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Unformatted text preview: T. Mitra, Fall 2006 Economics 617 Problem Set 1 [Due on Wednesday, September 6] 1. (a)Show, by using the definition of continuity, that the following func- tions are continuous on R + . (i) f : R + R + is defined by: f ( x ) = x for all x (ii) f : R + R + is defined by: f ( x ) = 1 + x for all x (iii) f : R + R + is defined by: f ( x ) = 1 / (1 + x ) for all x (iv) f : R + R + is defined by: f ( x ) = x/ (1 + x ) for all x (b) Draw the graph of f in each case to illustrate its key qualitative features. 2. 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 ) > 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 ) > for all x R ? Explain. 3. Here is the statement of Taylors theorem (up to the second term) that was discussed in class. Let f be a function from...
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