The mean value theorem and Taylor's theorem for fractional derivatives with Mittag-Leffler kernel

The Mean Value Theorem establishes a fundamental connection between the overall change in a quantity and its change at a specific instant. It formalizes the idea that average change over an interval must be reflected by instantaneous change at some point within that interval. When a function behaves smoothly across a range, the theorem guarantees that this connection always exists.This relationship is captured mathematically by the Mean Value Theorem, as stated below.The meaning of this result...

In calculus, the computation of the area under a continuous curve has been fundamentally simplified by applying the Fundamental Theorem of Calculus, Part 2. Rather than relying on the limiting process of summing infinitely many infinitesimal rectangles, this theorem permits direct evaluation using antiderivatives, thereby streamlining the process of definite integration.The Fundamental Theorem of Calculus, Part 2, states that if a function f(x) is continuous on a closed interval [a, b], then...

The Intermediate Value Theorem is a foundational result in calculus that guarantees the existence of solutions within certain intervals for continuous functions. Formally, the Intermediate Value Theorem states that if a function f is continuous on the closed interval [a, b], and if N is any value between f(a) and f(b), then there exists at least one c ∈ (a, b) such that f(c) = N. This theorem is instrumental in proving the existence of roots and in analyzing the behavior of continuous functions...

A Taylor series is a power series constructed to reproduce the local behavior of a smooth function about a chosen point, called the center. Its purpose is to encode the function’s value and successive derivatives into a structured expansion that captures local variation. The central task in its derivation is to find coefficients so that the series and the function share identical values and derivatives at the center.Consider a power series centered at x =...

The Taylor series provides a systematic method for approximating a smooth function by a polynomial that closely matches the function near a chosen point. This approach is particularly valuable in scientific and engineering contexts where functions may be difficult to evaluate directly, such as oscillatory voltages in alternating current (AC) circuits. Replacing complex functions with polynomial expressions simplifies computation while preserving essential local behavior. Taylor’s Theorem...

Taylor series provide a systematic way to represent a smooth function as an infinite polynomial centered at a chosen point. When that center is zero, the expansion is called a Maclaurin series. This form is especially useful because every derivative is evaluated at zero, which often makes the coefficients easier to compute. In applied mathematics and physics, such series are valuable because they replace complicated functions with polynomials that are easier to analyze and evaluate...