Osculating polynomial. (a) For some function f, you have a table of extended divided differences of the form f[...] f[d] i 0 1 2 3 Zi f[.] f[...] 5.0 5.0 [20] 5.0 f[21] [20,21] 6.0 4.0 5.0 -3.0 4.0 2.0 f[22, 23] [21, 22, 23] f[20, 21, 22, 23] Fill in the unknown entries in the table. (b) For the data in (a), what is the osculating polynomial p2(x) of degree at most 2 that satisfies
Osculating polynomial. (a) For some function f, you have a table of extended divided differences of the form f[...] f[d] i 0 1 2 3 Zi f[.] f[...] 5.0 5.0 [20] 5.0 f[21] [20,21] 6.0 4.0 5.0 -3.0 4.0 2.0 f[22, 23] [21, 22, 23] f[20, 21, 22, 23] Fill in the unknown entries in the table. (b) For the data in (a), what is the osculating polynomial p2(x) of degree at most 2 that satisfies
Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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![4. Osculating polynomial.
(a) For some function f, you have a table of extended divided differences of the form
i
f[.]
f[...]
ƒ [·,·,·]
ƒ[]
0
5.0
f[zo]
1
5.0
f[21] [20, 21]
2 6.0 4.0
5.0
3
4.0
-3.0
2.0 f[22, 23] [21, 22, 23] [20, 21, 22, 23]
Fill in the unknown entries in the table.
(b) For the data in (a), what is the osculating polynomial p2(x) of degree at most 2
that satisfies
P2(5.0) = f(5.0), p2(5.0) = f'(5.0), p2 (6.0) = f(6.0)?
(c) Suppose f'(4) = 3. Expand the divided difference table in (a) to find the oscu-
lating polynomial p4(x) of degree at most 4 that satisfies
P4 (5.0) = f(5.0), p4(5.0) = f'(5.0), p4(6.0) = f(6.0),
P4(4.0) = f(4.0), p₁ (4.0) = f'(4.0).
You do NOT need to simplify your polynomial.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3c842333-6f00-4a4b-9f61-7bda344711a4%2F6b2039d9-c986-4359-9a44-45e47d169a2f%2Ffcq1ur_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Osculating polynomial.
(a) For some function f, you have a table of extended divided differences of the form
i
f[.]
f[...]
ƒ [·,·,·]
ƒ[]
0
5.0
f[zo]
1
5.0
f[21] [20, 21]
2 6.0 4.0
5.0
3
4.0
-3.0
2.0 f[22, 23] [21, 22, 23] [20, 21, 22, 23]
Fill in the unknown entries in the table.
(b) For the data in (a), what is the osculating polynomial p2(x) of degree at most 2
that satisfies
P2(5.0) = f(5.0), p2(5.0) = f'(5.0), p2 (6.0) = f(6.0)?
(c) Suppose f'(4) = 3. Expand the divided difference table in (a) to find the oscu-
lating polynomial p4(x) of degree at most 4 that satisfies
P4 (5.0) = f(5.0), p4(5.0) = f'(5.0), p4(6.0) = f(6.0),
P4(4.0) = f(4.0), p₁ (4.0) = f'(4.0).
You do NOT need to simplify your polynomial.
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