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11Asn7 - ucsc supplementary notes ams/econ 11a Taylor...

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ucsc supplementary notes ams/econ 11a Taylor polynomials c 2008, Yonatan Katznelson 1. Introduction The most elementary functions are polynomials because they involve only the most basic arithmetic operations of addition and multiplication. Polynomials are also easy to differentiate, and their long term behavior is also very easy to understand. Mathematical modeling of economic phenomena, however, often leads to functions which are not polynomials, like exponential functions and logarithm functions, and combinations of functions that involve division, extracting roots, etc. Under certain conditions, it is possible to find polynomials that provide good approx- imations to more general functions. In the following sections I’ll outline one of the most basic and important ways of doing this for functions that are differentiable. I’ll focus on the first and second degree approximations (linear and quadratic), but, for completeness’ sake, I’ll briefly describe the general case as well. 2. Linear approximation. For a function f ( x ) that is differentiable at a point x = x 0 , we deduced the following approximation formula from the definition of the derivative, f ( x ) - f ( x 0 ) f 0 ( x 0 )( x - x 0 ) . (2.1) I mentioned that this approximation is accurate when the difference | x - x 0 | , is sufficiently small . Adding f ( x 0 ) to both sides of (2.1), gives a formula for approximating f ( x ) in the neighborhood of x 0 by a linear function, f ( x ) f ( x 0 ) + f 0 ( x 0 )( x - x 0 ) . (2.2) The linear function, T ( x ) = f ( x 0 )+ f 0 ( x 0 )( x - x 0 ), should look familiar to you, because its graph is precisely the tangent line to the graph y = f ( x ) at the point ( x 0 , f ( x 0 )), and the approximation (2.2) has a simple geometric interpretation. Namely, the tangent line y = T ( x ) is close to the graph y = f ( x ), when x is sufficiently close to x 0 . By close, I mean that the vertical distance , | T ( x ) - f ( x ) | , is small. Example 1. The graph of the function f ( x ) = - x 2 + 4 x + 25 - 2, and the graph of the linear approximation to this graph at the point (4 , 3), T ( x ) = 3 - 0 . 4( x - 4), are both displayed in Figure 1. A quick glance at the figure shows two things. First, the tangent line is very close to the graph when x is close to 4. Second, as x moves away from 4 (in either direction), the In mathematical terms, a ‘ neighborhood ’ of x 0 is an interval around x 0 of the form ( x 0 - δ, x 0 + δ ), where δ is a small, positive constant. 1
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 -0.8 0.8 1.6 2.4 3.2 4 4.8 y=T(x) y=f(x) Figure 1: Linear approximation to f ( x ) = - x 2 + 4 x + 25 - 2, centered at (4 , 3). vertical distance between the tangent line and the graph grows larger. The explanation for this first phenomenon has already been given, but we can also explain it as follows. The two functions, T ( x ) and f ( x ), have the same value and the same slope at x 0 = 4, i.e., T (4) = f (4) and T 0 (4) = f 0 (4). This means that their graphs emanate from the same point (4 , 3) and they both move in the same direction (same slope). So, for a while, the two graphs stay close to each other.
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