Question

Find a formula for f(x) given that f"(x) = sin x + cos x, f '(0) = 11, and f(0) = −2.

Answer 1

To find function f(x) when given f"(x), we integrate twice and use initial conditions f'(0) = 11 and f(0) = -2 to find the integration constants. The resulting function is f(x) = -sin x - cos x + 12x - 1.

Step-by-step explanation:

To find the function f(x) from its second derivative f"(x) = sin x + cos x, with initial conditions f'(0) = 11 and f(0) = -2, we need to integrate the second derivative twice.

1. First integration of f"(x) gives us the first derivative f'(x):

• ∫ (sin x + cos x) dx = -cos x + sin x + C1

To find the constant C1, we use the initial condition f'(0) = 11:

• -cos(0) + sin(0) + C1 = 11 → -1 + 0 + C1 = 11 → C1 = 12

So the first derivative is f'(x) = -cos x + sin x + 12.The second integration gives us f(x):

• ∫ (-cos x + sin x + 12) dx = -sin x - cos x + 12x + C2

To find C2, we use the initial condition f(0) = -2:

• -sin(0) - cos(0) + 12⋅(0) + C2 = -2 → -0 - 1 + 0 + C2 = -2 → C2 = -1

Finally, the function f(x) is f(x) = -sin x - cos x + 12x - 1.