Problem Set

Problem Set - Economics 2296: Managerial Economics Practice...

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Economics 2296: Managerial Economics Practice Questions Spring Term, 2007
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MATH AND CALCULUS REVIEW EXERCISES The following questions are to help you review exponentials, derivatives, partial derivatives, optimization, and constrained optimization. 1. a) What is 27 -1/3 equal to? b) What is X 1/2 X 1/3 equal to? c) What is (X 2 ) 3 equal to? d) What is the area of a triangle whose base is 5 metres and whose height is 20 metres? 2. What is the significance of the first order derivative of a function? 3. Find the first order and second order derivatives for each of the following functions: a) Y = 6; b) Y = 6X – 5; c) Y = 6X 3 + 3X 2 ; and d) Y = 6X 1/3 . 4. Using the product rule, find the first order derivative of the following functions: a) Y = 5X 4 (3X - 7); and b) Y = 3X 1/2 (2X 2 + 3). 5. a) A firm's total revenues are R = 200Q - 25Q 2 . What is the first order derivative (with respect to Q)? b) What is the meaning of this derivative? 6. a) Diagram the following function: Y = 5X 2 - 20X + 50. b) At what value of X does Y reach a minimum? c) What is the slope of the above function where it is minimized? 7. Find the partial derivatives of Z with respect to X and with respect to Y for each of the following functions. a) Z = 5X 1/2 Y 2/3 ; and b) Z = 5X(X + 2Y). 8. a) Find the value of X and Y which maximizes Z = XY subject to the constraint: 4X + 2Y = 40. b) Find the value of X and Y which minimizes Z = 10X + 50Y subject to the constraint: X 1/2 Y 2/5 = 100.
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SOLUTIONS TO MATH AND CALCULUS REVIEW EXERCISES 1. a) 1/3 b) X 5/6 c) X 6 d) Area = 1/2 X base X height = 50 metres 2 . 2. The first order derivative of a function is equal to the slope of that function. 3. a) d Y/ d X = 0, d 2 Y/ d X 2 = 0. b) d Y/ d X = 6, d 2 Y/ d X 2 = 0. c) d Y/ d X = 18X 2 + 6X, d 2 Y/ d X 2 = 36X + 6. d) d Y/ d X = 2X -2/3 , d 2 Y/ d X 2 = -4/3X -5/3 . 4. Under the product rule, given a function Y = V(X) W(X), then d Y/ d X = V( d W/ d X) + W( d V/ d X). a) Let V = 5X 4 and W = 3X - 7. d Y/ d X = 5X 4 (3) + (3X - 7)20X 3 = 75X 4 - 140X 3 . b) Let V = 3X 1/2 and W = 2X 2 + 3. d Y/ d X = 3X 1/2 (4X) + (2X 2 + 3)1.5X -1/2 = 15X 3/2 + 4.5X -1/2 . 5. a) d R/ d Q= 200 - 50Q. b) The derivative is known as marginal revenue, the additional revenue from producing an additional unit of output. 6. a)
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b) The function is minimized where X = 2. d Y/ d X = 10X - 20 = 0 or X = 2 c) Slope = 0. 7. a) d Z/ d X = (5/2)X -1/2 Y 2/3 ; d Z/ d Y = (10/3)X 1/2 Y -1/3 . b) d Z/ d X = 10X + 10Y; d Z/ d Y = 10X. 8. a) X = 5 and Y = 10. (See Notes on Solving Constrained Optimization Problems . ) The following shows the substitution approach. From the constraint, Y = 20 - 2X.
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This note was uploaded on 12/12/2009 for the course ECON 2296 taught by Professor Gray during the Spring '09 term at Langara.

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Problem Set - Economics 2296: Managerial Economics Practice...

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