// Numbas version: exam_results_page_options
{"name": "Patrick's copy of Find partial derivatives of $f(x,y)$ and identify its stationary points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "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>"}, "extensions": [], "functions": {}, "statement": "<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>", "preamble": {"js": "", "css": ""}, "advice": "<h4>a)</h4>\n<p>\\begin{align}<br/>\\partial f \\over \\partial x &amp;= \\simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))} \\\\[1em]<br/>\\partial f \\over \\partial y &amp;= \\simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})} <br/>\\end{align}</p>\n<h4>b)</h4>\n<p>$(a,b)$ is a stationary point for the function $f(x,y)$ if $f_x=0$ and $f_y=0$, where the partial derivatives are evaluated at $x=a$, $y=b$.</p>\n<p>So you have to make sure that both of these partial derivatives are $0$ at the stationary point.</p>\n<p>For this example we have from the above equations that:</p>\n<p>\\begin{align} <br/>\\simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))} &amp;= 0, &amp;&nbsp;\\mathbf{(1)}\\\\ <br/>\\simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})} &amp;= 0, &amp; \\mathbf{(2)}<br/>\\end{align}</p>\n<p>The left hand side of equation <strong>(1)</strong> 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:</p>\n<p>\\[ y=\\simplify[std]{{-a1}/{b1}*x}, \\text{ or } y= \\simplify[std]{{-c1}/{d1}*x} \\]</p>\n<h4>First case: $y= \\simplify[std]{{-a1}/{b1}*x}$</h4>\n<p>Substituting this into equation <strong>(2)</strong> gives:</p>\n<p>\\[\\simplify[std]{{b}/2*x^2-{2c*a1}/{b1}*x^2+{d}}=0 \\implies \\simplify[std]{{-b*b1+4*c*a1}/{2*b1}*x^2={d}}\\]</p>\n<p>Hence $x=\\var{m}$ or $x = \\var{-m}$. The corresponding stationary points are:</p>\n<p>\\[ \\left(\\var{m},\\simplify[std]{-{a1*m}/{b1}}\\right) \\text{ and } \\left(\\var{-m},\\simplify[std]{{a1*m}/{b1}}\\right) \\]</p>\n<h4>Second case: $y= \\simplify[std]{{-c1}/{d1}*x}$</h4>\n<p>Substituting this into equation <strong>(2)</strong> gives:</p>\n<p>\\[\\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>There can be no more stationary points as this equation has no real solution.</p>", "name": "Patrick's copy of Find partial derivatives of $f(x,y)$ and identify its stationary points", "variable_groups": [], "tags": [], "variables": {"c": {"group": "Ungrouped variables", "description": "", "name": "c", "definition": "b1*d1", "templateType": "anything"}, "d": {"group": "Ungrouped variables", "description": "", "name": "d", "definition": "-(b1*c1-3*a1*d1)/2*m^2", "templateType": "anything"}, "b1": {"group": "Ungrouped variables", "description": "", "name": "b1", "definition": "random(2,4,6)", "templateType": "anything"}, "c1": {"group": "Ungrouped variables", "description": "", "name": "c1", "definition": "if(b1*c2=3*a1*d1,c2+1,c2)", "templateType": "anything"}, "s5": {"group": "Ungrouped variables", "description": "", "name": "s5", "definition": "if(d*(-b*d1+4*c*c1)<=0,-1,1)", "templateType": "anything"}, "a1": {"group": "Ungrouped variables", "description": "", "name": "a1", "definition": "random(1..5)", "templateType": "anything"}, "c2": {"group": "Ungrouped variables", "description": "", "name": "c2", "definition": "random(1..4)", "templateType": "anything"}, "d1": {"group": "Ungrouped variables", "description": "", "name": "d1", "definition": "random(2,4,6)", "templateType": "anything"}, "ch": {"group": "Ungrouped variables", "description": "", "name": "ch", "definition": "if(a1*d1=b1*c1,0,1)", "templateType": "anything"}, "b": {"group": "Ungrouped variables", "description": "", "name": "b", "definition": "b1*c1+a1*d1", "templateType": "anything"}, "a": {"group": "Ungrouped variables", "description": "", "name": "a", "definition": "a1*c1", "templateType": "anything"}, "m": {"group": "Ungrouped variables", "description": "", "name": "m", "definition": "random(1..4)", "templateType": "anything"}}, "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "variablesTest": {"condition": "ch<>0\nand\ns5=-1", "maxRuns": 100}, "parts": [{"type": "gapfill", "scripts": {}, "prompt": "<p>Find the partial derivatives of $f$ with respect to $x$ and $y$.</p>\n<p>Note that if you want to enter a product of two 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>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"showpreview": true, "expectedvariablenames": ["x", "y"], "showFeedbackIcon": true, "vsetrange": [0, 1], "answer": "(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))", "vsetrangepoints": 5, "showCorrectAnswer": true, "type": "jme", "answersimplification": "std", "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "checkingtype": "absdiff", "marks": 2, "variableReplacements": [], "checkvariablenames": true}, {"showpreview": true, "expectedvariablenames": ["x", "y"], "showFeedbackIcon": true, "vsetrange": [0, 1], "answer": "((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})", "vsetrangepoints": 5, "showCorrectAnswer": true, "type": "jme", "answersimplification": "std", "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "checkingtype": "absdiff", "marks": 2, "variableReplacements": [], "checkvariablenames": true}]}, {"maxAnswers": 0, "displayColumns": 3, "maxMarks": 0, "displayType": "checkbox", "minAnswers": 0, "showFeedbackIcon": true, "minMarks": 0, "warningType": "none", "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        ", "distractors": ["", "", "", "", "", ""], "showCorrectAnswer": true, "type": "m_n_2", "scripts": {}, "variableReplacementStrategy": "originalfirst", "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>"], "matrix": [2, 2, 0, 0, 0, 0], "marks": 0, "variableReplacements": [], "shuffleChoices": true}], "ungrouped_variables": ["a", "c", "ch", "d", "m", "s5", "a1", "b", "b1", "c2", "c1", "d1"], "type": "question", "contributors": [{"name": "Patrick Joyce", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1674/"}]}]}], "contributors": [{"name": "Patrick Joyce", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1674/"}]}