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**1 - 2**of**2**### Families of Linear Functions and Their Envelopes

"... 2 (the upper branch of the hyperbola y 2 \Gamma x 2 = 1), i.e. the set of lines of the form y = ax + p 1 + a 2 . Figure 1 leads us to the conjecture that we may view this family of linear functions as the set of lines tangent to the graph of the upper half of the unit circle. In general we ..."

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2 (the upper branch of the hyperbola y 2 \Gamma x 2 = 1), i.e. the set of lines of the form y = ax + p 1 + a 2 . Figure 1 leads us to the conjecture that we may view this family of linear functions as the set of lines tangent to the graph of the upper half of the unit circle. In general we may view the family of linear functions generated by a differentiable function y = f(x) as the set of lines tangent to the graph of a differentiable function y = g(x). The function g(x) is called the envelope of the family of linear functions and our goal is to find the explict form of this function. We use the notation f(x) ! g(x) to denote that the envelope of the famil