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# X36 taylor's remainder theorem ii

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### X36 taylor's remainder theorem ii

1. 1. Taylor's Remainder Theorem II
2. 2. Taylor's Remainder Theorem II We state the general form of the Taylor's remainder formula.
3. 3. Taylor's Remainder Theorem II We state the general form of the Taylor's remainder formula. Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b]
4. 4. Taylor's Remainder Theorem II We state the general form of the Taylor's remainder formula. Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] a ( )[ ] b f(x) is infinitely differentiable in here
5. 5. Taylor's Remainder Theorem II We state the general form of the Taylor's remainder formula. Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and pn(x) = be the n'th Taylor-poly expanded around at a. a ( )[ ] b f(x) is infinitely differentiable in here Σk=0 n (x – a)k k! f(k)(a)
6. 6. Taylor's Remainder Theorem II We state the general form of the Taylor's remainder formula. Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and pn(x) = be the n'th Taylor-poly expanded around at a. Then there exists a "c" between a and b such that f(b) = pn(b) + (b – a)n+1 (n + 1)! f(n+1)(c) a ( )[ ] bc f(x) is infinitely differentiable in here Σk=0 n (x – a)k k! f(k)(a)
7. 7. or in full detail, f '(a)(b – a) f(2)(a) + 2!= f(a) + (b – a)2 f(b) .. f(n)(a) n! (b – a)n+ Taylor's Remainder Theorem II + Rn(b)
8. 8. or in full detail, where Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) is the Lagrange form of the Taylor-remainder. f '(a)(b – a) f(2)(a) + 2!= f(a) + (b – a)2 f(b) .. f(n)(a) n! (b – a)n+ Taylor's Remainder Theorem II + Rn(b)
9. 9. or in full detail, where Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) is the Lagrange form of the Taylor-remainder. f '(a)(b – a) f(2)(a) + 2!= f(a) + (b – a)2 f(b) .. f(n)(a) n! (b – a)n+ Again, we note the following Taylor's Remainder Theorem II + Rn(b)
10. 10. or in full detail, where Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) is the Lagrange form of the Taylor-remainder. f '(a)(b – a) f(2)(a) + 2!= f(a) + (b – a)2 f(b) .. f(n)(a) n! (b – a)n+ Again, we note the following * the theorem also works for the interval [b, a] Taylor's Remainder Theorem II + Rn(b)
11. 11. or in full detail, where Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) is the Lagrange form of the Taylor-remainder. f '(a)(b – a) f(2)(a) + 2!= f(a) + (b – a)2 f(b) .. f(n)(a) n! (b – a)n+ Again, we note the following * the theorem also works for the interval [b, a] * the value c changes if the value of b or n changes Taylor's Remainder Theorem II + Rn(b)
12. 12. or in full detail, where Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) is the Lagrange form of the Taylor-remainder. f '(a)(b – a) f(2)(a) + 2!= f(a) + (b – a)2 f(b) .. f(n)(a) n! (b – a)n+ Again, we note the following * the theorem also works for the interval [b, a] * the value c can't be easily determined, we just know there is at least one c that fits the description * the value c changes if the value of b or n changes Taylor's Remainder Theorem II + Rn(b)
13. 13. Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π 2 Taylor's Remainder Theorem II
14. 14. Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π 2 sin(x) cos(x) -sin(x) -cos(x) Taylor's Remainder Theorem II
15. 15. Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π 2 sin(x) cos(x) -sin(x) -cos(x) At x = π 2 , we get the sequence of coefficients 1, 0, -1, 0, 1, 0, -1, … Taylor's Remainder Theorem II
16. 16. Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π 2 sin(x) cos(x) -sin(x) -cos(x) At x = π 2 , we get the sequence of coefficients 1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is = 1P(x) Taylor's Remainder Theorem II
17. 17. Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π 2 sin(x) cos(x) -sin(x) -cos(x) At x = π 2 , we get the sequence of coefficients 1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is –= 1P(x) (x – π/2)2 2! Taylor's Remainder Theorem II
18. 18. Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π 2 sin(x) cos(x) -sin(x) -cos(x) At x = π 2 , we get the sequence of coefficients 1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is –= 1P(x) (x – π/2)2 2! 1(x – π/2)4 + 4! Taylor's Remainder Theorem II
19. 19. Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π 2 sin(x) cos(x) -sin(x) -cos(x) At x = π 2 , we get the sequence of coefficients 1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is –= 1P(x) (x – π/2)2 2! 1(x – π/2)4 + 4! – 1(x – π/2)6 6! .. 1(x – π/2)8 + 8! Taylor's Remainder Theorem II
20. 20. Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π 2 sin(x) cos(x) -sin(x) -cos(x) At x = π 2 , we get the sequence of coefficients 1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is –= 1P(x) (x – π/2)2 2! 1(x – π/2)4 + 4! – 1(x – π/2)6 6! .. or P(x) = Σ (-1)n(x – π/2)2n (2n)!n=0 n =∞ 1(x – π/2)8 + 8! Taylor's Remainder Theorem II
21. 21. Example: B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )π/2 Taylor's Remainder Theorem II
22. 22. Example: B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = ) The remainder Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) π/2 Taylor's Remainder Theorem II
23. 23. Example: B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = ) The remainder Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) = π/2 (b – )n+1 (n + 1)! f(n+1)(c) π 2 Taylor's Remainder Theorem II
24. 24. Example: B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = ) The remainder Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) = π/2 (b – )n+1 (n + 1)! f(n+1)(c) π 2 where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b. Taylor's Remainder Theorem II
25. 25. Example: B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = ) The remainder Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) = π/2 (b – )n+1 (n + 1)! f(n+1)(c) π 2 where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b. Since |f(n+1)(c)| < 1, Taylor's Remainder Theorem II
26. 26. Example: B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = ) The remainder Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) = π/2 (b – )n+1 (n + 1)! f(n+1)(c) π 2 where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b. Since |f(n+1)(c)| < 1, we've (b – )n+1 (n + 1)! f(n+1)(c) π 2 < (n + 1)! (b – )n+1π 2 Taylor's Remainder Theorem II
27. 27. Example: B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = ) The remainder Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) = π/2 (b – )n+1 (n + 1)! f(n+1)(c) π 2 where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b. Since |f(n+1)(c)| < 1, we've (b – )n+1 (n + 1)! f(n+1)(c) π 2 < (n + 1)! (b – )n+1π 2 Again, as n  we've (n + 1)! (b – )n+1π 2  0∞, Taylor's Remainder Theorem II
28. 28. Example: B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = ) The remainder Rn(b) = (b – a)n+1 (n + 1)! f(n+1)(c) = π/2 (b – )n+1 (n + 1)! f(n+1)(c) π 2 where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b. Since |f(n+1)(c)| < 1, we've (b – )n+1 (n + 1)! f(n+1)(c) π 2 < (n + 1)! (b – )n+1π 2 Again, as n  we've (n + 1)! (b – )n+1π 2  0∞, Hence Rn(b)  0 and that f(b) = P(b) for all b. Taylor's Remainder Theorem II
29. 29. Differentiation and Integration of Power Series The Taylor series of f(x) is the only power series that could be the same as f(x).
30. 30. Σk=0 ∞ Differentiation and Integration of Power Series Let f(x) = ck(x – a)k for all x in an open interval (a – R, a + R) for some R, then the series ck(x – a)k is the Taylor series P(x) of f(x). Σk=0 ∞ Theorem (Uniqueness theorem for Taylor series) : The Taylor series of f(x) is the only power series that could be the same as f(x).