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Topic: Application of derivative in commerce and economics Q 1. The average cost function AC for a commodity is given by

64 AC = x 100 in terms of output x. Find the total cost C x and M.C. as function of x. Find x for which AC increases. x 2 100 x 64, M.C. 2 x 100, x 8

Cx

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 by p 300 25x x 2 Find the (i) Average revenue (ii)Marginal revenue (iii)Marginal revenue when x = 2 Ans: i 300 25x x 2 ii 300 50x 3 x 2 iii Rs.388 ] Q 4.The total cost for a production and marketing activity is given by C x

3 2 x 4

7 x 27 . 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 C 0 60. also find the average cost.

0.25 2 x 2

Ans: C x

3x

Q 6. If p

100 x 2

60, AC

3 0.125x

60 ] x

6 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. =]

200 x 2

Q 7. If marginal cost of a firm is 3 x 2

2

6

2 x, find the cost

function, given C (0) = 0 [Ans: x 3 x 2 ] 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 R( x )

20x

1 2 x . Find the marginal 2

which A.C. is increasing. [Ans: MC = 4x – 11, AC increases for 0 x 5] Q 11. The fixed costs of a product are Rs. 20,484 and Rs. 1000 per units respectively. The demand function for the product is given by p = 4000 – 2x. 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

C( x )

ax

x b x c

c so that the

d ; a, b, c, d , 0, b

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 the demand curve is linear, determine (i) the demand function, (ii) the total revenue function and (iii) the marginal revenue function. [Ans: p

28 x , R( x ) 2

28x

x2

2

, M.R. = 28 – x ]

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

C( x )

x3 3

7 x 2 111x 50 , x 100 p find the profit

maximizing output. [Ans: x = 11] Q 15 If p a bx, prove that gradient of marginal revenue curve is twice that of average revenue curve. Q 16 If the variable cost function for a firm is given by C ( x ) x. loge x x 2 , find its average cost and marginal cost. [Ans: AC log e x x and MC 1 log e x 2 x] 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 P( x) 4000 150 x, find the break-even points. [Ans: 8 and 15] [I.S.C. 2007] Q 18. A monopolist’s demand function for one of its products is p = ax + b. He knows that he can sell 1400 units when the price is Rs.4 per unit and he can sell 1800 units at a price of Rs.2 per unit, find the marginal revenue function and the price per units when the marginal revenue is zero. [Ans: MR

25 x and x 2

50]

Q 19. The total cost and the total revenue of a company that revenue and average revenue when x = 10. Find the actual th revenue from selling the 15 item. [Ans: 15 and 10; Rs. 5.50] produces and sell ‘x’ units of a particular product are 2 respectively C ( x ) 5 x 350 and R( x ) 50x x . find (i) the Q 10. The average cost function of a product is given by 50 break-even values (ii) the value of x that gives a profit. A.C. 2 x 11 , where x is the number of units [Ans (i) x 10, 35 (ii)10 x 35] x produced. Find the marginal cost and range of the output for Q 20. A firm knows that demand function for its main product is linear. It is known that it can sell 3000 units at Rs.5 per unit 1|Page 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

(ii)Output for which AC increases. [I.S.C.2010] Q.27. Given the total cost function for x units of a commodity

x2 , MR 300

(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 producing and marketing x units of an item is given by

function.[Ans: R x

15 x

15

x , AR 150

15

x 300

as C( 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 ‘x’ units 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 2 5 Find (i)The average product is given by R( x ) 36x 3x revenue (ii)The marginal revenue. [Ans: AR

36 3x

5 , MR x

3

5

x2

]

Q 23. The cost of manufacturing ‘x’ units of a commodity is 3x 2 15x 27 (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 2 overheads material cost Rs.3 per item and labour cost x

90

AC

1 3 x 3

2 x 11

x

50 x

2

8 x 5 . Find

. 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. 2008] Q.29 The cost of manufacturing of certain items consisting of Rs 1600 as overheads, Rs 30 per item as the cost of the material and the labours cost Rs

x2 100

for x items produced.

How many item must be produced to have a minimum average cost? [I.S.C. 2009] Q.30 Given that the total cost function for x units of a 3

x for x items produced. Verify that the marginal cost and 3 x 2 7 x 16 commodity is: C ( x) 3 average cost are equal at the minimum point of average cost (i) Find the Marginal Cost (MC) [I.S.C. 2011] curve. (ii) Find the Average Cost (AC) Q.24. A company is selling a certain product. The demand (iii) Prove that : Marginal Average Cost function of the product is liner. The company can sell 2000 x(MC) - C(x) units when the price is Rs. 8 per units and when the price is Rs. (MAC) = 4 per units, it can sell 3000 units. Determine: x2 (i) the demand function [I.S.C.2005] Q.31 If total cost function is given by C( x) a bx cx2 , where (ii) the total revenue function ‘x’ is the quantity of output. Show that: [Ans: p 16 0.004 d 1 ( AC ) ( MC AC ) , MC is the marginal cost and AC is the 2

R( x )

16x 0.004x ]

dx

Q.25. A television manufacturer fined that the total cost for the production and marketing of x number of television sets 2 4200x 13500. Each product is sold for is: C ( x ) 300x

x

average cost.

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 by AC

x 5

36 in terms of output x. Find the: x

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

2|Page Mr. MANOJ KUMAR SHARMA (9719690895)

[I.S.C. 2012]