8/10/2019 Application of Derivative in Commerce and Economics

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1 | P a g e Mr. MANOJ KUMAR SHARMA (9719690895)

Topic: Application of derivative in commerce and economics

Q 1. The average cost function AC for a commodity is given by

AC =x

x64

100 in terms of output x. Find the total cost C

and M.C. as function of x. Find x for which AC increases.2

100 64, M.C. 2 100, 8x

C x x x x

Q 2. A company sells pencils at Rs.5 per unit. The fixed cost is

Rs.3750 and the variable cost is estimated to be 25% of the

total revenue. Determine the break-even point.

[Ans: x = 1000 units]

Q 3. The demand for a certain product is represented by225300 xxp Find the (i) Average revenue (ii)Marginal

revenue (iii)Marginal revenue when x = 2

Ans: 388.35030025300 22 Rsiiixxiixxi ]

Q 4.The total cost for a production and marketing activity is

given by 2774

3 2 xxCx . Find the level of output for

which MC =AC. [Ans: 6]

Q 5. The marginal cost of manufacturing a certain product is

given by 3+ 0.25x. Find the total cost, given that

.600C also find the average cost.

Ans:x

xxxCx60

125.03AC,602

25.03 2 ]

Q 6. If 6

2

100

x

p represent the demand function for a

product where p is the price per unit when x units are sold,

find the marginal revenue. [Ans: M.R. =]2

2006

2x

Q 7. If marginal cost of a firm is ,232 xx find the cost

function, given C (0) = 0 [Ans: ]23

xx

Q 8. The marginal cost of production of a commodity is 30 +

2x. It is known that fixed cost are Rs. 120. Find (i) the total cost

of producing 100 units and (ii) the cost of increasing output

from 100 to 200 units. .[Ans: (i) Rs. 13,120 , (ii) Rs. 33,000]

Q 9. The total revenue received from the sale of x units of a

product is given by .2

120 2)( xxR x Find the marginal

revenue and average revenue when x = 10. Find the actual

revenue from selling theth15 item. [Ans: 15 and 10; Rs. 5.50]

Q 10. The average cost function of a product is given by

,50

112..x

xCA where x is the number of units

produced. Find the marginal cost and range of the output for

which A.C. is increasing. [Ans: MC = 4x11, AC increases f

50 x ]Q 11. The fixed costs of a product are Rs. 20,484 and Rs. 10

per units respectively. The demand function for the produgiven by p = 40002x. Find the level of sales at which the

company expects to cover its costs. [Ans: 8 or 1250 units

Q 12. Verify for the cost function

cbdcbadcx

bxaxCx ,0,,,,;)( so that the

average and marginal cost curves fall continually with

increasing output.

Q 13. The demand per month for a commodity is 24 it the

price is Rs. 16 and 12 if the price is Rs. 22. Assuming that th

demand curve is linear, determine (i) the demand function

the total revenue function and (iii) the marginal revenuefunction.

[Ans: ,2

28 xp ,2

282

)(xxR x M.R. = 28x ]

Q 14. A firm has the following total cost and demand funct

5011173

23

)( xxx

Cx , px 100 find the prof

maximizing output. [Ans: x = 1

Q 15 If ,bxap prove that gradient of marginal reven

curve is twice that of average revenue curve.

Q 16 If the variable cost function for a firm is given by

,log. 2)( xxxC ex find its average cost and marginal co

[Ans: ]2log1andlog xxMCxxAC ee

Q 17. The fixed cost of a new product is Rs.18, 000 and the

variable cost per unit is Rs.550. If the demand function

is 4000 150 ,( )P xx find the break-even points.

[Ans: 8 and 15] [I.S.C. 2007

Q 18. A monopolists demand function for one of its produ

is p = ax + b. He knows that he can sell 1400 units whe

the price is Rs.4 per unit and he can sell 1800 units at a pri

of Rs.2 per unit, find the marginal revenue function and th

price per units when the marginal revenue is zero.

[Ans: ]502

25 xandxMR

Q 19. The total cost and the total revenue of a company th

produces and sell x units of a particular product are

respectively .50and35052

)()( xxRxC xx find (i) the

break-even values (ii) the value of x that gives a profit.

[Ans (i) 10, 35 (ii)10 35]x x

Q 20. A firm knows that demand function for its main prod

is linear. It is known that it can sell 3000 units at Rs.5 per u

8/10/2019 Application of Derivative in Commerce and Economics

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2 | P a g e Mr. MANOJ KUMAR SHARMA (9719690895)

and it can sell 1200 units when price is Rs 11 per unit.

Determine (i) the total revenue function.

(ii) The average revenue function

(iii) the marginal revenue

function.[Ans:2

15 , 15 , 15300 150 300

x x xR x MR AR

x]

Q 21. A company incurs Rs. 25, 000 as marketing cost for its

product and Rs. 15,200 as the rent of its office building. If xunits of the commodity are produced, the production cost per

units is Rs.8. If each units is sold at Rs.75, find the profit

function and break-even point. [Ans: x = 600]

Q 22. The total revenue received from the sell of x units of a

product is given by 53362

)( xxR x Find (i)The average

revenue (ii)The marginal revenue.

[Ans:x

xAR 5336 , ]53 2xMR

Q 23. The cost of manufacturing x units of a commodity is

27153 2 xx (i) Find the output when AC = MC (ii) Find the

output for which AC is increasing. [Ans: (i) x =3 (ii) x > 3]Q 24. The manufacturing cost of an item of Rs.2000 as

overheads material cost Rs.3 per item and labour cost90

2x

for x items produced. Verify that the marginal cost and

average cost are equal at the minimum point of average cost

curve.

Q.24. A company is selling a certain product. The demand

function of the product is liner. The company can sell 2000

units when the price is Rs. 8 per units and when the price is Rs.

4 per units, it can sell 3000 units. Determine:

(i)

the demand function [I.S.C.2005]

(ii)

the total revenue function

[Ans: 004.016p

]004.0162

)( xxR x

Q.25. A television manufacturer fined that the total cost for

the production and marketing of x number of television sets

is: .1350042003002

)( xxCx Each product is sold for

Rs.8400. Determine the break-even points [Ans: x = 5, 9]

[I.S.C. 2006]

Q.26. The average cost function AC for a commodity is given

by36

5AC xx

in terms of output x. Find the:

(i)Total cost and marginal cost as the functions of x.

(ii)Output for which AC increases. [I.S.C.2010

Q.27. Given the total cost function for x units of a commod

as1 3 2

8 5( ) 3

C x x xx

. Find

(i)the marginal cost function

(ii) the average cost function

(iii) the slope of the average cost function

Q.28 The average cost function associated with the produc

and marketing x units of an item is given by

502 11AC x

x. Find

(i)the total function and marginal function

(ii)the range of the values of the output x, for which AC

increasing. [I.S.C. 200

Q.29 The cost of manufacturing of certain items consisting

Rs 1600 as overheads, Rs 30 per item as the cost of the

material and the labours cost Rs2

100

x

for x items produced

How many item must be produced to have a minimum

average cost? [I.S.C. 200

Q.30 Given that the total cost function for x units of a

commodity is:3

2( ) 3 7 163

xC x x x

(i) Find the Marginal Cost (MC) [I.S.C. 201

(ii) Find the Average Cost (AC)

(iii)

Prove that : Marginal Average Cost

(MAC) =2

(MC) - C(x)x

x

Q.31 If total cost function is given by 2( ) ,xC a bx cx wh

x is the quantity of output. Show that:

1( ) ( )

dAC MC AC

dx x, MC is the marginal cost and AC is t

average cost. [I.S.C. 201