// Numbas version: exam_results_page_options
{"name": "Julie's Integration 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["c", "b", "b2", "ans1", "ans2", "t", "tol", "ans", "b1"], "name": "Julie's Integration 4", "tags": ["areas", "definite integration", "integration", "rebelmaths"], "advice": "\n  <p>First we observe that:\\[\\simplify[std]{int ({t}*exp({b}/{c}*x)+{1-t}*ln({b}x+{c}),x)={t*c}/{b}*exp({b}/{c}*x)+({1-t}/{b}*({b}x+{c})*(ln({b}x+{c})-1))+C}.\\]</p>\n    <p>Hence we have:</p>\n    <p>\\[\\begin{eqnarray*} \\simplify[std]{defint ({t}*exp({b}/{c}*x)+{1-t}*ln({b}x+{c}),x,{b1},{b2})}&amp;=&amp;\\left[\\simplify[std]{{t*c}/{b}*exp({b}/{c}*x)+({1-t}/{b}*({b}x+{c})*(ln({b}x+{c})-1))}\\right]_{\\var{b1}}^{\\var{b2}}\\\\&amp;=&amp;\\var{ans}\\end{eqnarray*}\\]</p>\n    <p>to 3 decimal places.</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Enter the area $A$ here to 3 decimal places:</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "ans+tol", "minValue": "ans-tol", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": "3", "type": "numberentry", "showPrecisionHint": false}], "extensions": [], "statement": "<p>Find the area $A$ of the shape bounded by the $x$-axis, the function $y=\\simplify[std]{{t}*exp({b}/{c}*x)+{1-t}*ln({b}x+{c})}$ and the lines $x=\\var{b1},\\;x=\\var{b2}$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"c": {"definition": "random(3..7 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..4 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "b2": {"definition": "b1+random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "ans1": {"definition": "precround(c*(exp(b*b2/c)-exp(b*b1/c))/b,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans1", "description": ""}, "ans2": {"definition": "precround(1/b*((b*b2+c)*(ln(b*b2+c)-1)-(b*b1+c)*(ln(b*b1+c)-1)),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans2", "description": ""}, "t": {"definition": "random(0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "ans": {"definition": "t*ans1+(1-t)*ans2", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "b1": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}}, "metadata": {"description": "<p>Finding areas under graphs using definite integration.</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}