Final answer:
To find f'(x) and f(x), we integrate f''(x)=cos(x) using the given initial conditions. The first derivative is f'(x)=sin(x)+7, and the original function is f(x)=-cos(x)+7x+14-7(π/2).
Step-by-step explanation:
The question is about finding the expressions for the first derivative f'(x) and the original function f(x) given the second derivative f''(x) = cos(x), along with initial conditions at x = π/2: f'(π/2) = 7 and f(π/2) = 14.
To find f'(x), we integrate the second derivative f''(x):
∫cos(x) dx = sin(x) + C
Using the initial condition f'(π/2) = 7, we solve for C:
sin(π/2) + C = 7
C = 7, since sin(π/2) = 1.
Therefore, the first derivative is f'(x) = sin(x) + 7.
Next, we repeat this process to find f(x). We integrate f'(x):
∫(sin(x) + 7) dx = -cos(x) + 7x + D
Using the initial condition f(π/2) = 14, we solve for D:
-cos(π/2) + 7(π/2) + D = 14
D = 14 - 7(π/2), since cos(π/2) = 0.
Therefore, the original function is f(x) = -cos(x) + 7x + 14 - 7(π/2).