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

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.

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

Copyright © Cengage Learning. All rights reserved.

9.10 Taylor and Maclaurin Series Colin Maclaurin

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.

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.

Infinite Series Copyright © Cengage Learning. All rights reserved.

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.

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.

Sect. 9-B LAGRANGE error or Remainder

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

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

MTH 253 Calculus (Other Topics) Chapter 11 – Infinite Sequences and Series Section 11.8 –Taylor and Maclaurin Series Copyright © 2009 by Ron Wallace, all.

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.

Limits Involving Infinity

Copyright © Cengage Learning. All rights reserved.

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

Algebra II Explorations Review ( )

Calculus II (MAT 146) Dr. Day Monday December 4, 2017

6.5/6.8 Analyze Graphs of Polynomial Functions

Section 11.3 – Power Series.

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.

The LaGrange Error Estimate

Section 8: Alternating Series

4.2 Properties of Polynomial Graphs

Polar Area Day 2 Section 10.5A Calculus BC AP/Dual, Revised ©2018

Taylor Polynomials & Approximation (9.7)

Calculus II (MAT 146) Dr. Day Wednesday May 2, 2018

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

Polynomial Approximations of Elementary Functions

Calculus II (MAT 146) Dr. Day Monday April 30, 2018

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

Taylor and MacLaurin Series

Clicker Question 1 What is the interval of convergence of A. (-, )

Section 11.4 – Representing Functions with Power Series

Section 11.3 Power Series.

Taylor and Maclaurin Series

Clicker Question 1 What is the interval of convergence of A. (-, )

Calculus II (MAT 146) Dr. Day Monday April 30, 2018

BC Calculus Number 6.

FP2 (MEI) Maclaurin Series

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

Taylor Series and Maclaurin Series

11.1 – Polynomial Approximations of Functions

Section 11.6 – Taylor’s Formula with Remainder

Introduction to Maclaurin Polynomials

11.10 – Taylor and Maclaurin Series

Section 2: Intro to Taylor Polynomials

Section 2.3 Calculus AP/Dual, Revised ©2017

Taylor Polynomials – Day 2

Polar Area Day 3 Section 10.5B Calculus BC AP/Dual, Revised ©2018

Section 4.2A Calculus AP/Dual, Revised ©2018

Second Fundamental Theorem of Calculus Day 2

9.7 Taylor Polynomials & Approximations

Calculus BC AP/Dual, Revised © : Hyperbolic Functions

9.2 Taylor Series.

Taylor and Maclaurin Series

Copyright © Cengage Learning. All rights reserved.

Lagrange Remainder.

Section 8.9: Applications of Taylor Polynomials

Presentation transcript:

Taylor Series – Day 2 Section 9.6 Calculus BC AP/Dual, Revised ©2014 viet.dang@humble.k12.tx.us 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

Definition of Nth degree Taylor Polynomial In a Maclaurin polynomial, it starts at zero. In a Taylor polynomial, it starts at one. 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations What Do You Notice? y1 = sin x y2 = 𝒙− 𝒙 𝟑 𝟑! + 𝒙 𝟓 𝟓! 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations Link http://www2.norwich.edu/frey/TaylorPolynomials/ 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations Example 1 Find the Maclaurin polynomial of degree n = 6 for f(x) = cos x. Then, use P6(x) to approximate the value of cos (0.1). 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations Example 1 Find the Maclaurin polynomial of degree n = 6 for f(x) = cos x. Then, use P6(x) to approximate the value of cos (0.1). 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations Example 2 Find the Taylor polynomial of degree n = 6 for f(x) = ex at c = 0 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations Example 3 Suppose that g is a function which as continuous derivatives and that g(2) = 3, g’(2) = –4, g’’(2) = 7, and g’’’(2) = –5 with a degree of 3 and for g to be centered at 2. 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations Your Turn Find the Taylor polynomial of degree n = 6 for f(x) = ln x at c = 1 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations Example 4a Using the MacLaurin polynomial for f(x) = ex, A) Find the third degree and B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations Example 4b Using the MacLaurin polynomial for f(x) = ex, A) Find the third degree and B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations Example 5a Suppose that the function f(x) is approximated near x = 0 by a third-degree Taylor Polynomial P3(x) = 2 – 5x2 + 8x3. A) Find the value of f(0), f ’(0), f ’’(0), and f ’’’(0) and B) does f have a local maximum, a local minimum, or neither at x = 0? Justify response. 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations Example 5b Suppose that the function f(x) is approximated near x = 0 by a third-degree Taylor Polynomial P3(x) = 2 – 5x2 + 8x3. A) Find the value of f(0), f ’(0), f ’’(0), and f ’’’(0) and B) does f have a local maximum, a local minimum, or neither at x = 0? Justify response. 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations

9.7 - Taylor Polynomials and Approximations Assignment Worksheet 12/4/2018 10:47 AM 9.7 - Taylor Polynomials and Approximations