91Ó°ÊÓ

The Fundamental Theorem of Calculus beautifully connects the concepts of differentiation and integration, two core operations in calculus. It tells us that if we have an integral function, say our arc length function \( s(x) \), its derivative is directly related to the integrand. In simpler terms, if you have \( s(x) = \int_{a}^{x} f(t) \, dt \), the derivative \( s'(x) \) is \( f(x) \). This principle plays a key role in solving our problem by providing the derivative of \( s(x) \) that yields \( s'(x) = \sqrt{3x+5} \). This derivative gives us the rate of change at any point \( x \) along the path, helping us to uncover the hidden function \( f(x) \) that describes the curve's trajectory.

Integral Calculus focuses on accumulation, change, and the areas under curves. In our exercise, it helps us find the arc length of a curve. The integral \( s(x) = \int_{0}^{x} \sqrt{3t+5} \, dt \) represents a running total - the arc length of the curve from \( x = 0 \) to \( x = x \). To find the function \( f(x) \), we take the derivative \( s'(x) \) and perform the inverse operation, which is integration.

• We calculate \( f(x) = \int \sqrt{3x+5} \, dx \) to 'reverse' the derivative back into the original function.
• This integration gives us a function that describes how the curve's length accumulates over a given range.

It's crucial to remember that integration introduces a constant \( C \) since there are multiple functions with the same derivative. This constant is determined by specific values, such as the \( y \)-intercept in our problem.

The Substitution Method is a powerful tool for solving more complicated integrals. It simplifies the integration process by changing variables, making it easier to integrate complex functions.
In our task, we face the integral \( \int \sqrt{3x + 5} \, dx \). A substitution is made by letting \( u = 3x + 5 \). This gives

• \( du = 3 \, dx \) or \( dx = \frac{du}{3} \).
• Now, our integral becomes \( \frac{1}{3} \int \sqrt{u} \, du \).

The integral of \( \sqrt{u} \, du \) is straightforward compared to the original form, making our calculations simpler. After integrating and substituting back \( u = 3x + 5 \), we acquire the function that accurately represents the curve's behavior. Substitution Method streamlines complex integrals into manageable tasks.

In the context of this problem, knowing that \( f(x) \) is an increasing function is pivotal.
An increasing function means that as \( x \) gets larger, \( f(x) \) also gets larger. This impacts how we interpret and solve the problem.

• First, it confirms that the arc length \( s(x) \) will continually grow as \( x \) increases, ensuring no 'unexpected turns' or decreases in the function.
• It aids in setting logical bounds for solutions. When solving for \( x \) given \( s(x) = 3 \), we can confidently navigate only positive solutions for \( x \).

Recognizing \( f \) as increasing guides us to accurately assess the curve's length and output with confidence that the calculated path from point to point follows a consistent upward trend. This nature of \( f \) underpins the predictability of the function values across our calculations.