Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) episode 2 part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 468 CHAPTER 8 Approximation Theory The polynomial of degree two or less that best uniformly approximates P3 x on 1 1 is p2 x P3 x - lf3 z J ELL J_ x _ _ nx2 . 13 1 3 _ 4 121 . ậ 11 2 ị 192 24 Ồ 6 4 192 8 24A However 1 3 -f2 x ịf x l l 2 i I which when added to the already accumulated error bound of exceeds the tolerance of . Consequently the polynomial of least degree that best approximates ex - Ẩ on 1 1 with an error bound of less than is P3W 192 x 24 6x3- -J Table lists the function and the approximating polynomials at various points in I 1 1 . Note that the tabulated entries for p2 are well within the tolerance of even ị though the error bound for p2 x exceeded the tolerance. -ị X ex Pi x P x p2 x EXERCISE SET 1. Use the zeros of T3 to construct an interpolating polynomial of degree two for the following functions on the interval 1 1 a. f x ex b. x sinx c. x ln x 2 d. fix - X4 2. Find a bound for the maximum error of the approximation in Exercise 1 on the interval . -1 1 . 3. Use the zeros of f4 to construct an interpolating polynomial of degree three for the functions in Exercise 1. 4. Repeat Exercise 2 for the approximations computed in Exercise 3. 5. Use the zeros of T3 and transformations of the given interval to construct an interpolating polynomial of degree two for the following functions a. 1 3 b. 0 2 c. x 2 COS X 4 I sin 2x 0 1 d. x xlnx 1 3 Rational Functio n Approxim a tio n 469 6. Use the zeros of Tị to consrruct an interpolating polynomial of degree three for the functions in Exercise 5. 7. Find the sixth Maclaurin polynomial for xe and use Chebyshev economization to obtain a lesser-degree polynomial approximation while keeping the error less than on 1 1 . 8. Find the sixth Maclaurin polynomial for sin X and use Chebyshev
