Drawing phase diagrams
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Please explain the steps and show the graphs. Thank you.
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The expert draws phase diagrams for graphs.
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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 ...
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