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Relationships and graphs

Suppose we consider an object identified with three parameters – call them A, B and C – and that each
parameter is dependent somehow on some other independent parameter called w. The relationships between A, B and C are determined by a set of rules. In this case, the rules are:
1. On a graph of B vs. w, the slope of B is given by the rate of change of A with respect to w. Further,
the value of B for any particular value of w is given by the area under the plot of A vs. w.
2. On a graph of C vs. w, the slope of C is given by the rate of change of B with respect to w. Further
the value of C for any particular value of w is given by the area under the plot of B vs.w.
3. The particular value of B at w = 0 is given by B^0 and the value of C at w = 0 is given by C^0.
Our task is divided into two parts. Using only the properties of the graphs mentioned in the rule set, we are to first show that the symbolic relationship for the behavior of B vs. w and C vs. w, for the conditions that B^0 =0 and C^0 =0 is given by B = A W and C = 1/2 A w^2. Note carefully that the results to be proven must involve properties of the graphs and not simply algebraic manipulations.
The second step is to relax the conditions on B^0 and C^0 to be any non-zero value and to show the relationships to then be B = B^0 + A w and C = C^0 + B^0 w + 1/2 Aw^2. Once again, note carefully that the results to be proven must involve properties of the graphs and not simply algebraic manipulations.

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