S ECT. 9-4 T AYLOR P OLYNOMIALS

Taylor Polynomials In this section we will be finding polynomial functions that can be used to approximate transcendental functions. If is a polynomial function used to approximate some other function, they must contain the same point with some x-value c. That means.

Taylor Polynomials To be a better approximation they should have the same slope at that point. This means. For even greater accuracy, and and so on.

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

Putting this together gives us the Taylor Polynomial Expansion: If f (x) has derivatives of all orders it can be approximated by the polynomial function shown This is called an n th degree or n th order Taylor Polynomial centered at c or expanded about c. Where the coefficient is given by: Taylor Polynomial Expansion

When the center is at c = o the Taylor polynomial is called a Maclaurin Polynomial which can be written as : By extending this pattern into an infinite series it becomes exactly correct instead of an approximation. Maclaurin Polynomial

When referring to Taylor polynomials, we can talk about number of terms, order or degree. This is a polynomial in 3 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.

1) Find the Maclaurin Polynomial of degree 5 for

2)Approximate the function by a Taylor Polynomial of degree 2 where a = 8

3) Approximate the function by a Maclaurin Polynomial of degree 4

Close approximation Using P 6 (x) we get an approximation for cos (0.1) ≈

H OME W ORK Page 658 # 1-4 (use graphing calculator) # 13, 15, 17, 19, 25, 26, 27, and 29