Now that you’ve found a polynomial to approximate your function, how good is your polynomial?
Find the 6th degree Maclaurin polynomial for x
xxf
sin
For what values of x does this polynomial best follow the curve? Where does the polynomial poorly follow the curve?
What are the limitations of graphically analyzing a Taylor polynomial?
Suppose that a function f(x) has derivatives at x = 0 given by the formula:
n
nn nf
3
!10
Write the first few terms of the Taylor series centered at x = 0 for this function.
Write the 4th degree Taylor polynomial for f centered at x = 0.
Estimate the error in using the 4th degree polynomial to approximate f(0.2).
Error Bounds for ALTERNATING Series
Example Write the 4th degree Maclaurin polynomial for:
xy cos
Show that this polynomial approximates cos(.9) to better than 1 part in 1000.
Example Consider the power series:
...21...642 12753 nn xnxxxxxf
What is the maximum error in truncating the function after the 4th term on the interval -.5 < x < .5?
Example Suppose that f is a function such that f(2)=3 and : nnf 22
Write the 3rd degree Taylor polynomial for f centered at x = 2.
Estimate f(2.1). What is the maximum difference between your estimate and the actual value of f(2.1)?
What is the 4th degree Maclaurin polynomial for ?2xey
Using the polynomial, estimate y(.2). How good is your estimate? Why we can’t we use our usual method to estimate the error?
Taylor’s Theorem
The difference between a function at x and it’s nth degree Taylor polynomial centered a is:
1
1
!1
nn
n axn
cfxR
for some c between x and a.
Taylor’s Theorem is an existence theorem. What does that mean?
What other existence theorems have we seen in Calculus?
Recall our 4th degree polynomial for and our estimate for y(.2).
2xey
Use Taylor’s Theorem to estimate the difference between our estimate and the true value of y(.2).
Lagrange Error Bound
Choose M to be at least as big as the maximum value of the n+1 derivative on the interval x to a.
1
1
n
xMxR
n
Example
Write the 3rd degree Taylor polynomial, P(x), for centered at x= 0.
xey 2
Estimate the error in using P(.2) to approximate . 4.0e