Question Number 159719 by tounghoungko last updated on 20/Nov/21
$$\:\:\:\:\:{F}\left({x}\right)=\:\mathrm{3cos}\:{x}\:+\:\mathrm{4sin}\:{x}\:,\:{F}^{\left(\mathrm{101}\right)} \left(\frac{\pi}{\mathrm{2}}\right)=? \\ $$
Answered by mathmax by abdo last updated on 20/Nov/21
$$\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)=\mathrm{3cos}\left(\mathrm{x}+\frac{\mathrm{n}\pi}{\mathrm{2}}\right)+\mathrm{4sin}\left(\mathrm{x}+\frac{\mathrm{n}\pi}{\mathrm{2}}\right) \\ $$$$\Rightarrow\mathrm{f}^{\left(\mathrm{101}\right)} \left(\frac{\pi}{\mathrm{2}}\right)=\mathrm{3cos}\left(\frac{\pi}{\mathrm{2}}+\frac{\mathrm{101}\pi}{\mathrm{2}}\right)+\mathrm{4sin}\left(\frac{\pi}{\mathrm{2}}+\frac{\mathrm{101}\pi}{\mathrm{2}}\right) \\ $$$$=\mathrm{3cos}\left(\frac{\mathrm{102}\pi}{\mathrm{2}}\right)+\mathrm{4sin}\left(\frac{\mathrm{102}\pi}{\mathrm{2}}\right) \\ $$$$=\mathrm{3cos}\left(\mathrm{51}\pi\right)+\mathrm{4sin}\left(\mathrm{51}\pi\right) \\ $$$$=\mathrm{3cos}\left(\pi+\mathrm{50}\pi\right)+\mathrm{4sin}\left(\pi+\mathrm{50}\pi\right) \\ $$$$=\mathrm{3cos}\left(\pi\right)+\mathrm{4sin}\left(\pi\right)=−\mathrm{3}+\mathrm{0}\:=−\mathrm{3} \\ $$
Commented by cortano last updated on 20/Nov/21
$${nice} \\ $$