Question

Given the table of values

x: 50 52 54 56

3√x : 3.684 3.732 3.779 3.825

Use Lagrange’s formula to find x when 3√x = 3.756,

Answer 1

$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} x: & 50 & 52 & 54 & 56 \\ \hline y=\sqrt[3]{x}: & 3.684 & 3.732 & 3.779 & 3.825\\ \hdashline \end{array}$

By Lagrange’s interpolation formula for inverse interpolation, we have:

$x=f^{-1}(y)= \frac{(y-y_1)(y-y_2)(y-y_3)}{(y_0-y_1)(y_0-y_2)(y_0-y_3)}x_0+ \frac{(y-y_0)(y-y_2)(y-y_3)}{(y_1-y_0)(y_1-y_2)(y_1-y_3)} x_1+ \frac{(y-y_0)(y-y_1)(y-y_3)}{(y_2-y_0)(y_2-y_1)(y_2-y_3)}x_2+ \frac{(y-y_0)(y-y_1)(y-y_2)}{(y_3-y_0)(y_3-y_1)(y_3-y_2)} x_3$

We put $y=3.756$ :

$x(3.756)=f^{-1}(3.756)= \frac{(3.756-3.732)(3.756-3.779)(3.756-3.825)}{(3.684-3.732)(3.684-3.779)(3.684-3.825)}50+ \frac{(3.756-3.684)(3.756-3.779)(3.756-3.825)}{(3.732-3.684)(3.732-3.779)(3.732-3.825)} 52+ \frac{(3.756-3.684)(3.756-3.732)(3.756-3.825)}{(3.779-3.684)(3.779-3.732)(3.779-3.825)}54+ \frac{(3.756-3.684)(3.756-3.732)(3.756-3.779)}{(3.825-3.684)(3.825-3.732)(3.825-3.779)} 56=-\frac{529}{8930}\cdot 50+\frac{1587}{2914}\cdot 52 +\frac{2592}{4465}\cdot 54-\frac{96}{1457}\cdot 56\approx 53.016$

Answer: $x\approx 53.016$ .