Interpolation

[amberjack]

Good evening all, i should resolve this exercise, the prof say that is more diffucult read it then resolve . In the same class of exercises i have resolved some problems with interpolation of Lagrange, Newton, Hermite...
Denoting with plus pij(x) the first degree interpolating polynomial in nodes xi and xj, show that

represents the second degree interpolating polynomial for the points x0, x1, xk. (All the letters/number after x should be subscripts).
I know that i have to do the determinant of the matrix that is: P01(x)(xk-x)-P0K(x)(x1-x)
Then i should calculate that expressions in x0, x1 and xk and show that the results are rispettivately y0, y1 and yk.
I don't understand what should i do .. i have tried to substitutes xk, x1 and x0 in each expression but i don't find a sense of what i do...or maybe is correct? I need a help..
Thanks in advance.

 

[amberjack]

Maybe i had to post in Calculus section?

 

[Deleted member 4993]

Good evening all, i should resolve this exercise, the prof say that is more diffucult read it then resolve . In the same class of exercises i have resolved some problems with interpolation of Lagrange, Newton, Hermite...
Denoting with plus pij(x) the first degree interpolating polynomial in nodes xi and xj, show that
View attachment 22604
represents the second degree interpolating polynomial for the points x0, x1, xk. (All the letters/number after x should be subscripts).
I know that i have to do the determinant of the matrix that is: P01(x)(xk-x)-P0K(x)(x1-x)
Then i should calculate that expressions in x0, x1 and xk and show that the results are rispettivately y0, y1 and yk.
I don't understand what should i do .. i have tried to substitutes xk, x1 and x0 in each expression but i don't find a sense of what i do...or maybe is correct? I need a help..
Thanks in advance.

I cannot read the following attachment:


Please write "darker" like following (your other attachment - this is legible)

 

[amberjack]

i have only substitute Xk, x1, x0 in the expression...its look too easy ..

 

[amberjack]

How can i do to show that the P01K(x) is the interpolating polynomial of second grade for the points x0, x1, xk? I don't have any other informations. It should not be a difficult question...