Finding F(5) Given The Second Derivative And Initial Conditions

Introduction

In the realm of calculus, differential equations play a pivotal role in modeling and understanding various phenomena across diverse fields, from physics and engineering to economics and biology. Among these equations, those involving higher-order derivatives, such as the second derivative, often present intriguing challenges and require sophisticated techniques to solve. This article delves into a specific problem that exemplifies the elegance and power of calculus in determining the behavior of a function given its second derivative and initial conditions. We embark on a journey to unravel the mystery surrounding the function f(x), armed with the knowledge that its second derivative, f''(x), is given by 4x + 8sin(x), and two initial conditions: f(0) = 4 and f'(0) = 4. Our ultimate goal is to compute the value of f(5), a task that necessitates a careful integration process and skillful application of the provided initial conditions.

This exploration is not merely a mathematical exercise; it mirrors real-world scenarios where we might know the rate of change of a rate of change (the second derivative) and some initial states, and we desire to predict the state of the system at a future time. Consider, for instance, the acceleration of a moving object, which is the second derivative of its position function. Knowing the acceleration and the initial position and velocity, we can determine the object's position at any given time. Similarly, in financial modeling, the rate of change of interest rates could be modeled as a second derivative, and with initial conditions, we could forecast future interest rate levels. Therefore, the techniques we will employ in this article have broad applicability beyond pure mathematics.

As we proceed, we will emphasize the step-by-step process of solving this problem, highlighting the underlying concepts and techniques. We will begin by integrating the second derivative to obtain the first derivative, introducing a constant of integration that we will subsequently determine using the initial condition f'(0) = 4. Next, we will integrate the first derivative to find the original function f(x), again encountering a constant of integration that we will resolve using the initial condition f(0) = 4. Finally, with the explicit form of f(x) in hand, we will evaluate it at x = 5 to arrive at our desired answer. Throughout this journey, we will emphasize the importance of careful calculation and attention to detail, as even a small error can propagate and lead to an incorrect result. By the end of this article, readers will not only understand how to solve this specific problem but also gain a deeper appreciation for the power and versatility of calculus in addressing a wide range of mathematical and scientific challenges. The journey into finding f(5) begins now.

Finding the First Derivative f'(x)

Our quest to determine the function f(5) begins with the pivotal step of finding the first derivative, f'(x). The first derivative, in calculus, represents the instantaneous rate of change of a function, providing crucial information about its slope and behavior. In our case, we are given the second derivative, f''(x) = 4x + 8sin(x), which is the rate of change of the first derivative. To recover f'(x), we must perform the inverse operation of differentiation, which is integration. This integration process will introduce a constant of integration, a common occurrence in indefinite integrals, which we will later determine using the provided initial condition f'(0) = 4.

The process of integration involves finding a function whose derivative matches the given expression. For f''(x) = 4x + 8sin(x), we will integrate term by term. The integral of 4x with respect to x is 2x², as the power rule of integration dictates that we increase the exponent by one and divide by the new exponent. Similarly, the integral of 8sin(x) with respect to x is -8cos(x), because the derivative of -8cos(x) is 8sin(x). It's crucial to remember the negative sign, as the derivative of cosine is negative sine. The constant multiple rule of integration allows us to pull the 8 outside the integral of sin(x), simplifying the process.

Thus, the indefinite integral of f''(x) yields f'(x) = 2x² - 8cos(x) + C, where C represents the constant of integration. This constant is a crucial element because there are infinitely many functions whose derivative is 4x + 8sin(x), differing only by a constant term. To pinpoint the specific f'(x) that satisfies our problem, we need to use the initial condition f'(0) = 4. This condition tells us that when x is 0, the value of the first derivative is 4. By substituting these values into our expression for f'(x), we can solve for C.

Substituting x = 0 and f'(0) = 4 into f'(x) = 2x² - 8cos(x) + C, we get 4 = 2(0)² - 8cos(0) + C. Since cos(0) = 1, the equation simplifies to 4 = -8 + C. Adding 8 to both sides, we find that C = 12. This constant of integration is the key to unlocking the specific first derivative that corresponds to our problem. With C determined, we can now write the explicit form of the first derivative as f'(x) = 2x² - 8cos(x) + 12. This expression represents the instantaneous rate of change of our function f(x) at any point x, and it serves as the foundation for our next step: finding the original function f(x) itself. The determination of f'(x) is a significant milestone in our journey toward f(5), and we now proceed to the next stage with a clear and accurate expression for the first derivative.

Unveiling the Original Function f(x)

Having successfully determined the first derivative, f'(x) = 2x² - 8cos(x) + 12, our attention now shifts to the crucial task of uncovering the original function, f(x). The original function represents the core relationship we are trying to understand, and it holds the key to answering our initial question: what is f(5)? To transition from the first derivative back to the function itself, we must once again employ the process of integration, applying it this time to f'(x). This step will mirror the previous one, introducing another constant of integration that we will resolve using the given initial condition f(0) = 4.

The integration process involves finding a function whose derivative is equal to f'(x). For f'(x) = 2x² - 8cos(x) + 12, we will integrate each term separately. The integral of 2x² with respect to x is (2/3)x³, obtained by increasing the exponent by one (from 2 to 3) and dividing by the new exponent. The integral of -8cos(x) with respect to x is -8sin(x), as the derivative of -8sin(x) is -8cos(x). Finally, the integral of 12 with respect to x is simply 12x, as the derivative of 12x is 12. As before, we must not forget to include the constant of integration, which we will denote as D to distinguish it from the previous constant C.

Thus, the indefinite integral of f'(x) yields f(x) = (2/3)x³ - 8sin(x) + 12x + D. This expression represents a family of functions, each differing by the constant D, all of which have the same first derivative. To pinpoint the specific function f(x) that satisfies our problem, we invoke the initial condition f(0) = 4. This condition tells us that when x is 0, the value of the function is 4. Substituting these values into our expression for f(x), we can solve for D.

Substituting x = 0 and f(0) = 4 into f(x) = (2/3)x³ - 8sin(x) + 12x + D, we get 4 = (2/3)(0)³ - 8sin(0) + 12(0) + D. Since sin(0) = 0, the equation simplifies to 4 = D. This elegantly simple result reveals the value of our constant of integration, allowing us to express the original function f(x) in its entirety. With D determined, we can now confidently write the explicit form of the function as f(x) = (2/3)x³ - 8sin(x) + 12x + 4. This expression encapsulates the relationship between x and f(x) that we have been seeking, and it is the culmination of our integration efforts. We are now poised to answer our original question, by evaluating f(x) at x = 5. The unveiling of f(x) marks a significant triumph in our journey, and we proceed to the final act with anticipation.

Evaluating f(5): The Final Calculation

With the explicit form of the function f(x) = (2/3)x³ - 8sin(x) + 12x + 4 firmly in hand, we arrive at the final and most satisfying step of our journey: evaluating f(5). This calculation will provide the numerical answer to our initial question, revealing the value of the function at x = 5. It is the culmination of our efforts in integrating the second derivative, applying initial conditions, and meticulously tracking constants of integration. Now, we substitute x = 5 into our expression for f(x) and perform the arithmetic to arrive at our solution.

Substituting x = 5 into f(x) = (2/3)x³ - 8sin(x) + 12x + 4, we obtain f(5) = (2/3)(5)³ - 8sin(5) + 12(5) + 4. This expression involves several terms, each of which we will evaluate carefully. First, (2/3)(5)³ = (2/3)(125) = 250/3. Next, we have -8sin(5), which requires us to compute the sine of 5 radians. It's crucial to ensure that our calculator is set to radian mode for this calculation. The value of sin(5) is approximately -0.958924. Therefore, -8sin(5) is approximately -8(-0.958924) ≈ 7.67139. Then, 12(5) = 60, and finally, we have the constant term 4.

Now, we add all these terms together: f(5) ≈ 250/3 + 7.67139 + 60 + 4. Converting 250/3 to a decimal gives us approximately 83.3333. Adding this to the other terms, we get f(5) ≈ 83.3333 + 7.67139 + 60 + 4 ≈ 155.00469. Rounding this to a reasonable number of decimal places, we can say that f(5) is approximately 155.005.

This numerical result represents the value of the function f(x) at x = 5, given its second derivative and the specified initial conditions. It is the tangible outcome of our step-by-step process, a testament to the power of calculus in solving problems involving rates of change and initial states. The careful execution of integration, the meticulous application of initial conditions, and the precise evaluation of the resulting expression have all contributed to this final answer. f(5) ≈ 155.005. This concludes our calculation and provides the solution to the problem at hand.

Conclusion

In this comprehensive exploration, we successfully navigated the intricacies of differential calculus to determine the value of f(5) for a function f(x) defined by its second derivative, f''(x) = 4x + 8sin(x), and the initial conditions f(0) = 4 and f'(0) = 4. The entire process, from the initial problem statement to the final numerical result, showcases the fundamental principles of calculus and their application in solving real-world problems. We began by recognizing the need to integrate the second derivative to obtain the first derivative, f'(x), a process that introduced a constant of integration. We then skillfully employed the initial condition f'(0) = 4 to resolve this constant, leading us to the explicit form of f'(x).

Next, we integrated the first derivative to uncover the original function, f(x), again encountering a constant of integration. The initial condition f(0) = 4 proved invaluable in determining this constant, allowing us to express f(x) in its complete form. With the explicit function in hand, we confidently evaluated f(5), arriving at the approximate numerical solution of 155.005. This final step underscored the importance of precise calculation and attention to detail, ensuring the accuracy of our result.

Throughout this journey, we have not only solved a specific mathematical problem but also reinforced the broader concepts of calculus. The interplay between derivatives and integrals, the significance of initial conditions, and the power of integration in recovering functions from their rates of change have been central themes. The techniques we have employed are widely applicable in various fields, from physics and engineering to economics and computer science, where differential equations and initial value problems are commonplace. This exploration serves as a testament to the versatility and elegance of calculus in modeling and understanding the world around us. The determination of f(5) is not just a numerical answer; it is a demonstration of the power of calculus to reveal the hidden relationships within mathematical systems. This article will help readers understand the step-by-step process, and the underlying concept and techniques. This knowledge can be applied to solve other mathematical and scientific challenges.