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.

Slides:

Advertisements

Similar presentations

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

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.

Power Series as Functions. Defining our Function In general, we can say that That is, if the series converges at x = a, then is a number and we can define.

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

9.3 Taylor’s Theorem: Error Analysis for Series

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.

4.5: Linear Approximations, Differentials and Newton’s Method Greg Kelly, Hanford High School, Richland, Washington.

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.

9.3 Taylor’s Theorem: Error Analysis for Series Tacoma Narrows Bridge: November 7, 1940 Greg Kelly, Hanford High School, Richland, Washington.

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.

3.3 Differentiation Rules Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

4.5: Linear Approximations, Differentials and Newton’s Method Greg Kelly, Hanford High School, Richland, Washington.

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.

Section 11.3 – Power Series.

Differentiation Rules

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.

9.2: 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.

Finding Common Maclaurin Series

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

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

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?

3.3 Differentiation Rules

6.2 Integration by Substitution M.L.King Jr. Birthplace, Atlanta, GA

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:

Applications of Taylor Series

3.3 Differentiation Rules

Rules for Differentiation

Section 2: Intro to Taylor Polynomials

how to represent certain types of functions as a power series

Taylor Polynomials – Day 2

4.8: Linear Approximations, Differentials and Newton’s Method

9.2 day 2 Maclaurin Series Liberty Bell, Philadelphia, PA

9.3 Taylor’s Theorem: Error Analysis for Series

4.5: Linear Approximations, Differentials and Newton’s Method

3.3 Rules for Differentiation

2.4 The Chain Rule (Part 2) Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002.

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.

9.2: 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.

U.S.S. Alabama 2.4 Chain Rule Mobile, Alabama

4.5 (part 1) Integration by Substitution

9.2 day 2 Maclaurin Series Liberty Bell, Philadelphia, PA

3.3 Differentiation Rules

Copyright © Cengage Learning. All rights reserved.

3.3 Differentiation Rules

Section 8.9: Applications of Taylor Polynomials

TAYLOR SERIES Maclaurin series ( center is 0 )

Finding Common Maclaurin Series

Presentation transcript:

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 of ideas regarding infinite series. Brook Taylor 1685 - 1731 Greg Kelly, Hanford High School, Richland, Washington

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

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

Our polynomial: has the form: This pattern occurs no matter what the original function was! or:

Maclaurin Series: (generated by f at ) 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 )

example:

The more terms we add, the better our approximation. Hint: On the TI-89, the factorial symbol is:

example: Rather than start from scratch, we can use the function that we already know:

example:

There are some Maclaurin series that occur often enough that they should be memorized. They are on your formula sheet.

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 x3 term drops out when using the third derivative. A recent AP exam required the student to know the difference between order and degree. This is also the 2nd order polynomial.

p The TI-89 finds Taylor Polynomials: taylor (expression, variable, order, [point]) F3 9 taylor taylor taylor p

p. 658 13 -29 odd A mathematician organizes a raffle in which the prize is an infinite amount of money paid over an infinite amount of time. Of course, with the promise of such a prize, his tickets sell like hot cake. When the winning ticket is drawn, and the jubilant winner comes to claim his prize, the mathematician explains the mode of payment: "1 dollar now, 1/2 dollar next week, 1/3 dollar the week after that..."