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{"name": "Josh's copy of Find stationary point of a curve and determine its nature", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "name": "Josh's copy of Find stationary point of a curve and determine its nature", "parts": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "prompt": "<p>Let $y = \\simplify{{a1}x^2 + {b1}x + {c1}}$.</p>\n<p>$\\dfrac{\\mathrm{d}y}{\\mathrm{d}x} = $ [[0]]</p>\n<p>Enter the coordinates of the stationary point of $y$: $\\big($ [[1]] $, $ [[2]] $\\big)$</p>\n<p>$\\dfrac{\\mathrm{d}^2y}{\\mathrm{d}x^2} = $ [[3]]</p>\n<p>What is the nature of the stationary point of $y$?</p>\n<p>[[4]]</p>", "marks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "gaps": [{"failureRate": 1, "variableReplacementStrategy": "originalfirst", "unitTests": [], "vsetRangePoints": 5, "useCustomName": false, "checkVariableNames": false, "answer": "2*{a1}*x+{b1}", "valuegenerators": [{"value": "", "name": "x"}], "marks": 1, "showCorrectAnswer": true, "showPreview": true, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "checkingType": "absdiff", "customName": "", "type": "jme", "vsetRange": [0, 1], "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true}, {"failureRate": 1, "marks": 1, "showPreview": false, "checkingType": "absdiff", "customName": "", "showFeedbackIcon": true, "type": "jme", "vsetRange": [0, 1], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "vsetRangePoints": 5, "answerSimplification": "fractionnumbers", "answer": "{sx1}", "valuegenerators": [], "showCorrectAnswer": true, "checkVariableNames": false, "checkingAccuracy": 0.001, "variableReplacements": [], "scripts": {}}, {"failureRate": 1, "marks": 1, "showPreview": false, "checkingType": "absdiff", "customName": "", "showFeedbackIcon": true, "type": "jme", "vsetRange": [0, 1], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "vsetRangePoints": 5, "answerSimplification": "fractionnumbers", "answer": "{sy1}", "valuegenerators": [], "showCorrectAnswer": true, "checkVariableNames": false, "checkingAccuracy": 0.001, "variableReplacements": [], "scripts": {}}, {"failureRate": 1, "variableReplacementStrategy": "originalfirst", "unitTests": [], "vsetRangePoints": 5, "useCustomName": false, "checkVariableNames": false, "answer": "{2a1}", "valuegenerators": [], "marks": 1, "showCorrectAnswer": true, "showPreview": true, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "checkingType": "absdiff", "customName": "", "type": "jme", "vsetRange": [0, 1], "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true}, {"variableReplacementStrategy": "originalfirst", "unitTests": [], "choices": ["<p>Maximum</p>", "<p>Minimum</p>", "<p>Point of inflection</p>"], "useCustomName": false, "displayType": "radiogroup", "shuffleChoices": false, "showCellAnswerState": true, "marks": 0, "showCorrectAnswer": true, "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "distractors": ["", "", ""], "customName": "", "type": "1_n_2", "matrix": [0, 1, 0], "minMarks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "displayColumns": 0, "showFeedbackIcon": true}], "customName": "", "type": "gapfill", "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "sortAnswers": false, "showFeedbackIcon": true}, {"variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "prompt": "<p>Let $z = \\simplify{{a2}x^2+{b2}x+{c2}}$.</p>\n<p>$\\dfrac{\\mathrm{d}z}{\\mathrm{d}x} = $ [[0]]</p>\n<p>Enter the coordinates of the stationary point of $z$: $\\big($ [[1]] $, $ [[2]] $\\big)$</p>\n<p>$\\dfrac{\\mathrm{d}^2z}{\\mathrm{d}x^2} = $ [[3]]</p>\n<p>What is the nature of the stationary point of $z$?</p>\n<p>[[4]]</p>", "marks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "gaps": [{"failureRate": 1, "variableReplacementStrategy": "originalfirst", "unitTests": [], "vsetRangePoints": 5, "useCustomName": false, "checkVariableNames": false, "answer": "2*{a2}x+{b2}", "valuegenerators": [{"value": "", "name": "x"}], "marks": 1, "showCorrectAnswer": true, "showPreview": true, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "checkingType": "absdiff", "customName": "", "type": "jme", "vsetRange": [0, 1], "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true}, {"failureRate": 1, "marks": 1, "showPreview": false, "checkingType": "absdiff", "customName": "", "showFeedbackIcon": true, "type": "jme", "vsetRange": [0, 1], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "vsetRangePoints": 5, "answerSimplification": "fractionnumbers", "answer": "{sx2}", "valuegenerators": [], "showCorrectAnswer": true, "checkVariableNames": false, "checkingAccuracy": 0.001, "variableReplacements": [], "scripts": {}}, {"failureRate": 1, "marks": 1, "showPreview": false, "checkingType": "absdiff", "customName": "", "showFeedbackIcon": true, "type": "jme", "vsetRange": [0, 1], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "vsetRangePoints": 5, "answerSimplification": "fractionnumbers", "answer": "{sy2}", "valuegenerators": [], "showCorrectAnswer": true, "checkVariableNames": false, "checkingAccuracy": 0.001, "variableReplacements": [], "scripts": {}}, {"failureRate": 1, "variableReplacementStrategy": "originalfirst", "unitTests": [], "vsetRangePoints": 5, "useCustomName": false, "checkVariableNames": false, "answer": "2{a2}", "valuegenerators": [], "marks": 1, "showCorrectAnswer": true, "showPreview": true, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "checkingType": "absdiff", "customName": "", "type": "jme", "vsetRange": [0, 1], "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true}, {"variableReplacementStrategy": "originalfirst", "unitTests": [], "choices": ["<p>Maximum</p>", "<p>Minimum</p>", "<p>Point of inflection</p>"], "useCustomName": false, "displayType": "radiogroup", "shuffleChoices": false, "showCellAnswerState": true, "marks": 0, "showCorrectAnswer": true, "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "distractors": ["", "", ""], "customName": "", "type": "1_n_2", "matrix": [1, 0, 0], "minMarks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "displayColumns": 0, "showFeedbackIcon": true}], "customName": "", "type": "gapfill", "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "sortAnswers": false, "showFeedbackIcon": true}, {"variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "prompt": "<p>Let $t = \\var{m}x^3+\\var{q}$.</p>\n<p>$\\dfrac{\\mathrm{d}t}{\\mathrm{d}x} = $ [[0]]</p>\n<p>Enter the coordinates of the stationary point of $t$: $\\big($ [[1]] $, $ [[2]] $\\big)$</p>\n<p>$\\dfrac{\\mathrm{d}^2t}{\\mathrm{d}x^2} = $ [[3]]</p>\n<p>What is the nature of the stationary point of $t$?</p>\n<p>[[4]]</p>", "marks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "gaps": [{"failureRate": 1, "variableReplacementStrategy": "originalfirst", "unitTests": [], "vsetRangePoints": 5, "useCustomName": false, "checkVariableNames": false, "answer": "3*{m}*x^2", "valuegenerators": [{"value": "", "name": "x"}], "marks": 1, "showCorrectAnswer": true, "showPreview": true, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "checkingType": "absdiff", "customName": "", "type": "jme", "vsetRange": [0, 1], "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true}, {"failureRate": 1, "marks": 1, "showPreview": false, "checkingType": "absdiff", "customName": "", "showFeedbackIcon": true, "type": "jme", "vsetRange": [0, 1], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "vsetRangePoints": 5, "answerSimplification": "fractionnumbers", "answer": "0", "valuegenerators": [], "showCorrectAnswer": true, "checkVariableNames": false, "checkingAccuracy": 0.001, "variableReplacements": [], "scripts": {}}, {"failureRate": 1, "marks": 1, "showPreview": false, "checkingType": "absdiff", "customName": "", "showFeedbackIcon": true, "type": "jme", "vsetRange": [0, 1], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "vsetRangePoints": 5, "answerSimplification": "fractionnumbers", "answer": "{q}", "valuegenerators": [], "showCorrectAnswer": true, "checkVariableNames": false, "checkingAccuracy": 0.001, "variableReplacements": [], "scripts": {}}, {"failureRate": 1, "variableReplacementStrategy": "originalfirst", "unitTests": [], "vsetRangePoints": 5, "useCustomName": false, "checkVariableNames": false, "answer": "6*{m}*x", "valuegenerators": [{"value": "", "name": "x"}], "marks": 1, "showCorrectAnswer": true, "showPreview": true, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "checkingType": "absdiff", "customName": "", "type": "jme", "vsetRange": [0, 1], "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true}, {"variableReplacementStrategy": "originalfirst", "unitTests": [], "choices": ["<p>Maximum</p>", "<p>Minimum</p>", "<p>Point of inflection</p>"], "useCustomName": false, "displayType": "radiogroup", "shuffleChoices": false, "showCellAnswerState": true, "marks": 0, "showCorrectAnswer": true, "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "distractors": ["", "", ""], "customName": "", "type": "1_n_2", "matrix": [0, 0, 1], "minMarks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "displayColumns": 0, "showFeedbackIcon": true}], "customName": "", "type": "gapfill", "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}, "sortAnswers": false, "showFeedbackIcon": true}], "variable_groups": [], "ungrouped_variables": ["a1", "sx1", "m", "sy1", "q", "sy2", "sx2", "a2", "b1", "b2", "c2", "c1"], "extensions": [], "functions": {}, "rulesets": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"sx1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "sx1", "definition": "-b1/(2*a1)"}, "a2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "definition": "random(-5..-2)"}, "b2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "definition": "random(-6..6 except (-1..1))"}, "sx2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "sx2", "definition": "-b2/(2*a2)"}, "m": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "definition": "random(2,3,4)"}, "c1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "definition": "random(-10..10 except (-1..1))"}, "b1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "definition": "random(-6..6 except (-1..1))"}, "c2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "definition": "random(-10..10 except (-1..1))"}, "sy2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "sy2", "definition": "a2*(sx2^2)+b2*sx2+c2"}, "a1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "definition": "random(2..5)"}, "sy1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "sy1", "definition": "a1*(sx1^2)+b1*sx1+c1"}, "q": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "q", "definition": "random(-10..10 except 0)"}}, "advice": "<h4>a)</h4>\n<p>$\\dfrac{\\mathrm{d}y}{\\mathrm{d}x} = \\simplify{{2a1}x+{b1}}$.</p>\n<p>To find the $x$-coordinate of the stationary point, solve $\\frac{\\mathrm{d}y}{\\mathrm{d}x} = 0$ for $x$:</p>\n<p>\\[ \\begin{align} \\frac{\\mathrm{d}y}{\\mathrm{d}x} = \\simplify{{2a1}x + {b1}} &amp;= 0&nbsp;\\\\ x &amp;= \\simplify[all,fractionNumbers]{{sx1}}&nbsp;\\end{align} \\]</p>\n<p>Find the $y$-coordinate by substituting this value of $x$ into the definition of $y(x)$:</p>\n<p>\\[\\begin{align} \\simplify[fractionnumbers]{y({sx1})} &amp;= \\simplify[basic,fractionnumbers]{{a1}{sx1}^2+{b1}{sx1}+{c1}} \\\\ &amp;= \\simplify[fractionnumbers]{{sy1}} \\end{align}\\]</p>\n<p>Finally, to determine the nature of the stationary point, look at $\\frac{\\mathrm{d}^2y}{\\mathrm{d}x^2}$ at $x = \\simplify[fractionnumbers]{{sx1}}$.</p>\n<p>\\[ \\frac{\\mathrm{d}^2y}{\\mathrm{d}x^2} = \\simplify{{2*a1}} \\]</p>\n<p>This is&nbsp;<strong>positive</strong>, so the stationary point is a&nbsp;<strong>minimum</strong>.</p>\n<h4>b)</h4>\n<p>$\\dfrac{\\mathrm{d}z}{\\mathrm{d}x} = \\simplify{{2a2}x+{b2}}$.</p>\n<p>To find the $x$-coordinate of the stationary point, solve $\\frac{\\mathrm{d}z}{\\mathrm{d}x} = 0$ for $x$:</p>\n<p>\\[ \\begin{align} \\frac{\\mathrm{d}z}{\\mathrm{d}x} = \\simplify{{2a2}x + {b2}} &amp;= 0&nbsp;\\\\ x &amp;= \\simplify[all,fractionNumbers]{{sx2}}&nbsp;\\end{align} \\]</p>\n<p>Find the $y$-coordinate by substituting this value of $x$ into the definition of $z(x)$:</p>\n<p>\\[\\begin{align} \\simplify[fractionnumbers]{z({sx1})} &amp;= \\simplify[basic,fractionnumbers]{{a2}{sx2}^2+{b2}{sx2}+{c2}} \\\\ &amp;= \\simplify[fractionnumbers]{{sy2}} \\end{align}\\]</p>\n<p>Finally, to determine the nature of the stationary point, look at $\\frac{\\mathrm{d}^2z}{\\mathrm{d}x^2}$ at $x = \\simplify[fractionnumbers]{{sx2}}$.</p>\n<p>\\[ \\frac{\\mathrm{d}^2z}{\\mathrm{d}x^2} = \\simplify{{2*a2}} \\]</p>\n<p>This is&nbsp;<strong>negative</strong>, so the stationary point is a&nbsp;<strong>maximum</strong>.</p>\n<h4>c)</h4>\n<p>$\\dfrac{\\mathrm{d}t}{\\mathrm{d}x} = \\simplify{{3m}x^2}$.</p>\n<p>To find the $x$-coordinate of the stationary point, solve $\\frac{\\mathrm{d}t}{\\mathrm{d}x} = 0$ for $x$:</p>\n<p>\\[ \\begin{align} \\frac{\\mathrm{d}t}{\\mathrm{d}x} = \\simplify{{3m}x^2} &amp;= 0&nbsp;\\\\ x &amp;= 0 \\end{align} \\]</p>\n<p>Find the $y$-coordinate by substituting this value of $x$ into the definition of $t(x)$:</p>\n<p>\\[\\begin{align} z(0) &amp;= \\simplify[basic,fractionnumbers]{{m}*0^3 + {q}} \\\\ &amp;= \\var{q} \\end{align}\\]</p>\n<p>Finally, to determine the nature of the stationary point, look at $\\frac{\\mathrm{d}^2t}{\\mathrm{d}x^2}$ at $x = 0$.</p>\n<p>\\[ \\frac{\\mathrm{d}^2t}{\\mathrm{d}x^2} = \\simplify{{6m}x} \\]</p>\n<p>\\[ \\left.\\frac{\\mathrm{d}^2t}{\\mathrm{d}x^2} \\right\\rvert_{x=0} = \\var{6m} \\times 0 = 0 \\]</p>\n<p>This is&nbsp;<strong>zero</strong>, so the stationary point is a&nbsp;<strong>point of inflection</strong>.</p>", "statement": "<p>For the following, find the stationary point and determine its nature. For your answers, where appropriate, write your solutions as fractions, NOT decimals, and cancel down where possible.</p>", "preamble": {"js": "", "css": ""}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Josh Lim", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2990/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Josh Lim", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2990/"}]}