Find the local linear approximation of f(x) = e x at x = 0. Find the local quadratic approximation of f(x) = e x at x = 0.

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Presentation transcript:

Find the local linear approximation of f(x) = e x at x = 0. Find the local quadratic approximation of f(x) = e x at x = 0.

we want to find a second degree polynomial of the form: atthat approximates the behavior of If we make, and the first, and the second, derivatives the same, then we would have a pretty good approximation.

Suppose we wanted to find a fourth degree polynomial of the form: atthat approximates the behavior of If we make, and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation.

If we plot both functions, we see that near zero the functions match very well!

This pattern occurs no matter what the original function was! Our polynomial: has the form: or:

Maclaurin Series: (generated by f at ) If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series: Taylor Series: (generated by f at )

Find the Taylor Series for centered at a =1 And determine the interval of convergence

If the limit of the ratio between consecutive terms is less than one, then the series will converge.

The interval of convergence is (0,2]. The radius of convergence is 1. If the limit of the ratio between consecutive terms is less than one, then the series will converge.

Write the Taylor Series for f(x) = cos x centered at x = 0.

The more terms we add, the better our approximation.

For use the Ratio Test to determine the interval of convergence.

If the limit of the ratio between consecutive terms is less than one, then the series will converge. The interval of convergence is

When referring to Taylor polynomials, we can talk about number of terms, order or degree. This is a polynomial with 3 positive terms. It is a 4th order Taylor polynomial, because it was found using the 4th derivative. It is also a 4th degree polynomial, because x is raised to the 4th power. The 3rd order polynomial for is, but it is degree 2. The x 3 term drops out when using the third derivative. This is also the 2nd order polynomial. A recent AP exam required the student to know the difference between order and degree.

Both sides are even functions. Cos (0) = 1 for both sides.

Both sides are odd functions. Sin (0) = 0 for both sides.

What is the interval of convergence for the sine series?

Power series for elementary functions Interval of Convergence