Taylor Theorem Revisited

Taylor Theorem Revisited - Chapter 01.07 Taylor Theorem...

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Chapter 01.07 Taylor Theorem Revisited After reading this chapter, you should be able to 1. understand the basics of Taylor’s theorem, 2. write transcendental and trigonometric functions as Taylor’s polynomial, 3. use Taylor’s theorem to find the values of a function at any point, given the values of the function and all its derivatives at a particular point, 4. calculate errors and error bounds of approximating a function by Taylor series, and 5. revisit the chapter whenever Taylor’s theorem is used to derive or explain numerical methods for various mathematical procedures. The use of Taylor series exists in so many aspects of numerical methods that it is imperative to devote a separate chapter to its review and applications. For example, you must have come across expressions such as + - + - = ! 6 ! 4 ! 2 1 ) cos( 6 4 2 x x x x (1) + - + - = ! 7 ! 5 ! 3 ) sin( 7 5 3 x x x x x (2) + + + + = ! 3 ! 2 1 3 2 x x x e x (3) All the above expressions are actually a special case of Taylor series called the Maclaurin series. Why are these applications of Taylor’s theorem important for numerical methods? Expressions such as given in Equations (1), (2) and (3) give you a way to find the approximate values of these functions by using the basic arithmetic operations of addition, subtraction, division, and multiplication. Example 1 Find the value of 25 . 0 e using the first five terms of the Maclaurin series. Solution The first five terms of the Maclaurin series for x e is ! 4 ! 3 ! 2 1 4 3 2 x x x x e x + + + + 01.07.1
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01.07.2 Chapter 01.07 ! 4 25 . 0 ! 3 25 . 0 ! 2 25 . 0 25 . 0 1 4 3 2 25 . 0 + + + + e 2840 . 1 = The exact value of 25 . 0 e up to 5 significant digits is also 1.2840. But the above discussion and example do not answer our question of what a Taylor series is. Here it is, for a function ( 29 x f ( 29 ( 29 ( 29 ( 29 ( 29 + + + + = + 3 2 ! 3 ! 2 h x f h x f h x f x f h x f (4) provided all derivatives of ( 29 x f exist and are continuous between x and h x + . What does this mean in plain English? As Archimedes would have said ( without the fine print ), “ Give me the value of the function at a single point, and the value of all (first, second, and so on) its derivatives, and I can give you the value of the function at any other point ”. It is very important to note that the Taylor series is not asking for the expression of the function and its derivatives, just the value of the function and its derivatives at a single point. Now the fine print : Yes, all the derivatives have to exist and be continuous between x (the point where you are) to the point, h x + where you are wanting to calculate the function at. However, if you want to calculate the function approximately by using the th n order Taylor polynomial, then th nd st n , .... , 2 , 1
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