Solutions To Mathematics Textbooks/Calculus (3rd) (0521867444)/Chapter 3

=Question 6 =

a)
Find $$ f_i(x) n-1 $$ degree polynomial where: $$ f_i(x_i) = 1$$ and $$ f_i(x_j) = 0$$

Noting that: $$ f_i(x) = x^{n-1} +... $$

is the same as $$ f_i(x) = \prod^n_{j=1} x_j + ... $$

Then:

$$ f_i(x) =\prod^n_{j=1} \frac{(x-x_j)}{(x_i-x_j)}$$ where $$ j != i $$

b)
Now find a polynomial function $$ f $$ of degree $$ n-1 $$ such that $$  f(x_i) = a_i $$

$$ f(x) = \sum^n_{i=1} a_i  f_i(x) $$ where $$ j != i $$

so: $$ f(x) = \sum^n_{i=1} a_i  \prod^n_{j=1} \frac{(x-x_j)}{(x_i-x_j)} $$ where $$ j != i $$

(Note that in this equation will always resulting in 0 unless x = x_i)