// Numbas version: exam_results_page_options
{"name": "Differentiation: coordinates of stationary points from a graph", "extensions": ["geogebra", "jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {"plotgraph": {"language": "javascript", "parameters": [["a", "number"], ["b", "number"], ["c", "number"], ["d", "number"]], "definition": "// This functions plots a cubic with coefficients a,b,c,d\n// It creates the board, sets it up, then returns an\n// HTML div tag containing the board.\n\n\n// Max and min x and y values for the axis.\nvar x_min = -6;\nvar x_max = 6;\nvar y_min = -10;\nvar y_max = 10;\n\n\n// First, make the JSXGraph board.\nvar div = Numbas.extensions.jsxgraph.makeBoard(\n  '500px',\n  '600px',\n {\n   boundingBox: [x_min,y_max,x_max,y_min],\n   axis: false,\n   showNavigation: true,\n   grid: true\n }\n);\n\n\n\n\n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar board = div.board; \n\n// create the x-axis.\nvar xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = board.create('ticks',[xaxis,1],{\n  drawLabels: true,\n  label: {offset: [-4, -10]},\n  minorTicks: 0\n});\n\n// create the y-axis\nvar yaxis = board.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nvar yticks = board.create('ticks',[yaxis,1],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n});\n\n\n\n\n// Plot the function.\n    board.create('functiongraph',\n             [function(x){ return a*x*x*x+b*x*x+c*x + d;},x_min,x_max]);\n\n\n\n\nreturn div;", "type": "html"}}, "variablesTest": {"condition": "max(abs(ymin),abs(ymax))<10", "maxRuns": 100}, "metadata": {"description": "<p>A cubic with a maximum and minimum point is given. Question is to determine coordinates of the minimum and maximum point. Non-calculator. Advice is given.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "variables": {"b": {"group": "Ungrouped variables", "description": "", "definition": "-(r1*a + 3*r2)/2", "name": "b", "templateType": "anything"}, "xmax": {"group": "Ungrouped variables", "description": "", "definition": "if(a=1, min(r1/3, r2/a), max(r1/3, r2/a))", "name": "xmax", "templateType": "anything"}, "d": {"group": "Ungrouped variables", "description": "", "definition": "random(-2..2)", "name": "d", "templateType": "anything"}, "r1": {"group": "Ungrouped variables", "description": "", "definition": "random(-1..1)*3", "name": "r1", "templateType": "anything"}, "a": {"group": "Ungrouped variables", "description": "", "definition": "random(-1..1 except 0)", "name": "a", "templateType": "anything"}, "ymax": {"group": "Ungrouped variables", "description": "", "definition": "a*(xmax)^3+b*(xmax)^2+c*(xmax) + d", "name": "ymax", "templateType": "anything"}, "c": {"group": "Ungrouped variables", "description": "", "definition": "r1*r2", "name": "c", "templateType": "anything"}, "ymin": {"group": "Ungrouped variables", "description": "", "definition": "a*(xmin)^3+b*(xmin)^2+c*(xmin) + d", "name": "ymin", "templateType": "anything"}, "xmin": {"group": "Ungrouped variables", "description": "", "definition": "if(a=1, max(r1/3, r2/a), min(r1/3, r2/a))", "name": "xmin", "templateType": "anything"}, "r2": {"group": "Ungrouped variables", "description": "", "definition": "random(-1..1 except 1/2*(a*r1/3 + mod(a*r1,2)))*2-mod(a*r1,2)", "name": "r2", "templateType": "anything"}}, "rulesets": {}, "variable_groups": [], "ungrouped_variables": ["a", "r1", "r2", "ymin", "ymax", "xmax", "xmin", "b", "c", "d"], "extensions": ["geogebra", "jsxgraph"], "parts": [{"variableReplacements": [], "type": "gapfill", "gaps": [{"notationStyles": ["plain"], "showCorrectAnswer": true, "maxValue": "xmax", "correctAnswerFraction": false, "unitTests": [], "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "marks": "1", "extendBaseMarkingAlgorithm": true, "minValue": "xmax"}, {"notationStyles": ["plain"], "showCorrectAnswer": true, "maxValue": "ymax", "correctAnswerFraction": false, "unitTests": [], "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "marks": "1", "extendBaseMarkingAlgorithm": true, "minValue": "ymax"}, {"notationStyles": ["plain"], "showCorrectAnswer": true, "maxValue": "xmin", "correctAnswerFraction": false, "unitTests": [], "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "marks": "1", "extendBaseMarkingAlgorithm": true, "minValue": "xmin"}, {"notationStyles": ["plain"], "showCorrectAnswer": true, "maxValue": "ymin", "correctAnswerFraction": false, "unitTests": [], "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "marks": "1", "extendBaseMarkingAlgorithm": true, "minValue": "ymin"}], "marks": 0, "prompt": "<p>{plotgraph(a,b,c,d)}</p>\n<p>Above is the graph of some function $f$.</p>\n<p>What are the coordinates of its maximum point? ([[0]],[[1]])</p>\n<p>What are the coordinates of its minimum point? ([[2]],[[3]])</p>", "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacementStrategy": "originalfirst"}], "tags": [], "advice": "<p>(i) A maximum point is a point where regardless if you move right or left, the height will decrease. &nbsp;A visual analogy would be a hill: if you're at the top of a hill, no matter which direction you go your height will decrease. &nbsp;So you're looking for a part of the graph which is 'like a hill', and in this graph the point is at $(\\var{xmax},&nbsp;\\var{ymax})$.</p>\n<p>(ii) A minimum point is the opposite of a maximum point (or an upside-down version of a maximum point, if you like). &nbsp;The analogy in this case would be a valley: no matter which direction you go your height will increase. &nbsp;In this graph, the minimum point is at $(\\var{xmin},&nbsp;\\var{ymin})$.</p>", "statement": "<p>Finding stationary&nbsp;points on a graph.</p>", "name": "Differentiation: coordinates of stationary points from a graph", "type": "question", "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}]}]}], "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}]}