1. Let f(r) 3, for each positive integer n, consider the regular partition0 ox2 < - *. < In-1 < In = 1of 0, 1] into n many equal-sized subintervals. Denote the corresponding Right-handRiemann sum byΣία) Δi=1(a) For each n, Rnrdx, but what is the smallest value of n for which Rn is withind? (Hint: find a formula for Rn in terms of n)0.1 of the exact value of(b) If a different choice for rin ri-1, X], for i = 1,2, ... , n is chosen to construct theRiemann sumΣ)Δni=1Idr, for a smaller valuecan the resulting sum be within 0.1 of the exact value ofof n than in part (a)? If so, give an example. If not, explain why

Question
Asked Oct 28, 2019
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1. Let f(r) 3, for each positive integer n, consider the regular partition
0 o
x2 < - *. < In-1 < In = 1
of 0, 1] into n many equal-sized subintervals. Denote the corresponding Right-hand
Riemann sum by
Σία) Δ
i=1
(a) For each n, Rn
rdx, but what is the smallest value of n for which Rn is within
d? (Hint: find a formula for Rn in terms of n)
0.1 of the exact value of
(b) If a different choice for rin ri-1, X], for i = 1,2, ... , n is chosen to construct the
Riemann sum
Σ)Δn
i=1
Idr, for a smaller value
can the resulting sum be within 0.1 of the exact value of
of n than in part (a)? If so, give an example. If not, explain why
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1. Let f(r) 3, for each positive integer n, consider the regular partition 0 o x2 < - *. < In-1 < In = 1 of 0, 1] into n many equal-sized subintervals. Denote the corresponding Right-hand Riemann sum by Σία) Δ i=1 (a) For each n, Rn rdx, but what is the smallest value of n for which Rn is within d? (Hint: find a formula for Rn in terms of n) 0.1 of the exact value of (b) If a different choice for rin ri-1, X], for i = 1,2, ... , n is chosen to construct the Riemann sum Σ)Δn i=1 Idr, for a smaller value can the resulting sum be within 0.1 of the exact value of of n than in part (a)? If so, give an example. If not, explain why

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Step 1

The problem concerns estimating errors in calculating the Right hand Riemann sums (to approximate definite integrals)

Step 2

a) Notation following the problem statement

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a) f(x) x over [0,1]. 1 Now,I =f(x)dx= 11 0.25 4 0 1 x ,k 0,1,2...,n; Ar = n n k xΣ f(%, R.- (A) 13 7

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Step 3

a) Determining the right ha...

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1)-Σfa. k R, (Ar)f (x 1 So, R nxn n(n+1) 4n 1 = 0.25(1+ W:

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