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

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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 0 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 0 R , and x 0 > x, we have f ( x 0 ) > f ( x ) . Suppose f : R R is differentiable on R . (i) If f 0 ( 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 0 ( x ) > 0 for all x R ? Explain. 4. [Applying Taylor’s theorem] Let f : R + R + be a continuous function on R + , which is twice continu- ously differentiable on R ++ . Assume that f (0) = 0 and f 0 ( x ) > 0 , f 00 ( x ) > 0 for all x > 0 . 1
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Let 0 < c < d. Show by using Taylor’s theorem that: ( i ) f ( d ) - f ( c ) d - c < f 0 ( d ) ( ii ) f ( d ) - f ( c ) d - c > f 0 ( c ) (1) Draw an appropriate graph of
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