Calculate the value of the first term of the Fourier series \\(\\frac{a_{0}}{2}\\).
\n\\(\\frac{a_{0}}{2}\\) = [[0]]
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\n\\( a_{\\var{k}}\\) = [[0]]
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\n\\( b_{\\var{k}}\\) = [[0]]
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\nAmplitude = [[0]]
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\n\\(f(t)=\\left[ \\begin{array}{cc}\\,\\,\\var{a}&\\,\\,-\\var{L}<t<-\\simplify{{L}/2}\\\\\\,\\,\\var{b}&\\,\\,-\\simplify{{L}/2}<t<\\simplify{{L}/2}\\\\\\,\\,\\var{c}&\\,\\,\\simplify{{L}/2}<t<\\var{L}\\end{array}\\right] \\,\\,\\,\\,f(t+\\simplify{2*{L}})=f(t)\\)
", "ungrouped_variables": ["a", "L", "k", "b", "c"], "variables": {"L": {"definition": "random(2..11#1)", "description": "", "group": "Ungrouped variables", "name": "L", "templateType": "randrange"}, "b": {"definition": "random(12..24#1)", "description": "", "group": "Ungrouped variables", "name": "b", "templateType": "randrange"}, "a": {"definition": "random(1..5#2)", "description": "", "group": "Ungrouped variables", "name": "a", "templateType": "randrange"}, "k": {"definition": "random(5..19#2)", "description": "", "group": "Ungrouped variables", "name": "k", "templateType": "randrange"}, "c": {"definition": "random(6..11#1)", "description": "", "group": "Ungrouped variables", "name": "c", "templateType": "randrange"}}, "extensions": [], "advice": "\\(f(t)=\\left[ \\begin{array}{cc}\\,\\,\\var{a}&\\,\\,-\\var{L}<t<-\\simplify{{L}/2}\\\\\\,\\,\\var{b}&\\,\\,-\\simplify{{L}/2}<t<\\simplify{{L}/2}\\\\\\,\\,\\var{c}&\\,\\,\\simplify{{L}/2}<t<\\var{L}\\end{array}\\right] \\,\\,\\,\\,f(t+\\simplify{2*{L}})=f(t)\\)
\n\\(\\frac{a_0}{2}=\\) the average value of the wave over one complete cycle \\(=\\frac{Area}{Base}\\)
\n\\(\\frac{a_0}{2}=\\frac{\\var{a}*\\simplify{{L}/2}+\\var{b}*\\var{L}+\\var{c}*\\simplify{{L}/2}}{\\simplify{2*{L}}}=\\simplify{({a}+2*{b}+{c})/4}\\)
\n\n\n\\(a_k=\\frac{1}{L}\\int_{-L}^{L}f(t)cos\\left(\\frac{{k}\\pi}{L}t\\right)dt\\)
\n\\(2L=\\simplify{2*{L}}\\implies L=\\var{L}\\)
\n\\(a_k=\\frac{1}{\\var{L}}\\left(\\int_{-\\var{L}}^{-\\frac{\\var{L}}{2}}\\var{a}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt+\\int_{-\\frac{\\var{L}}{2}}^{\\frac{\\var{L}}{2}}\\var{b}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt+\\int_{\\frac{\\var{L}}{2}}^{\\var{L}}\\var{c}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt\\right)\\)
\n\\(a_k=\\frac{1}{\\var{L}}\\left(\\frac{\\var{a}*\\var{L}}{{k}\\pi}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{-\\var{L}}^{-\\frac{\\var{L}}{2}}+\\frac{\\var{b}*\\var{L}}{{k}\\pi}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{-\\frac{\\var{L}}{2}}^{\\frac{\\var{L}}{2}}+\\frac{\\var{c}*\\var{L}}{{k}\\pi}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{\\frac{\\var{L}}{2}}^{\\var{L}}\\right)\\)
\n\\(a_k=\\frac{\\var{a}}{{k}\\pi}sin(-\\frac{{k}\\pi}{2})-\\frac{\\var{a}}{{k}\\pi}sin(-{k}\\pi)+\\frac{\\var{b}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})-\\frac{\\var{b}}{{k}\\pi}sin(-\\frac{{k}\\pi}{2})+\\frac{\\var{c}}{{k}\\pi}sin({k}\\pi)-\\frac{\\var{c}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})\\)
\n\\(a_k=-\\frac{\\var{a}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})+\\frac{\\var{b}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})+\\frac{\\var{b}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})-\\frac{\\var{c}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})\\)
\n\\(a_k=\\frac{\\simplify{-{a}+2{b}-{c}}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})\\)
\n\\(a_\\var{k}=\\simplify{(-{a}+2{b}-{c})/({k}*pi)sin({k}*pi/2)}\\)
\n\n\\(b_k=\\frac{1}{L}\\int_{-L}^{L}f(t)sin\\left(\\frac{{k}\\pi}{L}t\\right)dt\\)
\n\\(b_k=\\frac{1}{\\var{L}}\\left(\\int_{-\\var{L}}^{-\\frac{\\var{L}}{2}}\\var{a}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt+\\int_{-\\frac{\\var{L}}{2}}^{\\frac{\\var{L}}{2}}\\var{b}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt+\\int_{\\frac{\\var{L}}{2}}^{\\var{L}}\\var{c}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt\\right)\\)
\n\\(b_k=\\frac{1}{\\var{L}}\\left(-\\frac{\\var{a}*\\var{L}}{{k}\\pi}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{-\\var{L}}^{-\\frac{\\var{L}}{2}}-\\frac{\\var{b}*\\var{L}}{{k}\\pi}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{-\\frac{\\var{L}}{2}}^{\\frac{\\var{L}}{2}}-\\frac{\\var{c}*\\var{L}}{{k}\\pi}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{\\frac{\\var{L}}{2}}^{\\var{L}}\\right)\\)
\n\\(b_k=-\\frac{\\var{a}}{{k}\\pi}cos(-\\frac{{k}\\pi}{2})+\\frac{\\var{a}}{{k}\\pi}cos(-{k}\\pi)-\\frac{\\var{b}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})+\\frac{\\var{b}}{{k}\\pi}cos(-\\frac{{k}\\pi}{2})-\\frac{\\var{c}}{{k}\\pi}cos({k}\\pi)+\\frac{\\var{c}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})\\)
\n\\(cos(-\\theta)=cos(\\theta)\\)
\n\\(b_k=-\\frac{\\var{a}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})+\\frac{\\var{a}}{{k}\\pi}cos({k}\\pi)-\\frac{\\var{b}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})+\\frac{\\var{b}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})-\\frac{\\var{c}}{{k}\\pi}cos({k}\\pi)+\\frac{\\var{c}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})\\)
\n\\(b_k=\\frac{\\var{c}-\\var{a}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})+\\frac{\\var{a}-\\var{c}}{{k}\\pi}cos({k}\\pi)\\)
\n\\(b_\\var{k}=\\simplify{({c}-{a})/({k}*pi)cos({k}*pi/2)}+\\simplify{(-{c}+{a})/({k}*pi)cos({k}*pi)}\\)
\n\nRecall the amplitude of the \\(\\var{k}\\)th harmonic component is given by \\(\\sqrt{(a_\\var{k})^2+(b_\\var{k})^2}\\)
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", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "tags": [], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}