Consider the function [tex]$f(x)$[/tex] whose second derivative is [tex]$f^{\prime \prime}(x)=6 x+8 \sin (x)$[/tex]. If [tex]$f(0)=3$[/tex] and [tex]$f^{\prime}(0)=2$[/tex], what is [tex]$f(x)$[/tex]?
[tex]$f(x)= \square$[/tex]
Answer (2)
To find the function f ( x ) , we integrate the second derivative f^{ulletullet}(x) = 6x + 8\sin(x) to obtain the first derivative, apply initial conditions to find constants, and then integrate again to arrive at the final function. The complete function is f ( x ) = x 3 − 8 sin ( x ) + 10 x + 3 .
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Integrate the second derivative f ′′ ( x ) = 6 x + 8 sin ( x ) to find the first derivative f ′ ( x ) = 3 x 2 − 8 cos ( x ) + C 1 .
Use the initial condition f ′ ( 0 ) = 2 to solve for C 1 , obtaining f ′ ( x ) = 3 x 2 − 8 cos ( x ) + 10 .
Integrate the first derivative to find the function f ( x ) = x 3 − 8 sin ( x ) + 10 x + C 2 .
Use the initial condition f ( 0 ) = 3 to solve for C 2 , resulting in the final function f ( x ) = x 3 − 8 sin ( x ) + 10 x + 3 .
Explanation
Problem Analysis We are given the second derivative of a function f ( x ) as f ′′ ( x ) = 6 x + 8 sin ( x ) . We also have the initial conditions f ( 0 ) = 3 and f ′ ( 0 ) = 2 . Our goal is to find the function f ( x ) .
Finding the First Derivative First, we need to integrate f ′′ ( x ) to find f ′ ( x ) .
f ′ ( x ) = ∫ f ′′ ( x ) d x = ∫ ( 6 x + 8 sin ( x )) d x Integrating term by term, we get: f ′ ( x ) = 3 x 2 − 8 cos ( x ) + C 1 where C 1 is the constant of integration.
Using the Initial Condition for the First Derivative Now, we use the initial condition f ′ ( 0 ) = 2 to find C 1 :
2 = 3 ( 0 ) 2 − 8 cos ( 0 ) + C 1 = − 8 + C 1 Solving for C 1 , we get C 1 = 10 . Therefore, f ′ ( x ) = 3 x 2 − 8 cos ( x ) + 10
Finding the Function Next, we integrate f ′ ( x ) to find f ( x ) :
f ( x ) = ∫ f ′ ( x ) d x = ∫ ( 3 x 2 − 8 cos ( x ) + 10 ) d x Integrating term by term, we get: f ( x ) = x 3 − 8 sin ( x ) + 10 x + C 2 where C 2 is the constant of integration.
Using the Initial Condition to Find the Function Now, we use the initial condition f ( 0 ) = 3 to find C 2 :
3 = ( 0 ) 3 − 8 sin ( 0 ) + 10 ( 0 ) + C 2 = 0 + C 2 Solving for C 2 , we get C 2 = 3 . Therefore, f ( x ) = x 3 − 8 sin ( x ) + 10 x + 3
Final Answer Thus, the function f ( x ) is: f ( x ) = x 3 − 8 sin ( x ) + 10 x + 3
Examples
Imagine you are designing a suspension bridge. The shape of the bridge can be modeled by a function, and understanding its derivatives helps engineers analyze the forces acting on the bridge. If you know the second derivative (related to the curvature) and some initial conditions (like the height and slope at a certain point), you can determine the exact function that describes the bridge's shape, ensuring its stability and safety.
