Infinite Series 9 Copyright © Cengage Learning. All rights reserved.

Taylor Polynomials and Approximations Copyright © Cengage Learning. All rights reserved. 9.7

3 Find polynomial approximations of elementary functions and compare them with the elementary functions. Find Taylor and Maclaurin polynomial approximations of elementary functions. Use the remainder of a Taylor polynomial. Objectives

4 Polynomial Approximations of Elementary Functions

5 To find a polynomial function P that approximates another function f, begin by choosing a number c in the domain of f at which f and P have the same value. That is, The approximating polynomial is said to be expanded about c or centered at c. Geometrically, the requirement that P(c) = f(c) means that the graph of P passes through the point (c, f(c)). Of course, there are many polynomials whose graphs pass through the point (c, f(c)). Need to find a polynomial whose graph resembles the graph of f near this point. One way to do this is to impose the additional requirement that the slope of the polynomial function be the same as the slope of the graph of f at the point (c, f(c)). With these two requirements, you can obtain a simple linear approximation of f, as shown in Figure 9.10.

6 Example 1 For the function f(x) = e x, find a first-degree polynomial function P 1 (x) = a 0 + a 1 x whose value and slope agree with the value and slope of f at x = 0. Solution: Because f(x) = e x and f'(x) = e x, the value and the slope of f, at x = 0, are given by f(0) = e 0 = 1 and f'(0) = e 0 = 1. NOTE that P1 = tangent line at 0.

7 As x moves further away from 0, the accuracy of approximating by P3 decreases

8 My example:

9 Taylor and Maclaurin Polynomials

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11 Example 3 – A Maclaurin Polynomial for f(x) = e x Find the nth Maclaurin polynomial for f(x) = e x. Solution: The nth Maclaurin polynomial for f(x) = e x is given by

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13 CAS Implementation

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15 My example Maclaurin series for sin(x) about c=0.

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17 Ex 4: use polyn of ln(x) around c=1 Important properties, true for any Taylor polynomial:

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19 Remainder of a Taylor Polynomial

20 Remainder of a Taylor Polynomial An approximation technique is of little value without some idea of its accuracy. To measure the accuracy of approximating a function value f(x) by the Taylor polynomial P n (x), you can use the concept of a remainder R n (x), defined as follows. So, R n (x) = f(x) – P n (x). The absolute value of R n (x) is called the error associated with the approximation. That is, The next theorem gives a general procedure for estimating the remainder associated with a Taylor polynomial. This important theorem is called Taylor’s Theorem, and the remainder given in the theorem is called the Lagrange form of the remainder.

21 Important note!!! We can NOT find z, just find bounds for Rn Proof is very technical, see appendix A Rn looks exactly like next term of the Taylor series, but evaluated at an unknown point z.

22 I computed more Precise approx => Error is smaller

23 My example Use the 3 rd Maclaurin polynomial for arctan(x) around 0 to approximate arctan(0.2). Estimate the error and compare to exact error Note harder derivatives =P3(x) Max of 4 th derivative on [0,0.2] is achieved on the right corner of the domain at 0.2 => Take it as z – value. The exact error is smaller than Rmax as it should be

24 pattern of derivatives while loop to find Nmax

25 My example

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