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Introduction to Maclaurin Polynomials

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1 Introduction to Maclaurin Polynomials
Section 9.7 Calculus BC AP/Dual, Revised Β©2018 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

2 Β§9.7: Introduction to Maclaurin Polynomials
In the Calculator: On your calculator graph: 𝒀 𝟏 = 𝐬𝐒𝐧 𝒙 𝒀 𝟐 =π’™βˆ’ 𝒙 πŸ‘ πŸ‘! + 𝒙 πŸ“ πŸ“! βˆ’ 𝒙 πŸ• πŸ•! 𝑿 π’Žπ’Šπ’ =βˆ’πŸπŸŽ, 𝑿 π’Žπ’‚π’™ =𝟏𝟎 𝒀 π’Žπ’Šπ’ =βˆ’πŸ“, 𝒀 π’Žπ’‚π’™ =πŸ“ What do you notice? What is 𝒀 𝟏 𝟎 ? 𝒀 𝟐 𝟎 ? WAIT! There is more. 𝒀 𝟐 =π’™βˆ’ 𝒙 πŸ‘ πŸ‘! + 𝒙 πŸ“ πŸ“! βˆ’ 𝒙 πŸ• πŸ•! + 𝒙 πŸ— πŸ—! βˆ’ 𝒙 𝟏𝟏 𝟏𝟏! + 𝒙 πŸπŸ‘ πŸπŸ‘! 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

3 Brook Taylor and Colin Maclaurin
From Wikipedia 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

4 Brook Taylor and Colin Maclaurin
Discovered in early 1700’s by Colin Maclaurin (who was Scottish) as a way to estimate the graph without the use of a calculator such as (cosine 2), where 𝒄=𝟎 Graphs were polynomials and not necessarily with Euler’s Method, where it is done by several graphs Brook Taylor (from Middlesex, England SE London) took Maclaurin Polynomial and re-center 𝒄 to another point besides zero Taylor also discovered Integration by parts 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

5 Definition of Nth degree Maclaurin Polynomial
If 𝒇 has 𝒏 derivatives at 𝒙=𝒄, then the polynomial: 𝒏th-degree Maclaurin polynomial for 𝒇 at 𝒄=𝟎 is the equation: 𝑷 𝒏 𝒙 =𝒇 𝟎 + 𝒇 β€² 𝟎 π’™βˆ’πŸŽ + 𝒇 β€²β€² 𝟎 𝟐! 𝒙 𝟐 + 𝒇 β€²β€²β€² 𝟎 πŸ‘! 𝒙 πŸ‘ +… 𝒇 (𝒏) 𝟎 𝒏! 𝒙 𝒏 In a Maclaurin polynomial, the polynomial starts at zero (specific case of Taylor’s). For Taylor and Maclaurin polynomials, the approximate symbol β‰ˆ is used. 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

6 Β§9.7: Introduction to Maclaurin Polynomials
Example 1 Find the Maclaurin polynomial of degree 𝒏=πŸ’ for 𝒇 𝒙 = 𝒆 πŸ’π’™ . 𝒇 𝒙 𝒆 πŸ’π’™ 𝒇 𝟎 Maclaurin Poly. 𝑷 𝒙 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

7 Β§9.7: Introduction to Maclaurin Polynomials
Example 1 Find the Maclaurin polynomial of degree 𝒏=πŸ’ for 𝒇 𝒙 = 𝒆 πŸ’π’™ . 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

8 Β§9.7: Introduction to Maclaurin Polynomials
Example 2 Find the Maclaurin polynomial of degree 𝒏=πŸ” for 𝒇 𝒙 = 𝐜𝐨𝐬 𝒙. 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

9 Β§9.7: Introduction to Maclaurin Polynomials
Example 2 Find the Maclaurin polynomial of degree 𝒏=πŸ” for 𝒇 𝒙 = 𝐜𝐨𝐬 𝒙. 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

10 Β§9.7: Introduction to Maclaurin Polynomials
Your Turn Find the Maclaurin polynomial of degree 𝒏=πŸ“ for 𝒇 𝒙 =𝐬𝐒𝐧⁑𝒙. 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

11 Β§9.7: Introduction to Maclaurin Polynomials
Example 3 Find the Taylor polynomial of degree 𝒏=πŸ” for 𝒇 𝒙 = 𝒆 𝒙 at 𝒄=𝟎 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

12 Β§9.7: Introduction to Maclaurin Polynomials
Example 3 Find the Taylor polynomial of degree 𝒏=πŸ” for 𝒇 𝒙 = 𝒆 𝒙 at 𝒄=𝟎 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

13 Β§9.7: Introduction to Maclaurin Polynomials
Example 4 If the first five terms of the Taylor expansion for 𝒇 𝒙 about 𝒙=𝟎 are πŸ‘βˆ’πŸ•π’™+ πŸ“ 𝟐 𝒙 𝟐 + πŸ‘ πŸ’ 𝒙 πŸ‘ βˆ’πŸ” 𝒙 πŸ’ , then 𝒇 β€²β€²β€² 𝟎 = 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

14 Β§9.7: Introduction to Maclaurin Polynomials
Example 4 If the first five terms of the Taylor expansion for 𝒇 𝒙 about 𝒙=𝟎 are πŸ‘βˆ’πŸ•π’™+ πŸ“ 𝟐 𝒙 𝟐 + πŸ‘ πŸ’ 𝒙 πŸ‘ βˆ’πŸ” 𝒙 πŸ’ , then 𝒇 β€²β€²β€² 𝟎 = 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

15 AP Multiple Choice Practice Question 1 (non-calculator)
Let 𝑷 𝒙 =πŸ‘π’™ 𝟐 βˆ’πŸ“ 𝒙 πŸ‘ +πŸ• 𝒙 πŸ’ +πŸ‘ 𝒙 πŸ“ be the fifth-degree Maclaurin Polynomial for the function 𝒇 about 𝒙=𝟎. What is the value of 𝒇 β€²β€²β€² 𝟎 ? (A) βˆ’πŸ‘πŸŽ (B) βˆ’πŸπŸ“ (C) βˆ’πŸ“ (D) βˆ’ πŸ“ πŸ” 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

16 AP Multiple Choice Practice Question 1 (non-calculator)
Let 𝑷 𝒙 =πŸ‘π’™ 𝟐 βˆ’πŸ“ 𝒙 πŸ‘ +πŸ• 𝒙 πŸ’ +πŸ‘ 𝒙 πŸ“ be the fifth-degree Maclaurin Polynomial for the function 𝒇 about 𝒙=𝟎. What is the value of 𝒇 β€²β€²β€² 𝟎 ? Vocabulary Connections and Process Answer 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

17 Β§9.7: Introduction to Maclaurin Polynomials
Rules to Memorize Determine the Maclaurin Polynomials (where 𝒄=𝟎) for these seven functions. 𝐬𝐒𝐧 𝒙 β‰ˆ 𝐜𝐨𝐬 𝒙 β‰ˆ 𝒆 𝒙 β‰ˆ π₯𝐧 𝒙 β‰ˆ 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

18 Β§9.7: Introduction to Maclaurin Polynomials
Rules to Memorize Determine the Maclaurin Polynomials (where 𝒄=𝟎) for these six functions. E. 𝟏 𝒙 β‰ˆ F. 𝟏 𝒙+𝟏 β‰ˆ G. 𝐚𝐫𝐜𝐭𝐚𝐧 π’™β‰ˆ 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials

19 Β§9.7: Introduction to Maclaurin Polynomials
Assignment Worksheet 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials


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