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

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# Question 6

## a)

Find ${\displaystyle f_{i}(x)n-1}$ degree polynomial where: ${\displaystyle f_{i}(x_{i})=1}$ and ${\displaystyle f_{i}(x_{j})=0}$

Noting that: ${\displaystyle f_{i}(x)=x^{n-1}+...}$

is the same as ${\displaystyle f_{i}(x)=\prod _{j=1}^{n}x_{j}+...}$

Then:

${\displaystyle f_{i}(x)=\prod _{j=1}^{n}{\frac {(x-x_{j})}{(x_{i}-x_{j})}}}$ where ${\displaystyle j!=i}$

## b)

Now find a polynomial function ${\displaystyle f}$ of degree ${\displaystyle n-1}$ such that ${\displaystyle f(x_{i})=a_{i}}$

${\displaystyle f(x)=\sum _{i=1}^{n}a_{i}f_{i}(x)}$ where ${\displaystyle j!=i}$

so: ${\displaystyle f(x)=\sum _{i=1}^{n}a_{i}\prod _{j=1}^{n}{\frac {(x-x_{j})}{(x_{i}-x_{j})}}}$ where ${\displaystyle j!=i}$

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