Question #152967
Find by lagrange’s formula, the value of
U4 if U3 = 16 , U5 = 36 , U7 = 64 , U8 = 81 and U9 = 100
1
Expert's answer
2020-12-28T18:47:31-0500

Here the intervals are unequal.


x0=3,x1=5,x2=7,x3=8,x4=9y0=16,y1=36,y2=64,y3=81,y4=100\begin{matrix} x_0=3, & x_1=5, & x_2=7, & x_3=8, & x_4=9 \\ y_0=16, & y_1=36, & y_2=64, & y_3=81,& y_4=100 \end{matrix}

y=f(x)=(x+1)2y=f(x)=(x+1)^2

Put x=4x=4


f(4)=(4+1)2=25f(4)=(4+1)^2=25

By Lagrange’s interpolation formula we have

y=f(x)=(xx1)(xx2)(xx3)(xx4)(x0x1)(x0x2)(x0x3)(x0x4)×y0y=f(x)=\dfrac{(x-x_1)(x-x_2)(x-x_3)(x-x_4)}{(x_0-x_1)(x_0-x_2)(x_0-x_3)(x_0-x_4)}\times y_0

+(xx0)(xx2)(xx3)(xx4)(x1x0)(x1x2)(x1x3)(x1x4)×y1+\dfrac{(x-x_0)(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_0)(x_1-x_2)(x_1-x_3)(x_1-x_4)}\times y_1

+(xx0)(xx1)(xx3)(xx4)(x2x0)(x2x1)(x2x3)(x2x4)×y2+\dfrac{(x-x_0)(x-x_1)(x-x_3)(x-x_4)}{(x_2-x_0)(x_2-x_1)(x_2-x_3)(x_2-x_4)}\times y_2

+(xx0)(xx1)(xx2)(xx4)(x3x0)(x3x1)(x3x2)(x3x4)×y3+\dfrac{(x-x_0)(x-x_1)(x-x_2)(x-x_4)}{(x_3-x_0)(x_3-x_1)(x_3-x_2)(x_3-x_4)}\times y_3


+(xx0)(xx1)(xx2)(xx3)(x4x0)(x4x1)(x4x2)(x4x3)×y4+\dfrac{(x-x_0)(x-x_1)(x-x_2)(x-x_3)}{(x_4-x_0)(x_4-x_1)(x_4-x_2)(x_4-x_3)}\times y_4



Put x=4x=4


f(4)=(45)(47)(48)(49)(35)(37)(38)(39)×16f(4)=\dfrac{(4-5)(4-7)(4-8)(4-9)}{(3-5)(3-7)(3-8)(3-9)}\times 16

+(43)(47)(48)(49)(53)(57)(58)(59)×36+\dfrac{(4-3)(4-7)(4-8)(4-9)}{(5-3)(5-7)(5-8)(5-9)}\times 36

+(43)(45)(48)(49)(73)(75)(78)(79)×64+\dfrac{(4-3)(4-5)(4-8)(4-9)}{(7-3)(7-5)(7-8)(7-9)}\times 64

+(43)(45)(47)(49)(83)(85)(87)(89)×81+\dfrac{(4-3)(4-5)(4-7)(4-9)}{(8-3)(8-5)(8-7)(8-9)}\times 81

+(43)(45)(47)(48)(93)(95)(97)(98)×100+\dfrac{(4-3)(4-5)(4-7)(4-8)}{(9-3)(9-5)(9-7)(9-8)}\times 100

=25=25

U4=25U4=25



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