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{"name": "4. One Type Easier than the Other", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {"domain": {"definition": "var a = Numbas.jme.unwrapValue(scope.variables.a); // define random parameter r1\nvar b = Numbas.jme.unwrapValue(scope.variables.b); // define random parameter r1\nvar c = Numbas.jme.unwrapValue(scope.variables.c); // define random parameter r1\nvar n = Numbas.jme.unwrapValue(scope.variables.n); // define random parameter r1\n\nvar makeboard = Numbas.extensions.jsxgraph.makeBoard('250px','250px',\n               {boundingBox:[-b/10,1.5*c,1.5*b,-c/5], axis:true, grid:false}); // define how the graph is displayed\nvar board = makeboard.board; // generate the graph\n\nvar curve      = board.create('curve',\n                     [function(y){return y;},\n                      function(y){return a*Math.pow(y,1/n);},\n                      0, b]);\nvar curve_dash = board.create('curve',\n                     [function(y){return y;},\n                      function(y){return a*Math.pow(y,1/n);},\n                      b, 1.5*b], {dash:2});\nvar line      = board.create('curve',\n                     [function(y){return y;},\n                      function(y){return c;},\n                      0, b]);\nvar line_dash = board.create('curve',\n                     [function(y){return y;},\n                      function(y){return c;},\n                      b, 1.5*b], {dash:2});\n\n\nvar f = [];\nf.push(curve);\nf.push(line);\nvar area = board.create('curve', [[], []], {color: 'blue', opacity: 0.4});\narea.updateDataArray = function () {\n                      // start and end\n                      var left = 0, right = b,\n        x, y, curve;\n    x = [left];\n    y = [f[0].Y(left)];\n// go through f[0] forwards\n    curve = f[0];    \n    for (i = 0; i < curve.numberPoints; i++) {\n            x.push(curve.points[i].usrCoords[1]);\n            y.push(curve.points[i].usrCoords[2]);\n    }\n// walk backwards through f[1]\n    curve = f[1];\n    for (i = curve.numberPoints - 1; i >= 0; i--) {\n            x.push(curve.points[i].usrCoords[1]);\n            y.push(curve.points[i].usrCoords[2]);\n    }\n// close the curve\n    this.dataX = x;\n    this.dataY = y;\n};\n\nreturn makeboard; // update the board", "type": "html", "language": "javascript", "parameters": []}}, "parts": [{"expectedvariablenames": [], "showpreview": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "marks": 1, "vsetrangepoints": 5, "checkingtype": "absdiff", "showCorrectAnswer": true, "vsetrange": [0, 1], "variableReplacements": [], "showFeedbackIcon": true, "type": "jme", "checkvariablenames": false, "answer": "{num}/{den}(e^{m*c^(n+1)}-1)"}], "variable_groups": [{"name": "Combined variables", "variables": ["k1", "num", "den", "gcd"]}], "advice": "<p>We cannot integrate $\\simplify{{k}e^({m}y^{n+1})}$ with respect to $y$, so we must change type and&nbsp;integrate&nbsp;with respect to $x$.</p>\n<p>\\begin{align}<br/>&nbsp;I=\\int_0^\\var{c} \\int_0^{\\simplify{y^{n}/{a}}} &nbsp; &nbsp;\\var{k}e^\\simplify{{m}y^{n+1}}&nbsp; &nbsp;\\, \\mathrm{d}x \\, \\mathrm{d}y<br/>&amp;=\\int_0^\\var{c}&nbsp;&nbsp; \\var{k}xe^\\simplify{{m}y^{n+1}}&nbsp; &nbsp;\\bigg|_{x=0}^{x=\\simplify{y^{n}/{a}}} \\, \\mathrm{d}y \\\\<br/>&amp;=\\int_0^\\var{c}&nbsp; &nbsp;\\simplify{1/{a}y^{n}}\\cdot\\var{k}e^\\simplify{{m}y^{n+1}}&nbsp; &nbsp;\\, \\mathrm{d}y \\\\<br/>&amp;=\\int_0^\\var{c}&nbsp; &nbsp;\\simplify[fractionNumbers]{{k/a}}\\simplify{y^{n}}e^\\simplify{{m}y^{n+1}}&nbsp; &nbsp;\\, \\mathrm{d}y \\\\<br/>\\end{align}</p>\n<p>\\begin{align}<br/>\\mathrm{I}<br/>&amp;=\\simplify{1/({m*(n+1)}y^{n})}\\cdot\\simplify[fractionNumbers]{{k/a}}\\simplify{y^{n}}e^\\simplify{{m}y^{n+1}}&nbsp; &nbsp; \\bigg|_{y=0}^{y=\\var{c}} \\\\<br/>&amp;=\\simplify[fractionNumbers]{{k1}}e^\\simplify{{m}y^{n+1}}&nbsp; &nbsp;\\bigg|_{y=0}^{y=\\var{c}} \\\\<br/>&amp;=\\simplify[fractionNumbers]{{k1}}(\\simplify{e^{m*c^(n+1)}}-1) \\\\<br/>\\end{align}</p>", "preamble": {"css": "", "js": ""}, "metadata": {"description": "<p>I created this question, and every other question in Multiple Integration, for my dissertation `Computer-Aided Assessment of Multiple Integration'.</p>", "licence": "None specified"}, "tags": [], "name": "4. One Type Easier than the Other", "ungrouped_variables": ["a", "b", "c", "k", "m", "n"], "statement": "<p>Evaluate the integral</p>\n<p>\\[ \\int_0^\\simplify[fractionNumbers]{{b}}&nbsp;\\int_\\simplify{({a}x)^(1/{n})}^\\var{c}&nbsp; \\simplify{{k}e^({m}y^{n+1})}&nbsp; &nbsp;\\, \\mathrm{d}y \\, \\mathrm{d}x .\\]</p>\n<p>{domain()}</p>", "extensions": ["jsxgraph"], "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "variables": {"a": {"definition": "random(1..4#1)", "name": "a", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "n": {"definition": "random(1..3#1)", "name": "n", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "num": {"definition": "k/gcd", "name": "num", "group": "Combined variables", "description": "", "templateType": "anything"}, "den": {"definition": "a*m*(n+1)/gcd", "name": "den", "group": "Combined variables", "description": "", "templateType": "anything"}, "k1": {"definition": "k/(a*m*(n+1))", "name": "k1", "group": "Combined variables", "description": "", "templateType": "anything"}, "b": {"definition": "(c/a)^n", "name": "b", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "gcd": {"definition": "gcd(k,a*m*(n+1))", "name": "gcd", "group": "Combined variables", "description": "", "templateType": "anything"}, "c": {"definition": "random(1..2#1)", "name": "c", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "m": {"definition": "random(-3..3 except 0)", "name": "m", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "k": {"definition": "random(1..9#1)", "name": "k", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}}, "type": "question", "contributors": [{"name": "Nicholas Barker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1915/"}]}]}], "contributors": [{"name": "Nicholas Barker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1915/"}]}