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 concavity: 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. Firstorder equations whose righthand 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...
View
Full Document
 Spring '11
 Dr.Han
 Derivative, Convex function

Click to edit the document details