# 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.