Question

Please answer BOTH questions and provide a short explanation of how you got the answer.

Consider the function f(x) whose second derivative is f''(x) = 8x + 10sin(x). If f(0) = 2 and f'(0) = 2, what is f(2)?

Given that the graph of f(x)... (the rest of the text is not provided, but it seems like you may have more to include).

Answer 1

To find f(2), integrate the second derivative of f(x) twice and use the initial conditions to solve for the integration constants. The final equation is f(x) = x^3 - 10sin(x) + 2x + 2. Substituting x = 2 gives f(2) = 12 - 10sin(2).

Step-by-step explanation:

To find the value of f(2), we need to integrate the second derivative of f(x) twice. The first integration gives us the first derivative f'(x), and the second integration gives us f(x).

Given that f'(0) = 2, we can integrate the second derivative f''(x) = 8x + 10sin(x) to get f'(x). Integrating again will give us f(x).

After integrating f''(x), we obtain f'(x) = 4x^2 - 10cos(x) + C1. By substituting f'(0) = 2, we can solve for C1 and find that C1 = 2.

Next, we integrate f'(x) to get f(x). Using the initial condition f(0) = 2, we can solve for C2 and find that C2 = 2. Hence, f(x) = x^3 - 10sin(x) + 2x + 2.

Finally, substituting x = 2 into the equation, we find f(2) = 8 - 10sin(2) + 4 + 2 = 12 - 10sin(2).