What is Taylor approximation: Definition and 26 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. Z

    I Spivak, Ch. 20: Understanding a step in the proof of lemma

    In Chapter 20 of Spivak's Calculus is the lemma shown below (used afterward to prove Taylor's Theorem). My question is about a step in the proof of this lemma. Here is the proof as it appears in the book My question is: how do we know that ##(R')^{n+1}## is defined in ##(2)##? Let me try to...
  2. M

    State space controllers - find equations from differentials

    Summary:: This is similar to the examples of electrical circuit state space analysis, I have been trying to find the state space equations from the following non linear first order differentials but I keep getting stuck. Any help? A) Started off from non linear equations: $$y' =...
  3. B

    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 +...
  4. Jason Bennett

    (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)...
  5. Adgorn

    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...
  6. S

    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...
  7. U

    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 +...
  8. D

    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...
  9. Jozefina Gramatikova

    By calculating a Taylor approximation, determine K

    Homework Statement Homework Equations [/B]The Attempt at a Solution Can somebody explain to me how did we find the function in red? Thanks
  10. A

    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...
  11. Poetria

    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...
  12. Adgorn

    I Understanding the Taylor Expansion of a Translated Function

    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...
  13. S

    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...
  14. tanvi nautiyal

    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...
  15. E

    Calculus Taylor Approximation Proof

    1. The question is. Show that if |nx| <1, the following is exact up to (and including) the x^2 order. The hint giving says to use the Taylor Expansion for both sides of the equation2. (1+x)^n = e^n(x-(1/2)x^2) ; the n(x-(1/2)x^2) is all an exponent3. My first attempt was to take the taylor...
  16. Nugso

    Taylor Approximation: Show ∫f'(x)dx/f(x)=ln|f(x)|+C

    Homework Statement Show that ∫f'(x)dx/f(x) = ln|(f(x)|+C where f(x) is a differential function. Homework Equations First order Taylor approximation? f(x)=f(a)+f'(a)(x-a) The Attempt at a Solution Well, I'm not really sure how to approach the question. It's my Numerical...
  17. A

    Taylor Approximation: Error Calculation Tool?

    Often you use taylor series to approximate differential equations for easier solving. An example is the small angle approximation to the pendulum. My question is: Is there mathematical tool for calculating the error you make as time goes with such an approximation? Because I could Imagine it...
  18. I

    Taylor approximation of the Doppler Law for slow-moving emitters

    The relationship linking the emitted frequency Fe and the received frequency Fr is the Doppler Law: F_r = \sqrt \frac{1-\frac{v}{c}}{1-\frac{v}{c}} F_e The Taylor series for the function \sqrt\frac{1+x}{1-x} near x = 0 is 1+x+\frac{x^2}{2}+\frac{x^3}{3}+... On Earth, most objects travel...
  19. E

    Taylor Approximation (I think) on Transmission Coefficient

    Homework Statement I have this equation: T=(1+\frac{U_{0}^{2}}{4E(U_{0}-E)}sinh^{2}(2 \alpha L))^{-1} Where α is given by: \alpha = \sqrt{ \frac{2m(U_{0}-E)}{\hbar^{2}}} I have to show that in the limit αL>>1 my equation is approximately given by...
  20. R

    MATLAB Taylor Approximation

    Homework Statement find the 2nd, 3rd, and 6th degree taylor approximation of: f(x) = 10(x/2 -0.25)5 + (x-0.5)3 + 9(x-0.75)2-8(x-0.25)-1 for h = 0.1 to h = 1, with \Deltah = 0.05 and where xo=0; and x = h Homework Equations N.A The Attempt at a Solution I just need to...
  21. M

    Help with understanding taylor approximation

    Homework Statement I'm reading about fluid mechanics and in one of the examples they have approximated the velocity field. The field is two dimensional u = (u,v) I have never seen this before so cold someone tell me what it is called so I can look it up? The notes I am reading are hand...
  22. P

    Taylor approximation (probability)

    I have the following problem: Assume g is a (smooth enough) function, X a random variable and \varepsilon^h a sequence of random variables, whose moments converge to 0 as h goes to zero. I would then like to prove that \mathbb{E}\left|g(X+\varepsilon^h)-g(X)\right| converges to zero as well...
  23. M

    Very quick Taylor Approximation Question

    Homework Statement Let f(x) = sin x a) find p_6 (taylor polynomial 6th degree) for f at x = 0 b) How accurate is this on the interval [-1,1] Homework Equations The Attempt at a Solution I got p_6 = x + (x^3)/6 + (x^5)/120, which was correct as per the solution manual. My...
  24. E

    Taylor Approximation Proof for P(r) using Series Expansion

    [SOLVED] Taylor approximation Homework Statement I have an exact funktion given as: P(r)=1-e^{\frac{-2r}{a}}(1+\frac{2r}{a}+\frac{2r^2}{a^2}) I need to prove, by making a tayler series expansion, that: P(r)\approx \frac{3r^3}{4a^4} When r \prec \prec a The Attempt at a Solution...
  25. S

    Taylor Approximation Help - Find n Given x, a, ErrorBound

    Hi, I'm having trouble doing my work where I have to find the Taylor Approximation of function. My real work is the program this thing when the function, x, a, and ErrorBound is given. I don't knwo what to do with the ErrorBound to get n, where n is the number of terms. do i make any sense...
  26. L

    Confused about taylor approximation

    I am a bit confused about taylor approximation. Taylor around x_0 yields f(x) = f(x_0) + f'(x_0)(x-x_0) + O(x^2) which is the tangent of f in x_0, where f'(x) = f'(x_0) + f''(x_0)(x-x_0) + O(x^2) which adds up to f(x) &=& f(x_0) + (f'(x_0) + f''(x_0)(x-x_0) +...
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