Question

1. Denote I = ∫¹₀  (1/(1+4x²)) dx

a. Find the exact value of I (for example, by finding an antiderivative of the integrand).

b. For a generic positive integer n, we partition the interval [0, 1] into n equal subintervals [x₀, x₁], [x₁, x₂], . . . , [x_n-1, x_n]. Denote by L_n, R_n, M_n, Tn the Riemann sums corresponding to left-point, right-point, midpoint and trapezoid rule. Use sigma notation to write a formula for each L_n, R_n, M_n, T_n.

c. With the help of your calculator, compute L₄, R₄, M₄, T₄. Which of them is closest to I ?

d. Write Matlab codes to compute L_n, R_n, M_n, T_n when n = 8, 16, 32, 64.

e. Denote by e_n^(L)  = | L_n − I | the error term from left-point rule. We use similar notations for e_n^(R) , e_n^(M) , e_n^(T) . It is known that e_n^(L) , e_n^(R) ≤ (K(b − a)²)/2n , e_n^(M) ≤ (K^~(b − a)³)/24n² , e_n^(T) ≤ (K^~(b − a)³)/12n² where K = max_[a,b] |f'(x)| and K˜ = max_[a,b] |f''(x)|. Find n such that the left-point rule gives an error not exceeding e = 0.0001. The same question for the right-point, midpoint, trapezoid rule. Hint: you don’t need to find the exact values of K and K˜ . An upper bound for each of them would be sufficient for this problem.

Need help on e

Answer 1