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Lagrange.pdf
1.
๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐ถ๐ผร๐ ๐๐๐ฟ๐ผ๐๐๐๐ผ๐ด๐ฟ
2.
๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐ถ๐ผร๐ ๐๐๐ฟ๐ผ๐๐๐๐ผ๐ด๐ฟ ๐ถ๐๐๐ ๐๐ ๐ก๐ ๐๐
โ๐๐๐๐ ๐๐๐๐๐๐๐๐๐ ๐ก๐๐๐๐ ๐๐๐ ๐๐ข๐๐ก๐๐ ๐๐๐๐๐๐๐๐๐ ๐ฅ0, ๐ฅ1, โฆ , ๐ฅ๐, ๐๐๐ ๐ข๐ รบ๐๐๐๐ ๐๐๐๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐ โค ๐, ๐๐๐ ๐๐ ๐๐๐๐๐๐๐รณ๐ ๐๐ ๐๐ข๐: ๐๐ ๐ฅ๐ = ๐ ๐ฅ๐ ๐๐ ๐๐๐ ๐๐ข๐๐ก๐๐ ๐๐๐๐๐ .
3.
๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐ถ๐ผร๐ ๐๐๐ฟ๐ผ๐๐๐๐ผ๐ด๐ฟ ๐๐๐ ๐๐๐๐๐๐๐,
๐ ๐๐๐ ๐๐๐๐๐๐๐๐๐ ๐๐๐ ๐๐ข๐๐ก๐๐ ๐ฅ0 ๐ฆ ๐ฅ1 โถ
4.
๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐ถ๐ผร๐ ๐๐๐ฟ๐ผ๐๐๐๐ผ๐ด๐ฟ ๐ธ๐ ๐๐๐๐๐๐๐๐๐
๐๐ ๐๐๐ฆ๐๐ ๐๐๐๐๐ ๐๐ข๐ ๐๐๐ก๐๐๐๐๐๐ ๐ฅ0 ๐ฆ ๐ฅ1 ๐๐๐ ๐๐ข๐๐ก๐๐ : ๐๐ ๐ = 1, ๐๐ ๐๐๐๐๐, ๐ข๐ ๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐ ๐1 ๐ฅ : ๐1 ๐ฅ = ๐๐ฅ + ๐ ๐ถ๐๐ ๐๐๐ ๐๐๐๐๐๐๐๐๐๐๐ : 1) ๐1 ๐ฅ0 = ๐ ๐ฅ0 2) ๐1 ๐ฅ1 = ๐(๐ฅ1) ๐ธ๐ ๐๐๐๐๐, ๐ ๐ ๐ก๐๐๐๐ ๐ข๐ ๐ ๐๐ ๐ก๐๐๐ ๐๐ ๐๐๐ ๐๐๐ข๐๐๐๐๐๐๐ ๐๐๐ ๐๐๐ ๐๐๐รณ๐๐๐๐ก๐๐ , ๐ ๐ฆ ๐. ๐ ๐๐ ๐๐๐ฃ๐๐๐๐๐ ๐ ๐ ๐๐๐ก๐๐๐๐: ๐ = ๐ ๐ฅ1 โ ๐ ๐ฅ0 ๐ฅ1 โ ๐ฅ0 , ๐ = ๐ฅ1 โ ๐ ๐ฅ0 โ ๐ฅ0 โ ๐ ๐ฅ1 ๐ฅ1 โ ๐ฅ0 ๐น๐๐๐๐๐๐๐๐ก๐: ๐1 ๐ฅ = ๐ ๐ฅ1 โ๐ ๐ฅ0 ๐ฅ1โ๐ฅ0 โ ๐ฅ + ๐ฅ1โ๐ ๐ฅ0 โ๐ฅ0โ๐ ๐ฅ1 ๐ฅ1โ๐ฅ0 (1)
5.
๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐ถ๐ผร๐ ๐๐๐ฟ๐ผ๐๐๐๐ผ๐ด๐ฟ ๐ตรบ๐๐๐๐ ๐ ๐
๐ท๐๐๐๐๐ ๐ช๐๐๐๐๐๐ ๐๐ ๐ด๐๐๐๐ ๐๐๐๐ ๐ ๐ ๐๐ ๐ท๐๐๐๐๐๐๐๐ ๐ฐ๐๐๐๐๐๐๐๐๐๐๐ 2 1 3 2 4 3 Etc. โฎ ๐ธ๐ ๐๐๐๐๐, ๐๐๐๐ ๐ + 1 ๐๐ข๐๐ก๐๐ ๐๐๐๐๐๐๐๐๐ ๐๐ ๐๐๐ฆ๐๐ ๐๐๐๐๐ ๐๐๐ ๐๐๐๐๐๐๐๐๐ ๐๐๐ก๐๐๐๐๐๐๐๐ก๐ ๐๐ ๐.
6.
๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐ถ๐ผร๐ ๐๐๐ฟ๐ผ๐๐๐๐ผ๐ด๐ฟ ๐ด ๐๐๐๐๐๐
๐๐ข๐ ๐ ๐ ๐๐๐๐๐๐๐๐๐๐ ๐๐ข๐๐ฃ๐๐ ๐๐ข๐๐ก๐๐ ๐๐๐๐๐๐๐๐๐ , ๐๐ ๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐ข๐๐ก๐๐ ๐๐๐๐ ๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐ ๐๐๐ก๐๐๐๐๐๐๐๐ก๐๐ ๐๐ ๐๐๐ฆ๐๐. ๐ธ๐ ๐๐ ๐ ๐๐ข๐, ๐โ๐๐๐ ๐ ๐ ๐ฃ๐๐รก๐ ๐๐๐ ๐รฉ๐ก๐๐๐๐ ๐๐๐๐ ๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐ ๐๐๐ก๐๐๐๐๐๐๐๐ก๐๐ ๐๐ ๐๐๐๐๐๐ ๐ ๐๐๐๐๐๐๐.
7.
๐๐๐ฟ๐ผ๐๐๐๐ผ๐๐ ๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐๐๐ธ๐ ๐ธ๐
๐ฟ๐ด ๐น๐๐ ๐๐ด ๐ท๐ธ ๐ฟ๐ด๐บ๐ ๐ด๐๐บ๐ธ ๐ท๐ ๐๐ ๐๐ฅ๐๐๐๐ ๐รณ๐ 1 , ๐๐๐๐๐๐๐๐๐๐๐๐ ๐กรฉ๐๐๐๐๐๐ ๐ ๐ ๐๐ข๐๐๐ ๐๐๐๐ฃ๐๐ ๐ ๐๐ ๐๐๐๐๐: ๐1 ๐ฅ = ๐ฅ โ ๐ฅ1 ๐ฅ0 โ ๐ฅ1 โ ๐0 + ๐ฅ โ ๐ฅ0 ๐ฅ1 โ ๐ฅ0 โ ๐1 ๐ธ๐ ๐๐๐๐๐, ๐๐๐๐ ๐๐ ๐ ๐ข๐๐๐ ๐รณ๐ ๐๐๐ ๐๐๐๐๐ข๐๐ก๐, ๐๐โ ๐๐, ๐๐๐๐๐: ๐0 = ๐ฅ โ ๐ฅ1 ๐ฅ0 โ ๐ฅ1 ๐ฆ ๐1 = ๐ฅ โ ๐ฅ0 ๐ฅ1 โ ๐ฅ0
8.
๐๐๐ฟ๐ผ๐๐๐๐ผ๐๐ ๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐๐๐ธ๐ ๐ธ๐
๐ฟ๐ด ๐น๐๐ ๐๐ด ๐ท๐ธ ๐ฟ๐ด๐บ๐ ๐ด๐๐บ๐ธ ๐บ๐๐๐๐๐๐๐๐ง๐๐๐๐ ๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐ ๐, ๐ ๐ ๐ก๐๐๐๐ ๐๐ ๐๐ฅ๐๐๐๐ ๐รณ๐: ๐๐ ๐ฅ = เท 0 ๐ ๐๐ โ ๐๐ ๐๐๐๐๐๐๐๐๐ ๐๐ ๐ฟ๐๐๐๐๐๐๐ ๐๐ ๐ฅ = เท ๐=0 ๐โ ๐ ๐ ๐ฅ โ ๐ฅ๐ ๐ฅ๐ โ ๐ฅ๐ ๐น๐ข๐๐๐รณ๐ ๐๐ ๐น๐๐๐๐ ๐๐ ๐ฟ๐๐๐๐๐๐๐
9.
๐๐๐ฟ๐ผ๐๐๐๐ผ๐๐ ๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐๐๐ธ๐ ๐ธ๐
๐ฟ๐ด ๐น๐๐ ๐๐ด ๐ท๐ธ ๐ฟ๐ด๐บ๐ ๐ด๐๐บ๐ธ ๐ธ๐๐๐๐๐๐. ๐๐๐๐๐๐๐๐๐ ๐๐๐ก๐๐๐๐๐๐๐๐ก๐ ๐๐ ๐ฟ๐๐๐๐๐๐๐ ๐๐๐๐ ๐ = 2: ๐2 ๐ฅ = เท 0 2 ๐๐ โ ๐๐ = ๐0 โ ๐0 + ๐1 โ ๐1 + ๐2 โ ๐2 ๐ท๐๐๐๐, ๐๐ ๐ฅ = เท 0 ๐โ 0 2 ๐ฅ โ ๐ฅ๐ ๐ฅ0 โ ๐ฅ๐ = ๐ฅ โ ๐ฅ1 ๐ฅ0 โ ๐ฅ1 โ ๐ฅ โ ๐ฅ2 ๐ฅ0 โ ๐ฅ2 , ๐1 ๐ฅ = เท 0 ๐โ 1 2 ๐ฅ โ ๐ฅ๐ ๐ฅ1 โ ๐ฅ๐ = ๐ฅ โ ๐ฅ0 ๐ฅ1 โ ๐ฅ0 โ ๐ฅ โ ๐ฅ2 ๐ฅ1 โ ๐ฅ2 , ๐2 ๐ฅ = เท 0 ๐โ 2 2 ๐ฅ โ ๐ฅ๐ ๐ฅ2 โ ๐ฅ๐ = ๐ฅ โ ๐ฅ0 ๐ฅ2 โ ๐ฅ0 โ ๐ฅ โ ๐ฅ1 ๐ฅ2 โ ๐ฅ1
10.
๐๐๐ฟ๐ผ๐๐๐๐ผ๐๐ ๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐๐๐ธ๐ ๐ธ๐
๐ฟ๐ด ๐น๐๐ ๐๐ด ๐ท๐ธ ๐ฟ๐ด๐บ๐ ๐ด๐๐บ๐ธ ๐ธ๐๐๐๐๐๐. ๐๐๐๐๐๐๐๐๐ ๐๐๐ก๐๐๐๐๐๐๐๐ก๐ ๐๐ ๐ฟ๐๐๐๐๐๐๐ ๐๐๐๐ ๐ = 3: ๐3 ๐ฅ = เท 0 3 ๐๐ โ ๐๐ = ๐0 โ ๐0 + ๐1 โ ๐1 + ๐2 โ ๐2 + ๐3 โ ๐3 ๐ท๐๐๐๐, ๐๐ ๐ฅ = เท 0 ๐โ 0 3 ๐ฅ โ ๐ฅ1 ๐ฅ0 โ ๐ฅ1 โ ๐ฅ โ ๐ฅ2 ๐ฅ0 โ ๐ฅ2 โ ๐ฅ โ ๐ฅ3 ๐ฅ0 โ ๐ฅ3 , ๐1 ๐ฅ = เท 0 ๐โ 1 3 ๐ฅ โ ๐ฅ0 ๐ฅ1 โ ๐ฅ0 โ ๐ฅ โ ๐ฅ2 ๐ฅ1 โ ๐ฅ2 โ ๐ฅ โ ๐ฅ3 ๐ฅ1 โ ๐ฅ3 , ๐2 ๐ฅ = เท 0 ๐โ 2 3 ๐ฅ โ ๐ฅ0 ๐ฅ2 โ ๐ฅ0 โ ๐ฅ โ ๐ฅ1 ๐ฅ2 โ ๐ฅ1 โ ๐ฅ โ ๐ฅ3 ๐ฅ2 โ ๐ฅ3 , ๐3 ๐ฅ = เท 0 ๐โ 3 3 ๐ฅ โ ๐ฅ0 ๐ฅ3 โ ๐ฅ0 โ ๐ฅ โ ๐ฅ1 ๐ฅ3 โ ๐ฅ1 โ ๐ฅ โ ๐ฅ2 ๐ฅ3 โ ๐ฅ2
11.
๐๐๐ฟ๐ผ๐๐๐๐ผ๐๐ ๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐๐๐ธ๐ ๐ธ๐
๐ฟ๐ด ๐น๐๐ ๐๐ด ๐ท๐ธ ๐ฟ๐ด๐บ๐ ๐ด๐๐บ๐ธ ๐ธ๐๐๐๐๐๐. Con un polinomio interpolante en la forma de Lagrange, de primero y segundo grado evalรบe ln 2 basรกndose en los datos conocidos: ๐) ๐ = 1, ๐1 ๐ฅ = เท 0 1 ๐๐ โ ๐๐ = ๐0 โ ๐0 + ๐1 โ ๐1 ๐2 ๐ฅ = ๐ท๐๐๐๐, ๐๐ 2 โ ๐ ๐ฅ0 = เท 0 ๐โ 0 1 2 โ ๐ฅ๐ ๐ฅ0 โ ๐ฅ๐ = 2 โ 4 1 โ 4 โ 0 = 0, ๐1 2 โ ๐ ๐ฅ1 = เท 0 ๐โ 1 1 2 โ ๐ฅ๐ ๐ฅ1 โ ๐ฅ๐ = 2 โ 1 4 โ 1 1.386294 = 0.462098, ๐1 2 = 0 + 0.462098 = 0.462098 ๐ ๐๐ ๐ ๐๐ = ๐๐ ๐๐ 0 1 0 1 4 1.386294 2 6 1.791760
12.
๐๐๐ฟ๐ผ๐๐๐๐ผ๐๐ ๐ผ๐๐๐ธ๐ ๐๐๐ฟ๐ด๐๐๐ธ๐ ๐ธ๐
๐ฟ๐ด ๐น๐๐ ๐๐ด ๐ท๐ธ ๐ฟ๐ด๐บ๐ ๐ด๐๐บ๐ธ ๐) ๐ = 2, ๐2 ๐ฅ = เท 0 2 ๐๐ โ ๐๐ = ๐0 โ ๐0 + ๐1 โ ๐1 + ๐2 โ ๐2 ๐2 ๐ฅ = ๐๐ 2 โ ๐ ๐ฅ0 = เท 0 ๐โ 0 2 2 โ ๐ฅ๐ ๐ฅ0 โ ๐ฅ๐ = 2 โ 4 1 โ 4 โ 2 โ 6 1 โ 6 โ 0 = 0, ๐1 2 โ ๐ ๐ฅ1 = เท 0 ๐โ 1 2 2 โ ๐ฅ๐ ๐ฅ1 โ ๐ฅ๐ = 2 โ 1 4 โ 1 โ 2 โ 6 4 โ 6 1.386294 = 0.924196, ๐2 2 โ ๐ ๐ฅ2 = เท 0 ๐โ 2 2 2 โ ๐ฅ๐ ๐ฅ2 โ ๐ฅ๐ = 2 โ 1 6 โ 1 โ 2 โ 4 6 โ 4 โ 1.791760 = โ0.358352 ๐2 2 = 0 + 0.924196 โ 0.358352 = 0.565844 ๐ ๐๐ ๐ ๐๐ = ๐๐ ๐๐ 0 1 0 1 4 1.386294 2 6 1.791760
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