Home » Mathematics » Calculus BC » Lagrange Error

Connecting...

This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of Calculus BC
Bookmark & Share Embed

Share this knowledge with your friends!

Copy & Paste this embed code into your website’s HTML

Please ensure that your website editor is in text mode when you paste the code.
(In Wordpress, the mode button is on the top right corner.)
  ×
  • - Allow users to view the embedded video in full-size.
Since this lesson is not free, only the preview will appear on your website.

  • Discussion

  • Answer Engine

  • Study Guides

  • Practice Questions

  • Download Lecture Slides

  • Table of Contents

  • Related Books

Lecture Comments (1)

0 answers

Post by Brett Pienaar on April 17, 2024

0 < c < x    methinks

Lagrange Error

  • Lagrange error: difference between nth term partial sum and compete evaluation of series
  • Lagrange error bound:
  • Finding Lagrange error:
    • Differentiate function until derivative series pattern recognized
    • Apply to Lagrange formula for error bound

Lagrange Error

Find the Lagrange Error Remainder for appromixmating cosx with a MacLaurin series, with a constant c
    Determine the MacLaurin Series to determine fn(x)cosx = ∑n = 0∞ [(( − 1)nx2n)/(2n)!] fn(x) = [(( − 1)nx2n)/(2n)!]Rn(x) = [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] = [([(( − 1)n + 1x2(n + 1)(c)(x)(n + 2))/((2(n + 1) + 1)!)])/((n + 1)!)] = [(c( − 1)n + 1xn + 2(n + 1) + 2)/((n + 1)!(2(n + 1) + 1)!)]
    Q. Find the Lagrange Error Remainder for appromixmating cosx with a MacLaurin series, with a constant c

    *These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

      Answer

      Determine the MacLaurin Series to determine fn(x)cosx = ∑n = 0∞ [(( − 1)nx2n)/(2n)!] fn(x) = [(( − 1)nx2n)/(2n)!]Rn(x) = [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] = [([(( − 1)n + 1x2(n + 1)(c)(x)(n + 2))/((2(n + 1) + 1)!)])/((n + 1)!)] = [(c( − 1)n + 1xn + 2(n + 1) + 2)/((n + 1)!(2(n + 1) + 1)!)]

      Lagrange Error

      Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

      Related Books

      Related Books

      John Zhu

      John Zhu

      Lagrange Error

      Slide Duration:

      Table of Contents

      Section 1: Prerequisites
      Parametric Curves

      10m 10s

      Intro
      0:00
      Parametric Equations
      0:23
      Familiar Functions
      0:32
      Parametric Equation/ Function
      0:56
      Example 1: Graph Parametric Equation
      1:48
      Example 2
      4:30
      Example 3
      6:01
      Example 4
      7:12
      Example 5
      8:10
      Polar Coordinates

      14m 54s

      Intro
      0:00
      Polar Coordinates
      0:09
      Definition
      0:11
      Example
      0:23
      Converting Polar Coordinates
      1:49
      Example: Convert Polar Equation to Cartesian Coordinates
      3:06
      Example: Convert Polar Equation to Cartesian Coordinates
      5:21
      Example: Find the Polar Representation
      6:51
      Example: Find the Polar Representation
      8:39
      Example: Sketch the Graph
      10:02
      Vector Functions

      12m 5s

      Intro
      0:00
      Vector Functions
      0:11
      2 Parts
      0:18
      Example 1
      1:44
      Example 2
      3:52
      Example 3
      4:59
      Example 4
      6:03
      Example 5
      8:00
      Example 6
      9:23
      Example 7
      10:16
      Section 2: Differentiation
      Parametric Differentiation

      13m 15s

      Intro
      0:00
      Parametric Differentiation
      0:15
      Example 1
      1:16
      Example 2
      1:54
      Example 3
      3:15
      Example 4
      4:59
      Parametric Differentiation: Position, Speed & Acceleration
      7:45
      Example 5
      8:32
      Example 6
      10:37
      Polar Differentiation

      10m 14s

      Intro
      0:00
      Polar Differentiation
      0:11
      Goal
      0:13
      Method
      0:24
      Example 1
      0:52
      Example 2
      4:27
      Example 3
      7:03
      L'Hopital's Rule

      6m 43s

      Intro
      0:00
      L'Hopitals Rule
      0:09
      Conditions
      0:14
      Limit
      0:36
      Example 1
      1:21
      Example 2
      2:38
      Example 3
      4:09
      Example 4
      5:01
      Section 3: Integration
      Integration By Parts

      19m 4s

      Intro
      0:00
      Integration by Parts
      0:12
      Formula
      0:31
      How It's Derived From Product Rule
      1:09
      Integration by Parts: Rules to Follow
      3:24
      Recap of the Rules
      4:13
      Example 1
      4:29
      Example 2
      7:04
      Example 3
      8:40
      Example 4
      13:48
      Example 5
      15:36
      Integration By Partial Fractions

      22m 4s

      Intro
      0:00
      Integration by Partial Fractions: Goal
      0:08
      Example 1
      2:10
      Example 2
      7:10
      Example 3
      10:06
      Example 4
      14:23
      Example 5
      16:26
      Improper Integrals

      19m 1s

      Intro
      0:00
      Improper Integrals
      0:06
      3 Steps
      1:11
      Example 1
      1:46
      Example 2
      3:20
      Example 3
      6:05
      Example 4
      9:02
      Example 5
      11:21
      Example 6
      15:54
      Section 4: Applications of Integration
      Logistic Growth

      19m 16s

      Intro
      0:00
      Logistic Growth Function
      0:07
      Defining Variables
      0:40
      Equation Parts
      1:51
      Logistic Growth
      2:04
      Example 1
      2:59
      Example 2
      7:13
      Example 3
      11:29
      Example 4
      15:21
      Arc Length for Parametric & Polar Curves

      17m 40s

      Intro
      0:00
      Arc Length
      0:13
      Arc Length of a Normal Function
      0:24
      Example
      1:27
      Example 2: Arc Length for Parametric Curves
      3:31
      Example 3: Arc Length for Parametric Curves
      4:23
      Example 4: Arc Length for Parametric Curves
      8:05
      Example 5: Arc Length for Parametric Curves
      12:22
      Example 6: Arc Length for Polar Curves
      15:36
      Example 7: Arc Length for Polar Curves
      16:03
      Area for Parametric & Polar Curves

      4m 33s

      Intro
      0:00
      Area for Parametric Curves: Parametric Function
      0:10
      Example 1: Area for Parametric Curves
      0:35
      Area for Parametric Curves: Polar Function
      2:50
      Example 2: Area for Polar Curves
      3:18
      Section 5: Sequences, Series, & Approximations
      Definition & Convergence

      7m 10s

      Intro
      0:00
      Sequences: Definition
      0:09
      Definition
      0:31
      Example 1
      1:07
      Sequences: Convergence
      2:02
      Example 1
      2:52
      Example 2
      3:36
      Example 3
      4:47
      Example 4
      6:16
      Geometric Series

      10m 59s

      Intro
      0:00
      Geometric Series
      0:07
      Definition
      0:24
      Expanded Form
      0:39
      Convergence Rules
      1:00
      Example 1: Convergence
      1:22
      Example 2: Convergence
      2:36
      Example 3: Convergence
      3:45
      Sum of Series
      5:04
      Sum of First n Terms
      5:14
      Sum of Series
      5:24
      Example 1: Sum of Series
      5:39
      Example 2: Sum of Series
      7:15
      Example 3: Sum of Series
      8:24
      Example 4: Sum of Series
      9:36
      Harmonic & P Series

      3m 58s

      Intro
      0:00
      Harmonic Series
      0:08
      P-Series
      1:17
      Example 1: P-Series Test
      2:22
      Example 2: P-Series Test
      3:06
      Integral Test

      11m 31s

      Intro
      0:00
      Integral Test
      0:09
      Example 1
      0:54
      Example 2
      4:00
      Example 3
      7:59
      Example 4
      9:47
      Comparison Test

      7m 47s

      Intro
      0:00
      Comparison Test
      0:10
      Example 1
      1:07
      Example 2
      2:33
      Example 3
      4:20
      Example 4
      6:29
      Limit Comparison Test

      9m 40s

      Intro
      0:00
      Limit Comparison Test
      0:09
      Conditions
      0:31
      Example 1
      1:01
      Example 2
      2:53
      Example 3
      4:15
      Example 4
      6:19
      Ratio Test

      11m 20s

      Intro
      0:00
      Ratio Test
      0:09
      Example 1
      0:57
      Example 2
      2:55
      Example 3
      6:27
      Example 4
      8:36
      Alternating Series

      9m 11s

      Intro
      0:00
      Alternating Series
      0:08
      Convergence Test
      0:59
      Example 1
      1:27
      Example 2
      2:42
      Example 3
      4:57
      Example 4
      6:37
      Absolute Convergence

      7m 13s

      Intro
      0:00
      Absolute Convergence
      0:12
      Example 1
      0:52
      Example 2
      3:42
      Example 3
      5:21
      Power Series Convergence

      12m 52s

      Intro
      0:00
      Power Series
      0:09
      Definition
      0:19
      Radius & Interval of Convergence
      2:07
      Example 1
      2:28
      Example 2
      4:18
      Example 3
      6:20
      Example 4
      9:11
      Taylor Series

      15m 11s

      Intro
      0:00
      Taylor Series
      0:06
      MacLaurin Series
      0:45
      Example 1
      1:02
      Example 2
      2:45
      Example 3
      6:04
      Example 4
      8:23
      Example 5
      11:49
      Power Series Operations

      16m 40s

      Intro
      0:00
      Operations
      0:07
      Example 1: Substitution
      1:05
      Example 2: Substitution
      3:41
      Example 3: Differentiation/ Integration
      5:39
      Example 4: Differentiation/ Integration
      9:55
      Example 5: Multiplying/ Dividing
      12:32
      Example 6: Multiplying/ Dividing
      14:50
      Lagrange Error

      7m 9s

      Intro
      0:00
      Power Series: Lagrange Error
      0:06
      Lagrange Remainder
      0:21
      Lagrange Error Bound
      0:50
      Example 1
      1:06
      Example 2
      3:27
      Section 6: Practice Test
      AP Calc BC Practice Test Section 1: Multi Choice Part 1

      15m 45s

      Intro
      0:00
      Practice Problem 1
      0:10
      Practice Problem 2
      1:19
      Practice Problem 3
      2:33
      Practice Problem 4
      5:25
      Practice Problem 5
      6:32
      Practice Problem 6
      9:11
      Practice Problem 7
      11:31
      Practice Problem 8
      13:08
      Practice Problem 9
      14:16
      AP Calc BC Practice Test Section 1: Multi Choice Part 2

      21m 38s

      Intro
      0:00
      Practice Problem 10
      0:12
      Practice Problem 11
      3:03
      Practice Problem 12
      4:53
      Practice Problem 13
      6:56
      Practice Problem 14
      9:10
      Practice Problem 15
      12:17
      Practice Problem 16
      14:43
      Practice Problem 17
      16:18
      Practice Problem 18
      18:34
      AP Calc BC Practice Test Section 1: Multi Choice Part 3

      17m 31s

      Intro
      0:00
      Practice Problem 19
      0:09
      Practice Problem 20
      3:03
      Practice Problem 21
      4:07
      Practice Problem 22
      5:41
      Practice Problem 23
      8:48
      Practice Problem 24
      11:16
      Practice Problem 25
      12:53
      Practice Problem 26
      14:12
      Practice Problem 27
      15:21
      AP Calc BC Practice Test Section 1: Multi Choice Part 4

      14m 17s

      Intro
      0:00
      Practice Problem 28
      0:08
      Practice Problem 29
      1:19
      Practice Problem 30
      2:15
      Practice Problem 31
      4:19
      Practice Problem 32
      6:54
      Practice Problem 33
      8:13
      Practice Problem 34
      9:03
      Practice Problem 35
      10:21
      Practice Problem 36
      11:42
      AP Calc BC Practice Test Section 1: Multi Choice Part 5

      22m 45s

      Intro
      0:00
      Practice Problem 37
      0:10
      Practice Problem 38
      4:09
      Practice Problem 39
      7:21
      Practice Problem 40
      9:42
      Practice Problem 41
      11:35
      Practice Problem 42
      14:01
      Practice Problem 43
      15:50
      Practice Problem 44
      16:43
      Practice Problem 45
      19:35
      AP Calc BC Practice Test Section 1: Free Response Part 1

      9m 55s

      Intro
      0:00
      Practice Problem 1a
      0:08
      Practice Problem 1b
      1:30
      Practice Problem 1c
      3:08
      Practice Problem 2a
      3:50
      Practice Problem 2b
      5:40
      Practice Problem 2c
      7:24
      AP Calc BC Practice Test Section 1: Free Response Part 2

      10m 8s

      Intro
      0:00
      Practice Problem 3a
      0:09
      Practice Problem 3b
      3:01
      Practice Problem 4a
      3:55
      Practice Problem 4b
      5:43
      AP Calc BC Practice Test Section 1: Free Response Part 3

      13m 58s

      Intro
      0:00
      Practice Problem 5a
      0:07
      Practice Problem 5b
      2:21
      Practice Problem 5c
      4:21
      Practice Problem 6a
      6:16
      Practice Problem 6b
      8:57
      Practice Problem 6c
      11:03
      Educator®

      Please sign in to participate in this lecture discussion.

      Resetting Your Password?
      or
      Still having trouble? Contact us
      OR

      Start Learning Now

      Our free lessons will get you started (Adobe Flash® required).
       Get immediate access to our entire library.

      Membership Overview

      • Available 24/7. Unlimited Access to Our Entire Library.
      • Search and jump to exactly what you want to learn.
      • *Ask questions and get answers from the community and our teachers!
      • Practice questions with step-by-step solutions.
      • Download lecture slides for taking notes.
      • Track your course viewing progress.
      • Accessible anytime, anywhere with our Android and iOS apps.