Calculus Polynomial

the simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. important example in calculus taylor s theorem, states every differentiable function locally looks polynomial function, , stone–weierstrass theorem, states every continuous function defined on compact interval of real axis can approximated on whole interval closely desired polynomial function.

calculating derivatives , integrals of polynomial functions particularly simple. polynomial function

∑

i
=
0


n



a

i



x

i




{\displaystyle \sum _{i=0}^{n}a_{i}x^{i}}

the derivative respect x is

∑

i
=
1


n



a

i


i

x

i
−
1




{\displaystyle \sum _{i=1}^{n}a_{i}ix^{i-1}}

and indefinite integral is

∑

i
=
0


n





a

i



i
+
1




x

i
+
1


+
c
.


{\displaystyle \sum _{i=0}^{n}{a_{i} \over i+1}x^{i+1}+c.}