MATH 104 THE SOLUTIONS OF THE ASSIGNMENT

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1 MTH 4 THE SOLUTIONS OF THE SSIGNMENT Question9. (Page 75) Solve X = if = 8 and = 4 and write a system. X =, = 8 4 = *+ *4= = 8*+ 4*= For finding the system, we use ( ) = = 6= 5, 8 /5 /5 = = 5 8 8/5 /5 ( ) =, The system is =, /5 /5 = 8 /5 /5, 4 + = 8 = 6 Question 9. (Page 86) Solve for, if 7 7 = 6. (7 ) ( )(7) = 6 + = = ( )( 4) = So = or = 4. Page of 4

2 Question 4. (Page 86) Solve for, if 99 = 6 99 = 6 ll entries below the main diagonal are zeros. Thus, ()( )( ) = 6 ( ) = 6 = = ( 5)( + 4) =, So = 5 or = 4. Question 5. (Page 97) Solve the systems by using Cramer s rule. = y 8 = = (*)-(*(-))=9+= = = y = 8 8 = y = + y = 8 = = =, y y = = = Page of 4

3 Question 6. (Page 97) Solve the system by using Cramer`s rule. = 4 = = = = = y = + y z = y + 4z = + y + z = = (**) + (*4*) + (-**)-(-**)-(*4*)-(**) 4 = (**) + (*4*) + (-**)-(-**)-(*4*)-(**) 4 = (**) + (*4*) + (-**)-(-**)-(*4*)-(**) y = = z = = (**) + (**) + (**)-(**)-(**)-(**) z = = = = =, y y = = =, z z = = = Page of 4

4 Question 4. (age 7) Given the demand equation elasticity of demand when =9. q ( ) + =, determine the oint q dq η = =, we differentiate imlicitly for dq d qd d. dq q ( ) + = dq ( + )() + ( + )( ) = q q d dq q + + q + = d ( ) ( ) dq q ( + ) Thus =. d q( + ) Hence q +. q + q + q q( + ) ( ) ( ) ( ) η = = If =9, we fine q from the given equation: q (+ 9) = 9 9 q = q = Since q>. Thus ( 9) + η = 9 = =, 4 η =.4 < which is inelastic Page 4 of 4

5 Question 5. (age 7) The demand equation for a roduct is 6 ln(65 ). q = + a) Determine the oint elasticity of demand when =4, and classify the demand as elastic, inelastic, or of unit elasticity at this rice level. b) If the rice is lowered by % (from $4, to $,9), use the answer to art (a) to estimate the corresonding ercentage change in quantity sold. c) Will the changes in art (b) result in an increase or decrease in revenue? Elain. a) 6 q = + ln(65 ). q dq 6 η = = = d qd q 65 dq (6) 7 If =4, then q = + ln = 5, soη =,8 4 5 = = η =.8 > which is elastic. % increase in rice, results.8% decrease in demand. b)the ercentage change in q is (-). (-, 8) =7, 6%, so q increases by aroimately 7, 6%. c)lowering the rice increases revenue because demand is elastic. Page 5 of 4

6 Question. (age 78) Revenue The demand function for a monoolist s roduct is = 6 q If the monoolist wants to roduce at least units, but not more than units, how many units should be roduced to maimize total revenue? The revenue function is R ( q) q q 6 q = = and q ( ) 6 ( 6 ) / 6 q R q = q q q = q = 6 q R ( q) = ( 6 q) q =, R ( q) q q = 4 [,] So, there is no critical values on [ ] increasing on [, ]. = =, q = 4,. Since R ( q) units, but > on [, ] means ( ) R q is ( ) R = q = 6 = 5 R ( ) = q = 6 = So, R ( q) must have a maimum at q =. Page 6 of 4

7 Question4. (age 78) verage Cost C q q = is a cost function, find the average cost function. t what level of If roduction q is there a minimum average cost. The average cost is C C q + q+ = = =.q+ 5 + q q q. 5 dc dc C = =. =,.q dq q dq = =, q = ±, but q = ossible. So, q =. q From the second derivative test C ( q) =, ( ) So, when q =, the average cost is minimum. ( ) is not C = >, Relative minimum Page 7 of 4

8 Question5. (age 78) Profit The demand function for a monoolist s roduct is = 4 q and the average cost er unit for roducing q units is C = q+ 6 + q Where and c are in dollars er unit. Find the maimum rofit that the monoolist can achieve. R q = q = 4 q q = 4q q and The revenue function is ( ) ( ) C q = Cq = q + 6q+ The cost function is ( ) For the maimum rofit, MR ( ) 4 4 MR = R q = q = MC MC = q q = q+ 6, 6q = 4, q = 4 When q = 4, the rofit is maimum. The rofit function is ( ) ( ) ( ) P q = R q C q = 4q q q 6q = q + 4q nd the maimum rofit is ( ) ( ) ( ) P 4 = = 6 Page 8 of 4

9 5 Question 8. (Page 79) Given demand equation = ; q=. Determine whether q demand is elastic, is inelastic, or has unit elasticity for the indicated value of q. 5 = q 5 q q q 5 q 5 q η = = = ( ) = ( ) = d 5 q q 5 q 5 dq q Since η =, demand has unit elasticity when q=. Determine the following indefinite integrals. (age 749) Question. 4 + d d = d + d d = d + d d = + + ln c Question. e + + e+ d e e e = e e e e d e d d ed d e e+ = e + + e + eln + c e + Page 9 of 4

10 Evaluate the following definite integrals. (age 77) Quetion7. ( + ) ( ) y y dy y y dy + = y + y y y = y + y 9 4 = + + = + = Quetion. ( + ) d ( + ) = ( ) d d 4 9 = = = = = Page of 4

11 Question. 5 ed ed= e = e = e Question. ( Page 789) The demand equation for a roduct ( )( q ) suly equation is q= + =. a)verify, by substitution, that market equilibrium occurs when =, q=. b) Determine consumer s surlus under market equilibrium. + + = 8 a-) ( )( ) 8 = 8 b-) ( )( ) ( ) + = 4 + = q + = 8, + =, q+ = q+ 8 CS = dq = 8 ln ( q ) 4q q + + = 8 ln 4 8 ln = 8 ln 4 ( ) ( ) ( ) + + = 8 and the Question 59. ( Page 79) Marginal Revenue dr If marginal revenue is given by = q, determine the corresonding demand dq equation. r = q dq = dq q dq = q q + C when q =, then r =. Thus = + C so C =. Hence r = q q. Since r =. q r then = = q = q q Thus = q. Page of 4

12 Question 6. ( Page 79) Marginal Cost dc If marginal cost is given by q 7q 6 dq of roducing si units. ssume that costs are in dollars. q 7 c= ( q + 7q+ 6) dq = + q + 6q+ C. = + +, and fied costs are 5, determine the total cost When q =, then c = 5. Thus 5 = C so C = 5 q 7 = = 6, = $74. c q q When q thenc Question 6. ( Page 79) Marginal Revenue dr manufacturer s marginal-revenue function is dq = 75 q.q. If r is in dollars, find the increase in the manufacturer s total revenue if roduction is increased from to. q.q ( 75 q.q ) dq = 75 q = $9. Page of 4

13 Question 6. ( Page 94) Profit monoolist sells two cometitive roducts, and, for which the demand functions are q = + 4, q = + 6 If the constant average cost of roducing a unit of is 4 and a unit of is, how many units of and should be sold to maimize the monoolist s rofit? Revenue from = q. Revenue from = q. Total cost of roducing q units of and q units of is 4q + q. Total Profit=Total Revenue-Total Cost ( 4 ) P = q + q q + q P= P = = P = 6 + = Critical oint : 5 =, = 6 P P P = 4<, = <, = 6. t 5 = and = then 6 D 5, ( 4)( ) ( 6) 6 = = > thus relative maimum. 5 4 When = and = then q = and q =. 6 4 Thus to maimize rofit q =, q =. Page of 4

14 Question 5. ( Page 9) Maimizing Outut The roduction function for a firm is ( ) f lk, = l+ k l k. The cost to the firm of l and k is 4 and 8 er unit, resectively. If the firm wants the total cost of inut to be 88, find the greatest outut ossible, subject to this budget constraint. We maimize ( ) f lk, = l+ k l k subject to the constraint 4l+ 8k = 88. (,, λ) = + λ( ) F l k l k l k l k () ( ) ( ) Fl = l 4λ = Fk = 4k 8λ = Fλ = 4l 8k+ 88= Eliminate λ from ( ) and ( ) k = l. Substitute k l l = 8, k = 7.Therefore the greatest outut is f ( 8, 7) = 74 units. = into ( ) yields Page 4 of 4