Sect. 9-B LAGRANGE error or Remainder This template can be used as a starter file for presenting training materials in a group setting. Sections Right-click on a slide to add sections. Sections can help to organize your slides or facilitate collaboration between multiple authors. Notes Use the Notes section for delivery notes or to provide additional details for the audience. View these notes in Presentation View during your presentation. Keep in mind the font size (important for accessibility, visibility, videotaping, and online production) Coordinated colors Pay particular attention to the graphs, charts, and text boxes. Consider that attendees will print in black and white or grayscale. Run a test print to make sure your colors work when printed in pure black and white and grayscale. Graphics, tables, and graphs Keep it simple: If possible, use consistent, non-distracting styles and colors. Label all graphs and tables.

Lagrange or Taylor Polynomial Remainder If a non-alternating series is approximated, the method for finding the error is called the Lagrange Remainder or Taylor’s Theorem Remainder. 2

Taylor’s Theorem Remainder If f has derivatives of all orders in an open interval containing c then for each positive integer n and for each x in the interval 3

Taylor’s Theorem Remainder n is the degree of the Taylor Polynomial c is where it is centered x is the value we are attempting to approximate z is the x-value between x and c which makes a maximum. 4

1. Use a fifth degree Maclaurin polynomial to approximate then find the Lagrange remainder

Types of functions Case 1: Increasing function

Types of functions Case 2: decreasing function

Types of functions Case 3: increasing and decreasing function

Types of functions Case 4: Sine and Cosine You may know the maximum value for example: (sine and cosine functions have a maximum value of 1).

2. If is a decreasing function, find the error bound when a fifth degree Taylor Polynomial centered at x = 4 is used to approximate f(4.1). (set up but do not evaluate)

Approximate using a third degree Maclaurin polynomial

b) then use the Lagrange error bound to show that

4. Selected values of f and its first 4 derivatives are given in the table. The function f and its derivatives are decreasing on the interval 0<x<4 Write a third degree Taylor Polynomial for f about x = 3 and use it to approximate f(3.1) x f(x) f’(x) f’’(x) f’’’(x) f(4)(x) 3 12 -18 -38 -67 -17

4.Continued b) Use the Lagrange error bound to show that the third degree Taylor Polynomial for f about x = 3 approximates f(3.1) with an error less than 0.00008

5. The third degree Taylor Polynomial of f about x = -2 is given by: a) Find

5.Continued b) Does h have a relative max, relative min or neither at x = -2 ?

5. continued c) The fourth derivative of f satisfies the inequality on the interval Use the Lagrange error bound to show that

6. If and if x = 0.7 is the convergence interval for the power series centered at x = 0, find an upper limit for the error when the fourth-degree Taylor polynomial is used to approximate f(0.7)

Home Work Day 1 Worksheet 9-B Day 2 Worksheet Error Use a section header for each of the topics, so there is a clear transition to the audience. Day 1 Worksheet 9-B Day 2 Worksheet Error