Lecture

Derivability and Tangents

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Description

This lecture covers the concept of derivability, tangents, and the theorem of Rolle. It explains how to determine the slope of a tangent line and the linear approximation of a function. The lecture also discusses the theorem of finite increments and its application in calculating derivatives.

Official source

Ontological neighbourhood

Mathematics

Analysis: Calculus

Related concepts (32)

Linear approximation

In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations. Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case states that where is the remainder term.

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Rolle's theorem

In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero. The theorem is named after Michel Rolle.

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In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.

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