Lagrange: f(2.5)

Assumption. In the scan only the points x = 1, 2, 3, 4 with f = 1, 1.414, 1.732, 2.000 are readable (these are the values of √x). The x value in the last column is illegible (its f value, 2.6, does not fit √x at x = 5), so the formula is applied to these four points. Required: f(2.5).

Lagrange's formula for four points (cubic):

f(x) = Σ f(xᵢ)·Lᵢ(x), Lᵢ(x) = Π (x − xⱼ)/(xᵢ − xⱼ) (j ≠ i)

L₀(2.5) = (2.5−2)(2.5−3)(2.5−4) / [(1−2)(1−3)(1−4)] = (0.5)(−0.5)(−1.5)/(−6) = −0.0625
L₁(2.5) = (2.5−1)(2.5−3)(2.5−4) / [(2−1)(2−3)(2−4)] = (1.5)(−0.5)(−1.5)/(2)  =  0.5625
L₂(2.5) = (2.5−1)(2.5−2)(2.5−4) / [(3−1)(3−2)(3−4)] = (1.5)(0.5)(−1.5)/(−2)  =  0.5625
L₃(2.5) = (2.5−1)(2.5−2)(2.5−3) / [(4−1)(4−2)(4−3)] = (1.5)(0.5)(−0.5)/(6)   = −0.0625

f(2.5) = (−0.0625)(1) + (0.5625)(1.414) + (0.5625)(1.732) + (−0.0625)(2.000) = 1.5821

Answer: f(2.5) ≈ 1.5821 (the true √2.5 = 1.5811).

If the fifth point (5, 2.236) is also taken, the five-point formula gives f(2.5) ≈ 1.5815, which agrees to three decimal places.