Anti deravative

Question

Given f''(x)=−16sin(4x) and f'(0)=2 and f(0)=−4.

find f(pi/5)

Answer 1

Integrating:

f'(x) = 4 cos(4x) + c

f'(0)=2 ---> 4 +c = 2 --> c = -2

so f'(x) = 4 cos(4x) - 2

integrating again:

f(x) = sin(4x) - 2x + c

f(0)=-4 ---> c=-4

f(x) = sin(4x) - 2x - 4

check:

f'(x) = 4 cos(4x) - 2

f''(x) = -16 sin(4x)

yes it checks..

f(x) = f(x) = sin(4x) - 2x - 4

f(pi/5) = sin (4 *pi/5) - 2(pi/5) - 4

which numerically is APPROXIMATELY

= 0.58778525229247312916870595463907

-1.2566370614359172953850573533118 -4

=-4.6688518055111860036979978069211