Now%20that%20you

1
Now that youve found a polynomial to approximate
your function, how good is your polynomial?
Find the 6th degree Maclaurin polynomial for
For what values of x does this polynomial best
follow the curve? Where does the polynomial
poorly follow the curve?
2
What are the limitations of graphically analyzing
a Taylor polynomial?
3
Suppose that a function f(x) has derivatives at x
0 given by the formula
Write the first few terms of the Taylor series
centered at x 0 for this function.
4
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).
5
Error Bounds for ALTERNATING Series
6
Example
Write the 4th degree Maclaurin polynomial for
Show that this polynomial approximates cos(.9) to
better than 1 part in 1000.
7
Example
Consider the power series
What is the maximum error in truncating the
function after the 4th term on the interval -.5 lt
x lt .5?
8
Example
Suppose that f is a function such that f(2)3 and

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)?
9
What is the 4th degree Maclaurin polynomial for
?
Using the polynomial, estimate y(.2). How good
is your estimate? Why we cant we use our usual
method to estimate the error?
10
Taylors Theorem
The difference between a function at x and its
nth degree Taylor polynomial centered a is
for some c between x and a.
11
Taylors Theorem is an existence theorem. What
does that mean?
What other existence theorems have we seen in
Calculus?
12
Recall our 4th degree polynomial for
and our estimate for y(.2).
Use Taylors Theorem to estimate the difference
between our estimate and the true value of y(.2).
13
Lagrange Error Bound
Choose M to be at least as big as the maximum
value of the n1 derivative on the interval x to
a.
14
Example
Write the 3rd degree Taylor polynomial, P(x), for
centered at x 0.
Estimate the error in using P(.2) to approximate
.