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Economics Dynamics Problems 58

Economics Dynamics Problems 58 - isoclines Isoclines and...

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42 Economic Dynamics Figure 2.6. Hence, f ( x ) reaches a minimum at x = x where f ( x ) cuts the line y = ax / b . It must follow, then, that for x > x , f ( x ) is positively sloped. This can be verified immediately f ( x ) = ax bf ( x ) x > x implying ax b > f ( x ) or ax > bf ( x ) . . . f ( x ) > 0 All the analysis so far allows us to graph the properties, as shown in figure 2.6. The curve f ( x ) cuts the y -axis at a / b , declines and reaches a minimum where f ( x ) cuts the line y = ax / b , and then turns up. Although we cannot identify f ( x ) or the solution value of x , we do know that x is nonzero. But can we obtain additional information about the shape of f ( x )? Yes – if we consider
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Unformatted text preview: isoclines . Isoclines and direction Felds Given dy dx = ax − by then for every ( x , y )-combination this equation speciFes the slope at that point. A plot of all such slopes gives the direction Feld for the differential equation, and gives the ‘flow of solutions’. (The slopes at given points can be considered as small lines, like iron Flings, and if many of these are drawn the direction Feld is revealed – just like iron Flings reveal magnetic forces.) However, it is...
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