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A velocity function describes how the velocity of an object changes over time in a given direction. In this context, the velocity function is expressed as a rational function: \( v = \frac{t^2 + 14t + 27}{(2t+1)(t+5)^2} \). This equation provides the velocity at any specific time \( t \) along the object's path. The numerator \( t^2 + 14t + 27 \) reflects a quadratic relationship, meaning that the velocity changes in a non-linear way. Meanwhile, the denominator \((2t+1)(t+5)^2\) includes a linear and a squared factor, which further impacts how the velocity evolves over time.Understanding velocity functions is crucial, as they allow us to predict and understand the behavior of moving objects within mathematical and real-world applications. This particular function is complex, given its rational nature, indicating that the velocity does not change linearly with time.

The concept of distance traveled by an object is essential in physics and engineering. To find the distance traveled by an object using a velocity function, you must integrate the velocity over time. In this problem, the scenario requires calculating the distance the object travels within the first two seconds. Mathematically, this is represented using a definite integral from \( t = 0 \) to \( t = 2 \). This can be expressed as the integral of the velocity function:\[ \int_{0}^{2} \frac{t^2 + 14t + 27}{(2t+1)(t+5)^2} \, dt. \]Computing this integral involves summing up all the infinitesimal pieces of distance traveled in tiny time intervals. So essentially, the integral "adds up" all the pieces of velocity over the period from 0 to 2 seconds, giving the total distance traveled.

Partial fraction decomposition is a technique used to break down complex rational functions into simpler pieces, which makes integration more straightforward. In this exercise, the velocity function is quite complex:\[ \frac{t^2 + 14t + 27}{(2t+1)(t+5)^2}. \]To simplify this for integration, we express it as the sum of simpler fractions:\[ \frac{A}{2t+1} + \frac{B}{t+5} + \frac{C}{(t+5)^2}. \]Here, \( A \), \( B \), and \( C \) are unknown constants that can be determined through algebraic methods. By expanding, equating the original function to the expanded form, and solving a system of equations, these constants can be found. Understanding how to decompose a complex rational function into partial fractions can significantly simplify the integration process by transforming the function into more manageable parts.

A definite integral calculates the accumulated quantity of a given function over a specified interval. In this case, it represents the total distance traveled by integrating the velocity function from \( t = 0 \) to \( t = 2 \). The definite integral results in:\[ \left[ \frac{1}{2}\ln|2t+1| + 4\ln|t+5| - \frac{11}{t+5} \right]_{0}^{2}. \]After finding the antiderivative of the function, evaluation occurs by substituting the limits 2 and 0 into it and finding the difference. This method provides the net result directly without requiring unrelated constants of integration, due to the use of specific boundaries. Manipulating definite integrals is fundamental in applications across physics, engineering, and mathematics, as they help calculate areas, volumes, and other physical quantities.

Rational functions, such as the velocity function given in the problem, are mathematical expressions where both the numerator and denominator are polynomials. In this case, the rational function is:\( \frac{t^2 + 14t + 27}{(2t+1)(t+5)^2} \).Rational functions have well-defined rules and properties, such as vertical and horizontal asymptotes, which are valuable for understanding their behavior. The complexity in differentiating and integrating them often stems from their degree and the relative degrees of the polynomials in the numerator and denominator.Utilizing techniques like partial fraction decomposition can simplify these functions, making them easier to work with in calculus. Understanding rational functions is essential as they frequently appear in mathematical modeling, describing phenomena where relationship ratios are fundamental, such as in velocities, rates, and various physical properties.