(a) Lagrange gives $y(10)\approx14.67$. (b) IF $=x$; $y=\dfrac{x^4}{5}+\dfrac{c}{x}$.

(a) Lagrange's interpolation for $X:5,6,9,11$ and $Y:12,13,14,16$, find $y(10)$.

$y(x)=\sum_i y_i\prod_{j e i}\frac{x-x_j}{x_i-x_j}.$

With $x=10$:

$y(10)=12\cdot\frac{(10-6)(10-9)(10-11)}{(5-6)(5-9)(5-11)}+13\cdot\frac{(10-5)(10-9)(10-11)}{(6-5)(6-9)(6-11)}$

$+\,14\cdot\frac{(10-5)(10-6)(10-11)}{(9-5)(9-6)(9-11)}+16\cdot\frac{(10-5)(10-6)(10-9)}{(11-5)(11-6)(11-9)}.$

Evaluate each term:

$=12\cdot\frac{(4)(1)(-1)}{(-1)(-4)(-6)}+13\cdot\frac{(5)(1)(-1)}{(1)(-3)(-5)}+14\cdot\frac{(5)(4)(-1)}{(4)(3)(-2)}+16\cdot\frac{(5)(4)(1)}{(6)(5)(2)}$

$=12\cdot\frac{-4}{-24}+13\cdot\frac{-5}{15}+14\cdot\frac{-20}{-24}+16\cdot\frac{20}{60}$

$=2-\frac{13}{3}+\frac{35}{3}+\frac{16}{3}=2+\frac{-13+35+16}{3}=2+\frac{38}{3}=\frac{44}{3}\approx14.67.$

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