1. Calculus BC AP/Dual, Revised ©2015 9.7: Lagrange's Error Bound
Section 9.7
Calculus BC AP/Dual, Revised ©2015
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9.7: Lagrange's Error Bound

2. 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
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9.7: Lagrange's Error Bound

3. 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
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9.7: Lagrange's Error Bound

4. 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
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9.7: Lagrange's Error Bound

5. 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
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9.7: Lagrange's Error Bound

6. 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?
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9.7: Lagrange's Error Bound

7. 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.
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9.7: Lagrange's Error Bound

8. 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.
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9.7: Lagrange's Error Bound

9. 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.
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9.7: Lagrange's Error Bound

10. 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?
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9.7: Lagrange's Error Bound

11. 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?
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9.7: Lagrange's Error Bound

12. 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.
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9.7: Lagrange's Error Bound

13. 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.
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9.7: Lagrange's Error Bound

14. 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.
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9.7: Lagrange's Error Bound

15. 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.
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9.7: Lagrange's Error Bound

16. 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.
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9.7: Lagrange's Error Bound

17. 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.
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9.7: Lagrange's Error Bound

18. 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.
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9.7: Lagrange's Error Bound

19. 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.
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9.7: Lagrange's Error Bound

20. 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.
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9.7: Lagrange's Error Bound

21. 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
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9.7: Lagrange's Error Bound

22. 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.
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9.7: Lagrange's Error Bound

23. 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 )
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9.7: Lagrange's Error Bound

24. 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 )
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9.7: Lagrange's Error Bound

25. 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
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9.7: Lagrange's Error Bound

26. 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 .
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9.7: Lagrange's Error Bound

27. 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
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9.7: Lagrange's Error Bound

28. 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
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9.7: Lagrange's Error Bound

29. 9.7: Lagrange's Error Bound
Assignment
Worksheet
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9.7: Lagrange's Error Bound