# Since the same relation holds true for a variable y

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. Since the same relation holds true for a variable y , if y = f ( x ), we have a further way of defining the elasticity of y with respect to x : E = ( dy/y )/( dx/x ) = d ( ln y ) /d ( ln x ) . The second application of exponential functions concerns growth rates. In Module 9, we shall see how use of the exponential growth function X t = X 0 e gt (M.8.1) can actually simplify some calculations! Finding the Roots of Quadratic Equations : This note is a reminder that the roots of a quadratic equation in the form ax 2 + bx + c = 0 may be calculated using the following formula: x = b ± 2 b a 2 – 4 ac . (M.8.2) Recall that if the expression under the square root sign is negative, the roots of the equa- tion have an imaginary component. M8-2 MATH MODULE 8: SOME SPECIAL FUNCTIONS AND FORMULAS

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1.2 SOME SPECIAL FUNCTIONS The text contains extended discussions of some important functions that are widely used in consumption and production theory: the Cobb-Douglas, perfect complement, and perfect substitute functions. Cobb-Douglas functions are analyzed in Appendixes 3 and 9, especially on pages 592-3 and 636-8. Perfect complement and perfect substitute functions are discussed on pages 587-9 and 639. The importance of these functions is not primarily their realism, but rather the fact that they provide some simple yet strong cases that indicate the widely different outcomes that can occur in economic processes as a result of the nature of differing utility and production functions. Another function that rests on special assumptions is the constant-elasticity demand function, which is discussed on pages 594-6 of Appendix 4. In this section we provide the parallel analysis of constant-elasticity supply functions. The constant-elasticity sup- ply curve has exactly the same form as the constant-elasticity demand curve: Q = KP , or equivalently P = kQ 1/ , (M.8.3) where k = (1/ K ) 1/ is a positive number and is the constant price elasticity of supply.
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