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Unformatted text preview: If f ( x ) > 0 the graph is increasing. If f ( x ) < 0 the graph is going down. (3) The second derivative f 00 ( x ) gives information about concav-ity: If f 00 ( x ) > 0 the graph is concave upward. If f 00 ( x ) < 0 the graph is concave downward. The concepts of concavity (up or down) and monotonicity (going up or down) is illustrated in the following diagram. Figure 8. Concavity and Monotonicity 5.1. The phase diagram. First-order equations whose right-hand side does not depend on the independent variable are called autonomous. That is, autonomous equations are those of the form x = dx dt = f ( x ) (The independent variable is t )or y = dy dx = f ( y ) (The independent variable is x ). To draw a phase diagram of x = f ( x ), you would...
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This note was uploaded on 10/04/2011 for the course MAP 4231 taught by Professor Dr.han during the Spring '11 term at UNF.
- Spring '11