Page 883 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

2. is continuous on . Because is continuous for , it follows that at each point in the set ,
, …, we must have

(10)

Otherwise, the graphs of and would not join together to form a continuous curve at .

When we apply the interpolating property , it follows from 10 that , , 3, …,

, or from 4 that

(11)

3. is continuous on . Because is continuous for , it follows that
or, from 5,

(12)

4. is continuous on . Because is continuous for , it follows that

or, from 6,

(13)

Equations 8, 9, 11, 12, and 13 constitute a system of linear equations in the unknown coefficients , , , , ,

2, …, . Consequently, we need two more equations to determine these coefficients uniquely. Before obtaining these additional

equations, however, we can simplify our existing system by expressing the unknowns , , , and in terms of new unknown

quantities

and the known quantities

For example, from 6 it follows that

so