// Numbas version: exam_results_page_options
{"showstudentname": true, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>5 questions on definite integrals - integrate polynomials, trig functions and exponentials; find the area under a graph; find volumes of revolution.</p>"}, "navigation": {"reverse": true, "showresultspage": "oncompletion", "onleave": {"action": "none", "message": ""}, "preventleave": false, "showfrontpage": false, "allowregen": true, "browse": true}, "showQuestionGroupNames": false, "feedback": {"showactualmark": true, "feedbackmessages": [], "allowrevealanswer": true, "intro": "", "showtotalmark": true, "showanswerstate": true, "advicethreshold": 0}, "name": "Nick's copy of Definite Integration Practice", "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "duration": 0, "percentPass": 0, "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 1, "name": "Group", "questions": [{"name": "Jack's copy of Definite integration 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Jack Dunham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2484/"}], "parts": [{"showFeedbackIcon": true, "gaps": [{"allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "scripts": {}, "maxValue": "ans1+tol", "marks": 1, "customMarkingAlgorithm": "", "variableReplacements": [], "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "minValue": "ans1-tol", "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "unitTests": []}], "scripts": {}, "marks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "unitTests": [], "prompt": "\n      <p>\\[I=\\int_1^{\\var{b1}}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 2 decimal places.</p>\n      "}, {"showFeedbackIcon": true, "gaps": [{"allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "scripts": {}, "maxValue": "ans2+tol", "marks": 1, "customMarkingAlgorithm": "", "variableReplacements": [], "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "minValue": "ans2-tol", "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "unitTests": []}], "scripts": {}, "marks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "unitTests": [], "prompt": "<p>\\[I=\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\]</p>\n<p>$I=\\;\\;$[[0]]</p>\n<p>Input your answer to 2 decimal places.</p>"}, {"showFeedbackIcon": true, "gaps": [{"allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "scripts": {}, "maxValue": "ans3+tol", "marks": 1, "customMarkingAlgorithm": "", "variableReplacements": [], "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "minValue": "ans3-tol", "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "unitTests": []}], "scripts": {}, "marks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "unitTests": [], "prompt": "\n      <p>\\[I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 2 decimal places.</p>\n      "}, {"showFeedbackIcon": true, "gaps": [{"allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "scripts": {}, "maxValue": "ans4+tol1", "marks": 1, "customMarkingAlgorithm": "", "variableReplacements": [], "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "minValue": "ans4-tol1", "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "unitTests": []}], "scripts": {}, "marks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "unitTests": [], "prompt": "\n      <p>\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 4 decimal places.</p>\n      "}], "metadata": {"description": "<p>Evaluate $\\int_1^{\\,m}(ax ^ 2 + b x + c)^2\\;dx$, $\\int_0^{p}\\frac{1}{x+d}\\;dx,\\;\\int_0^\\pi x \\sin(qx) \\;dx$, $\\int_0^{r}x ^ {2}e^{t x}\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "ungrouped_variables": ["b2", "m3", "tans3", "s2", "ans2", "m2", "w", "n4", "ans3", "tans4", "tol1", "b4", "c1", "ans4", "b1", "ans1", "tans1", "m4", "a1", "t", "s7", "d1", "s6", "p", "tol"], "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"js": "", "css": ""}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {}, "statement": "<p>Evaluate the following definite integrals.</p>", "variables": {"b1": {"group": "Ungrouped variables", "name": "b1", "definition": "random(2..6)", "description": "", "templateType": "anything"}, "m2": {"group": "Ungrouped variables", "name": "m2", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "w": {"group": "Ungrouped variables", "name": "w", "definition": "random(0,1)", "description": "", "templateType": "anything"}, "s2": {"group": "Ungrouped variables", "name": "s2", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "tol1": {"group": "Ungrouped variables", "name": "tol1", "definition": "0.0001", "description": "", "templateType": "anything"}, "tans4": {"group": "Ungrouped variables", "name": "tans4", "definition": "(e^(p)*(p^2-2*p+2)-2)/(n4^3)", "description": "", "templateType": "anything"}, "tans1": {"group": "Ungrouped variables", "name": "tans1", "definition": "a1^2*(b1^5-1)/5+a1*c1*(b1^4-1)/2+(2*a1*d1+c1^2)*(b1^3-1)/3+c1*d1*(b1^2-1)+d1^2*(b1-1)", "description": "", "templateType": "anything"}, "b2": {"group": "Ungrouped variables", "name": "b2", "definition": "random(1..20)", "description": "", "templateType": "anything"}, "p": {"group": "Ungrouped variables", "name": "p", "definition": "n4*b4", "description": "", "templateType": "anything"}, "ans4": {"group": "Ungrouped variables", "name": "ans4", "definition": "precround(tans4,4)", "description": "", "templateType": "anything"}, "m3": {"group": "Ungrouped variables", "name": "m3", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "t": {"group": "Ungrouped variables", "name": "t", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "m4": {"group": "Ungrouped variables", "name": "m4", "definition": "2", "description": "", "templateType": "anything"}, "tans3": {"group": "Ungrouped variables", "name": "tans3", "definition": "if(w=0,((-1)^(m3)-1)/m3^2,-pi*(-1)^(m3)/m3)", "description": "", "templateType": "anything"}, "s7": {"group": "Ungrouped variables", "name": "s7", "definition": "1", "description": "", "templateType": "anything"}, "b4": {"group": "Ungrouped variables", "name": "b4", "definition": "s7*random(1,2,3)", "description": "", "templateType": "anything"}, "a1": {"group": "Ungrouped variables", "name": "a1", "definition": "random(1..7)", "description": "", "templateType": "anything"}, "d1": {"group": "Ungrouped variables", "name": "d1", "definition": "random(-9..9)", "description": "", "templateType": "anything"}, "ans3": {"group": "Ungrouped variables", "name": "ans3", "definition": "precround(tans3,2)", "description": "", "templateType": "anything"}, "s6": {"group": "Ungrouped variables", "name": "s6", "definition": "-1", "description": "", "templateType": "anything"}, "n4": {"group": "Ungrouped variables", "name": "n4", "definition": "s6*random(1,2,3)", "description": "", "templateType": "anything"}, "ans1": {"group": "Ungrouped variables", "name": "ans1", "definition": "precround(tans1,2)", "description": "", "templateType": "anything"}, "tol": {"group": "Ungrouped variables", "name": "tol", "definition": "0.01", "description": "", "templateType": "anything"}, "c1": {"group": "Ungrouped variables", "name": "c1", "definition": "t*random(1..9)", "description": "", "templateType": "anything"}, "ans2": {"group": "Ungrouped variables", "name": "ans2", "definition": "precround(ln(1+b2/m2),2)", "description": "", "templateType": "anything"}}, "advice": "\n  <p>a)<br/>\\[I=\\int_1^\\var{b1}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\] <br/>Expand the parentheses to obtain:</p>\n    <p>\\[\\begin{eqnarray*}I &amp;=&amp; \\int_1^\\var{b1} \\simplify[std]{{a1 ^ 2} * x ^ 4 + {2 * a1 * c1} * x ^ 3+ {c1 ^ 2+2*a1*d1} * x ^ 2 + {2 * c1 * d1} * x+ {d1 ^ 2} }\\;dx\\\\ &amp;=&amp;\\left[\\simplify[std]{{a1 ^ 2}/5 * x ^ 5 + {2 * a1 * c1}/4 * x ^ 4+ {c1 ^ 2+2*a1*d1}/3 * x ^ 3 + {2 * c1 * d1}/2 * x^2+ {d1 ^ 2}x }\\right]_1^\\var{b1}\\\\ &amp;=&amp;\\var{tans1}\\\\ \\\\&amp;=&amp;\\var{ans1}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]</p>\n    <p>b)<br/>\\[\\begin{eqnarray*}I&amp;=&amp;\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\\\ &amp;=&amp;\\left[\\ln(x+\\var{m2})\\right]_0^{\\var{b2}}\\\\ &amp;=&amp; \\ln(\\var{b2+m2})-\\ln(\\var{m2})\\\\ &amp;=&amp;\\ln\\left(\\frac{\\var{b2+m2}}{\\var{m2}}\\right)\\\\ &amp;=&amp;\\var{ans2}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]</p>\n    <p>c)<br/>\\[I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\] <br/>We use integration by parts.</p>\n    <p>Recall that:<br/>\\[\\int u\\frac{dv}{dx}\\;dx=uv-\\int \\frac{du}{dx}\\;v\\;dx\\]<br/>Here we set $u=x$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{ {w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x)}$</p>\n    <p>Hence \\[v=\\simplify[std]{({-w}/ {m3}) * Cos({m3} * x) + {1 -w} * (({1-w}/ {m3}) * Sin({m3} * x))}\\]</p>\n    <p>So \\[\\begin{eqnarray*} I&amp;=&amp;\\left[\\simplify[std]{{-w}*((x / {m3}) * Cos({m3} * x)) + {1 -w} * ((x / {m3}) * Sin({m3} * x))}\\right]_0^\\pi -\\int_0^\\pi\\simplify[std]{ ({ -w} / {m3} )* Cos({m3} * x) + {1 -w} * (1 / {m3} * Sin({m3} * x))}\\;dx\\\\ &amp;=&amp;\\simplify[std]{({-w*cos(m3*pi)})*({pi}/{m3})}-\\left[\\simplify[std]{{ -w} * (1 / {m3 ^ 2})* Sin({m3} * x) -({1 -w} * (1 / {m3 ^ 2}) * Cos({m3} * x))}\\right]_0^\\pi\\\\ &amp;=&amp; \\var{ans3}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]<br/>d)</p>\n    <p>\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]</p>\n    <p>Use integration by parts twice with $u=x^2$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{e^({n4}x)}\\Rightarrow v = \\simplify[std]{1/{n4}e^({n4}x)}$<br/>\\[\\begin{eqnarray*} I&amp;=&amp;\\left[\\simplify[std]{x^2/{n4}Exp({n4} * x)}\\right]_0^{\\var{b4}}+\\simplify[std]{2/{abs(n4)}DefInt(x*Exp({n4} * x),x,0,{b4})}\\\\ &amp;=&amp;\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\left[\\simplify[std]{x/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}+\\simplify[std]{1/{abs(n4)}DefInt(e^({n4}x),x,0,{b4})}\\right)\\\\ &amp;=&amp;\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\simplify[std]{{b4}/{n4}*e^{p}-1/{n4}}\\left[\\simplify[std]{1/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}\\right)\\\\ &amp;=&amp;\\simplify[std]{({b4 ^ 2} / {n4}) * Exp({p}) -(({2 * b4} / {n4 ^ 2}) * Exp({p})) + (2 / {n4 ^ 3}) * (Exp({p}) -1)}\\\\ &amp;=&amp;\\var{ans4}\\mbox{ to 4 decimal places} \\end{eqnarray*} \\]</p>\n  ", "tags": [], "type": "question"}, {"name": "Definite Integration 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "calculus", "definite integration", "integration", "integration by parts", "integration by parts twice"], "advice": "\n  <p>&nbsp;</p>\n    <p>b)<br/>\\[\\begin{eqnarray*}I&amp;=&amp;\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\\\ &amp;=&amp;\\left[\\ln(x+\\var{m2})\\right]_0^{\\var{b2}}\\\\ &amp;=&amp; \\ln(\\var{b2+m2})-\\ln(\\var{m2})\\\\ &amp;=&amp;\\ln\\left(\\frac{\\var{b2+m2}}{\\var{m2}}\\right)\\\\ &amp;=&amp;\\var{ans2}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]</p>\n    <p>&nbsp;</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      <p>\\[I=\\int_0^{\\var{b1}}\\simplify[std]{e^({a}x)}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans1-tol", "type": "numberentry", "maxvalue": "ans1+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>\\[I=\\int_0^{\\var{b2}}\\simplify[std]{1/({b}x+{m2})}\\;dx\\]</p>\n      <p>$I=\\;\\;$[[0]]</p>\n      <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans2-tol", "type": "numberentry", "maxvalue": "ans2+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>\\[I=\\int_0^{\\pi/2}\\simplify[std]{({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans3-tol", "type": "numberentry", "maxvalue": "ans3+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "<p>Evaluate the following definite integrals.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-2..2#0.5 except 0)", "name": "a"}, "m2": {"definition": "random(1..9)", "name": "m2"}, "b": {"definition": "random(2..5)", "name": "b"}, "w": {"definition": "random(0,1)", "name": "w"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "ans2": {"definition": "precround(1/b*(ln(1+b*b2/m2)),3)", "name": "ans2"}, "ans3": {"definition": "precround(tans3,3)", "name": "ans3"}, "b2": {"definition": "random(1..20)", "name": "b2"}, "b1": {"definition": "random(-1..2#0.5 except 0)", "name": "b1"}, "tol": {"definition": 0.001, "name": "tol"}, "t": {"definition": "random(1,-1)", "name": "t"}, "m3": {"definition": "random(2..9)", "name": "m3"}, "ans1": {"definition": "precround(tans1,3)", "name": "ans1"}, "c1": {"definition": "t*random(1..9)", "name": "c1"}, "tans1": {"definition": "(1/a)*(e^(a*b1)-1)", "name": "tans1"}, "tol1": {"definition": 0.0001, "name": "tol1"}, "tans3": {"definition": "1/m3*((1-w)*sin(m3*pi/2)-w*(cos(m3*pi/2)-1))", "name": "tans3"}, "d1": {"definition": "random(-9..9)", "name": "d1"}}, "metadata": {"notes": "\n    \t\t                \t\t<p><strong>3/07/1012:</strong></p>\n    \t\t                \t\t        <p>Added tags.</p>\n    \t\t                \t\t        <p>Checked calculations.</p>\n    \t\t                \t\t        <p>Left tolerances in, as easy to make minor errors in calculations.</p>\n    \t\t                \t\t        <p>Improved display in Advice.</p>\n    \t\t                \t\t        <p>Some superscripts e.g. the form a^\\var{b} in latex have to be written as a^{\\var{b}} as not displayed properly (if b has a second digit it slips down). Corrected.</p>\n    \t\t                \t\t        <p><strong>20/07/2012:</strong></p>\n    \t\t                \t\t        <p>Set new tolerace variables, tol=0.01, tol1=0.0001.</p>\n    \t\t                \t\t        <p>Can have expressions in Advice of the form $1\\times E$ where E is an expression. This can be remedied by rewriting - but later as not crucial.</p>\n    \t\t                \t\t        <p>Added description.</p>\n    \t\t                \t\t        <p></p>\n    \t\t                \t\t        <p><strong>25/07/2012:</strong></p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>Added tags.</p>\n    \t\t                \t\t        <p>A lot of work in this question - Perhaps it would be more managable broken down into two separate questions?</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>Question appears to be working correctly.</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \n    \t\t", "description": "<p>Evaluate $\\int_0^{\\,m}e^{ax}\\;dx$, $\\int_0^{p}\\frac{1}{bx+d}\\;dx,\\;\\int_0^{\\pi/2} \\sin(qx) \\;dx$.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Definite Integration: Calculating the area under a curve 1", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Gemma Crowe", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2440/"}, {"name": "Jack Dunham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2484/"}], "variables": {"a2": {"templateType": "anything", "name": "a2", "description": "", "definition": "1", "group": "b) quadratic. no neg region"}, "x31": {"templateType": "anything", "name": "x31", "description": "", "definition": "b3-random(2..3)", "group": "c quadratic. neg region"}, "x32": {"templateType": "anything", "name": "x32", "description": "", "definition": "b3", "group": "c quadratic. neg region"}, "x12": {"templateType": "anything", "name": "x12", "description": "", "definition": "random(1..4) + x11", "group": "linear graph (not used)"}, "x42": {"templateType": "anything", "name": "x42", "description": "", "definition": "b4", "group": "Ungrouped variables"}, "area3": {"templateType": "anything", "name": "area3", "description": "", "definition": "(x33^3/3 - 0.5*(a3+b3)*x33^2+a3*b3*x33)-2*(x32^3/3 - 0.5*(a3+b3)*x32^2+a3*b3*x32)+(x31^3/3 - 0.5*(a3+b3)*x31^2+a3*b3*x31)", "group": "c quadratic. neg region"}, "a3": {"templateType": "anything", "name": "a3", "description": "", "definition": "random(-3..-1)", "group": "c quadratic. neg region"}, "b4": {"templateType": "anything", "name": "b4", "description": "", "definition": "a4+random(2..3)", "group": "Ungrouped variables"}, "a1": {"templateType": "anything", "name": "a1", "description": "", "definition": "1", "group": "linear graph (not used)"}, "b3": {"templateType": "anything", "name": "b3", "description": "", "definition": "a3+random(3..4)", "group": "c quadratic. neg region"}, "c2": {"templateType": "anything", "name": "c2", "description": "", "definition": "random(1..4)", "group": "b) quadratic. no neg region"}, "x33": {"templateType": "anything", "name": "x33", "description": "", "definition": "x32+2", "group": "c quadratic. neg region"}, "x11": {"templateType": "anything", "name": "x11", "description": "", "definition": "random(1..2)", "group": "linear graph (not used)"}, "c4": {"templateType": "anything", "name": "c4", "description": "", "definition": "b4+2", "group": "Ungrouped variables"}, "a4": {"templateType": "anything", "name": "a4", "description": "", "definition": "random(-3..-2)", "group": "Ungrouped variables"}, "area2": {"templateType": "anything", "name": "area2", "description": "", "definition": "(a2*x22^3/3+c2*x22)-(a2*x21^3/3+c2*x21)", "group": "b) quadratic. no neg region"}, "x22": {"templateType": "anything", "name": "x22", "description": "", "definition": "random(3..5)+x21", "group": "b) quadratic. no neg region"}, "x21": {"templateType": "anything", "name": "x21", "description": "", "definition": "random(-4..-2)", "group": "b) quadratic. no neg region"}, "b1": {"templateType": "anything", "name": "b1", "description": "", "definition": "random(1..3)", "group": "linear graph (not used)"}, "x41": {"templateType": "anything", "name": "x41", "description": "", "definition": "a4", "group": "Ungrouped variables"}}, "preamble": {"js": "", "css": ""}, "functions": {"plotgraph": {"type": "html", "language": "javascript", "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 = -7;\nvar xmax = 7;\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: true\n }\n);\n\n\n\n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar brd = div.board; \n\n// create the x-axis.\nvar xaxis = brd.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = brd.create('ticks',[xaxis,1],{\n  drawLabels: true,\n  label: {offset: [-4, -10]},\n  minorTicks: 0\n});\n\n// create the y-axis\nvar yaxis = brd.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nyticks = brd.create('ticks',[yaxis,5],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 4\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) {\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 =  brd.create('curve', [dataX,dataY],{strokeWidth:0, fillColor:colour, fillOpacity:0.2});\n\nreturn shading;\n}\n\n\n//Define your functions\nvar f1 = function(x) {\n  return a*x+b;\n}\n\nvar f2 = function(x) {\n  return a*x*x + c;\n}\n\nvar f3 = function(x) {\n  return (x-a)*(x-b);\n}\n\nvar f4 = function(x) {\n  return 0.5*(x-a)*(x-b)*(x-c);\n}\n\n\n//Plot the graph and do shading\nswitch(q) {\n    case 1:\n        brd.create('functiongraph', [f1]);\n        shade(f1,x1,x2, 'red');\n        break;\n    case 2:\n        brd.create('functiongraph', [f2]);\n        shade(f2,x1,x2,'red');\n        break;\n    case 3:\n        brd.create('functiongraph', [f3]);\n        shade(f3,x1,x2,'red');\n        shade(f3,x2,x2+2,'green');\n        break;\n    case 4:\n        brd.create('functiongraph', [f4]);\n        shade(f4,x1,x2,'red');\n        shade(f4,x2,x2+2,'green');\n        break\n}\n\n\n\nreturn div;", "parameters": [["q", "number"], ["x1", "number"], ["x2", "number"], ["ymin", "number"], ["ymax", "number"], ["a", "number"], ["b", "number"], ["c", "number"]]}}, "ungrouped_variables": ["a4", "b4", "c4", "x41", "x42"], "rulesets": {}, "tags": [], "metadata": {"description": "<p>Graphs are given with areas underneath them shaded. The student is asked to select or enter the correct integral which calculates its area.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [{"name": "linear graph (not used)", "variables": ["x11", "x12", "a1", "b1"]}, {"name": "b) quadratic. no neg region", "variables": ["x21", "x22", "a2", "c2", "area2"]}, {"name": "c quadratic. neg region", "variables": ["a3", "b3", "x31", "x32", "x33", "area3"]}], "advice": "<p>Be sure to initally write down the integral expression for $A$ with the appropriate boundaries.</p>\n<p>\\[A=\\int_{a}^{b}{f(x)}dx\\]</p>\n<p>Double check your answer makes sense. Remember, areas below the $x$-axis are <strong>negative</strong>. Try and estimate whether most of the area is above or below the $x$-axis, from this you should be to check whether or not the sign of your answer is correct.</p>\n<p>For further information see Chapter 6 - Integration Notes.&nbsp;</p>", "statement": "<p>Find the following areas $A$ shown by the following shaded regions.&nbsp;Enter all values as fractions and&nbsp;<strong>not</strong> as decimals</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"type": "gapfill", "showCorrectAnswer": true, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>{plotgraph(2,x21,x22,-5,25,a2,0,c2)}</p>\n<p>This is the graph of the function $f(x) = \\simplify{{a2}*x^2+{c2}}$.</p>\n<p>$A=\\,\\,$[[0]]</p>", "marks": 0, "scripts": {}, "variableReplacements": [], "gaps": [{"type": "numberentry", "allowFractions": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": "3", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "area2", "mustBeReducedPC": 0, "variableReplacements": [], "unitTests": [], "minValue": "area2"}]}, {"type": "gapfill", "showCorrectAnswer": true, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>{plotgraph(3,x31,x32,-6,15,a3,b3,0)}</p>\n<p>This curve has equation $y = \\simplify{x^2-{a3+b3}*x + {a3*b3}}$.</p>\n<p>$A=\\,\\,$[[0]]</p>\n<p></p>\n<p></p>", "marks": 0, "scripts": {}, "variableReplacements": [], "gaps": [{"type": "numberentry", "allowFractions": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": "5", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "area3", "mustBeReducedPC": 0, "variableReplacements": [], "unitTests": [], "minValue": "area3"}]}], "type": "question"}, {"name": "Definite Integration: Calculating the area under a curve II. Needs integration by parts.", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Gemma Crowe", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2440/"}, {"name": "Jack Dunham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2484/"}], "parts": [{"gaps": [{"showPreview": true, "vsetRangePoints": 5, "answer": "{area1_exact}", "checkVariableNames": false, "showFeedbackIcon": true, "marks": "3", "scripts": {}, "failureRate": 1, "extendBaseMarkingAlgorithm": true, "unitTests": [], "expectedVariableNames": [], "vsetRange": [0, 1], "checkingType": "absdiff", "showCorrectAnswer": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "answerSimplification": "fractionNumbers", "type": "jme", "checkingAccuracy": 0.001, "customMarkingAlgorithm": ""}], "showFeedbackIcon": true, "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "sortAnswers": false, "showCorrectAnswer": true, "prompt": "<p>{plotgraph(1,0,x1,-2,ymax1,a1,0,0)}</p>\n<p>This is the graph of the function $f(x) = \\simplify[fractionNumbers]{x*e^({a1}*x)}$.</p>\n<p>$A=\\,\\,$&nbsp;[[0]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": ""}], "variable_groups": [{"variables": ["x1", "a1", "ymax1", "area1_exact", "area1"], "name": "(a)"}], "tags": [], "statement": "<p>Find the following area $A$ shown by the following shaded region.&nbsp;Enter all values as fractions and&nbsp;<strong>not</strong>&nbsp;decimals</p>", "advice": "<p>Be sure to initally write down the integral expression for $A$ with the appropriate boundaries.</p>\n<p>\\[A=\\int_{a}^{b}{f(x)}dx\\]</p>\n<p>For this question you will need to use integration by parts to integrate the function.&nbsp;</p>\n<p>Double check your answer makes sense. Remember, areas below the $x$-axis are&nbsp;<strong>negative</strong>. Try and estimate whether most of the area is above or below the $x$-axis, from this you should be to check whether or not the sign of your answer is correct.</p>\n<p>For further information see Chapter 6 - Integration Notes.&nbsp;</p>", "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Graphs are given with areas underneath them shaded. The student is asked to select or enter the correct integral which calculates its area.</p>"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": [], "preamble": {"css": "", "js": ""}, "functions": {"plotgraph": {"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 = -1;\nvar xmax = 7;\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: true\n }\n);\n\n\n\n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar brd = div.board; \n\n// create the x-axis.\nvar xaxis = brd.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = brd.create('ticks',[xaxis,1],{\n  drawLabels: true,\n  label: {offset: [-4, -10]},\n  minorTicks: 0\n});\n\n// create the y-axis\nvar yaxis = brd.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nyticks = brd.create('ticks',[yaxis,5],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 4\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) {\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 =  brd.create('curve', [dataX,dataY],{strokeWidth:0, fillColor:colour, fillOpacity:0.2});\n\nreturn shading;\n}\n\n\n//Define your functions\nvar f1 = function(x) {\n  return x*Math.exp(a*x);\n}\n\nvar f2 = function(x) {\n  return a*x*x + c;\n}\n\n\n\n//Plot the graph and do shading\nswitch(q) {\n    case 1:\n        brd.create('functiongraph', [f1]);\n        shade(f1,x1,x2, 'red');\n        break;\n    case 2:\n        brd.create('functiongraph', [f2]);\n        shade(f2,x1,x2,'red');\n        break;\n}\n\n\n\nreturn div;", "language": "javascript", "type": "html", "parameters": [["q", "number"], ["x1", "number"], ["x2", "number"], ["ymin", "number"], ["ymax", "number"], ["a", "number"], ["b", "number"], ["c", "number"]]}}, "variables": {"x1": {"group": "(a)", "templateType": "anything", "definition": "random(2..4)", "description": "", "name": "x1"}, "ymax1": {"group": "(a)", "templateType": "anything", "definition": "x1*e^(a1*x1)+5", "description": "", "name": "ymax1"}, "area1_exact": {"group": "(a)", "templateType": "anything", "definition": "expression((x1/a1-1/a1^2)+\"*e^(\"+a1*x1+\")+\"+1/a1^2)", "description": "", "name": "area1_exact"}, "a1": {"group": "(a)", "templateType": "anything", "definition": "random([0.1,0.2,0.25,0.5])", "description": "", "name": "a1"}, "area1": {"group": "(a)", "templateType": "anything", "definition": "(x1/a1-1/a1^2)*e^(a1*x1)+1/a1^2", "description": "", "name": "area1"}}, "type": "question"}]}], "type": "exam", "contributors": [{"name": "Nick Walker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2416/"}], "extensions": ["jsxgraph"], "custom_part_types": [], "resources": []}