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11Asn4

# 11Asn4 - ucsc supplementary notes ams/econ 11a Continuous...

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ucsc supplementary notes ams/econ 11a Continuous Functions, Smooth Functions and the Derivative c 2010 Yonatan Katznelson 1. Continuous functions One of the things that economists like to do with mathematical models is to extrapolate the general behavior of an economic system, e.g., forecast future trends, from theoretical assumptions about the variables and functional relations involved. In other words, they would like to be able to tell what’s going to happen next, based on what just happened; they would like to be able to make long term predictions; and they would like to be able to locate the extreme values of the functions they are studying, to name a few popular applications. Because of this, it is convenient to work with continuous functions. Definition 1. (i) The function y = f ( x ) is continuous at the point x 0 if f ( x 0 ) is defined and (1.1) lim x x 0 f ( x ) = f ( x 0 ) . (ii) The function f ( x ) is continuous in the interval ( a, b ) = { x | a < x < b } , if f ( x ) is continuous at every point in that interval. Less formally, a function is continuous in the interval ( a, b ) if the graph of that function is a continuous (i.e., unbroken) line over that interval. Continuous functions are preferable to discontinuous functions for modeling economic behavior because continuous functions are more predictable. Imagine a function modeling the price of a stock over time. If that function were discontinuous at the point t 0 , then it would be impossible to use our knowledge of the function up to t 0 to predict what will happen to the price of the stock after time t 0 . On the other hand, if the stock price is a continuous function of time, then we know that in the immediate vicinity of t 0 , the prices after t 0 should be close to the prices up to and including t 0 . As nice as the behavior of continuous functions is, compared to functions that are not continuous, it turns out that continuity by itself is not enough. Continuous functions can also display ‘erratic’ behavior, and sudden changes in direction. What we would really like to have are functions that change smoothly . 2. Smooth curves Consider the function f ( x ) = 11 x ( x 2 - x - 2) ( x - 2) 4 + 33 . It is a continuous function, defined for all real numbers x , and its graph is shown below in Figure 1. 1

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2 x0 y0 Figure 1. The graph of y = f ( x ). This graph certainly seems to be nice and smooth. But what does ‘smooth’ mean math- ematically? Smooth usually means no rough edges or corners. To see if we can translate this into more mathematical terms, We’ll focus on the point ( x 0 , f ( x 0 )) = ( x 0 , y 0 ) on the graph, and zoom in. x0 y0 x0 y0 Figure 2. Zooming in twice on ( x 0 , y 0 ). As you zoom in on something that is smooth, it begins to look flat (or straight) no matter how curved it was to begin with. The figure on the lefthand side of Figure 2 looks much straighter in the vicinity of the point ( x 0 , y 0 ) than in the original graph. When we zoom in again, this time on the circled portion of second graph, we get an arc that is almost straight in the immediate vicinity of ( x 0 , y 0 ), as illustrated in the graph on the righthand side of Figure 2.
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11Asn4 - ucsc supplementary notes ams/econ 11a Continuous...

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