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In calculus, differentiability is a crucial property of a function, indicating that it has a derivative at each point in its domain. When a function is differentiable over an interval, it means that the function's graph is smooth and has no sharp turns or cusps on that interval. Differentiability is essential for applying the Mean Value Theorem (MVT) as it requires the function to be both continuous on a closed interval and differentiable on an open interval. If a function is differentiable at a point, it implies the function is locally linear around that point, meaning it can be approximated by a tangent line in the neighborhood of the point. This property is vital for understanding changes in the function's value and applying various rules in calculus for solving real-world problems, like understanding motion or optimizing paths. In the problem at hand, the differentiability of the function ensures that the function can have a derivative at every point in the interval, enabling us to apply the MVT and connect changes in function values to the behavior of its derivatives.

The derivative of a function, often denoted as \( f'(x) \), represents the rate at which the function is changing at any given point. It is a measure of the function's sensitivity to change in its input, which is often visualized as the slope of the tangent line to the function's graph at a particular point. In the context of the Mean Value Theorem (MVT), the derivative is the central tool used to relate the change in function values over an interval to the instantaneous rate of change at a particular point within that interval. By knowing that the derivative \( f'(c) = \frac{f(x) - f(y)}{x - y} \), we can determine that the change in the function, \( f(b) - f(a) \), is essentially governed by this slope.Understanding derivatives allows us to predict how small changes in input \( x \) affect the output \( f(x) \), which is critical in fields ranging from physics, for instantaneous velocity, to business, for analyzing cost or revenue changes.

The sine function, \( \sin x \), is a fundamental trigonometric function that arises frequently in mathematics, particularly within geometry and calculus. It is periodic, repeating every \( 2\pi \), and is known for creating a smooth, wave-like graph. The sine function is differentiable everywhere on its domain, \( \mathbb{R} \), with a derivative \( f'(x) = \cos x \). The property of differentiability in sine allows us to apply calculus theorems, such as the Mean Value Theorem, to establish important relationships and proofs, like showing bounds on expressions involving sine values. In the given exercise, the sine function's differentiability and the bounded nature of its derivative (where \( |\cos x| \leq 1 \)) are used to show that the difference in sine values between any two points is constrained. This key property makes sine useful in establishing bounds and solving problems across varied scientific and engineering disciplines.

An inequality proof seeks to demonstrate that one mathematical expression is consistently greater than or less than another across its domain. In our exercise, we want to show that the absolute difference between \( f(x) \) and \( f(y) \), constrained by the Mean Value Theorem, leads to said inequality.The proof involves:

• Utilizing the Mean Value Theorem to link the difference \( |f(x) - f(y)| \) with the derivative \( |f'(c)| \, |x - y| \).
• Recognizing that if a derivative is bounded by an upper limit \( M \), so too is the change in the function values constrained as \( \leq M |x - y| \).

For the sine function, we've shown \( M = 1 \), thus proving that \( |\sin x - \sin y| \leq |x - y| \). These types of proofs establish a limit or range for change, which is incredibly useful in both theoretical and applied mathematics, such as setting acceptable error margins in models or predicting systems' behavior within certain constraints.