explain all these topics mentioned here---UNIT - IV Differential Calculus of Univariate Functions 06 Hours Main Topic-1: Mean Value Theorem: Rolle's Mean Value Theorem, Lagrange's Mean Value Theorem and Cauchy's Mean Value Theorem Main Topic-2: Introduction to Successive Differentiation Main Topic-3: Taylor's series expansion, McLaurin's Series expansion, Expansion of Standard functions.

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Welcome to Unit Four of Differential Calculus. This unit covers three fundamental topics that build upon basic differentiation. First, we study Mean Value Theorems, which connect function behavior over intervals to derivative values at specific points. Second, we explore Successive Differentiation, the process of finding higher order derivatives. Finally, we examine Taylor and Maclaurin Series, which represent functions as infinite polynomial expansions. Rolle's Mean Value Theorem is the first and simplest of the three mean value theorems. It states that if a function is continuous on a closed interval from a to b, differentiable on the open interval, and the function values at the endpoints are equal, then there must exist at least one point c between a and b where the derivative equals zero. Geometrically, this means there is a horizontal tangent line somewhere on the curve between the equal endpoints.