Lecture

Higher Derivatives, Taylor Introduction

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In course

MATH-101(de): Analysis I (German)

Es werden die Grundlagen der Analysis sowie der Differential- und Integralrechnung von Funktionen einer reellen Veränderlichen erarbeitet. ...

Description

This lecture covers the concept of higher derivatives, focusing on the introduction to Taylor development. It explains the construction of the derivative function, the closed interval, and the Taylor series. The lecture also delves into examples illustrating the Taylor development and the uniqueness of the Taylor polynomial. Additionally, it discusses the Taylor expansion around a point and the properties of the Taylor polynomial.

Instructors (3)

Joachim Krieger

Daniel Kressner

Tobias Johannes Schmid

Official source

Related concepts (31)

Derivative test

In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is proved mathematically by Fermat's theorem of stationary points. The first-derivative test examines a function's monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain.

Second derivative

In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation: where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change.

Generalizations of the derivative

In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. The Fréchet derivative defines the derivative for general normed vector spaces . Briefly, a function , an open subset of , is called Fréchet differentiable at if there exists a bounded linear operator such that Functions are defined as being differentiable in some open neighbourhood of , rather than at individual points, as not doing so tends to lead to many pathological counterexamples.

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function with respect to the variable is variously denoted by It can be thought of as the rate of change of the function in the -direction.

Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation.

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