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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
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Β§9.7: Introduction to Maclaurin Polynomials
In the Calculator: On your calculator graph: π π = π¬π’π§ π π π =πβ π π π! + π π π! β π π π! πΏ πππ =βππ, πΏ πππ =ππ π πππ =βπ, π πππ =π What do you notice? What is π π π ? π π π ? WAIT! There is more. π π =πβ π π π! + π π π! β π π π! + π π π! β π ππ ππ! + π ππ ππ! 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Brook Taylor and Colin Maclaurin
From Wikipedia 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Brook Taylor and Colin Maclaurin
Discovered in early 1700βs by Colin Maclaurin (who was Scottish) as a way to estimate the graph without the use of a calculator such as (cosine 2), where π=π Graphs were polynomials and not necessarily with Eulerβs Method, where it is done by several graphs Brook Taylor (from Middlesex, England SE London) took Maclaurin Polynomial and re-center π to another point besides zero Taylor also discovered Integration by parts 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Definition of Nth degree Maclaurin Polynomial
If π has π derivatives at π=π, then the polynomial: πth-degree Maclaurin polynomial for π at π=π is the equation: π· π π =π π + π β² π πβπ + π β²β² π π! π π + π β²β²β² π π! π π +β¦ π (π) π π! π π In a Maclaurin polynomial, the polynomial starts at zero (specific case of Taylorβs). For Taylor and Maclaurin polynomials, the approximate symbol β is used. 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Β§9.7: Introduction to Maclaurin Polynomials
Example 1 Find the Maclaurin polynomial of degree π=π for π π = π ππ . π π π ππ π π Maclaurin Poly. π· π 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Β§9.7: Introduction to Maclaurin Polynomials
Example 1 Find the Maclaurin polynomial of degree π=π for π π = π ππ . 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Β§9.7: Introduction to Maclaurin Polynomials
Example 2 Find the Maclaurin polynomial of degree π=π for π π = ππ¨π¬ π. 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Β§9.7: Introduction to Maclaurin Polynomials
Example 2 Find the Maclaurin polynomial of degree π=π for π π = ππ¨π¬ π. 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Β§9.7: Introduction to Maclaurin Polynomials
Your Turn Find the Maclaurin polynomial of degree π=π for π π =π¬π’π§β‘π. 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Β§9.7: Introduction to Maclaurin Polynomials
Example 3 Find the Taylor polynomial of degree π=π for π π = π π at π=π 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Β§9.7: Introduction to Maclaurin Polynomials
Example 3 Find the Taylor polynomial of degree π=π for π π = π π at π=π 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Β§9.7: Introduction to Maclaurin Polynomials
Example 4 If the first five terms of the Taylor expansion for π π about π=π are πβππ+ π π π π + π π π π βπ π π , then π β²β²β² π = 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Β§9.7: Introduction to Maclaurin Polynomials
Example 4 If the first five terms of the Taylor expansion for π π about π=π are πβππ+ π π π π + π π π π βπ π π , then π β²β²β² π = 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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AP Multiple Choice Practice Question 1 (non-calculator)
Let π· π =ππ π βπ π π +π π π +π π π be the fifth-degree Maclaurin Polynomial for the function π about π=π. What is the value of π β²β²β² π ? (A) βππ (B) βππ (C) βπ (D) β π π 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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AP Multiple Choice Practice Question 1 (non-calculator)
Let π· π =ππ π βπ π π +π π π +π π π be the fifth-degree Maclaurin Polynomial for the function π about π=π. What is the value of π β²β²β² π ? Vocabulary Connections and Process Answer 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Β§9.7: Introduction to Maclaurin Polynomials
Rules to Memorize Determine the Maclaurin Polynomials (where π=π) for these seven functions. π¬π’π§ π β ππ¨π¬ π β π π β π₯π§ π β 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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Β§9.7: Introduction to Maclaurin Polynomials
Rules to Memorize Determine the Maclaurin Polynomials (where π=π) for these six functions. E. π π β F. π π+π β G. ππ«ππππ§ πβ 2/1/2019 4:25 AM Β§9.7: Introduction to Maclaurin Polynomials
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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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