Question

f"(x)=senx+cosx; f(0)=3; f'(0)=4

Answer 1

$Fun\c{c}\~ao\ derivada\ segunda:\\\\ f''_{(x)}= \sin x + \cos x\\\\\\ Integrando\ para\ encontrar\ a\ derivada\ primeira:\\\\ f'_{(x)}=\int (\sin x + \cos x).dx\\\\ f'_{(x)}=-\cos x + \sin x+constante\\\\\\ Para\ f'_{(0)}=4\\\\ f'_{(0)}=-\cos 0 + \sin 0+constante=4\\\\ f'_{(0)}=-1 + 0+constante=4\\\\ constante=4+1\\\\ constante = 5\\\\\\ Fun\c{c}\~ao\ completa:\\\\ f'_{(x)}=-\cos x + \sin x+5$


$Integrando\ novamemente\ para\ encontrar\ a\ fun\c{c}\~ao\ inicial:\\\\ f_{(x)}= \int (-\cos x + \sin x+5). dx\\\\ f_{(x)}= -\sin x -\cos x+5\dfrac{x^{0+1}}{0+1}+constante\\\\ f_{(x)}= -\sin x -\cos x+5x+constante\\\\\\ Para\ f_{(0)}=3:\\\\ f_{(0)}= -\sin 0 -\cos 0+5(0)+constante=3\\\\ f_{(0)}= 0 -1+0+constante=3\\\\ -1+constante=3\\\\ constante=3+1\\\\ constante = 4\\\\\\ Fun\c{c}\~ao\ completa:\\\\ \boxed{f_{(x)}= -\sin x -\cos x+5x+4}$


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Bons estudos!