Calculus BC AP/Dual, Revised ©2015 9.7: Lagrange's Error Bound Section 9.7 Calculus BC AP/Dual, Revised ©2015 viet.dang@humble.k12.tx.us 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Question How can we approximate 𝐬𝐢𝐧 𝟏 without a calculator? Look at the graph and guess Compare it to 𝐜𝐨𝐬 𝝅 𝟔 = 𝟑 𝟐 Tangent line approximation Euler’s method Taylor polynomial of degree less than 1 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Equation When a Taylor polynomial, 𝑷 𝒏 𝒙 , centered at 𝒙=𝒄 is used to approximate a function, 𝒇 𝒙 , at a value 𝒙=𝒂 near the center, use the concept of a remainder as follows: If Exact Value (Function) = Polynomial Approximation + Remainder: Then, Remainder = Exact Value (Function) – Polynomial Approximation 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Equation When a Taylor polynomial, 𝑷 𝒏 𝒙 , centered at 𝒙=𝒄 is used to approximate a function, 𝒇 𝒙 , at a value 𝒙=𝒂 near the center, use the concept of a remainder as follows: Then, Remainder = Exact Value (Function) – Polynomial Approximation 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

Lagrange’s Error (known as Taylor’s Theorem) When a Taylor or Maclaurin polynomial is to approximate a function, an error will always be present Basic Formula: Error = 𝑹 𝒏 𝒙 = 𝒇 𝒙 − 𝑷 𝒏 𝒙 = 𝒇 𝒏+𝟏 𝒛 𝒏+𝟏 ! 𝒙−𝒄 𝒏+𝟏 LaGrange’s Error Bound: If a function 𝒇 is differentiable through the (𝒏+𝟏)𝒕𝒉 term in an interval that contains the center, 𝒄, then each 𝒙 in that interval exists from 𝒙,𝒄 or 𝒄,𝒙 that helps maximizes 𝒇 𝒏+𝟏 𝒇 𝒏+𝟏 𝒛 is the max value of 𝒏+𝟏 derivative *DO NOT FIND Z* (𝒛 is the 𝒙-value on the interval where the number is as large as it can be) This error bound is supposed to tell you how far off from the real number 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 1 Let 𝒇 be a function with 5 derivatives in the interval, 𝟐, 𝟑 and assume that 𝒇 𝟓 𝒙 <𝟎.𝟐 for all 𝒙 in the interval 𝟐, 𝟑 . If a fourth degree Taylor Polynomial for 𝒇 is at 𝒄=𝟐 is used to estimate 𝒇 𝟑 : How accurate is this approximation? Round to 4 decimal places. Suppose that 𝑷 𝟒 𝟑 =𝟏.𝟕𝟔𝟑. Use your answer from (a) to find an interval in which 𝒇 𝟑 must reside. Could 𝒇 𝟑 equal 𝟏.𝟕𝟔𝟖? Why or why not? Could 𝒇 𝟑 equal 𝟏.𝟕𝟔𝟒? Why or why not? 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 1a Let 𝒇 be a function with 5 derivatives in the interval, 𝟐, 𝟑 and assume that 𝒇 𝟓 𝒙 <𝟎.𝟐 for all 𝒙 in the interval 𝟐, 𝟑 . If a fourth degree Taylor Polynomial for 𝒇 is at 𝒄=𝟐 is used to estimate 𝒇 𝟑 (a) How accurate is this approximation? Round to 4 decimal places. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 1a Let 𝒇 be a function with 5 derivatives in the interval, 𝟐, 𝟑 and assume that 𝒇 𝟓 𝒙 <𝟎.𝟐 for all 𝒙 in the interval 𝟐, 𝟑 . If a fourth degree Taylor Polynomial for 𝒇 is at 𝒄=𝟐 is used to estimate 𝒇 𝟑 . (a) How accurate is this approximation? Round to 4 decimal places. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 1b Let 𝒇 be a function with 5 derivatives in the interval, 𝟐, 𝟑 and assume that 𝒇 𝟓 𝒙 <𝟎.𝟐 for all 𝒙 in the interval 𝟐, 𝟑 . If a fourth degree Taylor Polynomial for 𝒇 is at 𝒄=𝟐 is used to estimate 𝒇 𝟑 (b) Suppose that 𝑷 𝟒 𝟑 =𝟏.𝟕𝟔𝟑. Use your answer from (a) to find an interval in which 𝒇 𝟑 must reside. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 1c Let 𝒇 be a function with 5 derivatives in the interval, 𝟐, 𝟑 and assume that 𝒇 𝟓 𝒙 <𝟎.𝟐 for all 𝒙 in the interval 𝟐, 𝟑 . If a fourth degree Taylor Polynomial for 𝒇 is at 𝒄=𝟐 is used to estimate 𝒇 𝟑 (c) Could 𝒇 𝟑 equal 𝟏.𝟕𝟔𝟖? Why or why not? 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 1d Let 𝒇 be a function with 5 derivatives in the interval, 𝟐, 𝟑 and assume that 𝒇 𝟓 𝒙 <𝟎.𝟐 for all 𝒙 in the interval 𝟐, 𝟑 . If a fourth degree Taylor Polynomial for 𝒇 is at 𝒄=𝟐 is used to estimate 𝒇 𝟑 (d) Could 𝒇 𝟑 equal 𝟏.𝟕𝟔𝟒? Why or why not? 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Your Turn (calc) The function 𝒇 has derivatives of all orders for all real numbers 𝒙. Assume that 𝒇 𝟐 =𝟔, 𝒇 ′ 𝟐 =𝟒, 𝒇 ′′ 𝟐 =−𝟕, and 𝒇 ′′′ 𝟐 =𝟖. (a) Write the third-degree Taylor polynomial for 𝒇 about 𝒙=𝟐, and use it to approximate 𝒇 𝟐.𝟑 . (b) The fourth derivative of 𝒇 satisfies the inequality | 𝒇 𝟒 (𝒙)|≤𝟗 for all 𝒙 in the closed interval 𝟐, 𝟐.𝟑 . Use this information to find a bound for the error in the approximation of 𝒇 𝟐.𝟑 found in part (a) to find an interval 𝒂, 𝒃 such that 𝒂≤𝒇 𝟐.𝟑 ≤𝒃. (c) Could 𝒇 𝟐.𝟑 equal 𝟔.𝟗𝟐𝟐? Explain why or why not. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Your Turn A The function 𝒇 has derivatives of all orders for all real numbers 𝒙. Assume that 𝒇 𝟐 =𝟔, 𝒇 ′ 𝟐 =𝟒, 𝒇 ′′ 𝟐 =−𝟕, and 𝒇 ′′′ 𝟐 =𝟖. (a) Write the third-degree Taylor polynomial for 𝒇 about 𝒙=𝟐, and use it to approximate 𝒇 𝟐.𝟑 . Round to 4 decimal places. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Your Turn B The function 𝒇 has derivatives of all orders for all real numbers 𝒙. Assume that 𝒇 𝟐 =𝟔, 𝒇 ′ 𝟐 =𝟒, 𝒇 ′′ 𝟐 =−𝟕, and 𝒇 ′′′ 𝟐 =𝟖. (b) The fourth derivative of 𝒇 satisfies the inequality |𝒇 𝟒 (𝒙)|≤𝟗 for all 𝒙 in the closed interval 𝟐, 𝟐.𝟑 . Use this information to find a bound for the error in the approximation of 𝒇 𝟐.𝟑 found in part (a) to find an interval 𝒂, 𝒃 such that 𝒂≤𝒇 𝟐.𝟑 ≤𝒃. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Your Turn B The function 𝒇 has derivatives of all orders for all real numbers 𝒙. Assume that 𝒇 𝟐 =𝟔, 𝒇 ′ 𝟐 =𝟒, 𝒇 ′′ 𝟐 =−𝟕, and 𝒇 ′′′ 𝟐 =𝟖. (b) The fourth derivative of 𝒇 satisfies the inequality |𝒇 𝟒 (𝒙)|≤𝟗 for all 𝒙 in the closed interval 𝟐, 𝟐.𝟑 . Use this information to find a bound for the error in the approximation of 𝒇 𝟐.𝟑 found in part (a) to find an interval 𝒂, 𝒃 such that 𝒂≤𝒇 𝟐.𝟑 ≤𝒃. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Your Turn C The function 𝒇 has derivatives of all orders for all real numbers 𝒙. Assume that 𝒇 𝟐 =𝟔, 𝒇 ′ 𝟐 =𝟒, 𝒇 ′′ 𝟐 =−𝟕, and 𝒇 ′′′ 𝟐 =𝟖. (c) Could 𝒇 𝟐.𝟑 equal 𝟔.𝟗𝟐𝟐? Explain why or why not. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 2 (Non-Calc) Given a Maclaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 and graph: Write the fourth-degree Maclaurin polynomial for 𝒇 𝟏 and use polynomial to approximate 𝒆 when the Lagrange error bound for the maximum error is 𝒙 ≤𝟏. Use your answer from (a) to find an interval 𝒂,𝒃 such that 𝒂≤𝒆≤𝒃. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 2a Given a Maclaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 . Write the fourth-degree Maclaurin polynomial for 𝒇 𝟏 and use polynomial to approximate 𝒆 when the Lagrange error bound for the maximum error is 𝒙 ≤𝟏. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 2b Given a Maclaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 . (b) Use your answer from (a) to find an interval 𝒂,𝒃 such that 𝒂≤𝒆≤𝒃. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 2b Given a Maclaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 . (b) Use your answer from (a) to find an interval 𝒂,𝒃 such that 𝒂≤𝒆≤𝒃. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Your Turn Given 𝒇 𝒙 =𝐬𝐢𝐧⁡𝒙 Find the Maclaurin polynomial of the degree of 𝒏=𝟓. Then, approximate 𝐬𝐢𝐧⁡𝟏. Use Taylor’s Theorem to find the maximum error for the approximation. Give three decimal places. (Use 𝟎, 𝟏 ) Find an interval of 𝒂, 𝒃 such that 𝒂≤𝐬𝐢𝐧⁡𝟏≤𝒃 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Your Turn A Given 𝒇 𝒙 =𝐬𝐢𝐧⁡𝒙 Find the Maclaurin polynomial of the degree of 𝒏=𝟓. Then, approximate 𝐬𝐢𝐧⁡𝟏. 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Your Turn B Given 𝒇 𝒙 = 𝐬𝐢𝐧 𝒙 (B) Use Taylor’s Theorem to find the maximum error for the approximation. Give three decimal places. (Use 𝟎, 𝟏 ) 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Your Turn B Given 𝒇 𝒙 = 𝐬𝐢𝐧 𝒙 (B) Use Taylor’s Theorem to find the maximum error for the approximation. Give three decimal places. (Use 𝟎, 𝟏 ) 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Your Turn C Given 𝒇 𝒙 = 𝐬𝐢𝐧 𝒙 (C) Find an interval of 𝒂, 𝒃 such that 𝒂≤𝐬𝐢𝐧⁡𝟏≤𝒃 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 3 Let 𝒇 be the function given by 𝒇 𝒙 =𝐬𝐢𝐧 𝟓𝒙+ 𝝅 𝟑 and let 𝑷 𝒙 be the third degree Taylor Polynomial for 𝒇 about 𝒙=𝟎 Find 𝑷 𝒙 Use Lagrange’s Error bound to show that 𝒇 𝟏 𝟏𝟓 −𝑷 𝟏 𝟏𝟓 < 𝟏 𝟏𝟐𝟎𝟎 . 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 3a Let 𝒇 be the function given by 𝒇 𝒙 =𝐬𝐢𝐧 𝟓𝒙+ 𝝅 𝟑 and let 𝑷 𝒙 be the third degree Taylor Polynomial for 𝒇 about 𝒙=𝟎 Find 𝑷 𝒙 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Example 3b Let 𝒇 be the function given by 𝒇 𝒙 =𝐬𝐢𝐧 𝟓𝒙+ 𝝅 𝟑 and let 𝑷 𝒙 be the third degree Taylor Polynomial for 𝒇 about 𝒙=𝟎 (b) Use Lagrange’s Error bound to show that 𝒇 𝟏 𝟏𝟓 −𝑷 𝟏 𝟏𝟓 < 𝟏 𝟏𝟐𝟎𝟎 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound

9.7: Lagrange's Error Bound Assignment Worksheet 11/28/2018 11:06 PM 9.7: Lagrange's Error Bound