9.7 Taylor Polynomials & Approximations Objectives: Find Taylor and Maclaurin polynomials Use Taylor & Maclaurin Polynomials to approximate Find the error of Taylor & Maclaurin approximations ©2004Roy L. Gover (www.mrgover.com)

Definition A polynomial in x is any expression in the form Where n is a nonnegative integer and coefficients are complex numbers

Example The following are polynomials: 1st degree 2nd degree 3rd degree

Important Idea Polynomial functions can be used to approximate other functions such as: The approximation can be as accurate as we choose.

Example Find a polynomial, P(x), that has the same value & slope as f(x) at x=0 Approximate at values close to x=0 P(0)=f(0)=1 Slope of P(x)=f’(0)=1

Example P(0)=f(0)=1 Approximate at values close to x=0 Slope of P(x)=f’(0)=1 therefore Remember

Example Approximate at values close to x=0 x f(x) P(x) -.25 . 25

Definition x P(x) converges to f(x) at x=0. P(x) is centered at 0 1 is a 1st degree approximation

Important Idea As x moves either side of center (x=0), the approximation of f(x) by P(x) decreases in accuracy x c

Important Idea P(x) is an accurate approximationof f(x) at values of x close to c x c

Example Find such that: Find a more accurate approximation of centered at x=0.

Example Find a more accurate approximation of centered at x=0. is a 2nd degree approximation

Example Find a more accurate approximation of centered at x=0. Begins to diverge Notice how the polynomial more closely fits

Analysis x f(x) P2(x) -.5 .6065 .625 -.25 .7788 .7813 1 .25 1.284 1.281 .5 1.649 1.625 A 2nd degree approx. is more accurate for more values of x

Example Find an even more accurate approx. for is a 3rd degree approximation Notice how the polynomial fits even more closely

Analysis Do you see a pattern?

Important Idea

Try This Graph a 5th degree polynomial approximation for using your graphing calculator. Use the table feature of your calculator to find the approximation for and compare with the calculator value for e.

Solution looks good

Solution per approx. 2.7183 per calculator

Important Idea Can be generalized to center at values other than 0

Important Idea is the nth polynomial approx. for centered at c.

Important Idea Generalizing even more, the following polynomial can be used to approx any function at any center:

Important Idea We just need a way to find the values of the coefficients for each we wish to approx.

Definition The nth Taylor polynomial for f at c:

Definition When referring to Taylor polynomials, we can talk about number of terms, order or degree. For Example: centered at 0.

Definition 1 2 3 This Taylor Polynomial has 3 terms

Definition It is a 4th order Taylor polynomial, because it was found using the 4th derivative. (The 1st and 3rd derivative is 0)

Definition It is a 4th degree Taylor polynomial, because the highest degree term is

Definition The Maclaurin Polynomial is the Taylor Polynomial with c=0

Important Idea The value of f and its first n derivatives must agree with the value of P and its first n derivatives at x=c. See p. 598

Example Find the Taylor Polynomials for and for centered at c=1 Steps: 1. Find the coefficients 2. Substitute into Taylor

Try This Find the Taylor Polynomial centered at c=1 for

Example Use the Taylor Polynomial centered at c=1 for to approximate ln (1.1) Ln(1.1) approx 0.09531

Example Compare the graphs of with &

Try This Find the Maclaurin 6th degree approximation, , for . Approximate and compare with the calculated value .

Important Idea x’s close to c Better approximations occur for: Higher degree polynomials

Try This Which will give the better approximation of ln (1.1)? The 4th Taylor Polynomial centered at c=1 and x=1.1 for ln(x) The 4th Maclaurin Polynimial (centered at c=0) and x=.1 for ln(1+x) See Example 7, page 653 Approximations are the same

Definition The Remainder of a Taylor polynomial, is the difference between the exact value and the Taylor approximation. Exact Value Approx. Value

Important Idea If you evaluate a Taylor polynomial to n terms as an approximation for f(x) centered at c, then the difference between the actual and approximate values, ,for f(x) is related to the n+1 term-as follows:

Definition The Error of a Taylor approximation is: where: and z is in [c,x] Lagrange Remainder (Error)

Definition This is the n+1 term of the Taylor polynomial Choose that gives the max. value for Maclaurin c=0.

Example Find the 3rd degree Maclaurin polynomial for . Use to approximate sin(.01) and determine the accuracy of your approximation. P 603,ex 8

Important Idea When finding the Lagrange Form of the Remainder, you should not try to find the exact value of z. Rather, you should try to find a z that gives you an upper bound for

Definition Lagrange Remainder: The error of an nth polynomial approximation is less than the absolute value of the maximum of the n+1 term of the polynomial approximation.

Example Estimate the error (Lagrange form of the remainder) when is replaced by its Maclaurin polynomial: for x=0.1

Example Use the 3rd degree Taylor polynomial centered at c=1 to approximate . Estimate the error.

Try This Find the 4rd degree Maclaurin polynomial for . Use to approximate cos(.02) and determine the accuracy of your approximation. where