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 the formula for f(x), we integrate the second derivative, then determine the values of the constant of integration using the given conditions. Finally, we integrate the first derivative and obtain the formula f(x) = -sin x - cos x + 12x - 1.

Step-by-step explanation:

To find a formula for f(x), given f''(x) = sin x + cos x, f'(0) = 11, and f(0) = −2, we need to integrate the given second derivative and find the first derivative. Then, we can integrate the first derivative and find the original function f(x).

Step-by-step solution:

1. Integrate f''(x) to find f'(x): ∫(sin x + cos x) dx = -cos x + sin x + C, where C is the constant of integration.
2. Use the given condition f'(0) = 11 to determine the value of the constant C: -cos 0 + sin 0 + C = -1 + 0 + C = C = 12.
3. Integrate f'(x) to find f(x): ∫(-cos x + sin x + 12) dx = -sin x - cos x + 12x + D, where D is another constant of integration.
4. Use the given condition f(0) = -2 to determine the value of the constant D: -sin 0 - cos 0 + 12(0) + D = 0 - 1 + 0 + D = D = -1.
5. The formula for f(x) is therefore: f(x) = -sin x - cos x + 12x - 1.