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Question Number 189865 by uchihayahia last updated on 23/Mar/23

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