]</p>\n  ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "variables": {"a": {"definition": "a1*c1", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "b1*d1", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "ch": {"definition": "if(a1*d1=b1*c1,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "ch", "description": ""}, "d": {"definition": "-(b1*c1-3*a1*d1)/2*m^2", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "m": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "s5": {"definition": "if(d*(-b*d1+4*c*c1)<=0,-1,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "a1": {"definition": "random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "other": {"definition": "if(s5=-1,'There can be no more stationary points as we cannot solve this.', if(ch=0,' we get the same stationary points',' You solve this getting 2 values of $x$ and then you obtain two more stationary points (which are not on the list you choose from).'))", "templateType": "anything", "group": "Ungrouped variables", "name": "other", "description": ""}, "b1": {"definition": "random(2,4,6)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "c2": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}, "c1": {"definition": "if(b1*c2=3*a1*d1,c2+1,c2)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "b": {"definition": "b1*c1+a1*d1", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "check": {"definition": "if(ch=0, 'The two expressions for y in terms of x are the same and we will get exactly the same so ignore the rest!',' ')", "templateType": "anything", "group": "Ungrouped variables", "name": "check", "description": ""}, "d1": {"definition": "random(2,4,6)", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "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>&nbsp;</p>\n            \t\t", "description": "<p>Find the stationary points of the function: $f(x,y)=a x ^ 3 + b x ^ 2 y + c y ^ 2 x + dy$ by choosing from a list of points.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Robert Anderson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/809/"}]}]}], "contributors": [{"name": "Robert Anderson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/809/"}]}