i know b_n =0, but a_0 =(4/3)F_0 (according to solution) my answer is a_0 =(8/3)F_0 where did i did wrong how to find answer (Fourier transformation) like picture below (in the comment? f(t)= { ((((3F_0 )/t_0 )t,0≤t≤(t_0 /3))),((F_0 , (t_0 /3)≤t≤((2t_0 )/3))),((((-3F_0 )/t_0 )t+3F_0 ,((2t_0 )/3)≤t≤t_0 )) :} a_n =(2/t_0 )∫_0 ^t_0 f(t) cos(nωt)dt a_n =(2/t_0 )∫_0 ^t_0 f(t) cos(nωt)dt a_0 =(2/t_0 )∫_0 ^t_0 f(t) dt ∫_0 ^((4t_0 )/3) f(t) dt =∫_0 ^(t_0 /3) ((3F_0 )/t_0 )tdt+∫_(t_0 /3) ^((2t_0 )/3) F_0 dt+∫_((2t_0 )/3) ^t_0 ((-3F_0 )/t_0 )t+3F_0 dt =[((3F_0 )/t_0 )t]_0 ^(t_0 /3) +[F_0 t]_(t_0 /3) ^((2t_0 )/3) +[((-3F_0 )/t_0 )t+3F_0 t]_((2t_0 )/3) ^t_0 =F_0 +(1/3)F_0 t_0 −F_0 +F_0 t_0 =(4/3) F_0 t_0 so a_0 =(2/t_0 )×(4/3)F_0 t_0 =(8/3)F


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Question Number 189865 by uchihayahia last updated on 23/Mar/23
$i know b_n =0, but a_0 =(4/3)F_0 (according to solution) my answer is a_0 =(8/3)F_0 where did i did wrong how to find answer (Fourier transformation) like picture below (in the comment? f(t)= { ((((3F_0 )/t_0 )t,0≤t≤(t_0 /3))),((F_0 , (t_0 /3)≤t≤((2t_0 )/3))),((((-3F_0 )/t_0 )t+3F_0 ,((2t_0 )/3)≤t≤t_0 )) :} a_n =(2/t_0 )∫_0 ^t_0 f(t) cos(nωt)dt a_n =(2/t_0 )∫_0 ^t_0 f(t) cos(nωt)dt a_0 =(2/t_0 )∫_0 ^t_0 f(t) dt ∫_0 ^((4t_0 )/3) f(t) dt =∫_0 ^(t_0 /3) ((3F_0 )/t_0 )tdt+∫_(t_0 /3) ^((2t_0 )/3) F_0 dt+∫_((2t_0 )/3) ^t_0 ((-3F_0 )/t_0 )t+3F_0 dt =[((3F_0 )/t_0 )t]_0 ^(t_0 /3) +[F_0 t]_(t_0 /3) ^((2t_0 )/3) +[((-3F_0 )/t_0 )t+3F_0 t]_((2t_0 )/3) ^t_0 =F_0 +(1/3)F_0 t_0 −F_0 +F_0 t_0 =(4/3) F_0 t_0 so a_0 =(2/t_0 )×(4/3)F_0 t_0 =(8/3)F_0$
$i k n o w b_{n} = 0, b u t a_{0} = \frac{4}{3} F_{0} (a c c o r d i n g$ $t o s o l u t i o n) m y a n s w e r i s a_{0} = \frac{8}{3} F_{0}$ $w h e r e d i d i d i d w r o n g h o w t o$ $f i n d a n s w e r (F o u r i e r t r a n s f o r m a t i o n)$ $l i k e p i c t u r e b e l o w (i n t h e c o m m e n t ?$ $f (t) = {\begin{cases} \frac{3 F_{0}}{t_{0}} t, 0 ⩽ t ⩽ \frac{t_{0}}{3} \\ F_{0}, \frac{t_{0}}{3} ⩽ t ⩽ \frac{2 t_{0}}{3} \\ \frac{- 3 F_{0}}{t_{0}} t + 3 F_{0}, \frac{2 t_{0}}{3} ⩽ t ⩽ t_{0} \end{cases}$ $a_{n} = \frac{2}{t_{0}} \int_{0}^{t_{0}} f (t) c o s (n ω t) d t$ $a_{n} = \frac{2}{t_{0}} \int_{0}^{t_{0}} f (t) c o s (n ω t) d t$ $a_{0} = \frac{2}{t_{0}} \int_{0}^{t_{0}} f (t) d t$ $\int_{0}^{\frac{4 t_{0}}{3}} f (t) d t = \int_{0}^{\frac{t_{0}}{3}} \frac{3 F_{0}}{t_{0}} t d t + \int_{\frac{t_{0}}{3}}^{\frac{2 t_{0}}{3}} F_{0} d t + \int_{\frac{2 t_{0}}{3}}^{t_{0}} \frac{- 3 F_{0}}{t_{0}} t + 3 F_{0} d t$ $= {[\frac{3 F_{0}}{t_{0}} t]}_{0}^{\frac{t_{0}}{3}} + {[F_{0} t]}_{\frac{t_{0}}{3}}^{\frac{2 t_{0}}{3}} + {[\frac{- 3 F_{0}}{t_{0}} t + 3 F_{0} t]}_{\frac{2 t_{0}}{3}}^{t_{0}}$ $= F_{0} + \frac{1}{3} F_{0} t_{0} - F_{0} + F_{0} t_{0}$ $= \frac{4}{3} F_{0} t_{0}$ $s o a_{0} = \frac{2}{t_{0}} \times \frac{4}{3} F_{0} t_{0} = \frac{8}{3} F_{0}$
Commented by uchihayahia last updated on 23/Mar/23


Commented by uchihayahia last updated on 23/Mar/23

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