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Problem 2. In this problem, we will practice a bit with numerical integration. We take the following facts for granted: for any a

Problem 2. In this problem, we will practice a bit with numerical integration. We take the following facts for granted: for any a

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 2. In this problem, we will practice a bit with numerical integration. We take the following facts for granted: for any a <b, any f: [a, b] → R, and any n ≥ 1, and RIGHT(ƒ,n) – [*° f(x)dx ≤ 2 1(b-a)² n Compute SIMP(f,n) - [ f(x)dx ≤ max f(i)(x)\, [ab] max f'(x)| z€[a,b] (b-a)5 180n4 SIMP(f,n): (S) where RIGHT(f, n) is the right hand Riemann sum with n equal intervals and SIMP (f, n) is the approximation of f f(x)dx using Simpson's rule with n equal intervals. Here f(iv) is the fourth derivative of f. (i) Suppose that f is a third degree polynomial f(x) = a3x³ + a₂x² + a₁ + ao. Argue that, for any n, a, and b, (L) f(x)dr. (ii) Show that the same is not true for RIGHT(f, n) by giving an example of a third degree polynomial f, a natural number n, and endpoints a < b so that RIGHT (f, n) + f(z)dr. Hint: linear functions are third degree polynomials and n = 1 is a natural number... (iii) For the remaining parts, use a = 0, b = 2, and f(x)= -4. RIGHT(f, 4) and SIMP (f, 4). You may do this by hand, by writing code, or by using an online calculator, but you must document your computations (either via screenshots or by writing down your work).
Transcribed Image Text:Problem 2. In this problem, we will practice a bit with numerical integration. We take the following facts for granted: for any a <b, any f: [a, b] → R, and any n ≥ 1, and RIGHT(ƒ,n) – [*° f(x)dx ≤ 2 1(b-a)² n Compute SIMP(f,n) - [ f(x)dx ≤ max f(i)(x)\, [ab] max f'(x)| z€[a,b] (b-a)5 180n4 SIMP(f,n): (S) where RIGHT(f, n) is the right hand Riemann sum with n equal intervals and SIMP (f, n) is the approximation of f f(x)dx using Simpson's rule with n equal intervals. Here f(iv) is the fourth derivative of f. (i) Suppose that f is a third degree polynomial f(x) = a3x³ + a₂x² + a₁ + ao. Argue that, for any n, a, and b, (L) f(x)dr. (ii) Show that the same is not true for RIGHT(f, n) by giving an example of a third degree polynomial f, a natural number n, and endpoints a < b so that RIGHT (f, n) + f(z)dr. Hint: linear functions are third degree polynomials and n = 1 is a natural number... (iii) For the remaining parts, use a = 0, b = 2, and f(x)= -4. RIGHT(f, 4) and SIMP (f, 4). You may do this by hand, by writing code, or by using an online calculator, but you must document your computations (either via screenshots or by writing down your work).
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