// Numbas version: exam_results_page_options
{"name": "Calculate Riemann sums of a quadratic", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tans2,3)", "description": "", "name": "ans2"}, "ans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tans3,3)", "description": "", "name": "ans3"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-a+random(4..8#2 except a)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+b", "description": "", "name": "c"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..100#2)", "description": "", "name": "n"}, "tans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a+b)^3*(n+1)(4n-1)/(48*n^2)", "description": "", "name": "tans2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "name": "a"}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tans1,3)", "description": "", "name": "ans1"}, "tans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a+b)^3/6", "description": "", "name": "tans1"}, "tans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a+b)^3*(4*n+1)(n-1)/(48*n^2)", "description": "", "name": "tans3"}}, "ungrouped_variables": ["a", "c", "b", "ans1", "ans2", "ans3", "n", "tol", "tans1", "tans3", "tans2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Calculate Riemann sums of a quadratic", "functions": {"riemann": {"type": "html", "language": "javascript", "definition": "\n\t\t\tvar f = function(x){ return a*b+(b-a)*x-x*x; };\n\t\t\tvar m = (a+b)*(a+b)/4\n\t\t\tvar div = Numbas.extensions.jsxgraph.makeBoard('200px','200px', {axis:false,ticks:false,showNavigation:false, boundingbox:[Math.min(-a-2,-2),m+2,Math.max(b+2,2),-2]});\n\t\t\t\n\t\t\tvar brd=div.board;\n\t\t\tvar xaxis=brd.create('line',[[0,0],[1,0]],{fixed:true,strokeColor:'black'});\n\t\t\tvar yaxis=brd.create('line',[[0,0],[0,1]],{fixed:true,strokeColor:'black'});\n\t\t\tvar plot=brd.create('functiongraph',[f,-a-2,b+2]);\n\t\t\tvar txt1=brd.create('text',[-a-0.5,-0.5,'p']);\n\t\t\tvar txt2=brd.create('text',[b+0.5,-0.5,'q']);\n\t\t\tvar txt3=brd.create('text',[(b-a)/2,-0.5,'c']);\n\t\t\treturn div;\n\t\t\t", "parameters": [["a", "number"], ["b", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "-a", "minValue": "-a", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "b", "minValue": "b", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n\t\t\t<p>Find the points $x=p,\\;x=q,\\;p \\leq q$ where the graph of $f$ cuts the $x$-axis.</p>\n\t\t\t<p>$p=\\;$[[0]] &nbsp; &nbsp; &nbsp;$q=\\;$[[1]]</p>\n\t\t\t", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans1+tol", "minValue": "ans1-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "(b-a)/2", "minValue": "(b-a)/2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "(a+b)^2/4", "minValue": "(a+b)^2/4", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n\t\t\t<p>Using elementary calculus find the area below the curve and above the interval $[p,q]$.</p>\n\t\t\t<p>Area= [[0]] &nbsp;(enter to 3 decimal places).</p>\n\t\t\t<p>Also find the point $x=c$ at which the function attains its maximum value over the interval $[p,q]$.</p>\n\t\t\t<p>$c=\\;$[[1]]</p>\n\t\t\t<p>Maximum value $f(c)=\\;$[[2]]</p>\n\t\t\t", "showCorrectAnswer": true, "marks": 0}, {"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "({(c * (c + 2) * (2 * c  - 1))} / 12)", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 2, "vsetrangepoints": 5}, {"answer": "({(c * (c - 2) * (2 * c + 1))} / 12)", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t<p>Let $p$ and $q$ be as above.</p>\n\t\t\t<p>Compute the upper and lower sums for the function $f$ over the interval $[p,q]$ &nbsp;using the partition:</p>\n\t\t\t<p>\\[\\{p,\\;p+1,\\;\\ldots,\\;q\\}\\] with subintervals of length 1.</p>\n\t\t\t<p>Upper sum = [[0]]</p>\n\t\t\t<p>Lower sum = [[1]]</p>\n\t\t\t<p></p>\n\t\t\t<p>You will find useful formulae on clicking Show steps.</p>\n\t\t\t<p></p>\n\t\t\t", "steps": [{"type": "information", "prompt": "<p>\\[\\begin{eqnarray}<br/>\\sum_{r=1}^n r&amp;=&amp;\\frac{n(n+1)}{2}\\\\<br/>\\sum_{r=1}^n r^2&amp;=&amp;\\frac{n(n+1)(2n+1)}{6}<br/>\\end{eqnarray}\\]</p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans3+tol", "minValue": "ans3-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n\t\t\t<p>Let $p$ and $c$ be as above.</p>\n\t\t\t<p>Consider the partition of the interval $[p,c]$ into $\\var{n}$ subintervals each of length &nbsp;$\\displaystyle \\frac{c-p}{\\var{n}}$:</p>\n\t\t\t<p>\\[\\Delta=\\left\\{p,\\;p+\\frac{c-p}{\\var{n}},\\;p+\\frac{2(c-p)}{\\var{n}},\\;\\ldots,c\\right\\}\\]</p>\n\t\t\t<p>Find the upper sum and lower sums for $f$ &nbsp;corresponding to this partition.</p>\n\t\t\t<p>Upper sum = [[0]] (enter to 3 decimal places)</p>\n\t\t\t<p>Lower sum = [[1]] (enter to 3 decimal places)</p>\n\t\t\t", "showCorrectAnswer": true, "marks": 0}], "statement": "\n\t<p>Let $f:\\mathbb{R}\\rightarrow \\mathbb{R}$ be defined by $f(x)=\\simplify{{a*b}+{b-a}*x-x^2}$.</p>\n\t<p>{Riemann(a,b)}</p>\n\t", "tags": ["checked2015", "MAS2224"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}