9.2 Taylor Series.

Slides:

Advertisements

Similar presentations

9.2 day 2 Finding Common Maclaurin Series Liberty Bell, Philadelphia, PA.

Advertisements

Taylor Series Section 9.2b.

Taylor’s Theorem Section 9.3a. While it is beautiful that certain functions can be represented exactly by infinite Taylor series, it is the inexact Taylor.

Power Series is an infinite polynomial in x Is a power series centered at x = 0. Is a power series centered at x = a. and.

Section 9.2a. Do Now – Exploration 1 on p.469 Construct a polynomial with the following behavior at : Since, the constant coefficient is Since, the coefficient.

Taylor Series (11/12/08) Given a nice smooth function f (x): What is the best constant function to approximate it near 0? Best linear function to approximate.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 1.

Taylor Series. Theorem Definition The series is called the Taylor series of f about c (centered at c)

9.2 Taylor Series Quick Review Find a formula for the nth derivative of the function.

Additional Topics in Differential Equations

SECOND-ORDER DIFFERENTIAL EQUATIONS Series Solutions SECOND-ORDER DIFFERENTIAL EQUATIONS In this section, we will learn how to solve: Certain.

Infinite Series Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 17 Second-Order Differential Equations.

Find the local linear approximation of f(x) = e x at x = 0. Find the local quadratic approximation of f(x) = e x at x = 0.

Polynomial Approximations BC Calculus. Intro: REM: Logarithms were useful because highly involved problems like Could be worked using only add, subtract,

Consider the following: Now, use the reciprocal function and tangent line to get an approximation. Lecture 31 – Approximating Functions

Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=ex. Graph f and your approximation function for a graphical comparison.

9.7 and 9.10 Taylor Polynomials and Taylor Series.

TAYLOR AND MACLAURIN  how to represent certain types of functions as sums of power series  You might wonder why we would ever want to express a known.

Taylor and Maclaurin Series Lesson Convergent Power Series Form Consider representing f(x) by a power series For all x in open interval I Containing.

MATH 6B CALCULUS II 11.3 Taylor Series. Determining the Coefficients of the Power Series Let We will determine the coefficient c k by taking derivatives.

The Convergence Problem Recall that the nth Taylor polynomial for a function f about x = x o has the property that its value and the values of its first.

9.3 Taylor’s Theorem Quick Review Tell whether the function has derivatives of all orders at the given values of a.

Remainder Theorem. The n-th Talor polynomial The polynomial is called the n-th Taylor polynomial for f about c.

Taylor and MacLaurin Series Lesson 8.8. Taylor & Maclaurin Polynomials Consider a function f(x) that can be differentiated n times on some interval I.

In this section, we will investigate how we can approximate any function with a polynomial.

Wednesday, April 6MAT 146. Wednesday, April 6MAT 146.

The following table contains the evaluation of the Taylor polynomial centered at a = 1 for f(x) = 1/x. What is the degree of this polynomial? x T(x) 0.5.

S ECT. 9-4 T AYLOR P OLYNOMIALS. Taylor Polynomials In this section we will be finding polynomial functions that can be used to approximate transcendental.

Copyright © Cengage Learning. All rights reserved.

Lecture 25 – Power Series Def: The power series centered at x = a:

Calculus II (MAT 146) Dr. Day Thursday, Dec 8, 2016

Chapter 9 Power Series Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Section 11.3 – Power Series.

Taylor series in numerical computations (review)

Starting problems Write a tangent line approximation for each:

PROGRAMME 12 SERIES 2.

Taylor Series Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of.

Sum and Difference Identities for the Sin Function

Finding Common Maclaurin Series

how to represent certain types of functions as sums of power series

Connor Curran, Cole Davidson, Sophia Drager, Avash Poudel

Remainder of a Taylor Polynomial

Taylor and MacLaurin Series

Section 11.4 – Representing Functions with Power Series

Section 11.3 Power Series.

Taylor Series Revisited

Taylor and Maclaurin Series

Taylor Series – Day 2 Section 9.6 Calculus BC AP/Dual, Revised ©2014

FP2 (MEI) Maclaurin Series

Find the Taylor series for f (x) centered at a = 8. {image} .

Taylor Series and Maclaurin Series

Applications of Taylor Series

11.1 – Polynomial Approximations of Functions

11.10 – Taylor and Maclaurin Series

Section 2: Intro to Taylor Polynomials

how to represent certain types of functions as a power series

Taylor Polynomials – Day 2

9.2 day 2 Maclaurin Series Liberty Bell, Philadelphia, PA

Finding Common Maclaurin Series

MACLAURIN SERIES how to represent certain types of functions as sums of power series You might wonder why we would ever want to express a known function.

9.7 Taylor Polynomials & Approximations

9.2 day 2 Maclaurin Series Liberty Bell, Philadelphia, PA

Taylor and Maclaurin Series

Copyright © Cengage Learning. All rights reserved.

Section 8.9: Applications of Taylor Polynomials

TAYLOR SERIES Maclaurin series ( center is 0 )

Finding Common Maclaurin Series

Presentation transcript:

9.2 Taylor Series

Quick Review

Quick Review Solutions

What you’ll learn about Constructing a Series Series for sin x and cos x Beauty Bare Maclaurin and Taylor Series Combining Taylor Series Table of Maclaurin Series … and why The partial sums of a Taylor series are polynomials that can be used to approximate the function represented by the series.

Example Constructing a Power Series for sin x

Taylor Series Generated by f at x=0 (Maclaurin Series)

Example Approximating a Function near 0

Taylor Series Generated by f at x=a

Example A Taylor Series at x = 1

Example A Taylor Polynomial for a Polynomial

Example A Taylor Polynomial for a Polynomial

Maclaurin Series

Maclaurin Series