# taylor approximation Definition and Topics - 11 Discussions

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.

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1. ### 4th order Taylor approximation

So I just followed Taylor's formula and got the four derivatives at p = 0 ##f^{(0)}(p) = (1 + \frac {p^2} {m^2c^2})^{\frac 1 2} ## ##f^{(0)}(0) = 1 ## ## f^{(1)}(p) = \frac {p} {m^2c^2}(1 + \frac {p^2} {m^2c^2})^{\frac {-1} 2} ## ## f^{(1)}(0) = 0 ## ## f^{(2)}(p) = \frac {1} {m^2c^2}(1 +...
2. ### (Physicist version of) Taylor expansions

3) Taylor expansion question in the context of Lie algebra elements: Consider some n-dimensional Lie group whose elements depend on a set of parameters \alpha =(\alpha_1 ... \alpha_n) such that g(0) = e with e as the identity, and that had a d-dimensional representation D(\alpha)=D(g( \alpha)...
3. ### Limit of the remainder of Taylor polynomial of composite functions

Since $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(x)} {x^n}=0,$$ ##P_{n,0,g}(x)## contains only terms of degree ##\geq 1## and ##R_{n,0,g}## approaches ##0## as quickly as ##x^n##, I can most likely prove this using ##\epsilon - \delta## arguments, but that seems overly complicated. I also can't...
4. ### Lagrange error bound inequality for Taylor series of arctan(x)

The error ##e_{n}(y)## for ##\frac{1}{1-y}## is given by ##\frac{1}{(1-c)^{n+2}}y^{n+1}##. It follows that ##\frac{1}{1+y^2}=t_n(-y^2)+e_n(-y^2)## where ##t_n(y)## is the Taylor polynomial of ##\frac{1}{1-y}##. Taking the definite integral from 0 to ##x## on both sides yields that...
5. ### A Taylor expansion for a nonlinear system and Picard Iterations

Hello guys I struggle since yesterday with the following problem I am reading the book "Elements of applied bifurcation theory" by Kuznetsov . At one point he has the following Taylor expansion of a nonlinear system with respect to x=0 where ##x\in \mathbb(R)^n## \dot{x} = f(x) = \Lambda x +...
6. ### Derivative of expanded function wrt expanded variable?

Homework Statement If I have the following expansion f(r,t) \approx g(r) + \varepsilon \delta g(r,t) + O(\varepsilon^2) This means for other function U(f(r,t)) U(f(r,t)) = U( g(r) + \varepsilon \delta g(r,t)) \approx U(g) + \varepsilon \delta g \dfrac{dU}{dg} + O(\varepsilon^2) Then up to...
7. ### Solving Schrodinger's Equation with a weak Imaginary Potential

Homework Statement A particle of energy E moves in one dimension in a constant imaginary potential -iV where V << E. a) Find the particle's wavefunction \Psi(x,t) approximating to leading non-vanishing order in the small quantity \frac{V}{E} << 1. b) Calculate the probability current density...
8. ### Approximating square root of 2 (Taylor remainder)

Homework Statement [/B] Use the Taylor remainder theorem to give an expression of ##\sqrt 2 - P_3(1)## P_3(x) - the degree 3 Taylor polynomial ##\sqrt {1+x}## in terms of c, where c is some number between 0 and 1 Find the maximum over the interval [0, 1] of the absolute value of the...
9. ### I Taylor expansion of f(x+a)

I recently found out the rule regarding the Taylor expansion of a translated function: ##f(x+h)=f(x)+f′(x)⋅h+\frac 1 2 h^ 2 \cdot f′′(x)+⋯+\frac 1 {n!}h^n \cdot f^n(x)+...## But why exactly is this the case? The normal Taylor expansion tells us that ##f(x)=f(a)+f'(a)(x-a)+\frac 1...
10. ### Approximating a spring constant for an air leg

Hi all, In short: For an air leg or air spring, there is a method using a Taylor approximation to find the spring constant for very small displacements, but I can't seem to figure out how it works. I've learned that air legs don't follow Hooke's law very much at all, except for when the...
11. ### I Second order Taylor approximation

Hello, Can someone explain this to me? In the above case ct=yt-gt I tried to solve it as a three variable taylor approximation but got a few extra terms that weren't included in the above. So I am a little confused now. I only need to understand how the first line was derived because I get...