all numerical analysis problems

Numerical analysis I, Spring 2024 01:640:373 1 February 21, 2024 Homework assignment 6 Problems 2,4 (4.4) from the textbook[10 points] Use the Composite Trapezoidal rule and Composite Simpson’s rule with the indicated values n to approximate the following integrals. R 0.5 (a) −0.5 (cos(x))2 dx, n = 4, (b) R 0.5 2 Problem 12 (4.4) from the textbook[10 points] −0.5 x log(x + 1) dx, n = 6. Determine the values of n and h required to approximate Z π x2 cos(x) dx 0 to within 10−4 . Use Composite Trapezoidal rule and Composite Simpson’s rule. 3 Problem 2 (d) (4.5) from the textbook[10 points] Use the Romberg integration to compute R3,3 for the integral Z 2e e 4 1 dx. x log(x) Problem 6 (4.5) from the textbook[10 points] R1 Use Romberg integration to approximate the integral −1 (cos(x))2 dx to within 10−6 . Compute the Romberg table until either |Rn−1,n−1 − Rn,n | < 10−6 or n = 10. Compare the result with the actual value of the integral. 5 Problem 10 (4.5) from the textbook[10 points] Romberg integration is used to approximate Z 1 0 x2 dx. x3 + 1 If R1,1 = 0.25 and R2,2 = 0.2315 what is R2,1 ? 1 6 Sample exam question[10 points] You are given the following table of the values of some unknown function f . x 3.2 3.3 3.4 3.5 3.6 3.7 f (x) 0.2 0.4 ? 0.9 ? 1.7 (a) Use Trapezoidal Rule to approximate the value of Z 3.3 xf (x) dx. 3.2 (b) One applied Three–Points Midpoint Rule with h = 0.1 to approximate f ′ (3.3). The result was 1.0. Find f (3.4). R 3.7 (c) One applied Simpson’s Rule to approximate 3.5 f (x) dx. The result was 0.5. Find f (3.6). 7 Programming exercise[15 points] Write the program which compute the approximate value of the length of the curve given by the parametric equation γ(t) = (x(t), y(t)), t ∈ [a, b]. Suppose that the functions x, y, and a, b are given. Apply your program to find the approximate the length of: (a) segment γ(t) = (t, t), t ∈ [0, 1], (b) parabola γ(t) = (t, t2 ), t ∈ [0, 1], (c) circle γ(t) = (sin(t), cos(t)), t ∈ [0, 2π], (d) cycloid γ(t) = (1 − sin(t), 1 − cos(t)), t ∈ [0, 2π]. 2