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Question Number 159719 by tounghoungko last updated on 20/Nov/21

F(x)= 3cos x + 4sin x , F^((101)) ((π/2))=?

$$\:\:\:\:\:{F}\left({x}\right)=\:\mathrm{3cos}\:{x}\:+\:\mathrm{4sin}\:{x}\:,\:{F}^{\left(\mathrm{101}\right)} \left(\frac{\pi}{\mathrm{2}}\right)=? \\ $$

Answered by mathmax by abdo last updated on 20/Nov/21

f^((n)) (x)=3cos(x+((nπ)/2))+4sin(x+((nπ)/2)) ⇒f^((101)) ((π/2))=3cos((π/2)+((101π)/2))+4sin((π/2)+((101π)/2)) =3cos(((102π)/2))+4sin(((102π)/2)) =3cos(51π)+4sin(51π) =3cos(π+50π)+4sin(π+50π) =3cos(π)+4sin(π)=−3+0 =−3

$$\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)=\mathrm{3cos}\left(\mathrm{x}+\frac{\mathrm{n}\pi}{\mathrm{2}}\right)+\mathrm{4sin}\left(\mathrm{x}+\frac{\mathrm{n}\pi}{\mathrm{2}}\right) \\ $$$$\Rightarrow\mathrm{f}^{\left(\mathrm{101}\right)} \left(\frac{\pi}{\mathrm{2}}\right)=\mathrm{3cos}\left(\frac{\pi}{\mathrm{2}}+\frac{\mathrm{101}\pi}{\mathrm{2}}\right)+\mathrm{4sin}\left(\frac{\pi}{\mathrm{2}}+\frac{\mathrm{101}\pi}{\mathrm{2}}\right) \\ $$$$=\mathrm{3cos}\left(\frac{\mathrm{102}\pi}{\mathrm{2}}\right)+\mathrm{4sin}\left(\frac{\mathrm{102}\pi}{\mathrm{2}}\right) \\ $$$$=\mathrm{3cos}\left(\mathrm{51}\pi\right)+\mathrm{4sin}\left(\mathrm{51}\pi\right) \\ $$$$=\mathrm{3cos}\left(\pi+\mathrm{50}\pi\right)+\mathrm{4sin}\left(\pi+\mathrm{50}\pi\right) \\ $$$$=\mathrm{3cos}\left(\pi\right)+\mathrm{4sin}\left(\pi\right)=−\mathrm{3}+\mathrm{0}\:=−\mathrm{3} \\ $$

Commented by cortano last updated on 20/Nov/21

nice

$${nice} \\ $$

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