How large should we take n in order to guarantee that the trapezoidal and midpoint rule approximations for [² (²) Solution If f(x) = 1/₁ |f"(x)| = 2 S = 11/13 We take K = 2, a = 1, and b = 2. Accuracy to within 0.00002 means that the size of the error should be less than 0.00002. Therefore, we chose n so that n²> 12n² Solving the inequality for n, we get ])² or then f'(x) = 2 2 n> < 0.00002. 12(0.00002) n> 1 ✓0.00012 Thus, n = For the same accuracy with the midpoint rule we choose n so that ≈ 91.29. Z , and f"(x) = 1 . Since 1 ≤ x ≤ 2, we have ≤ 1, so < 0.00002, (rounded up to the nearest integer) will ensure the desired accuracy. 24n² which gives the following. (Round your answer up to the nearest integer.) 1 0.00024 dx are accurate to within 0.00002?
How large should we take n in order to guarantee that the trapezoidal and midpoint rule approximations for [² (²) Solution If f(x) = 1/₁ |f"(x)| = 2 S = 11/13 We take K = 2, a = 1, and b = 2. Accuracy to within 0.00002 means that the size of the error should be less than 0.00002. Therefore, we chose n so that n²> 12n² Solving the inequality for n, we get ])² or then f'(x) = 2 2 n> < 0.00002. 12(0.00002) n> 1 ✓0.00012 Thus, n = For the same accuracy with the midpoint rule we choose n so that ≈ 91.29. Z , and f"(x) = 1 . Since 1 ≤ x ≤ 2, we have ≤ 1, so < 0.00002, (rounded up to the nearest integer) will ensure the desired accuracy. 24n² which gives the following. (Round your answer up to the nearest integer.) 1 0.00024 dx are accurate to within 0.00002?
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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Transcribed Image Text:How large should we take n in order to guarantee that the trapezoidal and midpoint rule approximations for
Solution
If f(x) = then f'(x) =
-1
X
|f"(x)|
2
or
>
2
=
12n²
Solving the inequality for n, we get
3
2
n>
≤
< 0.00002.
12(0.00002)
n>
We take K = 2, a = 1, and b = 2. Accuracy to within 0.00002 means that the size of the error should be less than 0.00002. Therefore, we chose n so that
1
0.00012
333
=
≈ 91.29.
I
< 0.00002,
and f"(x)
=
Thus, n =
For the same accuracy with the midpoint rule we choose n so that
1
. Since 1 ≤ x ≤ 2, we have
(rounded up to the nearest integer) will ensure the desired accuracy.
24n²
which gives the following. (Round your answer up to the nearest integer.)
1
0.00024
[² (²) ₁
X
≤ 1, so
dx are accurate to within 0.00002?
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