Question

Find the first derivative f' and the original function f. f"(x) = cos(x), f' (π/2) = 7, f(π/2) = 4.

(Use symbolic notation and fractions where needed.)

Answer 1

To find the first derivative f', we integrate the second derivative f"(x) = cos(x) and use the initial condition f'(π/2) = 7, finding f'(x) = sin(x) + 6. For the original function f, we integrate f' and use the condition f(π/2) = 4, resulting in f(x) = -cos(x) + 6x + 4 - 3π.

Step-by-step explanation:

To find the first derivative f' given the second derivative f"(x) = cos(x), we integrate the second derivative:

⎇Integrate f"(x) = cos(x) to find f'(x): f'(x) = ∫ cos(x) dx = sin(x) + C

Use the condition f'(π/2) = 7 to solve for C: sin(π/2) + C = 7 ⇒ C = 7 - 1 = 6. So, f'(x) = sin(x) + 6

Next, to find the original function f(x), we integrate f'(x):

Integrate f'(x) = sin(x) + 6 to get f(x): f(x) = -cos(x) + 6x + D

Use the condition f(π/2) = 4 to solve for D: -cos(π/2) + 6(π/2) + D = 4 ⇒ 0 + 3π + D = 4 ⇒ D = 4 - 3π

Therefore, the original function is f(x) = -cos(x) + 6x + 4 - 3π.