cal9 - Chapter Nine The Taylor Polynomial 9.1 Introduction...

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9.1 Chapter Nine The Taylor Polynomial 9.1 Introduction Let f be a function and let F be a collection of "nice" functions. The approximation problem is simply to find a function g F that is "close" to the given function f . There are two issues immediately. How is the collection F selected, and what do we mean by "close"? The answers depend on the problem at hand. Presumably we want to do something to f that is difficult or impossible (This might be something as simple as finding f x ( ) for some x. ). The collection F would thus consist of functions to which it is easy to do that which we wish to do to f . Our measure of how close one function is to another would try to reflect the closeness of the results of our operations. Now, what are we talking about here. Suppose, for example, we wish to find f x ( ) . Our collection F of functions should include functions that are easy to evaluate at x , and two function would be "close" simply if there values are close. We might, for instance, want to evaluate sin x for all x is some interval I . The collection F could be a collection of second degree polynomials. The approximation problem is then to find elements of F that make the "distance" max{|sin ( )|: } x p x x I - as small as possible. Similarly, we might want to find the integral of some function f over an interval I . Here we would want F to consist of functions easily integrated and measure the distance between functions by the difference of their integrals over I . In the previous chapter, we found the "best" straight line approximation to a set of data points. In that case, the collection
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cal9 - Chapter Nine The Taylor Polynomial 9.1 Introduction...

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