Problem_Set_1 - T. Mitra, Fall 2008 Economics 6170 Problem...

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Unformatted text preview: T. Mitra, Fall 2008 Economics 6170 Problem Set 1 [Due on Wednesday, September 10] 1. [Limit and Continuity] Let f : R + R + be defined by: f ( x ) = 1 / (1 + x ) for all x Use the definition of continuity of a function to show that f is continuous on R + . 2. [Applying the Chain Rule of Differentiation] Let a > 0 be a positive real number (constant). Consider the function f : R R defined by: f ( x ) = a x for all x R Obtain the derivative of f by using the chain-rule of differentiation (and the differentiation formula for the exponential function). 3. [Applying the Mean Value Theorem] We say that a function f : R R is increasing on R if whenever x,x R , and x > x, we have f ( x ) > f ( x ) . Suppose f : R R is differentiable on R . (i) If f ( x ) > 0 for all x R , use the Mean Value Theorem to show that f is increasing on R . (ii) If f is increasing on R , does it follow that f ( x ) > 0 for all x R ?...
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This note was uploaded on 10/03/2009 for the course ECON 6170 taught by Professor Mitra during the Fall '08 term at Cornell University (Engineering School).

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Problem_Set_1 - T. Mitra, Fall 2008 Economics 6170 Problem...

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