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3. (Groups C and F) Let f(x) = x². Complete the following steps to evaluate Darboux sums. (a) For each n E N, consider the partition Pn = Determine M, and m; on any subinterval Recall that Σ i=1 1 n+ 1 2n 1 (b) Show that U(Pn, f) = } (" + ¹) (²^2 + ¹). = (e) Show that n(n+1) (2n + 1) 6 [ (d) Evaluate lim U(Pn, f) and lim L(Pn, f). n→∞ n→∞ 2 n-1 = {0, 1, ², "=¹,1} ₁ with Axi = η η 2n 1 (c) It can be shown that L(Pn, f) - 1/2 (^= ¹) (²²2² =²¹) = 3 n 2n x² dx = 1 - 1 3 n 2 r1 S 2 x²dx using upper and lower for each i=1,2,..., n. n (try this if you have time).

3. (Groups C and F) Let f(x) = x². Complete the following steps to evaluate Darboux sums. (a) For each n E N, consider the partition Pn = Determine M, and m; on any subinterval Recall that Σ i=1 1 n+ 1 2n 1 (b) Show that U(Pn, f) = } (" + ¹) (²^2 + ¹). = (e) Show that n(n+1) (2n + 1) 6 [ (d) Evaluate lim U(Pn, f) and lim L(Pn, f). n→∞ n→∞ 2 n-1 = {0, 1, ², "=¹,1} ₁ with Axi = η η 2n 1 (c) It can be shown that L(Pn, f) - 1/2 (^= ¹) (²²2² =²¹) = 3 n 2n x² dx = 1 - 1 3 n 2 r1 S 2 x²dx using upper and lower for each i=1,2,..., n. n (try this if you have time).

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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3. (Groups C and F) Let ƒ(x) = x². Complete the following steps to evaluate Darboux sums. (a) For each n € N, consider the partition Pn = Determine M, and m; on any subinterval 1 1 2n 1 (b) Show that U(Pn, f) = U (P₁, ƒ) = ² ( n + ¹ ) ( ²n + 3 n 2n Recall that n £²- Σ;2 i=1 (e) Show that = (c) It can be shown that L(Pn, f) 1 n(n + 1)(2n + 1) 6 (d) Evaluate lim U(Pn, f) and lim_L(Pn, f). n→∞ x² dx = 1 3 0, 7 n→∞ 12 n 2 η η - 1 " n - 1 n - n 9 S 1} 1 with Axi 1 1 2n-1 - L(Pn, f) = (n = ¹) (2²7 ²2 n (77²) (try this if you have time). 3 n 2n x²dx using upper and lower = 1 for each i = 1, 2, ..., n. n
Transcribed Image Text:3. (Groups C and F) Let ƒ(x) = x². Complete the following steps to evaluate Darboux sums. (a) For each n € N, consider the partition Pn = Determine M, and m; on any subinterval 1 1 2n 1 (b) Show that U(Pn, f) = U (P₁, ƒ) = ² ( n + ¹ ) ( ²n + 3 n 2n Recall that n £²- Σ;2 i=1 (e) Show that = (c) It can be shown that L(Pn, f) 1 n(n + 1)(2n + 1) 6 (d) Evaluate lim U(Pn, f) and lim_L(Pn, f). n→∞ x² dx = 1 3 0, 7 n→∞ 12 n 2 η η - 1 " n - 1 n - n 9 S 1} 1 with Axi 1 1 2n-1 - L(Pn, f) = (n = ¹) (2²7 ²2 n (77²) (try this if you have time). 3 n 2n x²dx using upper and lower = 1 for each i = 1, 2, ..., n. n
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