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Exploring Implicit Functions: Derivatives, Frontiers, Theorems

Learn about implicit functions, derivatives, and the Production Possibility Frontier in mathematics, with insights on solving and understanding these concepts. Discover the Implicit Function Theorem's importance in various applications.

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Exploring Implicit Functions: Derivatives, Frontiers, Theorems

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  1. Implicit Functions • The equation y = mx + b is an explicit function • y is considered the dependent variable • x is the independent variable • The equation f(x,y,m,b) = 0 is an implicit function • The relationships between the variables and the parameters are implicitly present in the equation but not explicitly stated

  2. Derivatives from Implicit Functions • In many circumstances, it will be helpful to compute derivatives directly from implicit functions • If f(x,y)=0, then its total differential of f1dx + f2dy = 0 • Thus,

  3. Production Possibility Frontier • Earlier example: 2x2 + y2 = 225 • Can be rewritten: f(x,y) = 2x2 + y2 - 225 = 0 • Then, the opportunity cost trade-off between x and y is Because fx = 4x and fy = 2y

  4. Implicit Function Theorem • It may not always be possible to solve implicit functions of the form g(x,y)=0 for unique explicit functions of the form y = f(x) • Mathematicians have derived the necessary conditions • In many economic applications, these conditions are the same as the second-order conditions for a maximum (or minimum)

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