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# HW1Sol - Fall 2010 IEOR 160 Industrial Engineering...

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Fall 2010 IEOR 160 September 9, 2010 Page 1 of 2 HOMEWORK 1 SOLUTIONS Chapter 12.1 3. (a) - xe - x + e - x (b) 2 x ( x 2 +1) - 2 x 3 ( x 2 +1) 2 (c) 3 e 3 x (d) - 6(3 x + 2) - 3 (e) 3 x 5. ln(1 + h ) = h - h 2 2 + 2 p - 3 3! h 3 for p [1 , 1 + h ]. 6. (a) f 0 ( p ) < 0 if a price increase lowers demand. Thus we expect to ﬁnd f 0 ( p ) < 0. (b) Let r ( p ) = pf ( p ). If r 0 ( p ) < 0, then price decrease increases revenue. Now r 0 ( p ) = pf 0 ( p ) + f ( p ) < 0 if p dq dp < - q which implies that p q dq dp = E < - 1. (c) If - 1 < E < 0, then r 0 ( p ) > 0, so a price cut will decrease revenue. 8. (a) The cost of producing x ’th unit is c 0 ( x ) = k (1 - b ) x - b which is a decreasing function of x . (b) If total amount produced is x , then production cost per unit is kx - b . If total amount produced is 2 x , then production cost per unit is k (2 x ) - b . Thus doubling the amount produced reduces cost per unit to 100(2 - b )% of what it was previously. Chapter 12.2

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HW1Sol - Fall 2010 IEOR 160 Industrial Engineering...

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