a
Calculate the Taylor polynomials centered at 0,
n f (k) (0) k Tn (x)=∑k=0 k! (x) ,
for f = cos(x) for the following 4 values of ,x
1 2 3 4
x∈ , , , which is equivalent to 10 10 10 10 j x = , for j =1,2,3,4.
10
For each value of , j find the smallest integer such thatn
j j −12
Tn −cos <10 .
10 10
Calculate the "exact" value using np.cos(x). Present your results in a table and discuss them. How do your errors compare to the error bounds for Taylor polynomials that we discussed in class? Remember that the error for a Taylor polynomial is given by its remainder term
(n+1)
f (c) n+1
Rn (x)= f x( )−Tn (x)= (x−a)
(n+1 !)
