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CALCULUS

• Integration is reverse of Differentiation

• Derivative of Function f(x) = f’(x) 

•

Integral of Function f’(x) = f(x) 

• Integral represents area under Curve

• Integration : defined between Limits is Definite Integration

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CALCULUS• Marginal Cost is given as f’(x) = 3x2 + 2x - 1

• Cost =

= x3 + x2 - x + k 

When x= 0 , Constant of Integration = 0

Cost = x3 + x2 - x

Cost of 2 Units = 10

dx x x )1223(

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CALCULUS• Marginal Cost Function of a Firm = (logx)2

• Cost Function =

• Integrating by Parts

• C= x (logx)2 -

• C = x (logx)2 - 2{xlogx - x}C = x(logx)2 - { 2xlogx}+ 2x 

2)(log  x

dx x x

 x.log.2

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CALCULUS

• If Principal P is paid in the bank every year,

The Interest Rate Compounded every year is i %For a period of n years

A =

If Rs. 48000 is earned uniformly throughout each year

for 5 years with Rate of Interest 12 % . Determine Future

Value of Earnings at end of 5 years

n

indneP

0

..

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CALCULUS

• P = 48000

• i = 0.12

• N = 5 years

• A =

• A =

• A = 48000 { (e 0.12n)/0.12 }

• A = 48000 { (e0.6 / 0.12) - (1/0.12)}

• A = 328844

n

indneP

0

..

ne

12.05

0

.)48000(

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CALCULUS• A Bank pays 6 % Interest per annum compounded

continuously. If a person places Rs. 1000 in the bank each

year , how much will be in his account after 3 years.

Ans :

P = 1000

i = 6 %n = 3 years

A =

A =

dneP

n

in..0

dnen ..1000

5

0

06.0

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CALCULUS• A = 1000 ( e 0.06n / 0.06)5

0

A = Rs. 3286.67

Principal explains the money deposited in the bank every year

Number of years indicate the amount of time money is kept

in the bank 

Rate of Interest indicates the Discounting Interest in the bank 

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CALCULUS• A Sinking Fund is created for redemption of a debenture

of Rs. 100000 at end of 20 years. How much money

should be provided out of profits each year for sinking fundif the Investment Fund can earn interest 9 % per annum.

compounded continuosly.

Ans :

A = 100000 n = 20 years i = 9 %

A =

A =

dneP

n

in..

0

dneP

n

n..

0

09.0

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CALCULUS

•

100000 =

• P = 1782. 50

• P indicates the Principal that should be provided each year

for sinking fund

•

A indicates the Amount

• i indicates the Interest Rate

dxeP

n

 x

..0

09.0

10

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CALCULUS• A machine costs Rs. 55000 and its effective life is

estimated to be 10 years. If Scrap Value is Rs. 5000

only. What should be retained out of profits at end

of each year to accumulate an Interest of 11 % in a manner

that new machine is purchased at same price after 10 years.

Ans : Net Amount of Machine after 10 years

= Total Cost of Machine - Amount Realized From Scrap

= 55000 - 5000

= 50,000

A = n

indneP

0

.

11

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CALCULUS

• 50,000 =

50,000 =

50,000 = P. {(e0.11x)/ 0.11 )}010

Principal = 2744. 28

dneP

n

in..0

dxePx ..

10

0

11.0

12