Antiderivatives and Indefinite Integration

Question

Consider the function f(x) whose second derivative is f''(x) = 7x+7sin(x). If f(0) = 3 and f'(0)= 4, what is f(3)?

Answer 1

f^' = integral of f'' so that f'(0) = 4

f^' = 7x²/2 - 7cosx + C

Use condition to solve for C:

4 = -7 + C C = 11

f = integral of f' with f(0) = 3

f = 7x³/6 - 7sinx + 11x + c

3 = c

f(x) = 7x³/6 - 7sinx + 11x + 3

Plug in 3 in order to get f(3)

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