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Mohammed Twin
Mohammed Twin

Taylor Polynomials And Approximations Homework


How to Use Taylor Polynomials and Approximations to Solve Homework Problems




Taylor polynomials and approximations are powerful tools that can help you solve various homework problems in calculus, physics, engineering, and other fields. They allow you to approximate complicated functions with simpler ones that are easier to work with. In this article, we will explain what Taylor polynomials and approximations are, how to find them, and how to use them to solve homework problems.




Taylor Polynomials And Approximations Homework



What are Taylor Polynomials and Approximations?




A Taylor polynomial is a polynomial function that approximates another function near a given point. A Taylor approximation is the value of a Taylor polynomial at a given point. For example, suppose you have a function f(x) and you want to approximate it near x = a. You can find a Taylor polynomial P(x) of degree n such that P(a) = f(a), P'(a) = f'(a), P''(a) = f''(a), ..., P^(n)(a) = f^(n)(a), where P^(n) denotes the nth derivative of P. The value of P(x) at x = a is called the Taylor approximation of f(x) at x = a.


The idea behind Taylor polynomials and approximations is that if two functions have the same value and the same derivatives up to a certain order at a point, then they are very close to each other near that point. The higher the degree of the Taylor polynomial, the better the approximation. However, finding higher degree Taylor polynomials can be tedious and time-consuming, so sometimes it is enough to use lower degree ones for homework purposes.


How to Find Taylor Polynomials and Approximations?




To find a Taylor polynomial of degree n for a function f(x) near x = a, you need to use the following formula:


P(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^(n)(a)(x-a)^n/n!


This formula is called the Taylor series expansion of f(x) at x = a. It is an infinite series that converges to f(x) as n goes to infinity. However, for practical purposes, we only use the first few terms of the series as our Taylor polynomial.


To find a Taylor approximation of f(x) at x = b, you simply need to plug in x = b into the Taylor polynomial P(x). For example, suppose you want to find a Taylor approximation of sin(x) at x = pi/6 using a Taylor polynomial of degree 3 near x = 0. You can use the following steps:


  • Find the first four derivatives of sin(x): sin'(x) = cos(x), sin''(x) = -sin(x), sin'''(x) = -cos(x), sin''''(x) = sin(x).



  • Evaluate these derivatives at x = 0: sin(0) = 0, cos(0) = 1, -sin(0) = 0, -cos(0) = -1.



  • Plug these values into the Taylor series formula: P(x) = 0 + 1(x-0) + 0(x-0)^2/2! + -1(x-0)^3/3!.



  • Simplify the expression: P(x) = x - x^3/6.



  • Plug in x = pi/6 into P(x): P(pi/6) = pi/6 - (pi/6)^3/6.



  • Calculate the result: P(pi/6) 0.5 - 0.0073 0.4927.



This is the Taylor approximation of sin(pi/6) using a Taylor polynomial of degree 3 near x = 0.