Find f if f"...?

Question

find f if f"(theta) = sin(theta) + cos(theta), f(0)=8, and f'(0)=6.

Answer 1

f '' = sin x + cos x

so...

f ' = -cos x + sin x + c
(plug in the f'(0) = 6 to find c...)
6 = - cos 0 + sin 0 + c
6 = - 1 + 0 + c
7 = c

so....
f ' = -cos x + sin x + 7

f = -sin x - cos x + 7x + c
(plug in f(0) = 8)
8 = -sin 0 - cos 0 + 7(0) + c
8 = 0 - 1 + 0 + c
9 = c

so...
f = -sin x - cos x + 7x + 9

Answer 2

f'' = sin(theta) + cos(theta)

Take the antiderivative:

f' = -cos(theta) + sin(theta) + C

Given that f'(0) = 6, solve for C:

-cos(0) + sin(0) + C = 6
-1 + 0 + C = 6
C = 7

So we know that f'(theta) is:

f'(theta) = -cos(theta) + sin(theta) + 7

Take the antiderivative of that:

f(theta) = -sin(theta) - cos(theta) + 7*theta + C

Given that f(0) = 8, solve for C:

f(0) = -sin(0) - cos(0) + 7*0 + C
8 = 0 - 1 + 0 + C
C = 9

So the answer is:

f(theta) = -sin(theta) - cos(theta) + 7*theta + 9