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Solutions - Introduction to Mathematical Methods for...

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Introduction to Mathematical Methods for Economics and Management, 2003 Week 1 Solutions to Worksheet 1 Review of Algebra Question 4. Simplify: (a) 6 a 4 b × 4 b ÷ 8 ab 3 c Solution: 6 a 4 b × 4 b ÷ 8 ab 3 c = 3 a 3 bc (b) p 3 x 3 y ÷ 27 xy Solution: p 3 x 3 y ÷ 27 xy = x 3 (c) ¡ 2 x 3 ¢ 3 × ¡ xz 2 ¢ 4 Solution: ¡ 2 x 3 ¢ 3 × ¡ xz 2 ¢ 4 = 8 x 13 z 8 Question 8. Write as a single logarithm: (a) 2 log a (3 x ) + log a x 2 Solution: 2 log a (3 x ) + log a x 2 = log a ¡ 9 x 4 ¢ (b) log a y 3 log a z Solution: log a y 3 log a z = log a ¡ y z 3 ¢ Question 9. Solve the following equations: (a) 5 (2 x 9) = 2 (5 3 x ) Solution: 5 (2 x 9) = 2 (5 3 x ) , Solution is: x = 55 16 (b) 1 + 6 y 8 = 1 Solution: 1 + 6 y 8 = 1 , Solution is: y = 5 (c) z 0 . 4 = 7 Solution: z 0 . 4 = 7 , Solution is: z = 7 5 2 = 129 . 64 (d) 3 2 t 1 = 4 Solution: 3 2 t 1 = 4 , Solution is: t = (ln 3+ln 4) 2 ln 3 = 1 . 130 9 Question 10. Solve these equations for x, in terms of the parameter a : (a) ax 7 a = 1 Solution: ax 7 a = 1 , Solution is: x = ½ © 1 a (7 a + 1) ª if a 6 = 0 if a = 0 (b) 5 x a = x a 1
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Solution: 5 x a = x a , Solution is: x = ( n a 1 a +5 o if a 6 = 1 5 if a = 1 5 (c) log a (2 x + 5) = 2 Solution: log a (2 x + 5) = 2 , Solution is: x = 1 2 a 2 5 2 Question 11. Make Q the subject of: P = q a Q 2 + b Solution: P = q a Q 2 + b > Q = p a P 2 b Question 12. Solve the equations: (a) 7 2 x 2 = 5 x Solution: 7 2 x 2 = 5 x , Solution is: x = 7 2 , 1 (b) y 2 + 3 y 0 . 5 = 0 Solution: y 2 + 3 y 0 . 5 = 0 , Solution is: y = 3 . 158 3 , 0 . 158 31 (c) | 1 z | = 5 Solution: | 1 z | = 5 , Solution is: z = 4 , 6 Question 13. Solve the simultaneous equations: (a) 2 x y = 4 and 5 x = 4 y + 13 Solution: 2 x y = 4 5 x = 4 y + 13 , Solution is: [ x = 1 , y = 2] (b) y = x 2 + 1 and 2 y = 3 x + 4 Solution: y = x 2 + 1 2 y = 3 x + 4 , Solution is: [ x = 2 , y = 5] , £ x = 1 2 , y = 5 4 ¤ Question 14. Solve the inequalities: (a) 2 y 7 3 Solution: 2 y 7 3 , Solution is: y ( −∞ , 5] (b) 3 z > 4 + 2 z Solution: 3 z > 4 + 2 z , Solution is: z ¡ −∞ , 1 3 ¢ (c) 3 x 2 < 5 x + 2 Solution: 3 x 2 < 5 x + 2 , Solution is: x ¡ 1 3 , 2 ¢ Solutions to Worksheet 2 Lines and Graphs Question 6. 2
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Sketch the graph of y = 3 x x 2 +4 , and hence solve the inequality 3 x x 2 < 4 . Solution: y ( x ) = 3 x x 2 + 4 y ( x ) 5 2.5 0 -2.5 -5 0 -10 -20 -30 x y 3 x x 2 < 4 , Solution is: x ( −∞ , 1) (4 , ) Question 7. Draw a diagram to represent the inequality 3 x 2 y < 6 . Solution: 3 x 2 y < 6 or y > 3 2 x 3 First let us sketch the graph y ( x ) = 3 2 x 3 y ( x ) 5 2.5 0 -2.5 -5 2.5 0 -2.5 -5 -7.5 -10 x y The area above the straight line y ( x ) = 3 2 x 3 represents the inequality in question. Question 8. 3
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Electricity costs 8p per unit during the daytime and 2p per unit if used at night. The quarterly charge is £10. A consumer has £50 to spend on electricity for the quarter. (a) What is his budget constraint? Solution: 8 Day + 2 Night 4000 or 4 Day + Night 2000 (b) Draw his budget set (with daytime units as "good 1" on the horizontal axis).
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