Please read attachment. - Answer each question with a minimum of 50 words per question:Discuss the advantages and disadvantages for using polynomials for approximation. Document Preview: Answer each...

1)
1)
We can write any function as a polynomial expression of degree n. Where n is the highest power of the variable x involved. For an ex. for any cubic function it will have the degree of 3.
So P(x) =
r
n
r
r
x
a
å
=
0
Where
r
a
is being called the co-efficient of the polynomial which may attain real as well as complex number value.
There are many functions which initially may not look like polynomials but we may express it in a polynomial using certain approximation and it is being called the polynomial approximation.
Suppose we take the example of Sin(x). It’s a trigonometrically function not a polynomial but we may write Sin(x) as
=
)
(
x
Sin
n
n
x
a
x
a
x
a
a
+
+
+
+
..
..........
2
2
1
0
Now Using the Taylor’s Series we can express Sin(x) as
.....
..........
!
5
!
3
)
(
5
3
+
+
-
=
x
x
x
x
Sin
Now if we take only first three terms then it becomes the polynomial of degree 5.
There are many approximation techniques available. Some of them are mentioned below:
a) Taylor’s approximation
b) Least square approximation
c) Chebyshev Polynomial
d) Hermite approximation
e) Legendre Polynomial
Advantages of Polynomial...