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Parametric equations are a way of expressing spatial coordinates as continuous functions of a third variable, typically denoted as a parameter such as \(t\). These types of equations are particularly useful when dealing with curves in a coordinate plane. For instance, when we have a curve like a circle or semicircle, parametric equations allow us to describe every point on this curve succinctly. In our problem, the semicircle and quarter circle are represented by the parametric equations \(x = 2\cos t\) and \(y = 2\sin t\). These equations are derived from the trigonometric identities of the unit circle, scaled by a factor matching the circle's radius. This allows us to systematically examine every point along the circle by adjusting the parameter \(t\), which varies between specified limits to cover the desired portion of the circle. In doing so, parametric equations provide a comprehensive way to explore figures that are not conveniently described by traditional \(y = f(x)\) functions.

A semicircle represents half of a circle. In mathematical terms, if a circle is expressed as \(x^2 + y^2 = r^2\), then a semicircle with the upper half above the x-axis is described with \(y \geq 0\). When using parametric equations, we can represent this semicircle with \(x = r\cos t\) and \(y = r\sin t\) for \(0 \leq t \leq \pi\). For our specific exercise, the semicircle has a radius of 2, so the parametric equations become \(x = 2\cos t\) and \(y = 2\sin t\). This covers all points from the leftmost boundary (\(-2, 0\)) to the rightmost boundary (\(2, 0\)), tracing the semicircle in the positive y-direction. Understanding this geometric interpretation is crucial for analyzing functions and identifying extremum values on curved paths like semicircles.

A quarter circle is simply one-fourth of a full circle. In our exploration, the relevant quarter circle lies in the first quadrant with both \(x \geq 0\) and \(y \geq 0\). Mathematically, if a circle is described by the equation \(x^2 + y^2 = r^2\), a quarter circle can be described using similar parametric equations to the semicircle, adjusted for a reduced range for \(t\), specifically: \(x = r\cos t\), \(y = r\sin t\) with \(0 \leq t \leq \frac{\pi}{2}\). In this exercise, with a radius of 2, the parametric forms are \(x = 2\cos t\) and \(y = 2\sin t\). This setup assists in visiting all points within the first quadrant, starting at the rightmost point on the x-axis and sweeping up to the topmost point on the y-axis. This quarter circle representation is essential in our analysis when determining extrema within a specific circular segment.

Differentiation is a powerful mathematical tool used to find rates of change and identify points of extrema, which are maxima or minima, on curves. By taking the derivative of a function, we can determine its slope at any given point. In the exercise, finding the derivative of the parametrized function helps us locate these points of interest. For each parametric form, like \(f(t) = 2(\cos t + \sin t)\), differentiation provides the derivative \(f'(t) = 2(-\sin t + \cos t)\). Setting the derivative equal to zero allows us to find critical points, potential locations of the function's extrema. Here, solving \(-\sin t + \cos t = 0\) identifies that \(t = \frac{\pi}{4}\) is a critical point, which we use to evaluate the original function and determine its maximum and minimum values within the specified domain. This process shows the importance of differentiation in analyzing parametric curves and their associated functions.