Applications of algebraic K-theory to algebraic geometry and by Bloch S.J., et al. (eds.)

By Bloch S.J., et al. (eds.)

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8. Table 0-3 Example 3 Let y = 2x2 +2x — 1, the quadratic in Example 1. 4. In Example 1, we found algebraically the roots to be the irrational numbers ( - 1 + V^)/2 and ( - 1 - \/3 )/2. 366 ( « means approximately equal). These values are rational approximations to the irrational values, but closer than can be read from a graph (Figure 0-4). The graph of y = ax2 + bx + c, where a, b, and c are real numbers, and a > 0, is always a curve with a shape of this one, called a parabola, symmetric with respect to an axis parallel to the y axis, extending indefinitely upward more and more steeply.

80 60 40 20 \ 28 32 36 40 44 48 52 Fig. 1-4 20 28 62 54 36 40 28 32 44 48 52 Fig. 1-4' 37 1 Functional relationships A principal concern of human beings in general, and of business people in particular, is to attempt to predict the future. Hence, extrapolation from sets of data is of great interest, but often unreliable, on account of unforeseen factors that keep things from going on as they did in the past. This is illustrated by the following problem. 4 The number of overseas telephone calls originating in the United States (N million per year) varied with the time (t years after 1950) as in Table 1-5.

3 X - 2 = 1 0 ~ 3 J C - 1 2 = 0. Let y = 3x —12. We see in Table 0-1 that for an increase of one unit in x there is an increase of three units in y, and if we look at the formula we see that this is true for any increase of one unit in x. This means that the graph is a straight line, of "slope" 3, as in Figure 0-1. The line crosses the x axis at (4,0); that is, x = 4 is the solution of 3x -12 = 0. , free of x\ and a¥^0. 7). " Remember that in the ordinary graphical system there is a one-to-one correspondence between the points of the plane and the ordered pairs of real numbers; for example, (1,2) and (2,1) represent different points.

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