// Numbas version: exam_results_page_options
{"name": "Contour integral of a complex function II", "extensions": [], "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"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "c1"}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(exp(-b1)*sin(c1),3)", "description": "", "name": "v"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "b1"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(exp(-a1)-exp(-b1)*cos(c1),3)", "description": "", "name": "u"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "a1"}}, "ungrouped_variables": ["a1", "u", "b1", "tol", "v", "c1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Contour integral of a complex function II", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "u+tol", "minValue": "u-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "v+tol", "minValue": "v-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$I=$[[0]]$+i$[[1]]. &nbsp;(Enter your answers to 3d.p.)</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Find the value of the integral</p>\n    <p>\\[I=\\int_C{\\!\\mathrm{e}^{-z}\\,\\mathrm{d}z},\\]</p>\n    <p>along any path $C$ from $z=\\var{a1}$ to $z=\\simplify{{b1}+{c1}*i}$.</p>", "tags": ["checked2015", "MAS2103"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/7/2012:</strong></p>\n<p>Added tags.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Contour integral of $\\mathrm{e}^{-z}$ along any path.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>This is an exercise in integration, so</p>\n    <p>\\[\\begin{align}I&amp;=\\int_C{\\!\\mathrm{e}^{-z}\\,\\mathrm{d}z}\\\\&amp;=\\int_{\\var{a1}}^{\\simplify{{b1}+{c1}*i}}{\\!\\mathrm{e}^{-z}\\,\\mathrm{d}z}\\\\&amp;=-\\left[\\mathrm{e}^{-z}\\right]_{\\var{a1}}^{\\simplify{{b1}+{c1}*i}}\\\\&amp;=\\mathrm{e}^{\\var{-a1}}-\\mathrm{e}^{\\simplify{{-b1}+{-c1}*i}}.\\end{align}\\]</p>\n    <p>This can be rewritten as</p>\n    <p>\\[I=\\mathrm{e}^{\\var{-a1}}-\\mathrm{e}^{\\var{-b1}}\\biggl(\\cos(\\var{c1})-i\\sin(\\var{c1})\\biggr)=\\simplify{{u}+{v}*i}\\;\\text{to 3d.p.}\\]</p>", "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/"}]}