klinik matematik pengurusan open university … · the demand function for product a produced by...

MATHEMATICS FOR MANAGEMENT

QUESTIONS AND ANSWERS

P a g e | 1 Prepared by Ezaidin bin Norman Tutor BBMP1103

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KLINIK MATEMATIK PENGURUSAN BBMP 1103 OPEN UNIVERSITY MALAYSIA (OUM) PEPERIKSAAN AKHIR SEMESTER SEPTEMBER 2011 BAHAGIAN A 1.

a) Given that 52 3 xy . Find dx

dy.

b) Given that 52 13 xxf . Find ''f .

Answer:

a) 52 3

dx

dyx

dx

dy

26x

b) 743 xdx

dy

Let 43 xxg then 3' xg and 7n . Therefore

6

17

1

4321

3437

'.'

x

x

xgxgnxyn

543378

31643126

64321

'.1''

x

x

x

xgnxgnxy

2. Find the following:

a) dxx

xx

3

b) dxxx 212 3

Answer: a)




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cxx

cxx

x

x

dxx

xx

dxx

xxxx

3

12

2

2

2

3

1

12

1

1

1

..

b)

cxxxx

cxxxx

dxxxx

dxxxx

22

1

5

2

21113

4

14

2

242

242

245

111314

34

34

3. The demand function for Product A produced by Hup Seng Sdn Bhd is given by

qp 3200 where p is the price per unit in ringgit (RM) and q is the quantity

demanded per week. a) Determine the revenue function b) If 50 units of Product A are sold per week, what is the average revenue? Answer: a)

23200

3200

)(

qq

qq

pqqr

b)




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50

50

500,7000,10

50

50350200

3200

2

2

RM

q

qq

q

qRqR

4. Fauziah deposits RM120,000 in a bank which gives an interest rate of 8% compounded

semi annually. How much will Fauziah get when she withdraws all her savings at the end of 5 years? Answers:

31.629,177

04.1000,120

2

08.01000,120

1

10

25

RMS

S

S

k

rPS

nk

5. Given 223 432, yyxxyxf . Find

a) xf

b) yf

c) xyf

Answer:

a) xyxf x 66 2

b) yxf y 83 2

c) xf xy 6




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BAHAGIAN B 1. a) The cost function of Product B is given by 6004802.0)( 2 qqqC .

i) Determine the average cost function of Product B. ii) What is the average cost to produce 20 units of product B?

b) The revenue of Product R is given by 2329)( qqqR and its average cost function is

given by q

qC60

5)(

i) Determine the total cost function ii) Determine the profit function iii) Determine the quantity of Product R that will maximise the profit iv) Calculate the maximum profit

Answers: a) i) Average Cost Function

qq

q

qq

q

qCqC

6004802.0

6004802.0 2

ii) Average Cost To Produce 20 units

40.78

20

600482002.0

6004802.020

RM

qqC

b)

i) Total Cost Function

605

605

q

qq

qqCqC

ii) Total Profit Function




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1

60324

605329

605329

2

2

2

qq

qqq

qqq

qCqRTPF

iii) Quantity of Product R that will maximise the profit

To maximise the profit, 0

dq

dand 0

2

2

qd

d

4

624

60324 2

q

qdq

d

qqdq

d

6

60324 2

2

2

qqqd

d

Therefore, 4q is maximised

iv) Calculate the maximum profit

108

604896

6043424

603244

2

2

RM

qqTPF

2. a) Differentiate the following:

i) Given that 3212 xxy . Use Product Rule to find dx

dy.

ii) Given that 12

13 2

x

xy . Use Quotient Rule to find

dx

dy.

iii) Given that 12 xy . Use Chain Rule to find dx

dy.

b) The cost function of a product is given by 100323 qqC .

i) Determine the marginal cost function. ii) Find the rate of change in the cost for producing 3 units of the product.

Answer:

a) 3212 xxy

Let 12 xxg and 32 xxh




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Then 2' xg and 2' 23 xxh

Therefore;

122322

231222

231222

23

23

23

'''

xxx

xxx

xxx

xhxgxgxhxf

b) 12

13 2

x

xy

Let 13 2 xxg and 12 xxh

Then xxg 6' and 2' xh

Therefore;

22

2

22

2

2

2

'''

12

16

12

6112

12

23612

x

x

x

xx

x

xxx

xh

xhxgxgxhxf

c) 12 xy

2

1

12 xy

Step 1: Introduce one new variable, u so that du

dy and

dx

du are easy to calculate.

Let 12 xu then 2

1

uy

Step 2: Calculate du

dy and

dx

du

When 12 xu and 2

1

uy

Then 2dx

du and 2

1

2

1

udu

dy

Step 3: Use the Chain Rule to calculate dx

dy




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1

2

1

2

1

'

22

1

u

u

dx

du

du

dy

dx

dyxy

Step 4: Calculate dx

dy into expressions of x .

Substitute 12 xu into dx

dy, gives

2

1

2

1

2

1

12

1

12

x

x

udx

dy

b) 100323 qqC .

i) Determine the marginal cost function.

Let 32 qxg then 2' xg and 3n

Therefore;

2

13

'1'

326

2323

q

q

xgxgnxyn

ii) Find the rate of change in the cost for producing 3 units of the product.

486

816

3326

3263

2

2'

qy

3. a) Find the values of the following:

i) dxxx 2

0112

ii) dxx

xx

2

1 4

24

b) The marginal cost function of a product is given by 32' 2 qqC and its fixed cost is

RM500.




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i) Determine the cost function. ii) How much is the cost of producing 3 units of the product?

Answers:

a) i) dxxx 2

0112

3

4

0223

16

002

10

3

222

2

12

3

2

2

1

3

2

10

1

1112

2

12

122

2323

2

0

23

2

0

101112

2

0

2

2

0

2

xxx

xxx

dxxx

dxxxx

ii) dxx

xx

2

1 4

24

2

1

1

11

2

12

1122

12

11

11

11

2

1

12

2

2

1 2

xx

x

dxx

b) 32' 2 qqC and its fixed cost is RM500.

i) Determine the cost function.




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1

50033

2

500312

2

32

32'

3

12

2

2

xq

xq

dxq

qqC

ii) How much is the cost of producing 3 units of the product?

527

5003333

2

50033

23

3

3

xqC




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BAHAGIAN C 1. a) Given the function 56382, 22 yyxxyxf

i) Find the critical point of this function. ii) Determine whether this point is a maximum or minimum. iii) Find the maximum or minimum value of this function.

b) A firm produces its product at two plants: A and B. The quantity produced in Plant A is x unit

and the quantity produced in Plant B is y units. In order to minimise its production cost, the

total quantity produced must be 100 units. The cost of producing these products is given by

10001571.0, 2 yxxyxC . Find the quantity to be produced at the respective

plant in order to minimise the cost. Answers:

a) Given the function 56382, 22 yyxxyxf

i) Find the critical point of this function.

STEP 1: Derive first degree differentiation

xf = 56382 22 yyxx

0,0,0,8,22 1112

dx

df

dx

df

dx

dfx

dx

dfx

dx

df

84 xf x

yf = 56382 22 yyxx

0,6,6,0,0dy

df

dy

dfy

dy

df

dy

df

dy

df

66 yf y

STEP 2: Obtain the critical points

2

84

xf x

1

66

yf y

The critical point is 1,2

ii) Determine whether this point is a maximum or minimum.

STEP 1: Obtain xxf , yyf , xyf to determine maximum or minimum point.




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84 xf x

4xxf

66 yf y

6yyf

84 xf x

0xyf

STEP 2: 2,,, bafbafbafM xyyyxx

24

0642

0M the point 1,2 is a minimum point

iii) Find the maximum or minimum value of this function.

STEP 1: To obtain the minimum value, substitute the values of x and y into the function

1,2 f = 56382 22 yyxx

= 51613282222

= 6

The minimum value 1,2 f is therefore 6 .

2. a) Find the area between the curve 23xy , the straight line 183 xy and the x -axis.

b) The demand function and supply function of a product is given by 2400 qp and

10020 qp respectively. Find the consumer’s surplus at market equilibrium.

Answers:

a) Find the area between the curve 23xy , the straight line 183 xy and the x -axis.

Step 1: Sketch the two graphs

23xy

x -2 -1 0 1 2 y 12 3 0 3 12




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183 xy

If 18,0 yx

If 6,0 xy

Step 2: Obtain the intersection points between graphs

2,3

6

1833

3183

2

2

2

xx

xx

xx

xx

Substitute the values of x into 23xy If 27,3 yx 27,3

If 12,2 yx 12,2 Step 3: Graph above minus graph below

2

2

2

6

3318

3318

xx

xx

xx

Step 4: Determine the integration and obtain its value

x 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 0

y

3

6

9

12

15

18

183 xy

23xy




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1

6

125

2

27

3

22

33

13

2

1362

3

12

2

126

3

1

2

16

12116

6

3232

2

3

32

2

3

1211

2

3

2

xxx

xxx

dxxx

b) The demand function and supply function of a product is given by 2400 qp and

10020 qp respectively. Find the consumer’s surplus at market equilibrium.

Step 1: Sketch the graphs in the first quadrant only

2400 qp

q -2 -1 0 1 2 p 396 399 400 399 396

10020 qp

If 10,0 qp

If 100,0 pq

q

p

10 10

300

100

400

0




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r 2

01

1

10,30

01030

30020

40010020

10020400

2

2

2

qq

qq

qq

qq

qq

Step 2: Obtain the market equilibrium point

Substitute the values of 10q into 10020 qp

300

1001020

10020

p

qp

Hence, 300,10 is the market equilibrium point.

Step 3: Find the consumer’s surplus

3

2000

30003

10004000

300103

110400

30003

1400

300012

400

30010400

10

0

3

10

0

3

10

0

12

10

0

2

qq

qq

dqqCS




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klinik matematik pengurusan open university … · the demand function for product a produced by...

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