Taylor Polynomials – Day 2 Section 9.7 Calculus BC AP/Dual, Revised ©2018 viet.dang@humbleisd.net 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 1 Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. Solve for 𝒇 𝟒 and 𝒇′′ 𝟒 Write the second-degree Taylor polynomial for 𝒇′ about 4 and use it to approximate 𝒇 ′ 𝟒.𝟑 Write the fourth-degree Taylor polynomial for 𝒈 𝒙 = 𝟒 𝒙 𝒇 𝒕 𝒅𝒕 at 𝒙=𝟒. Can the exact value of 𝒇 𝟑 be determined from the information given? Justify your answer. 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 1a Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. Solve for 𝒇 𝟒 and 𝒇′′ 𝟒 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 1b Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. (b) Write the second-degree Taylor polynomial for 𝒇′ about 4 and use it to approximate 𝒇 ′ 𝟒.𝟑 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 1b Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. (b) Write the second-degree Taylor polynomial for 𝒇′ about 4 and use it to approximate 𝒇 ′ 𝟒.𝟑 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 1c Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. (c) Write the fourth-degree Taylor polynomial for 𝒈 𝒙 = 𝟒 𝒙 𝒇 𝒕 𝒅𝒕 at 𝒙=𝟒. 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 1c Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. (c) Write the fourth-degree Taylor polynomial for 𝒈 𝒙 = 𝟒 𝒙 𝒇 𝒕 𝒅𝒕 at 𝒙=𝟒. 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 1d Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. (d) Can the exact value of 𝒇 𝟑 be determined from the information given? Justify your answer. 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 2 Suppose that the function 𝒇 𝒙 is approximated near 𝒙=𝟎 by a third-degree Taylor Polynomial 𝑷 𝟑 𝒙 =𝟐−𝟓 𝒙 𝟐 +𝟖 𝒙 𝟑 . Find the value of 𝒇 𝟎 , 𝒇 ′ 𝟎 , 𝒇 ′′ 𝟎 , and 𝒇 ′′′ 𝟎 Does 𝒇 have a local maximum, a local minimum, or neither at 𝒙=𝟎? Justify response. 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 2 Suppose that the function 𝒇 𝒙 is approximated near 𝒙=𝟎 by a third-degree Taylor Polynomial 𝑷 𝟑 𝒙 =𝟐−𝟓 𝒙 𝟐 +𝟖 𝒙 𝟑 . Find the value of 𝒇 𝟎 , 𝒇 ′ 𝟎 , 𝒇 ′′ 𝟎 , and 𝒇 ′′′ 𝟎 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 2b Suppose that the function 𝒇 𝒙 is approximated near 𝒙=𝟎 by a third-degree Taylor Polynomial 𝑷 𝟑 𝒙 =𝟐−𝟓 𝒙 𝟐 +𝟖 𝒙 𝟑 . (b) Does 𝒇 have a local maximum, a local minimum, or neither at 𝒙=𝟎? Justify response. 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 3 Using the MacLaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 : Find the third-degree polynomial Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 3a Using the MacLaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 , Find the third degree polynomial 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 3b Using the MacLaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 , B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 3b Using the MacLaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 , B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Example 3b Using the MacLaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 , B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Your Turn Using the MacLaurin polynomial for 𝒇 𝒙 =𝐬𝐢𝐧 𝒙: Find the third-degree polynomial Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝒙 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Your Turn Using the MacLaurin polynomial for 𝒇 𝒙 =𝐬𝐢𝐧 𝒙: Find the third degree polynomial 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Your Turn Using the MacLaurin polynomial for 𝒇 𝒙 =𝐬𝐢𝐧 𝒙, B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝒙 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

AP Multiple Choice Practice Question 1 (non-calculator) The coefficient of 𝒙 𝟑 in the Taylor series for 𝒆 𝟑𝒙 about 𝒙=𝟎 is (A) 𝟏 𝟑 (B) 𝟏 𝟐 (C) 𝟑 𝟐 (D) 𝟗 𝟐 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

AP Multiple Choice Practice Question 1 (non-calculator) The coefficient of 𝒙 𝟑 in the Taylor series for 𝒆 𝟑𝒙 about 𝒙=𝟎 is Vocabulary Connections and Process Answer 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2

§9.7: Taylor Polynomials Day 2 Assignment Worksheet 2/24/2019 7:01 AM §9.7: Taylor Polynomials Day 2