Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

X36 taylor's remainder theorem ii

326 views

Published on

taylor's remainder theorem ii

Published in: Education
  • Be the first to comment

  • Be the first to like this

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).

×