// Numbas version: exam_results_page_options
{"name": "blackfyre's copy of Functions: approximate area under graph", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<p>See Lecture 4.1, Workshop 4.2, and the previous questions in this online exercise.</p>\n<p>(If you have studied integration before, it would be worthwhile to attempt the appropriate integration and compare the exact area with this estimated value).</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "x0", "x", "fx", "total"], "functions": {"plotgraph": {"language": "javascript", "parameters": [["a", "number"], ["b", "number"], ["c", "number"], ["num", "number"]], "type": "html", "definition": "// Shading under a graph! This functions plots a graph of y = a(x-r1)(x-r2)\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 xmin = -2;\nvar xmax = 8;\nvar ymin = -10;\nvar ymax = 100;\n\n// First, make the JSXGraph board.\nvar div = Numbas.extensions.jsxgraph.makeBoard(\n  '500px',\n  '500px',\n {\n   boundingBox: [xmin,ymax,xmax,ymin],\n   axis: false,\n   showNavigation: false,\n   grid: false\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: 1\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,10],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n});\n\n\n\n// This function shades in the area below the graph of f\n// between the x values x1 and x2\n\nvar shade = function(f,x1,x2,colour,opacity) {\n  var dataX1 = [x1,x1];\n  var dataY1 = [0,f(x1)];\n\n  var dataX2 = [];\n  var dataY2 = [];\n  for (var i = x1; i <= x2; i = i+0.1) {\n      dataX2.push(i);\n      dataY2.push(f(i));\n  }\n\n  var dataX3 = [x2,x2];\n  var dataY3 = [f(x2),0];\n\n  dataX = dataX1.concat(dataX2).concat(dataX3);\n  dataY = dataY1.concat(dataY2).concat(dataY3);\n\nvar shading =  board.create('curve', [dataX,dataY],{strokeWidth:0, fillColor:colour, fillOpacity:opacity});\n\nreturn shading;\n}\n\n\n//Define your functions\nvar f = function(x) {\n  return a*(x-b)*(c-x);\n}\n\n\nvar f1 = function(x) {\n  return a*(c-4-b)*(c-c+4);\n}\n\nvar f2 = function(x) {\n  return a*(c-3-b)*(c-c+3);\n}\n\nvar f3 = function(x) {\n  return a*(c-2-b)*(c-c+2);\n}\n\nvar f4 = function(x) {\n  return a*(c-1-b)*(c-c+1);\n}\n\n\n\n//Plot the graph and do shading\nboard.create('functiongraph', [f]);\n\n\n\nswitch(num) {\n  case 1:\n    shade(f,c-4,c, 'red',0.2);\n    break;\n  case 2:\n    for (i = 0; i < 8; i++) { \n      var g = function(x) {\n        return a*(c-(4-0.5*i)-b)*(c-(c-(4-0.5*i)));  //defining horizontal function for ith column\n      }\n      if (i%2 == 0) {   \n      shade(g,c-(4-0.5*i),c-(4-0.5*(i+1)), 'blue',0.2);  //shading ith column blue if i even\n      } else {\n      shade(g,c-(4-0.5*i),c-(4-0.5*(i+1)), 'green',0.6);  // shading ith column green if i odd\n      }\n    } \n    board.create('ellipse',[[c-4,f(c-4)+0.995],[c-4,f(c-4)-0.995],[c-4,f(c-4)+1]],{color:'black'});\n    //brd.create('circle',[[c-4,f(c-4)],0.1],{color:'black'});  //adding a point\n    board.create('text',[c-4,a*(c-4-b)*(c-c+4)+3,'A']);\n    board.create('ellipse',[[c-3.5,f(c-3.5)+0.995],[c-3.5,f(c-3.5)-0.995],[c-3.5,f(c-3.5)+1]],{color:'black'});\n    board.create('text',[c-3.5,a*(c-3.5-b)*(c-c+3.5)+3,'B']);\n    break;\n}\n        \n\n\nreturn div;"}}, "variables": {"fx": {"name": "fx", "definition": "vector([a*(x[0]-b)*(c-x[0]),\n  a*(x[1]-b)*(c-x[1]),\n  a*(x[2]-b)*(c-x[2]),\n  a*(x[3]-b)*(c-x[3]),\n  a*(x[4]-b)*(c-x[4]),\n  a*(x[5]-b)*(c-x[5]),\n  a*(x[6]-b)*(c-x[6]),\n  a*(x[7]-b)*(c-x[7])])\n\n  ", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "total": {"name": "total", "definition": "sum(fx)*0.5", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "x": {"name": "x", "definition": "[x0,x0+0.5,x0+1,x0+1.5,x0+2,x0+2.5,x0+3,x0+3.5]", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a": {"name": "a", "definition": "random(3)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b": {"name": "b", "definition": "random(-5..-2)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "x0": {"name": "x0", "definition": "random(1..2)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c": {"name": "c", "definition": "x0+4", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "parts": [{"scripts": {}, "type": "gapfill", "showFeedbackIcon": true, "gaps": [{"markPerCell": false, "numRows": 1, "showFeedbackIcon": true, "correctAnswer": "matrix([x[0],fx[0]])", "variableReplacements": [], "allowResize": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "showCorrectAnswer": true, "scripts": {}, "type": "matrix", "tolerance": 0, "numColumns": "2", "allowFractions": false, "correctAnswerFractions": false}, {"markPerCell": false, "numRows": 1, "showFeedbackIcon": true, "correctAnswer": "matrix([x[1],fx[1]])", "variableReplacements": [], "allowResize": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "showCorrectAnswer": true, "scripts": {}, "type": "matrix", "tolerance": 0, "numColumns": "2", "allowFractions": true, "correctAnswerFractions": false}, {"minValue": "{total}", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": "3", "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "maxValue": "{total}", "allowFractions": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "mustBeReduced": false}, {"minValue": "", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 1, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "maxValue": "", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "mustBeReduced": false}, {"markPerCell": false, "numRows": 1, "showFeedbackIcon": true, "correctAnswer": "", "variableReplacements": [], "allowResize": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "showCorrectAnswer": true, "scripts": {}, "type": "matrix", "tolerance": 0, "numColumns": 1, "allowFractions": false, "correctAnswerFractions": false}, {"displayColumns": 0, "showFeedbackIcon": true, "choices": ["Choice 1", "Choice 2", "Choice 3"], "distractors": ["", "", ""], "shuffleChoices": false, "variableReplacementStrategy": "originalfirst", "minMarks": 0, "showCorrectAnswer": true, "scripts": {}, "type": "1_n_2", "displayType": "radiogroup", "marks": 0, "maxMarks": 0, "matrix": [0, 0, 0], "variableReplacements": []}], "prompt": "<p>Below is the graph of the function $v(t) =\\simplify{{a}(t-{b})*({c}-t)}$.&nbsp; We would like to calculate the area of the shaded region.</p>\n<p>{plotgraph(a,b,c,1)}</p>\n<p></p>\n<p>$v(t) = \\simplify{{a}(t-{b}*({c}-t))}$</p>\n<p></p>\n<p>You will learn how to do this in Maths 2 (using so-called integration), but for now we will just <span style=\"text-decoration: underline;\">estimate</span> the area. We do this by calculating the area of the columns, as in the diagram below:</p>\n<p>{plotgraph(a,b,c,2)}</p>\n<p></p>\n<p>To save you scrolling, the function was&nbsp;$v(t) =\\simplify{{a}(t-{b})*({c}-t)}$.</p>\n<p>(i) What are the coordinates of the points A and B?</p>\n<p>A. [[0]]</p>\n<p>B. [[1]]</p>\n<p></p>\n<p>(ii) By completing a table like the one below, determine the total area of the columns.</p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>Width</td>\n<td>Height</td>\n<td>Area</td>\n</tr>\n<tr>\n<td>Column 1</td>\n<td></td>\n<td></td>\n<td></td>\n</tr>\n<tr>\n<td>Column 2</td>\n<td></td>\n<td></td>\n<td></td>\n</tr>\n<tr>\n<td>Column 3</td>\n<td></td>\n<td></td>\n<td></td>\n</tr>\n<tr>\n<td>Column 4</td>\n<td></td>\n<td></td>\n<td></td>\n</tr>\n<tr>\n<td>Column&nbsp;5</td>\n<td></td>\n<td></td>\n<td></td>\n</tr>\n<tr>\n<td>Column&nbsp;6</td>\n<td></td>\n<td></td>\n<td></td>\n</tr>\n<tr>\n<td>Column&nbsp;7</td>\n<td></td>\n<td></td>\n<td></td>\n</tr>\n<tr>\n<td>Column&nbsp;8</td>\n<td></td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p>Total area of columns&nbsp; = [[2]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0}], "name": "blackfyre's copy of Functions: approximate area under graph", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Graphs are given with areas underneath them shaded. The area of the shaded regions are given and from this the value of various integrals are to be deduced.</p>"}, "preamble": {"css": "", "js": ""}, "variable_groups": [], "tags": [], "statement": "<p>This is a calculator question.</p>", "extensions": ["jsxgraph"], "rulesets": {}, "type": "question", "contributors": [{"name": "blackfyre blackfyre", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2008/"}]}]}], "contributors": [{"name": "blackfyre blackfyre", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2008/"}]}