What is the trapezoidal rule primarily used for in numerical methods?

The trapezoidal rule is fundamentally used in numerical methods to approximate definite integrals. It operates by dividing the area under a curve into trapezoids rather than using rectangles, which provides a more accurate estimate of the area when considering the shape of the function being integrated. By using the endpoints of the interval, the trapezoidal rule takes into account the linear approximation between these points, making it effective for estimating the integral of a continuous function over a specified interval.

The trapezoidal rule is particularly advantageous because it is relatively simple to implement and requires only evaluations of the function at a limited number of points. This method is part of a broader class of numerical integration techniques aimed at finding approximate solutions when exact solutions are difficult or impossible to obtain through analytical methods.

In the context of the other options, assessing the stability of numerical solutions pertains more to methods and techniques in numerical analysis that ensure reliable results over iterative methods. Calculating eigenvalues of matrices is addressed through different numerical algorithms, such as the QR algorithm or power iteration. Deriving differential equations is a separate area of study that deals with the formation of equations governing dynamic systems and is not directly related to numerical integration techniques like the trapezoidal rule.