Calculus 1 Initial Value

Question

Find f(x) if f″(x)=sin⁡(x)+cos⁡(x), f′(0)=5, and f(0)=10

Answer 1

f'(x) = ∫f"(x)dx = -cosx + sinx + C

Since f'(0) = 5, we have -1 + 0 + C = 5. So, C = 6

f'(x) = -cosx + sinx + 6

f(x) = ∫f'(x)dx = -sinx - cosx + 6x + K

Since f(0) = 10, we get 0 - 1 + 0 + K = 10. So, K = 11

Therefore, f(x) = -sinx - cosx + 6x + 11