Now that you’ve found a polynomial to approximate your function, how good is your polynomial? Find the 6 th degree Maclaurin polynomial for For what values.

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Presentation transcript:

Now that you’ve found a polynomial to approximate your function, how good is your polynomial? Find the 6 th degree Maclaurin polynomial for 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: Write the first few terms of the Taylor series centered at x = 0 for this function.

Write the 4 th degree Taylor polynomial for f centered at x = 0. Estimate the error in using the 4 th degree polynomial to approximate f(0.2).

Error Bounds for ALTERNATING Series

Example Write the 4 th degree Maclaurin polynomial for: Show that this polynomial approximates cos(.9) to better than 1 part in 1000.

Example Consider the power series: What is the maximum error in truncating the function after the 4 th term on the interval -.5 < x <.5?

Example Suppose that f is a function such that f(2)=3 and : Write the 3 rd 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 4 th degree Maclaurin polynomial for ? 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 n th degree Taylor polynomial centered a is: 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 4 th degree polynomial for and our estimate for y(.2). 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.

Example Write the 3 rd degree Taylor polynomial, P(x), for centered at x= 0. Estimate the error in using P(.2) to approximate.