APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS

Application of Calculus in Commerce an Economics 41 APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS æ We have learnt in calculus that when 'y' is a function of '', the erivative of y w.r.to i.e. y ö ç çè ø measures the instantaneous rate of change of y with respect to. In Economics an commerce we come across many such variables where one variable is a function of the other. For eample, the quantity emane can be sai to be a function of price. Supply an price or cost an quantity emane are some other such variables. Calculus helps us in fining the rate at which one such quantity changes with respect to the other. Marginal analysis in Economics an Commerce is the most irect application of ifferential calculus. In this contet, ifferential calculus also helps in solving problems of fining maimum profit or minimum cost etc., while integral calculus is use to fin he cost function when the marginal cost is given an to fin total revenue when marginal revenue is given. In this lesson, we shall stuy about the total, average or marginal functions an the optimisation problems. OBJECTIVES After stuying this lesson, you will be able to : efine Total Cost, Variable Cost, Average Cost, Marginal Cost, Total Revenue, Marginal Revenue an Average Revenue; fin marginal cost an average cost when total cost is given; fin marginal revenue an average revenue when total revenue is given; fin optimum profit an minimum total cost uner given conitions; an fin total cost/ total revenue when marginal cost/marginal revenue are given, uner given conitions. 1

Application of Calculus in Commerce an Economics EXPECTED BACKGROUND KNOWLEDGE Derivative of a function Integration of a function 41.1 BASIC FUNCTIONS Before stuying the application of calculus, let us first efine some functions which are use in business an economics. OPTIONAL - II 41.1.1 Cost Function The total cost C of proucing an marketing units of a prouct epens upon the number of units (). So the function relating C an is calle Cost-function an is written as C = C (). The total cost of proucing units of the prouct consists of two parts (i) (ii) Fie Cost Variable Cost i.e. C () = F + V () Fie Cost : The fie cost consists of all types of costs which o not change with the level of prouction. For eample, the rent of the premises, the insurance, taes, etc. Variable Cost : The variable cost is the sum of all costs that are epenent on the level of prouction. For eample, the cost of material, labour cost, cost of packaging, etc. 41.1. Deman Function An equation that relates price per unit an quantity emane at that price is calle a eman function. If 'p' is the price per unit of a certain prouct an is the number of units emane, then we can write the eman function as = f(p) or p = g () i.e., price (p) epresse as a function of. 41.1. Revenue function If is the number of units of certain prouct sol at a rate of Rs. 'p' per unit, then the amount erive from the sale of units of a prouct is the total revenue. Thus, if R represents the total revenue from units of the prouct at the rate of Rs. 'p' per unit then R= p. is the total revenue Thus, the Revenue function R () = p.. =.p () 41.1.4 Profit Function The profit is calculate by subtracting the total cost from the total revenue obtaine by selling units of a prouct. Thus, if P () is the profit function, then 41.1.5 Break-Even Point P() = R() C() Break even point is that value of (number of units of the prouct sol) for which there is no 1

profit or loss. i.e. At Break-Even point P ( ) = 0 or R() - C() = 0 i.e. R() = C() Let us take some eamples. Application of Calculus in Commerce an Economics Eample 41.1 For a new prouct, a manufacturer spens Rs. 1,00,000 on the infrastructure an the variable cost is estimate as Rs.150 per unit of the prouct. The sale price per unit was fie at Rs.00. Fin (i) Cost function (ii) Revenue function (iii) Profit function, an (iv) the break even point. Solution : (i) Let be the number of units prouce an sol, then cost function C ( ) = Fie cost + Variable Cost (ii) Revenue function = p. = 00 = 1,00,000 + 150 (iii) Profit function P ( ) = R() -C() (iv) At Break-Even point P() = 0 50-100,000 = 0 = 00 - (100,000+ 150) = 50-100,000 100,000 = = 000 50 Hence = 000 is the break even point. i.e. When 000 units of the prouct are prouce an sol, there will be no profit or loss. Eample 41. A Company prouce a prouct with Rs 18000 as fie costs. The variable cost is estimate to be 0% of the total revenue when it is sol at a rate of Rs. 0 per unit. Fin the total revenue, total cost an profit functions. Solution : (i) Here, price per unit (p) = Rs. 0 Total Revenue R ( ) = p. = 0 where is the number of units sol. (ii) Cost function 0 C() = 18000+ R() 100 (iii) Profit function P() = R() -C() 0 = 18000+ 0 100 = 18000+ 6 14

Application of Calculus in Commerce an Economics = 0 - ( 18000+ 6) = 14 18000 - Eample 41. A manufacturing company fins that the aily cost of proucing items of a prouct is given by C() = 10+ 7000 (i) (ii) If each item is sol for Rs. 50, fin the minimum number that must be prouce an sol aily to ensure no loss. If the selling price is increase by Rs. 5 per piece, what woul be the break even point. Solution : (i) Here, R() = 50 an C() = 10+ 7000 \ P() = 50-10-7000 For no loss P() = 0 = 140-7000 Þ 140-7000= 0 or = 50 Hence, to ensure no loss, the company must prouce an sell at least 50 items aily. (ii) When selling price is increase by Rs. 5 per unit, R() = (50 + 5) = 85 OPTIONAL - II \ P() = 85 -(10+ 7000) = 175-7000 At Break even point P() = 0 Þ 175-7000= 0 7000 = = 40 175 CHECK YOUR PROGRESS 41.1 1. The fie cost of a new prouct is Rs. 18000 an the variable cost per unit is Rs. 550. If eman function p() = 4000-150, fin the break even values.. A company spens Rs. 5000 on infrastructure an the variable cost of proucing one item is Rs. 45. If this item is sol for Rs. 65, fin the break-even point.. A television manufacturer fin that the total cost of proucing an selling television sets is ( ) C = 50 + 000+ 4750. Each prouct is sol for Rs. 6000. Determine the break even points. 4. A company sells its prouct at Rs.60 per unit. Fie cost for the company is Rs.18000 an the variable cost is estimate to be 5 % of total revenue. Determine : (i) the total revenue function (ii) the total cost function (iii) the breakeven point. 15

Application of Calculus in Commerce an Economics 5. A profit making company wants to launch a new prouct. It observes that the fie cost of the new prouct is Rs. 5000 an the variable cost per unit is Rs. 500. The revenue function for the sale of units is given by ( ) R = 5000-100. Fin : (i) Profit function (ii) break even values (iii) the values of that result in a loss. 41.1.6 Average an Marginal Functions If two quantities an y are relate as y = f (), then the average function may be efine f as ( ) an the marginal function is the instantaneous rate of change of y with respect to. i.e. y Marginal function is or ( f( ) ) Average Cost : Let C = C() be the total cost of proucing an selling units of a prouct, C then the average cost (AC) is efine as AC = Thus, the average cost represents per unit cost. Marginal Cost : Let C = C() be the total cost of proucing units of a prouct, then the marginal cost (MC), is efine to be the rale of change of C () with respect to. Thus C MC = or ( C( ) ). Marginal cost is interprete as the approimate cost of one aitional unit of output. For eample, if the cost function is C = 0. + 5, then the marginal cost is MC= 0.4 \ The marginal cost when 5 units are prouce is [ ] ( ) ( ) MC = 5 = 0.4 5 = i.e. when prouction is increase from 5 units to 6, then the cost of aitional unit is approimately Rs.. However, the actual cost of proucing one more unit after 5 units is C(6) C(5) = Rs.. Eample 41.4 The cost function of a firm is given by C= + - 5. Fin (i) the average cost (ii) the marginal cost, when = 4 Solution : (i) C + 5 AC = = 5 = + 1-16

Application of Calculus in Commerce an Economics At 4 5 AC= 4 + 1-4 =, ( ) = 9-1.5= 7.75 (ii) MC = ( C) = 4 + 1 \ MC at = 4= 44 ( ) + 1= 16+ 1= 17 OPTIONAL - II Eample 41.5 Show that he slope of average cost curve is equal to 1 (MC AC) for the total cost function C= a + b + c+. Solution : Cost function C= a + b + c+ Average cost Marginal cost C AC = = a + b + c + MC = (C) = a + b + c Slope of AC curve = (AC) = a + b + c + \ slope of AC curve = a+ b- = a+ b- 1 é ù = a + b - êë úû 1 é æ ö ù = a + b+ c - a + b+ c+ ê ç ë è øú û slope of AC curve ê( ) Eample 41.6 1 = MC AC [ ] If the total cost function C of a prouct is given by æ+ 7ö C= ç + 7 çè+ 5ø Prove that he marginal cost falls continuously as the output increases. æ æ Solution : Here + 7ö + 7ö C= ç + 7 = 7 çè+ 5ø + ç çè + 5 ø 17

\ Application of Calculus in Commerce an Economics ( + ) ( + ) ( + ) C 5 7 7.1 = ( + 5) é ù + 17+ 5- -7 = ê ( + 5) ú ë û é ù + 10+ 5 = ê ( + 5) ú ë û é ù ( + 5) + 10 = ê ( + 5) ú ë û \ MC = é = 1 + ê ë é 1 + ê ë 10 ( + 5) It is clear that when increases ( ) 5 ecreases. 10 ù ú û ( + 5) ù ú û 10 + increases an so (+ 5) Thus, the marginal cost falls continuously as the output increases. ecreases an hence MC 41. AVERAGE REVENUE AND MARGINAL REVENUE We have alreay learnt that total revenue is the total amount receive by selling items of the prouct at a price 'p' per unit. Thus, R = p. Average Revenue : If R is the revenue obtaine by selling units of the prouct at a price 'p' per unit, then the term average revenue means the revenue per unit, an is written as AR. R \ AR = But R = p.. \ p. AR= = p Hence, average revenue is the same as price per unit. Marginal Revenue : The marginal revenue (MR) is efine as the rate of change of total revenue with respect to the quantity emane. 18

Application of Calculus in Commerce an Economics = or R \ MR ( R ) The marginal revenue is interprete as the approimate revenue receive from proucing an selling one aitional unit of the prouct. Eample 41.7 R() = 1+ + 6. Fin The total revenue receive from the sale of units of a prouct is given by (i) the average revenue (ii) the marginal revenue (iii) marginal revenue at = 50 (iv) the actual revenue from selling 51st item OPTIONAL - II R 1+ + 6 Solution : (1) Average revenue AR= = (ii) Marginal revenue 6 = 1+ + MR = (R) = 1+ 4. (iii) [ MR] = 50 = 1 + 4(50) = 1 (iv) The actual revenue receive on selling 51st item Eample 41.8 = R(51) -R(50) é ù é ù ( ) ( ) ( ) ( ) = ê1 51 + 51 + 6ú- ê1 50 + 50 + 6ú ë û ë û = 1( 51-50) + 51 é -50 ù êë úû = 1+ 101 = 1+ 0= 14 The eman function of a prouct for a manufacturer is p () = a + b He knows that he can sell 150 units when the price is Rs.5 per unit an he can sell 1500 units at a price of Rs.4 per unit. Fin the total, average an marginal revenue functions. Also fin the price per unit when the marginal revenue is zero. Solution :Here, p ( ) = a+ b an when = 150, p = Rs 5 \ 5 = 150 a + b...(i) an when = 1500, p = Rs 4 \ 4 = 1500 a + b...(ii) 19

Solving (i) an (ii) we get Application of Calculus in Commerce an Economics 1 a =-, b= 10 50 \ Deman function is epresse as ( ) \ Total Revenue Average Revenue an Marginal revenue Now, when MR = 0 we have p = 10-50 R = p. = 10-50 p= 10-50 MR= 10- = 10-50 15 10- = 0 \ = 150 15 Thus, i.e., price per unit is Rs. 5. 150 p = 10 = 5 50 Thus, at a price of Rs 5 per unit the marginal revenue vanishes. Eample 41.9 The eman function of a monopolist is given by p = 1500. Fin : (i) the revenue function, (ii) the marginal revenue function (iii) the MR when = 0 Solution : p = 1500 (i) \ Revenue function R = p. = 1500 (ii) Marginal revenue MR = (R) = 1500-4- (iii) [ MR] 0 1500 80 100 0 = = - - = Note : In the absence of any competition, the business is sai to be operate as monopoly business, an the businessman is sai to be a monopolist. Thus, in case of a monopolist, the price of the prouct epens upon the number of units prouce an sol. CHECK YOUR PROGRESS 41. 1. The total cost C() of a company is given as C() = 1000+ 5 + where is the output. Determine : (i) the average cost (ii) the marginal cost (iii) the marginal cost when 15 units are prouce, an (iv) the actual cost of proucing 15th unit. 0

Application of Calculus in Commerce an Economics. The cost function of a firm is given by C= + + 4. Fin (i) The average cost (ii) the marginal cost, (iii) Marginal cost, when = 5.. The total cost function of a firm is given as C() = 0.00-0.04 + 5+ 1500 where is the output. Determine : (i) the average cost (ii) the marginal average cost (MAC) (iii) the marginal cost (iv) the rate of change of MC with respect to 4. The average cost function (AC) for a prouct is given by OPTIONAL - II 5000 AC= 0.006-0.0-0 +, where is the output. Fin (i) the marginal cost function (ii) the marginal cost when 50 units are prouce. 5. The total cost function for a company is given by level of output for which MC = AC. C() = - 7+ 7. Fin the 4 6. The eman function for a monopolist is given by =100 4p, where is the number of units of prouct prouce an sol an p is the price per unit. Fin : (i) total revenue function (ii) average revenue function (iii) marginal revenue function an (iv) price an quantity at which MR = 0. 7. A firm knows that the eman function for one of its proucts is linear. It also knows that it can sell 1000 units when the price is Rs.4 per unit an it can sell 1500 units when the price is Rs. per unit. Determine : (i) the eman function (ii) the total revenue function (iii) the average revenue function (iv) the marginal revenue function. 8. The eman function for a prouct is given by 5 p =. Show that the marginal rev- + enue function is a ecreasing function. Minimization of Average cost or total cost an Maimization of total revenue, the total profit. We know that if C = C () is the total cost function for units of a prouct, then the average cost (AC) is given by C() AC = In Economics an Commerce, it is very important to fin the level of output for which the average cost is minimum. Using calculus, this can be calculate by solving (AC) = 0 an to get that value of for which ( AC) > 0. 1

Application of Calculus in Commerce an Economics Similarly, when we are intereste to fin the level of output for which the total revenue is maimum. R 0 we solve [ ( )] = an fin that value of for which [ ( ) ] R < 0. Similarly we can fin the value of for maimum profit by solving [ ( )] that value of for which [ ( ) ] P < 0. P = 0an to fin Eample 41.10 The manufacturing cost of an item consists of Rs.6000 as over heas, material cost Rs. 5 per unit an labour cost Rs. for units prouce. Fin how many units must be 60 prouce so that the average cost is minimum. Solution : Total cost C() = 6000+ 5 + 60 6000 \ AC = + 5+ 60 AC =- 6000 + 1 60 Now, ( ) \ ( ) 6000 1 AC = 0 Þ - + = 0 60 = 600,00 = 600 1000 ( AC) =+ > 0 at = 600 Hence AC is minimum when = 600 Eample 41.11 The total cost function of a prouct is given by 615 C() = - + 15750 + 18000, where is the number of units prouce. Determine the number of units that must be prouce to minimize the total cost. Solution : We have, ( ) 615 C = - + 15750+ 18000 \ [ C ( ) ] = 615 + 15750

Application of Calculus in Commerce an Economics é C ( ) ù= 0 ë û Þ - 615+ 15750= 0 or, or, ( -175)( - 0) = 0 This gives = 175, = 0 ( ) é ë C ù= û 6-615 So, C() is minimum when 175 units are prouce. - 05+ 550= 0, which is positive at = 175 Eample 41.1 The eman function for a manufacturer's prouct is = 70-5p, where is the number of units an 'p' is the price per unit. At what value of will there be maimum revenue? What is the maimum revenue? Solution : Deman function is given as = 70-5p This gives, \ R( ) 70- p = 5 70- = p. = 5 é R 1 70 ë ù= û 5 - ( ) [ ] OPTIONAL - II é R ( ) ù= 0 ë û Gives = 5 1 Now, ér( ) ù ( ) 0 ë û = - < 5 \ for maimum revenue, = 5 an Maimum revenue ( ) ( ) 70 5 5 = Rs. = Rs. 45 5 Eample 41.1 A company charges Rs.700 for a raio set on an orer of 60 or less sets. The charge is reuce by Rs.10 per set for each set orere in ecess of 60. Fin the largest size orer company shoul allow so as to receive a maimum revenue. Solution : Let be the number of sets orere in ecess of 60. i.e. number of sets orere = ( 60 + ) \ Price per set = Rs. ( 700-10) \ Total revenue R = ( 60+ )( 700-10)

Application of Calculus in Commerce an Economics R ( 60 )( 10) ( 700 10) 1 = + - + - =-600-10+ 700-10 = 100-0 R 0 5 = Þ = R 0 0 =- < \ For maimum revenue, the largest size of orer = ( 60 + 5 ) sets = 65 sets Eample 41.14 The cost function for units of a prouct prouce an sol by a company is C() = 50+ 0.005 an the total revenue is given as R = 4. Fin how many items shoul be prouce to maimize the profit. What is the maimum profit? Solution : C() = 50+ 0.005 an R () = 4 \ Profit function P ( ) = R( ) -C ( ) \ ( ) = 4-50-0.005 é P ( ) ù = 4-0.010 ë û é P ù = 0 ë û gives 4-0.01= 0 or ( ) ù ép an ë û 0.01 0 =- < \ For maimum profit, = 400 an maimum profit ( ) 4 = = 400.01 5 = Rs. 4 400 50 400 400 1000 = Rs. [ 1600 50 800 ] = Rs.550 Eample 41.15 A firm has foun from past eperience that its profit in terms of number of units prouce, is given by Compute : P() =- + 79 + 700, 0 5. 4

(i) Application of Calculus in Commerce an Economics (ii) the value of that maimizes the profit, an the profit per unit of the prouct, when this maimum level is achieve. Solution : ( ) \ ( ) P =- + 79+ 700 é P ù =- + 79 ë \ ( ) û é P ù = 0 Þ = 7 ë û ( ) ép ù ë û =- < 0 \ For maimum profit, = 7 (ii) ( ) \ Profit per unit P =- + 79+ 700 700 = + 79 + 79 700 = Rs. 79 + + 7 = Rs. [ 4 + 79 + 100 ] = Rs. [ 89 4 = Rs.586 ] OPTIONAL - II CHECK YOUR PROGRESS 41. 1. The cost of manufacturing an item consists of Rs.000 as over heas, material cost Rs. 8 per item an the labour cost for items prouce. Fin how many items must be 0 prouce to have the average cost as minimum.. The cost function for a firm is given by 1 C= 00-10 +, where is the output. Determine : (i) (ii) (iii) the output at which marginal cost is minimum, the output at which average cost is minimum, an the output at which average cost is equal to the marginal cost.. If C= 0.01 + 5+ 100 is the cost function for items of a prouct. At what level of prouction,, is there minimum average cost? What is this minimum average cost? 5

Application of Calculus in Commerce an Economics 4. A television manufacturer prouces sets per week so that the total cost of prouction is given by the relation C() = - 195 + 6600+ 15000. Fin how many television sets must be manufacture per week to minimize the total cost. 80-5. The eman function for a prouct markete by a company is p =, where is 4 the number of units an p is the price per unit. At what value of will there be maimum revenue? What is this maimum revenue? 6. A company charges Rs. 15000 for a refrigerator on orers of 0 or less refrigerators. The charge is reuce on every set by Rs.100 per piece for each piece orere in ecess of 0. Fin the largest size orer the company shoul allow so as to receive a maimum revenue. 7. A firm has the following eman an the average cost-functions: = 480-0p an AC= 10 + 15 Determine the profit maimizing output an price of the monopolist. æ 8. A given prouct can be manufacture at a total cost ö C= Rs. + 100+ 40, where ç çè100 ø æ ö is the number of units prouce. If p= Rs. 00 ç - çè 400ø is the price at which each unit can be sol, then etermine for maimum profit. 41. APPLICATION OF INTEGRATION TO COMMERCE AND ECONOMICS We know that marginal function is obtaine by ifferentiating the total function. Now, when Marginal function is given an initial values are given, then total function can be obtaine with the help of integration. 41..1 Determination of cost function If C enotes the total cost an ( ) ( ) C MC = is the marginal cost, then we can write C = C = MC + k, where k is the constant of integration, k, being the constant, is the fie cost. Eample 41.16 The marginal cost function of manufacturing units of a prouct is 5 + 16. The total cost of proucing 5 items is Rs. 500. Fin the total cost function. Solution :Given, MC= 5+ 16- \ ( ) = ( + - ) ò C 5 16 6

Application of Calculus in Commerce an Economics ( ) When = 5, C() = C(5) = Rs. 500 = 5+ 16 - + k C = 5+ 8 - + k or, 500= 5+ 00-15+ k This gives, k = 400 OPTIONAL - II \ ( ) Eample 41.17 C = 5+ 8 - + 400 The marginal cost function of proucing units of a prouct is given by MC =. Fin the total cost function an the average cost function if the fie cost + 500 is Rs. 1000. MC = Solution : + 500 \ Let + 500= t \ C ( ) C( ) = ò + k + 500 ( ) When = 0, C(0) = Rs 1000 Þ = t t tt = ò + k t ò C = t+ k= t + k = + 500 + k \ 1000= 500+ k= 50+ k or, k = 950 \ ( ) C = + 500 + 950 500 950 AC= 1+ + Eample 41.18 The marginal cost (MC) of a prouct is given to be a constant multiple of number of units () prouce. Fin the total cost function, if fie cost is Rs.5000 an the cost of proucing 50 units is Rs. 565. Solution : Here MCµ i.e MC= k 1 ( k 1 is a constant) C \ k 1 = Þ C= ò k 1 + k 7

\ C= k + k 1 since fie cost = Rs 5000 \ = 0 Þ C = 5000 Þ k = 5000 Now cost of proucing 50 units is Rs 565 \ 1 500 565= k + 5000 Þ 65= 150k1 Þ k1 Application of Calculus in Commerce an Economics 1 = HenceC= + 5000, is the require cost function. 4 41.. Determination of Total Revenue Function If R() enotes the total revenue function an MR is the marginal revenue function, then MR = [ R() ] R = ò MR + k Where k is the constant of integration. \ ( ) ( ) R() Also, when R () is known, the eman function can be foun as p = Eample 41.19 The marginal revenue function of a commoity is given as MR= 1- + 4. Fin the total revenue an the corresponing eman function. Solution : MR= 1- + 4 \ ( ) ò R = 1- + 4 + k R = 1- + [constant of integration is zero in this case] \ Revenue function is given by R = 1+ - Since = 0,R = 0 k = 0 \ R p= = 1+ - is the eman function. Eample 41.0 The marginal revenue function for a prouct is given by 6 MR = 4. ( ) Fin the total revenue function an the eman function. 8

Application of Calculus in Commerce an Economics Solution : 6 MR= -4 ( - ) é ù 6 6 \ R = ò ( ) - 4 =- - 4+ k ê - ú - ë û = 0, R = 0 Þ k = \ Now, 6 R =- -4-, which is the require revenue function. - R 6 p= =- -4- ( - ) OPTIONAL - II 6 =- - -4 ( - ) -6- + 6 = -4 ( - ) - = - 4= -4 - - \ The eman function is given by p= -4. - CHECK YOUR PROGRESS 41.4 1. The marginal cost of prouction is MC= 0-0.04+ 0.00, where is the number of units prouce. The fie cost is Rs. 7000. Fin the total cost an the average cost function.. The marginal cost function of manufacturing units of a prouct is given by. MC= - 10+. The total cost of proucing one unit of the prouct is Rs.7. Fin the total an average cost functions.. The marginal cost function of a commoity is given by 14000 MC= 7+ 4 an the fie cost is Rs. 18000. Fin the total cost an average cost of proucing units of the prouct. 4. If the marginal revenue function is eman function. 4 MR= -1, fin the total revenue an the ( + ) 9

5. If MR= 0-5+, fin total revenue function. 6. If MR= 14-6+ 9, fin the eman function. Application of Calculus in Commerce an Economics LET US SUM UP Cost function of proucing an selling units of a prouct epens upon. C () = Fie cost + Variable cost. Deman function written as p = f ( ) or = f (p) where p is the price per unit, an number of units prouce. Revenue function, is the money erive from sale of units of a prouct. \ R() = p.. Profit function =R() - C(). i.e. P() = R() -C() Break even point is that value of for which P () = 0 C Average cost AC = Marginal cost MC= C( ) R Average revenue AR= = p Marginal revenue = MR = (R) é ë ù û For minimization of AC, solve (AR) = 0 an then fin that value of for which (AR) > 0. R() 0 fin for which n orer erivative is negative. For maimization of R() or P(X), solve ( ) = or ( ) ò C = MC + k ( ) ( ) 1 ò R = MR + k ( ) ( ) P() = 0 an then SUPPORTIVE WEB SITES http : // www.wikipeia.org http : // mathworl.wolfram.com 0

Application of Calculus in Commerce an Economics TERMINAL EXERCISE 1. A profit making company wants to launch a new prouct. It observes that the fie cost of the new prouct is Rs.7500 an the variable cost is Rs.500. The revenue receive on sale of units is 500-100. Fin : (i) profit function (ii) break even point.. A company pai Rs. 16100 towars rent of the builing an interest on loan. The cost of proucing one unit of a prouct is Rs. 0. If each unit is sol for Rs. 7, fin the break even point.. A company has fie cost of Rs. 6000 an the cost of proucing one unit is Rs.0. If each unit sells for Rs. 4, fin the breakeven point. 4. A company sells its prouct for Rs. 10 per unit. Fie costs for the company are Rs. 5000 an the variable costs are estimate to run 0 % of total revenue. Determine : (i) the total revenue function (ii) total cost function an (iii) quantity the company must sell to cover the fie cost. 5. The fie cost of a new prouct is Rs. 0000 an the variable cost per unit is Rs. 800. If the eman function is p() = 4500-100fin the break even values. 6. If the total cost function C of a prouct is given by OPTIONAL - II æ+ 7ö C= ç. Prove that the marginal cost falls continuously as the output increases. çè + 5 ø 6 7. The average cost function (AC) for a prouct is given by AC= + 5+, where is the output. Fin the output for which AC is increasing an the output for which AC is ecreasing with increasing output. Also fin the total cost C an the marginal cost MC. 8. A firm knows that the eman function for one of its proucts in linear. It also knows that it can sell 1000 units when the price is Rs. 4 per unit, an it can sell 1500 units when the price is Rs. per unit. Fin : (i) the eman function (ii) the total revenue function (iii) the average revenue function an (iv) the marginal revenue function. 6 9. The average cost function AC for a prouct is given by AC= + 5 +,¹ 0. Fin the total cost an marginal cost functions. Also fin MC when = 10. 10. The eman function for a prouct is given as p= 0+ - 5, where is the number of units emane an p is the price per unit. Fin (i) Total revenue (ii) Marginal revenue (iii) MR when =. 11. For the eman function -5 p =, show that the marginal revenue function is an in- + creasing function. 1

Application of Calculus in Commerce an Economics 1. The eman function for a prouct is given as = 4- p, where is the number of units emane at a price of p per unit. Fin : (i) the Revenue function R in terms of p (ii) the price an number of units emane for which revenue is maimum. 1. The cost function C of a firm is given as 1 C= 100-10 +, Calculate : (i) output, at which the marginal cost is minimum. (ii) output, at which the average cost is minimum. (iii) output, at which the average cost is equal to the marginal cost. 8000 14. The profit of a monopolist is given by p() = -. Fin the value of for which 500+ the p () is maimum. Also fin the maimum profit. 15. The marginal cost of proucing units of a prouct is given bymc= + 1. The cost of proucing units is Rs. 7800. Fin the cost function. 16. The marginal revenue function for a firm is given by MR= - + 5. + + ( ) Show that the eman function is p= + 5. + 17. The cost function of proucing units of a prouct is given by C() = a+ b. where a, b are positive. Using erivatives show that the average an marginal cost curves fall continuously with increasing output. 18. A manufactur's marginal revenue function is given by MR= 75-- 0.. Fin the increase in the manufacturer's total revenue if the prouction is increase from 10 to 0 units.

Application of Calculus in Commerce an Economics ANSWERS OPTIONAL - II CHECK YOUR PROGRESS 41.1 1. = 15, = 8. = 150. = 5, 5 4. R () = 60, C ( ) = 18000 + 15, = 400 5. P() = 4500-100 - 5000, = 10, = 5, < 10,> 5. CHECK YOUR PROGRESS 41. 1. (i) 1000 AC= + 5+ (ii) MC= 5+ 4, (iii) 85 (iv) 8. (i). (i) 4 AC= + + (ii) MC= 4+ (iii) 1500 AC= 0.00-0.04+ 5+ 1500 (ii) MAC= 0.004-0.04- (iii) MC= 0.006-0.08+ 5 (iv) ( MC ) = 0.01-0.08 4. (i) MC= 0.018-0.04-0 (ii) [ MC] 50 = 1 5. = 6 6. (i) (iii) R = 5- (ii) AR= 5-4 4 MR= 5- (iv) = 50, p = 1.5 7. (i) = 000-50p (ii) R = 8-50 (iii) AR= 8- (iv) MR= 8-50 15

CHECK YOUR PROGRESS 41. Application of Calculus in Commerce an Economics 1. 00. (i) = 10 (ii) = 15 (iii) =15. (i) = 100, Rs. 7 4. = 110 5. 40, 400 6. 85 7. 60, 5 8. 4000 CHECK YOUR PROGRESS 41.4 1.. 7000 0-0.0 + 0.001 + 7000; 0-0.0+ 0.001 + 7 C= - 5 + + 7, AC= - 5+ +. C= 4000 7+ 4+ 10000, 4. 4 4 R = -, p= -1 6+ 9 6+ 9 4000 10000 AC= 7+ 4+ 5. 5 R = 0- + 6. p= 14- + TERMINAL EXERCISE 1. P() = 000-100 - 7500; = 5, 15.. 00. 000 4. R() = 10, C() = + 5000, = 50 5. = 1, 5 7. > 6,0< < 6,C = + 5+ 6, MC= + 5 8. p= 8-, R = 8-, AR= 8-, 50 50 50 9. + 5+ 6, + 5, 5 10. R = 0+ - 5, MR= 0+ 4-15, 177 1 1. R ( 4p p ) = -, p = 6, = 4 MR= 8-15 1. (i) = 10 (ii) = 15 (iii) = 0, = 15 14. 1500, 4500 5/ / 116888 + - + + 18. Rs. 1900 5 15 15. ( 1) ( 1) 4