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11Bsn2 - ams/econ 11b supplementary notes ucsc Linear and...

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ams/econ 11b supplementary notes ucsc Linear and quadratic Taylor polynomials for functions of several variables. c 2010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is one of the most important applications of differential calculus to economics. In general, there are two steps to this process: (i) finding the the points where extreme values may occur, and (ii) analyzing the behavior of the function near these points, to determine whether or not extreme values actually occur there. For a function of one variable, we saw that step (i) consisted of finding the critical points of the function, and step (ii) consisted of using the first or second derivative test to analyze the behavior of the function near the point. We use the same two steps to find the critical points and classify the critical values of functions of several variables, and to understand how the procedure generalizes, we need to first understand the linear and quadratic Taylor’s polynomials for functions of several variables, and the approximations that they provide. Comment : I will assume throughout this note that all the functions being discussed have continuous first, second and third order derivatives (or partial derivatives). This assumption justifies the claims made below about the accuracy of the approximations. 1. The one-variable case. To makes sense of Taylor polynomials in several variables, we first recall what they look like for functions of one variable. 1.1 The first order (linear) approximation The first order Taylor polynomial for y = f ( t ), centered at t 0 is the linear function T 1 ( t ) = f ( t 0 ) + f 0 ( t 0 ) · ( t - t 0 ) . (1) With this definition of T 1 , the approximation f ( t ) T 1 ( t ) , (2) is very good if t is sufficiently close to t 0 . This approximation is also called the tangent line approximation because the graph y = T 1 ( t ) is the tangent line to the graph y = f ( t ) at the point ( t 0 , f ( t 0 )). 1.2 The second order (quadratic) approximation The second order Taylor polynomial for the function y = f ( t ), centered at t 0 is the quadratic function T 2 ( t ) = f ( t 0 ) + f 0 ( t 0 )( t - t 0 ) + f 00 ( t 0 ) 2 ( t - t 0 ) 2 . (3) For a more thorough discussion of Taylor polynomials for functions of one variable, please see SN 7 on the review page of the 11A website: http://people.ucsc.edu/ ˜ yorik/11A/review.htm . The error of approximation | f ( t ) - T 1 ( t ) | is less than a multiple of | t - t 0 | 2 . As t approaches t 0 , the squared difference | t - t 0 | 2 goes to 0 very rapidly. 1
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You should note that T 2 ( t ) = T 1 ( t )+ f 00 ( t 0 ) 2 ( t - t 0 ) 2 . As for the linear approximation, the quadratic approximation f ( t ) T 2 ( t ) . (4) is very good if t is sufficiently close to t 0 . In fact, the quadratic approximation is typically much better than the linear approximation once t is very close to t 0 . § At this point the key question is: Why do the Taylor polynomials T 1 ( t ) and T 2 ( t ) do such a good job of approximating the original function f ( t ) in the neighborhood of t 0 ?
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