// Numbas version: exam_results_page_options
{"name": "Fourier series", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Fourier series", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Consider the following periodic function</p>\n<p>\\[f(t)=\\left\\{\\matrix{\\simplify{{p}*t}&amp;\\text{if }0\\leq t&lt;\\pi\\\\\\var{q}&amp;\\text{if }\\pi\\leq t&lt;2\\pi}\\right.\\]</p>\n<p>where $f(t+2\\pi)=f(t)$.</p>", "advice": "<p>First calculate $a_0$ as the mean value of $f(t)$ over one period:</p>\n<p>\\[a_0=\\frac{1}{2}\\times\\var{p}\\times\\frac{\\pi}{2}+\\frac{1}{2}\\times\\var{q}=\\var{a0}\\]</p>\n<p>Now calculate the coefficients $a_n$:</p>\n<p>\\[a_n=\\frac{1}{\\pi}\\int_0^{2\\pi}f(t)\\cos(nt)\\;dt=\\frac{\\var{p}}{\\pi}\\int_0^\\pi t\\cos(nt)\\;dt+\\frac{\\var{q}}{\\pi}\\int_\\pi^{2\\pi}\\cos(nt)\\;dt\\]</p>\n<p>The first integral is performed by parts:</p>\n<p>\\[\\frac{\\var{p}}{\\pi}\\int_0^\\pi t\\cos(nt)\\;dt=\\frac{\\var{p}}{n\\pi}\\left[t\\sin(nt)\\right]_0^\\pi-\\frac{\\var{p}}{n\\pi}\\int_0^\\pi\\sin(nt)\\;dt=\\frac{\\var{p}}{n\\pi}\\left[t\\sin(nt)\\right]_0^\\pi+\\frac{\\var{p}}{n^2\\pi}\\left[\\cos(nt)\\right]_0^\\pi\\]</p>\n<p>Therefore</p>\n<p>\\[a_n=\\frac{\\var{p}}{n\\pi}\\left[t\\sin(nt)\\right]_0^\\pi+\\frac{\\var{p}}{n^2\\pi}\\left[\\cos(nt)\\right]_0^\\pi+\\frac{\\var{q}}{n\\pi}\\left[\\sin(nt)\\right]_\\pi^{2\\pi}\\]</p>\n<p>When $n$ is odd,</p>\n<p>\\[a_n=\\frac{\\var{p}}{n\\pi}\\left(0-0\\right)+\\frac{\\var{p}}{n^2\\pi}\\left(-1-1\\right)+\\frac{\\var{q}}{n\\pi}\\left(0-0\\right)=\\simplify{-2*{p}/(n^2*pi)}\\]</p>\n<p>And when $n$ is even,</p>\n<p>\\[a_n=\\frac{\\var{p}}{n\\pi}\\left(0-0\\right)+\\frac{\\var{p}}{n^2\\pi}\\left(1-1\\right)+\\frac{\\var{q}}{n\\pi}\\left(0-0\\right)=0\\]</p>\n<p>Therefore we obtain the following coefficients:</p>\n<p>\\[a_1=\\simplify{-2*{p}/pi}=\\var{a1}\\qquad a_2=0\\qquad a_3=\\simplify{-2*{p}/(3^2*pi)}=\\var{a3}\\]</p>\n<p>Now calculate the coefficients $b_n$:</p>\n<p>\\[b_n=\\frac{1}{\\pi}\\int_0^{2\\pi}f(t)\\sin(nt)\\;dt=\\frac{\\var{p}}{\\pi}\\int_0^\\pi t\\sin(nt)\\;dt+\\frac{\\var{q}}{\\pi}\\int_\\pi^{2\\pi}\\sin(nt)\\;dt\\]</p>\n<p>The first integral is performed by parts:</p>\n<p>\\[\\frac{\\var{p}}{\\pi}\\int_0^\\pi t\\sin(nt)\\;dt=-\\frac{\\var{p}}{n\\pi}\\left[t\\cos(nt)\\right]_0^\\pi+\\frac{\\var{p}}{n\\pi}\\int_0^\\pi \\cos(nt)\\;dt=-\\frac{\\var{p}}{n\\pi}\\left[t\\cos(nt)\\right]_0^\\pi+\\frac{\\var{p}}{n^2\\pi}\\left[\\sin(nt)\\right]_0^\\pi\\]</p>\n<p>Therefore</p>\n<p>\\[b_n=-\\frac{\\var{p}}{n\\pi}\\left[t\\cos(nt)\\right]_0^\\pi+\\frac{\\var{p}}{n^2\\pi}\\left[\\sin(nt)\\right]_0^\\pi+\\frac{\\var{-q}}{n\\pi}\\left[\\cos(nt)\\right]_\\pi^{2\\pi}\\]</p>\n<p>When $n$ is odd,</p>\n<p>\\[b_n=-\\frac{\\var{p}}{n\\pi}\\left(-\\pi-0\\right)+\\frac{\\var{p}}{n^2\\pi}\\left(0-0\\right)+\\frac{\\var{-q}}{n\\pi}\\left(1--1\\right)=\\simplify{{p}/n-2*{q}/(n*pi)}\\]</p>\n<p>And when $n$ is even,</p>\n<p>\\[b_n=-\\frac{\\var{p}}{n\\pi}\\left(\\pi-0\\right)+\\frac{\\var{p}}{n^2\\pi}\\left(0-0\\right)+\\frac{\\var{-q}}{n\\pi}\\left(1-1\\right)=\\simplify{-{p}/n}\\]</p>\n<p>Therefore we obtain the following coefficients:</p>\n<p>\\[b_1=\\simplify{{p}-2*{q}/pi}=\\var{b1}\\qquad b_2=\\simplify{-{p}/2}=\\var{b2}\\qquad b_3=\\simplify{{p}/3-2*{q}/(3*pi)}=\\var{b3}\\]</p>\n<p>The Fourier series for $f(t)$ (to the third harmonic) is:</p>\n<p>\\[f(t)=a_0+a_1\\cos(t)+a_2\\cos(2t)+a_3\\cos(3t)+b_1\\sin(t)+b_2\\sin(2t)+b_3\\sin(3t)\\]</p>\n<p>On substituting the values of the coefficients $a_n$ and $b_n$ this becomes:</p>\n<p>\\[f(t)=\\simplify{{a0}+{a1}*cos(t)+{a2}*cos(2t)+{a3}*cos(3t)+{b1}*sin(t)+{b2}*sin(2t)+{b3}*sin(3t)}\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p": {"name": "p", "group": "Setup", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Setup", "definition": "random(-3..3 except [0,p])", "description": "", "templateType": "anything", "can_override": false}, "a0": {"name": "a0", "group": "an coefficients", "definition": "(p*pi+2*q)/4", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "bn coefficients", "definition": "p-2*q/pi", "description": "", "templateType": 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