Question

Given three data points: (x0, f(x0)), (x1, f(x1)), and (x2, f(x2)), prove the expression for the coefficient b2 in the quadratic interpolation is:________

Answer 1

attached below is the prove

Explanation:

In order to prove the expression for coefficient b
`_(2)` we have to find b
`_(0)` and b
`_(1)`

F(x) =
`b_(0) + b_(1) ( x - x_(0) ) + b_(2) ( x - x_(0) ) (x - x_(1) )`

i) Determine b
`_(0)`

at x = x
`_(0)`

b
`_(0)` = f(x
`_(0)` )

ii) Determine b
`_(1)`

at x =
`x_(1)`

f (
`x_(1)`) = f (
`x_(0)` ) + b
`_(1)`
`(x_(1) - x_(0) )` + 0

`b_(1) = (f(x_(1) )-f(x_(0) ))/((x_(1)-x_(0) ) )`

prove of the expression for the coefficient B2 in the quadratic interpolation

attached below is the detailed prove