Here the intervals are unequal.
x0=3,y0=16,x1=5,y1=36,x2=7,y2=64,x3=8,y3=81,x4=9y4=100
y=f(x)=(x+1)2 Put x=4
f(4)=(4+1)2=25
By Lagrange’s interpolation formula we have
y=f(x)=(x0−x1)(x0−x2)(x0−x3)(x0−x4)(x−x1)(x−x2)(x−x3)(x−x4)×y0
+(x1−x0)(x1−x2)(x1−x3)(x1−x4)(x−x0)(x−x2)(x−x3)(x−x4)×y1
+(x2−x0)(x2−x1)(x2−x3)(x2−x4)(x−x0)(x−x1)(x−x3)(x−x4)×y2
+(x3−x0)(x3−x1)(x3−x2)(x3−x4)(x−x0)(x−x1)(x−x2)(x−x4)×y3
+(x4−x0)(x4−x1)(x4−x2)(x4−x3)(x−x0)(x−x1)(x−x2)(x−x3)×y4
Put x=4
f(4)=(3−5)(3−7)(3−8)(3−9)(4−5)(4−7)(4−8)(4−9)×16
+(5−3)(5−7)(5−8)(5−9)(4−3)(4−7)(4−8)(4−9)×36
+(7−3)(7−5)(7−8)(7−9)(4−3)(4−5)(4−8)(4−9)×64
+(8−3)(8−5)(8−7)(8−9)(4−3)(4−5)(4−7)(4−9)×81
+(9−3)(9−5)(9−7)(9−8)(4−3)(4−5)(4−7)(4−8)×100
=25 U4=25
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