Using Lagrange's interpolation formula, find y at x=1 given the points (0,1),(2,9),(3,19).

Exercises · Q9

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Setting up the three terms ( $x_0=0,y_0=1$ ; $x_1=2,y_1=9$ ; $x_2=3,y_2=19$ ; evaluating at $x=1$ )

\text{Term 1: } 1\cdot\frac{(1-2)(1-3)}{(0-2)(0-3)}=1\cdot\frac{(-1)(-2)}{(-2)(-3)}=1\cdot\frac26=\frac13

\text{Term 2: } 9\cdot\frac{(1-0)(1-3)}{(2-0)(2-3)}=9\cdot\frac{(1)(-2)}{(2)(-1)}=9\cdot\frac{-2}{-2}=9

\text{Term 3: } 19\cdot\frac{(1-0)(1-2)}{(3-0)(3-2)}=19\cdot\frac{(1)(-1)}{(3)(1)}=-\frac{19}3

Summing

$y=\frac13+9-\frac{19}3=\frac{1-19}3+9=\frac{-18}3+9=-6+9=3$ …

Q.Using Lagrange's interpolation formula, find $y$ at $x=1$ given the points $(0,1),(2,9),(3,19)$ .