Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Transcribed Image Text:Scroll to the bottom of this question to see a table defining two vectors, U and V, each with 31 components.
(Note:
The table has been carefully prepared to make it easy to copy-paste into a spreadsheet. Computer assistance is recommended in this problem.)
30
(a) In high-dimensional geometry, given vectors of compatible dimensions are combined to produce a number called the dot product. For vectors with 31 components, like U and V here, the definition of their dot product is U.V-U.VI.
Evaluate the dot product: U. V=
Now consider the generic integral I =
u(x₁) = U₁, v(2₁) = V₁
link the numbers given in the table with the values of u and at the equally-spaced nodes
₁0+iAz, where Az =
b-a
30
(b) Suppose I is defined using a = 0 and b = 1. Find the approximations for I indicated below.
The right-endpoint Riemann sum with n = 30 subintervals: Ra
The Trapezoidal approximation with = 30 subintervals: To
(c) Suppose I is defined using a 1 and 6-6. Find the approximations for I indicated below.
The Trapezoidal approximation with n= 15 subintervals: T₁s =
The approximation from Simpson's Rule with n=10 subintervals: 510 =
0
Here are the definitions for U = (Ue,U₁,...,U30) and V = (V, V₁,..., V₁0).
4
1
2
3
4
7
-1
-1
8
6
-3
-3
7
4
5 27
5
6
#
6
-3
6
7 7 6
8
7
3
9
10
-[u(z)u(z) dir, where the key equations
7
11
11
2 9
12
-50
13 8 5
14 69
15 -20
16 3 -3
17 8 -3
18 7 -1
19 9
-1
20 3
1
21 -2 3
22 7 6
23 -2
3
24 2 7
25 7
-1
26 -4 2
27 -5 9
28 3 7
29 0 -1
30 4 -1
Ⓡ
#
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