Consider the function f ( x ) whose second derivative is f ' ' ( x ) = 7 x + 4 sin ( x ) . If f ( 0 ) = 2 and f ' ( 0 ) = 2 , what is f ( 5 ) ?

Question

Consider the function  whose second derivative is . If  and , what is ? 

Answer 1

This is called an initial value problem. You're give the derivative and must integrate to find the original function, solving for the constant(s) of integration, c, using the given initial conditions.

f''(x) = 7x + 4sin(x)

Integrate to get,

f'(x) = (7/2)x² - 4cos(x) + c1

At this point we solve for the first c (which I labeled as c1 to differentiate it from the other constant of integration we will get later). It's given that f'(0) = 2. This means when x = 0, f' = 2

f'(0) = 2 = (7/2)(0)² - 4cos(0) + c1

2 = 0 - 4 + c1

c1 = 6

Plug that back into f'(x), and then integrate one more time.

f'(x) = (7/2)x² - 4cos(x) + 6

f(x) = (7/6)x³ - 4sin(x) + 6x + c2

It's given that f(0) = 2. Again this means that when x = 0, f = 2

f(0) = 2 = (7/6)(0)³ - 4sin(0) + 6(0) + c2

2 = 0 - 0 + 0 + c2

c2 = 2

Plugging that back into f(x), we get our original function.

f(x) = (7/6)x³ - 4sin(x) + 6x + 2

Often this is the last step for these types of questions, but this particular question asks for f(5), so we have to plug in x = 5.

f(5) = (7/6)(5)³ - 4sin(5) + 6(5) + 2

f(5) = (1067/6) - 4sin(5)

f(5) ≈ 181.669