Articles containing proofs | Theorems in real analysis | Theorems in calculus | Approximations
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the kth-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k 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. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory. Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formulas to accurately compute values of many transcendental functions such as the exponential function and trigonometric functions.It is the starting point of the study of analytic functions, and is fundamental in various areas of mathematics, as well as in numerical analysis and mathematical physics. Taylor's theorem also generalizes to multivariate and vector valued functions. (Wikipedia).
In this video, I give a very neat and elegant proof of Taylor’s theorem, just to show you how neat math can be! It is simply based on repeated applications of the fundamental theorem of calculus. Enjoy! Note: The thumbnail is taken from https://i.redd.it/kv7lk5kn31e01.jpg
From playlist Calculus
Free ebook http://tinyurl.com/EngMathYT A lecture showing how to compute Taylor polynomials. Plenty of examples are discussed and solved. Such ideas are used in approximation of functions and are seen in university mathematics.
From playlist A second course in university calculus.
Taylor's Theorem with Remainder
This videos shows how to determine the error when approximating a function value with a Taylor polynomial. http://mathispower4u.yolasite.com/
From playlist Infinite Sequences and Series
Maclaurin series and applications
Free ebook http://bookboon.com/en/learn-calculus-2-on-your-mobile-device-ebook Basic example on Maclaurin series and some applications. In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's deriv
From playlist A second course in university calculus.
One topic in basic calculus that you may not have seen before that is that of the Taylor expansion of a function. It is a series that can be used in stead of the actual function around a certain x-value for easier calculations.
From playlist Biomathematics
How to Compute a Maclaurin Polynomial
Free ebook http://bookboon.com/en/learn-calculus-2-on-your-mobile-device-ebook What is a Maclaurin polynomial? In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point
From playlist A second course in university calculus.
Math 031 041917 Taylor's Theorem and the Lagrange Remainder Theorem (no sound)
Motivation: how do you know of the Taylor series converges back to the original function? Statement of Taylor's Theorem (+ Lagrange Remainder Formula). Example application: showing that the Taylor series for the sine recovers the sine (at x = 1; then for general x). Same application for
From playlist Course 3: Calculus II (Spring 2017)
Math 031 Spring 2018 043018 Lagrange Remainder Theorem
Definition of Taylor polynomial; of remainder (error). Statement of Lagrange Remainder Theorem. Example.
From playlist Course 3: Calculus II (Spring 2018)
Taylor polynomials + functions of two variables
Download the free PDF http://tinyurl.com/EngMathYT This is a basic tutorial on how to calculate a Taylor polynomial for a function of two variables. The ideas are applied to approximate a difficult square root. Such concepts are seen in university mathematics.
From playlist Several Variable Calculus / Vector Calculus
Multivariable Taylor Polynomials
Free ebook http://tinyurl.com/EngMathYT A lecture on how to calculate Taylor polynomials and series for functions of two variables. Such ideas are useful in approximation of functions. We show where the polynomial representation comes from.
From playlist Mathematics for Finance & Actuarial Studies 2
What is Taylor's theorem? - Week 6 - Lecture 5 - Sequences and Series
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From playlist Ohio State: Calculus Two with Jim Fowler: Sequences and Series | CosmoLearning Mathematics
Oxford Calculus: Taylor's Theorem Explained with Examples and Derivation
University of Oxford mathematician Dr Tom Crawford derives Taylor's Theorem for approximating any function as a polynomial and explains how the expansion works with two detailed examples. Test yourself with some exercises on Taylor's Theorem with this FREE worksheet in Maple Learn: https
From playlist Oxford Calculus
The Proof of TAYLOR'S THEOREM Ms. Michael Never Taught You
In this video, we look at a very elementary proof of Taylor's theorem that I stumbled upon while going for a walk the other week. The proof uses only the fundamental theorem of calculus is such an easy proof that we could've all discovered it if we had gone on enough walks. Timestamps: 0:
From playlist Calculus
Taylor's Theorem for Remainders
Calculus: Given a Taylor polynomial for a function f(x) with n+1 derivatives, Taylor's Theorem gives us a method for estimating the error from the actual value. The example of f(x) = x^5 + 1 is given. For more videos like this one, please visit the Calculus playlists at this channel.
From playlist Calculus Pt 6: Sequences and Series
Train your logical thinking skills and learn how to deal with complex numbers by trying out Brilliant! =D https://brilliant.org/FlammableMaths Subscribe to @FlammysWood to see your dad working his wood :^D https://www.youtube.com/watch?v=FQAk0TtI9LE Handcrafted products, puzzles and more
From playlist Taylor Series
The history of Taylor Series and Maclaurin Series including the works of de Lagny, Halley, Gregory, and Madhava using primary sources whenever possible. Lesson also presents the Taylor Theorem along with visualizations of James Gregory's equations. Finally the video discusses the time peri
From playlist Numerical Methods
Calculus BC - Unit 5 Lesson 2: Lagrange Error Bound
Calculus BC - Taylor's Remainder Theorem and the Lagrange Error Bound
From playlist AP Calculus BC
Real Analysis - Part 47 - Proof of Taylor's Theorem
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From playlist Real Analysis
Math 131 Fall 2018 102418 Taylor's Theorem; Introduction to Sequences
Sketch of proof of L'Hopital's Rule. Taylor's theorem: definition of Taylor polynomial. Proof of Taylor's theorem. Introduction to sequences. Definition of convergence of a sequence (in a metric space). Example. Implications of convergence to a point: every neighborhood of the point
From playlist Course 7: (Rudin's) Principles of Mathematical Analysis (Fall 2018)
Lecture 20: Taylor's Theorem and the Definition of Riemann Sums
MIT 18.100A Real Analysis, Fall 2020 Instructor: Dr. Casey Rodriguez View the complete course: http://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP61O7HkcF7UImpM0cR_L2gSw We study Taylor’s theorem, essentially a di
From playlist MIT 18.100A Real Analysis, Fall 2020