91Ó°ÊÓ

Integration is the reverse process of differentiation. When we have a function's derivative and we need to find the original function, we use integration. This is often called finding the "antiderivative." In the context of this exercise, we started with a derivative, 6x - 1, and needed to find the function f(x) that produces this derivative.

Antiderivatives are calculated using the power rule for integration, similar to how you might differentiate a function. For example, when integrating a term like x^n, we increase the power by one and then divide by the new power: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

• Start by identifying each term in the expression.
• Apply the integration rule to each term separately.
• Don't forget the constant of integration, C, which we'll discuss next.

When performing indefinite integration, we always add a constant, C, to our final result. This constant of integration acknowledges that there are infinitely many antiderivatives. Why? Because when you differentiate a constant, it becomes zero. Hence, when working backward through integration, any constant could have been in the original function.

In the exercise, after integrating the derivative of f's function, we got 3x^2 - x + C. Here, C could be any real number. However, to find the specific function that passes through the point (2,7), we needed to determine the exact value of C. Using the given point, we solved for C to ensure f(x) meets the required condition at x = 2. This calculation gave us C = -3.

The derivative of a function represents the rate of change of that function concerning its variable. It is a foundational concept in calculus, determining how a function behaves as its input changes. In this exercise, the derivative given was f'(x) = 6x - 1, suggesting that the function's rate of change varies linearly with x.

What does this mean? For every increase in x, the function's value changes according to 6x - 1.

• At any point, x, the slope or steepness of the function is defined by f'(x).
• Finding the antiderivative of f'(x) gives insight into how the function is structured.
• The method of integration lets us reverse the process of differentiation, revealing the broader function rather than just its rate of change.

A function in calculus is a rule or relationship that assigns each input exactly one output. In this problem, our task was to find such a function, f(x), using its derivative and a specific point through which its graph passes. This is a common task in calculus, figuring out the most specific function possible based on given constraints.

A complete function embodies its entire behavior, not just a snapshot like a derivative. It helps us predict the behavior at any point in its domain. When given a point, for example (2, 7), it acts as a crucial condition, enabling us to solve for any unknown constants, thereby pinning down one precise function from the infinite possibilities.

Thus, the function determined from our exercise, f(x) = 3x^2 - x - 3, is the one that satisfies both its derivative form and the condition f(2) = 7.