// Numbas version: exam_results_page_options
{"name": "Fourier Series square wave function.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {}, "advice": "<p>&nbsp; \\(f(t)=\\left[\\begin{array}{cc}\\,\\,0 &amp;\\,\\,-\\pi&lt;t&lt;0\\\\\\,\\,\\var{a}&amp;\\,\\,\\,\\,\\pi/2&lt;t&lt;\\pi/2\\\\\\,\\,0&amp;\\,\\,\\,\\,\\pi/2&lt;t&lt;\\pi\\end{array}\\right] \\)</p>\n<p style=\"padding-left: 30px;\">\\(a_0=\\frac{1}{\\pi}\\int_{\\frac{-\\pi}{2}}^{\\frac{\\pi}{2}}3dx\\) the average value of the wave over one complete cycle</p>\n<p style=\"padding-left: 30px;\">As the function is even</p>\n<p style=\"padding-left: 30px;\">\\(b_n=0\\)&nbsp;</p>\n<p>&nbsp; &nbsp; &nbsp; \\(a_n=\\frac{1}{\\pi}\\int_{-\\pi}^{\\pi} f(t)\\cos\\left(nx\\right)dx\\)</p>\n<p>\\(a_n=\\frac{1}{\\pi}\\left(\\int_{-\\pi}^{-\\frac{\\pi}{2}}0\\sin\\left({n}x\\right)dx+\\int_{\\frac{-\\pi}{2}}^{\\frac{\\pi}{2}}\\var{a}\\sin\\left({n}x\\right)dx+\\int_{\\frac{\\pi}{2}}^{\\pi}0\\sin\\left({n}x\\right)dx\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(a_n=\\frac{1}{\\pi}\\left(\\frac{\\var{2} *\\var{a} (\\sin (\\frac{\\pi}{2} n)) }{n}\\right)\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>If \\(n\\) is an&nbsp;even&nbsp;number \\(sin(n\\frac{\\pi}{2})=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(b_n=0\\)</p>\n<p>If \\(n=1, 5, 9, ...\\) is then \\(sin(n\\frac{\\pi}{2})=1\\)</p>\n<p style=\"padding-left: 30px;\">\\(b_n=\\frac{1}{\\pi}\\left(\\frac{\\var{2} *\\var{a}&nbsp; }{n}\\right)\\)</p>\n<p>If \\(n=3, 7, 11, ...\\) is an even&nbsp;number \\(sin(n\\frac{\\pi}{2})=-1\\)</p>\n<p style=\"padding-left: 30px;\">\\(b_n=-\\frac{1}{\\pi}\\left(\\frac{\\var{2} *\\var{a}&nbsp; }{n}\\right)\\)</p>", "name": "Fourier Series square wave function.", "ungrouped_variables": ["a", "k"], "extensions": [], "rulesets": {}, "variables": {"k": {"group": "Ungrouped variables", "definition": "random(2..7#2)", "templateType": "randrange", "name": "k", "description": ""}, "a": {"group": "Ungrouped variables", "definition": "random(1..6#1)", "templateType": "randrange", "name": "a", "description": ""}}, "statement": "<p>Given the function:</p>\n<p style=\"padding-left: 30px;\">&nbsp; \\(f(t)=\\left[\\begin{array}{cc}\\,\\,0 &amp;\\,\\,-\\pi&lt;t&lt;0\\\\\\,\\,\\var{a}&amp;\\,\\,\\,\\,\\pi/2&lt;t&lt;\\pi/2\\\\\\,\\,0&amp;\\,\\,\\,\\,\\pi/2&lt;t&lt;\\pi\\end{array}\\right] \\)</p>\n<p></p>", "preamble": {"css": "", "js": ""}, "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Calculating particular&nbsp;harmonic components of a Fourier series expansion.</p>"}, "variable_groups": [], "parts": [{"prompt": "<p>Calculate the value&nbsp;for the trigonometric Fourier coefficient \\(b_{k}\\).</p>\n<p>\\(b_{k}\\) =&nbsp;[[0]]</p>", "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": false, "extendBaseMarkingAlgorithm": true, "marks": 0, "gaps": [{"mustBeReducedPC": 0, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "scripts": {}, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "maxValue": "0", "marks": "2", "unitTests": [], "allowFractions": false, "mustBeReduced": false, "correctAnswerStyle": "plain", "variableReplacements": [], "minValue": "0"}], "unitTests": [], "sortAnswers": false, "variableReplacements": []}, {"vsetRange": [0, 1], "prompt": "<p>Determine an expression for the trigonometric Fourier coefficient \\(a_{0}\\), and hence evaluate \\(a_{0}\\).&nbsp;</p>\n<p>&nbsp;\\(a_{0}\\) =</p>", "showPreview": true, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "type": "jme", "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "marks": "2", "unitTests": [], "failureRate": 1, "expectedVariableNames": [], "checkVariableNames": false, "variableReplacements": [], "answer": "{a}"}, {"vsetRange": [0, 1], "prompt": "<p>Determine an expression for the trigonometric Fourier coefficient \\(a_{1}\\), and hence evaluate \\(a_{1}\\).&nbsp;</p>\n<p>&nbsp;\\(a_{1}\\) =</p>", "showPreview": true, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "type": "jme", "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "marks": "2", "unitTests": [], "failureRate": 1, "expectedVariableNames": [], "checkVariableNames": false, "variableReplacements": [], "answer": "2*{a}/pi"}, {"vsetRange": [0, 1], "prompt": "<p>Determine an expression for the trigonometric Fourier coefficient \\(a_{n}\\), and hence evaluate \\(a_{n}\\), for n=1,5,....&nbsp;</p>\n<p>&nbsp;\\(a_{n}\\) =</p>", "showPreview": true, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "type": "jme", "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "marks": "2", "unitTests": [], "failureRate": 1, "expectedVariableNames": [], "checkVariableNames": false, "variableReplacements": [], "answer": "2*{a}/(n pi)"}, {"vsetRange": [0, 1], "prompt": "<p>Determine an expression for the trigonometric Fourier coefficient \\(a_{n}\\), and hence evaluate \\(a_{n}\\), for n=3,7,....&nbsp;</p>\n<p>&nbsp;\\(a_{n}\\) =</p>", "showPreview": true, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "type": "jme", "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "marks": "2", "unitTests": [], "failureRate": 1, "expectedVariableNames": [], "checkVariableNames": false, "variableReplacements": [], "answer": "-2*{a}/(n pi)"}, {"prompt": "<p>Determine an expression for the trigonometric Fourier coefficient \\(a_{n}\\), and hence evaluate \\(a_{n}\\), for n=2,4,....&nbsp;</p>\n<p>&nbsp;\\(a_{n}\\) =</p>\n<p>[[0]]</p>", "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "marks": 0, "gaps": [{"vsetRange": [0, 1], "showPreview": true, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "type": "jme", "scripts": {}, "showFeedbackIcon": true, "marks": "2", "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "answer": "0", "unitTests": [], "failureRate": 1, "expectedVariableNames": [], "checkVariableNames": false, "variableReplacements": []}], "unitTests": [], "sortAnswers": false, "variableReplacements": []}, {"minMarks": 0, "prompt": "<p>Is the function even or odd?</p>", "choices": ["Even", "Odd", "Neither even or odd"], "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "scripts": {}, "distractors": ["", "", ""], "showFeedbackIcon": true, "showCellAnswerState": true, "matrix": ["2", "0", 0], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "displayColumns": 0, "shuffleChoices": false, "marks": 0, "unitTests": [], "maxMarks": 0, "displayType": "radiogroup", "variableReplacements": []}], "type": "question", "contributors": [{"name": "John Butler", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/555/"}, {"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "John Butler", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/555/"}, {"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}