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Drawing phase diagrams

Please explain the steps and show the graphs. Thank you.

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is the rate of change of with respect to time.
• If then increases with time.
• If then decreases with time.
• if then remains constant.

is the rate of change of with respect to time.
• If then (the rate of change of x) increases with time.,
• If then decreases with time.
• if then remains constant (that is the rate of change of x remains the same)

Let's draw on the same graph , represented by the blue line, and represented by the red line:

A region where the function is "concave down" is an interval where the function attains a local maxima. In a maxima the function goes from an increase (positive ) to a decrease (negative )

A region where the function is "concave up" is an interval where the function attains a local minima. In a minima the function goes from a decrease (negative ) to an increase (positive )
If we assume is continuous, the it implies that in both maxima and minima points we must have
In between two local extremas we must have a point where the concavity of the function changes from up to down, or from down to up.
This means that at that point the rate ...

Solution Summary

The expert draws phase diagrams for graphs.

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