Section 8.9: Applications of Taylor Polynomials

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Presentation transcript:

Section 8.9: Applications of Taylor Polynomials Practice HW from Stewart Textbook (not to hand in) p. 628 # 1-21 odd

Taylor Polynomials In this section, we use Taylor polynomials to approximate a given function near a point x = a.

Definition The Taylor polynomial at a function at x = a is given by where

Example 1: Find the Taylor (Maclaurin) polynomial for at a = 0 and n = 2 and use it to approximate . Solution:

Note: In general, the higher the degree of the Taylor polynomial and the closer we are to the point x = a that the Taylor polynomial is centered at, the better the approximation

Example 2: Find the Taylor (Maclaurin) polynomial for at a = 0 and n = 4 and use it to approximate . Solution:

The following table illustrates the accuracy of the Taylor polynomials and to near x = 0. x -8 -1 -0.1 -0.01 0.01 0.1 1 5 0.0003 0.367879 .904837 0.9900498 1.01005016 1.1051709 2.7188281 148.413 25 0.5 0.905 0.99005 1.01005 1.105 2.5 18.5 110.3 0.375 0.904837 1.1051708 2.7083333 65.375

As can be seen, the closer the value of x is nearer to zero, the better the approximation the Taylor polynomials provided for the function. This is further illustrated by the following graphs:

Note: The following Maple commands can be used to find the 2nd and 4th degree Taylor polynomials for near x = 0. > with(Student[Calculus1]): > TaylorApproximation(exp(x), x = 0, order = 2); > TaylorApproximation(exp(x), x = 0, order = 4);

Example 3: Find the Taylor polynomial for the function at a = 1 for n = 3. Solution:

Error of Approximation Can be used to determine how close a Taylor polynomial is to . We define the function , which represents the error between and , as follows:

Estimation of Error – Taylor’s Inequality If for ( ), then for

Note: It is many times useful to use Maple to help determine the error of approximation.

Example 4: For , use Maple to Approximate f by a Taylor polynomial of degree n = 3 centered at a = 1. b. Use Taylor’s inequality to estimate the accuracy of the approximation of for . Solution: (In typewritten notes)