Let f(x) = x defined on the closed interval [0, 4]. (a) For the partition P = RS(f,P,S) = 8. (b) For the partition P = RS(f,P,S) = 9. (c) For the partition P = sampling point S for P so that RS(f,P,S) (d) Describe a partition P = (xo, x1,..., x100) of [0, 4] and a list of sampling points S for P such that RS(f,P, S) < 0.01. (0, 1, 2, 3, 4) of [0, 4], find a list of sampling points S ($1, 82, 83, 84) for P such that (0, 1, 2, 3, 4) of [0, 4], find a list of sampling points S = (81, 82, 83, 84) for P such that (0,4) of [0, 4] and any real number y such that 0 < y < 8, show that there is a list of = y. (Note that the list S will have just one member.)
Let f(x) = x defined on the closed interval [0, 4]. (a) For the partition P = RS(f,P,S) = 8. (b) For the partition P = RS(f,P,S) = 9. (c) For the partition P = sampling point S for P so that RS(f,P,S) (d) Describe a partition P = (xo, x1,..., x100) of [0, 4] and a list of sampling points S for P such that RS(f,P, S) < 0.01. (0, 1, 2, 3, 4) of [0, 4], find a list of sampling points S ($1, 82, 83, 84) for P such that (0, 1, 2, 3, 4) of [0, 4], find a list of sampling points S = (81, 82, 83, 84) for P such that (0,4) of [0, 4] and any real number y such that 0 < y < 8, show that there is a list of = y. (Note that the list S will have just one member.)
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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use method in example 35.2 to solve a b c d
![Example 35.2. Let f(x) = x² and consider the closed interval [3, 7].
Let P be the partition (3, 3.2, 5.1, 6, 6.8, 7) (so n = 5 here). Note
that ||P|| = 1.9.
Let S be the sampling points (3.15, 4, 5.5, 6, 7). Then
RS(f,P,S) = 3.15°(.2)+4²(1.9)+5.5°(.9)+6°(.8)+7°(.2) = 98.2095 (exactly).
This number approximates the area under y = x², above the T-axis,
between a = 3 and x = 7. Of course, fromn calculus we know that the
actual value of this area is f x² dx =
73
33
* = 105.3.
3
3
We can now define f(x) dx. Loosely speaking, it is the number I
so that if we consider partitions P of very small norm, then RS(f, P,S)
is very close to I for all S. Such a number need not exist, however,
and so f(x) dx is sometimes undefined.
SO](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba5a3306-2973-4825-a0c7-f3a4b662c857%2F93f8d39d-b30e-4a22-b380-a385642b77f1%2Fvsoynmg_processed.png&w=3840&q=75)
Transcribed Image Text:Example 35.2. Let f(x) = x² and consider the closed interval [3, 7].
Let P be the partition (3, 3.2, 5.1, 6, 6.8, 7) (so n = 5 here). Note
that ||P|| = 1.9.
Let S be the sampling points (3.15, 4, 5.5, 6, 7). Then
RS(f,P,S) = 3.15°(.2)+4²(1.9)+5.5°(.9)+6°(.8)+7°(.2) = 98.2095 (exactly).
This number approximates the area under y = x², above the T-axis,
between a = 3 and x = 7. Of course, fromn calculus we know that the
actual value of this area is f x² dx =
73
33
* = 105.3.
3
3
We can now define f(x) dx. Loosely speaking, it is the number I
so that if we consider partitions P of very small norm, then RS(f, P,S)
is very close to I for all S. Such a number need not exist, however,
and so f(x) dx is sometimes undefined.
SO
![(1) Let f(x) = x defined on the closed interval [0, 4].
(a) For the partition P =
RS(f,P,S) = 8.
(b) For the partition P =
RS(f,P,S) = 9.
(c) For the partition P =
sampling point S for P so that RS(f,P,S)
(d) Describe a partition P =
(0, 1, 2, 3, 4) of [0, 4], find a list of sampling points S =
($1, 82, 83, 84) for P such that
(0, 1, 2,3, 4) of [0, 4], find a list of sampling points S = (s1, $2, $3, 84) for P such that
(0, 4) of [0, 4] and any real number y such that 0 < y < 8, show that there is a list of
= y. (Note that the list S will have just one member.)
(x0, x1,..., x100) of [0,4] and a list of sampling points S for P such that RS(f,P,S) <
0.01.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba5a3306-2973-4825-a0c7-f3a4b662c857%2F93f8d39d-b30e-4a22-b380-a385642b77f1%2Fhfgg6mf_processed.png&w=3840&q=75)
Transcribed Image Text:(1) Let f(x) = x defined on the closed interval [0, 4].
(a) For the partition P =
RS(f,P,S) = 8.
(b) For the partition P =
RS(f,P,S) = 9.
(c) For the partition P =
sampling point S for P so that RS(f,P,S)
(d) Describe a partition P =
(0, 1, 2, 3, 4) of [0, 4], find a list of sampling points S =
($1, 82, 83, 84) for P such that
(0, 1, 2,3, 4) of [0, 4], find a list of sampling points S = (s1, $2, $3, 84) for P such that
(0, 4) of [0, 4] and any real number y such that 0 < y < 8, show that there is a list of
= y. (Note that the list S will have just one member.)
(x0, x1,..., x100) of [0,4] and a list of sampling points S for P such that RS(f,P,S) <
0.01.
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