91Ó°ÊÓ

Integration is a fundamental concept in calculus that helps in determining the original function given its derivative. In this problem, we were provided with the derivative expression \( f'(x) = 8x^3 + 12x + 3 \). Our task was to find the original function \( f(x) \). To achieve this, we used the technique of antidifferentiation, also known simply as integration.

• When integrating, consider each term separately. Here, for the term \( 8x^3 \), we increase the exponent by one to get \( x^4 \) and then divide by the new exponent, resulting in \( 2x^4 \).
• Similarly, for \( 12x \), adjust the power to get \( x^2 \) and divide by 2, resulting in \( 6x^2 \).
• The constant term, \( 3 \), integrates to \( 3x \).

Finally, always remember to add a constant of integration, denoted here as \( C \). This constant is essential as integration represents a family of functions, and \( C \) allows us to pinpoint the specific function that satisfies any given initial conditions.

The concept of an initial value problem in calculus is used to determine a particular solution from a family of solutions. After integrating the given derivative \( f'(x) \), our general solution was \( f(x) = 2x^4 + 6x^2 + 3x + C \). Here, \( C \) represents an unknown constant.

Our problem included an initial condition, \( f(1) = 6 \), which is crucial for finding the exact value of \( C \). We substitute \( x = 1 \) into our general solution to match the condition:

• Substitute \( x = 1 \) into the equation: \( 6 = 2(1)^4 + 6(1)^2 + 3(1) + C \).
• Simplifying the expression confirms: \( 6 = 11 + C \).
• Solving for \( C \) gives: \( C = -5 \).

By solving the initial value problem, we adjusted \( C \) accordingly, resulting in the particular solution: \( f(x) = 2x^4 + 6x^2 + 3x - 5 \). This process confirms the unique function that fulfills both the derivative equation and the initial condition.

Polynomial functions, such as the one we derived in this problem, are expressions involving powers of \( x \) multiplied by coefficients. Our task involved stepping back from its derivative \( f'(x) \) to find the original function \( f(x) \). The function we identified was \( f(x) = 2x^4 + 6x^2 + 3x - 5 \).
Polynomial functions are incredibly useful in mathematical modelling and problem-solving due to their straightforward properties.

• The degree of the polynomial refers to the highest power of \( x \) present in the function. Here, our polynomial \( 2x^4 + 6x^2 + 3x - 5 \) is a quartic polynomial since the highest power is 4.
• Polynomial functions are continuous and smooth, making them predictable and relatively easy to differentiate and integrate, as seen in our solution.
• They consist of simple arithmetic operations, which contribute to their flexibility in both analysis and algebraic manipulations.

Thus, understanding the nature of polynomial functions allows deeper insight into their behavior and applications to real-world scenarios, effectively extending our approach to solving calculus problems.