9.7 Taylor Series. Brook Taylor 1685 - 1731 Taylor Series Brook Taylor was an accomplished musician and painter. He did research in a variety of areas,

Slides:

Advertisements

Similar presentations

Section 11.6 – Taylor’s Formula with Remainder

Advertisements

Taylor Series Section 9.2b.

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

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.

Calculus I – Math 104 The end is near!. Series approximations for functions, integrals etc.. We've been associating series with functions and using them.

AP Calculus BC Monday, 07 April 2014 OBJECTIVE TSW (1) find polynomial approximations of elementary functions and compare them with the elementary functions;

Copyright © Cengage Learning. All rights reserved.

9.10 Taylor and Maclaurin Series Colin Maclaurin

Calculus I – Math 104 The end is near!. 1. Limits: Series give a good idea of the behavior of functions in the neighborhood of 0: We know for other reasons.

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

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

11.8 Power Series 11.9 Representations of Functions as Power Series Taylor and Maclaurin Series.

Infinite Series Copyright © Cengage Learning. All rights reserved.

Taylor’s Polynomials & LaGrange Error Review

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

Do Now: Find both the local linear and the local quadratic approximations of f(x) = e x at x = 0 Aim: How do we make polynomial approximations for a given.

Now that you’ve found a polynomial to approximate your function, how good is your polynomial? Find the 6 th degree Maclaurin polynomial for For what values.

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’s Theorem: Error Analysis for Series. Taylor series are used to estimate the value of functions (at least theoretically - now days we can usually.

Section 9.7 Infinite Series: “Maclaurin and Taylor Polynomials”

In section 11.9, we were able to find power series representations for a certain restricted class of functions. Here, we investigate more general problems.

Copyright © Cengage Learning. All rights reserved.

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.

In this section we develop general methods for finding power series representations. Suppose that f (x) is represented by a power series centered at.

Sect. 9-B LAGRANGE error or Remainder

This is an example of an infinite series. 1 1 Start with a square one unit by one unit: This series converges (approaches a limiting value.) Many series.

Section 9.7 – Taylor Theorem. Taylor’s Theorem Like all of the “Value Theorems,” this is an existence theorem.

Maclaurin and Taylor Polynomials Objective: Improve on the local linear approximation for higher order polynomials.

9.3 Taylor’s Theorem: Error Analysis yes no.

Taylor series are used to estimate the value of functions (at least theoretically - now days we can usually use the calculator or computer to calculate.

Section 3.9 Linear Approximation and the Derivative.

Chapter 10 Power Series Approximating Functions with Polynomials.

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

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

Infinite Series 9 Copyright © Cengage Learning. All rights reserved.

Convergence of Taylor Series Objective: To find where a Taylor Series converges to the original function; approximate trig, exponential and logarithmic.

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.

In the special case c = 0, T (x) is also called the Maclaurin Series: THEOREM 1 Taylor Series Expansion If f (x) is represented by a power series.

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

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

Section 11.3 – Power Series.

The Taylor Polynomial Remainder (aka: the Lagrange Error Bound)

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.

9.7: 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.

The LaGrange Error Estimate

9.2: 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.

Taylor and Maclaurin Series

11.8 Power Series.

Taylor Polynomials & Approximation (9.7)

Calculus BC AP/Dual, Revised © : Lagrange's Error Bound

Remainder of a Taylor Polynomial

Polynomial Approximations of Elementary Functions

For the geometric series below, what is the limit as n →∞ of the ratio of the n + 1 term to the n term?

Section 11.3 Power Series.

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

Warm up In terms of letters and variables write the general form of the first 5 polynomials. For example: the first degree polynomial is represented by:

11.1 – Polynomial Approximations of Functions

Section 11.6 – Taylor’s Formula with Remainder

Taylor’s Theorem: Error Analysis for Series

Section 2: Intro to Taylor Polynomials

9.2: 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.

9.7 Taylor Polynomials & Approximations

Copyright © Cengage Learning. All rights reserved.

Lagrange Remainder.

TAYLOR SERIES Maclaurin series ( center is 0 )

Presentation transcript:

9.7 Taylor Series

Brook Taylor 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 ideas regarding infinite series.

Identify this graph:What if we zoom out a bit? And a bit more? What the?! The actual function is:

Suppose we wanted to find a fifth degree polynomial of the form: at that approximates the behavior of If we make, and the first, second, third, fourth, and fifth derivatives the same, then we would have a pretty good approximation.

0 1 0

0 1 0

0 1

0 1

P(x) is the fifth degree polynomial that has the same first through fifth derivatives at x = 0 that f(x) = sin x has. We call P(x) the 5 th degree Taylor Polynomial of f(x) at x = 0.

If we plot both functions, we see that near zero the functions match very well!

This pattern occurs no matter what the original function was! Our polynomial: has the form: or: This is called a Taylor Polynomial. If we continue this pattern infinitely, we get a Taylor Series. When the series is centered at x = 0, it is called a Maclaurin Series.

If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series: Taylor Series: (generated by f at ) Maclaurin Series: (generated by f at )

Example: Generate the Maclaurin Series for The formula for Maclaurin Series is:

The more terms we add, the better our approximation.

example:

When referring to Taylor polynomials, we can talk about number of terms, order or degree. This is a polynomial in 3 terms. It is a 4th order Taylor polynomial, because it was found using the 4th derivative. It is also a 4th degree polynomial, because x is raised to the 4th power. The 3rd order polynomial for is, but it is degree 2. The x 3 term drops out when using the third derivative. This is also the 2nd order polynomial. A recent AP exam required the student to know the difference between order and degree.

A Taylor Polynomial is used to approximate a function. A Taylor Series is equal to a function.

In practice, we usually use a Taylor Polynomial to approximate a function so we also want to know the error in using the approximation. The error (also known as the remainder, R n (x)) is the difference between the actual value of the function, f(x), and the polynomial of degree n, P n (x).

Taylor’s Theorem specifies that the remainder R n (x) (also called the Lagrange Error or Lagrange form of the remainder) is given by the formula: This looks a lot like the next term in the Taylor Series except that it is not evaluated at a but at some number z which is between x and a. z z

Taylor’s Theorem guarantees that there exists a value of z between a and x such that: It usually isn’t possible to find the value of but you can often find an upper bound on it. This will be called max

Example: Therefore the Lagrange error term