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Unformatted text preview: Chapter 12 Approximation methods 12.1 Introduction In this chapter we explore a few techniques for finding approximate solutions to problems of great practical significance. The techniques here described are linked by a number of common features; most notably, all are based on exploiting the fact that a tangent line is a good (local) approximation to the behaviour of a function (at least close to the point of tangency). Most of these methods are numerical, i.e. involve calculations with real numbers. Here is where the features of a spreadsheet can help significantly, to cut out the drudgery of hand calculations or even to avoid the need for a calculator 1 . The first method, that of linear approximation has been discussed before, and is a direct appli- cation of the tangent line as such an approximation. We will illustrate how this approximation can lead to simple one-step computation of rough values that a function of interest takes. A second technique described here, Newtons method, is used to find precise decimal approxi- mations to zeros of a function: recall these are places where a function crosses the x-axis, i.e. where f ( x ) = 0. The method gives an important example of an iteration scheme : that is, a recipe that is repeated (several times) to generate successively finer approximations. A third technique is applied to calculating numerical solutions of a differential equation. This method, called Eulers method, uses the initial condition and the differential equation to compute approximate values of the solution step by step, starting with the initial time and incrementally computing the solution value for each of many small time steps. While some of these techniques have been superseded by improved (graphics) calculators, or mathematical software, the concepts behind the methods are still fundamental. Also important is understanding the limitations of such methods, since each relies on certain assumptions and underlying concepts. 12.2 Linear approximation We have already encountered the idea that the tangent line approximates the behaviour of a func- tion. In this technique, the approximation is used to generate rough values of a function close to 1 Labs 5 and 6 of this calculus course are based on methods taught in this chapter, namely on Eulers method for solving differential equations, and Newtons method for finding roots. The students will see, perhaps for the first time, the idea of iterations in such examples. v.2005.1 - September 4, 2009 1 Math 102 Notes Chapter 12 some point at which the value of the function and of its derivative are known, or easy to calculate. Below we illustrate the idea of linear approximation to the function y = f ( x ) = x....
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This note was uploaded on 05/19/2011 for the course MATH 102 Math 102 taught by Professor Allard during the Winter '09 term at The University of British Columbia.
- Winter '09