If C is the cost of producing an output x, then marginal cost function MC = dc/dx

**Cost functions from marginal cost functions**

If *C* is the cost of producing an output *x*, then
marginal cost function *MC* = dc/dx

Using integration, as the reverse process of differentiation, we
obtain,

Cost function *C *=* *âˆ«* *(* MC *)* dx *+* k*

Where *k* is the constant of integration which is to be
evaluated,

Average cost function *AC *= C/X, *x* â‰
0

**Example 3.9**

The marginal cost function of manufacturing *x* shoes is 6 + 10*x* âˆ’ 6*x*^{2} .
The cost producing a pair of shoes is â‚¹12. Find the total and average cost
function.

*Solution:*

Given,

Marginal cost *MC* = 6 +
10*x* âˆ’ 6*x*^{2}

*C *=* *âˆ«* MC dx *+* k*

= âˆ«( 6 +
10*x* âˆ’ 6*x*^{2} ) *dx* + *k*

= 6*x* + 5*x*^{2} âˆ’ 2*x* ^{3} + *k (1)*

*When x= 2 , C *=* *12* *(given)

12 = 12 +
20 âˆ’16 + *k *

*k*=-4

*C *=* *6*x *+* *5*x*^{2}* *âˆ’* *2*x*^{3}* *â€“* *4

**Example 3.10**

A company has determined that the marginal cost function for a
product of a particular commodity is given by *MC* = 125 + 10*x* âˆ’ *x*^{2}/9 where
C rupees is the cost of 9 producing *x* units of the commodity. If the
fixed cost is â‚¹250 what is the cost of producing 15 units.

*Solution:*

**Example 3.11**

The marginal cost function *MC* = 2 + 5*e ^{x}* (i)
Find

*Solution:*

**Rate of growth or sale**

If the rate of growth or sale of a function is a known function of
*t* say *f*(*t*) where *t* is a time measure, then total
growth (or) sale of a product over a time period *t* is given by,

Total sale =

**Example 3.12**

The rate of new product is given by *f* (*x*) = 100 âˆ’ 90 *e*^{âˆ’}* ^{x}* where

*Solution:*

**Example 3.13**

A company produces 50,000 units per week with 200 workers. The
rate of change of productions with respect to the change in the number of
additional labour *x* is represented as 300 âˆ’ 5*x* ^{2/3} . If 64 additional
labours are employed, find out the additional number of units, the company can
produce.

*Solution:*

Let *p* be the additional product produced for additional of *x*
labour,

âˆ´ The number of
additional units produced 16128

Total number of units produced by 264 workers

50,000 + 16,128 = 66128 units

**Example 3.14**

The rate of change of sales of a company after an advertisement
campaign is represented as, *f* (*t* ) = 3000*e* ^{âˆ’}^{0.3t} where *t*
represents the number of months after the advertisement. Find out the total
cumulative sales after 4 months and the sales during the fifth month. Also find
out the total sales due to the advertisement campaign *e *^{âˆ’}^{1 . 2}* *=* *0.3012,* e*^{âˆ’}^{1.5}* *=* *0.2231* *.

*Solution:*

**Example 3.15**

The price of a machine is 6,40,000 if the rate of cost saving is
represented by the function *f*(*t*) = 20,000 *t*. Find out the
number of years required to recoup the cost of the function.

*Solution:*

Saving Cost *S*(*t*)
=* *âˆ«^{t}_{0}* *20000*t dt*

= 10000 *t*^{2}

To recoup the total price,

10000 *t*^{2} = 640000

*t*^{2}* *= 64

*t *= 8

When *t* = 8 years, one can recoup the price.

Tags : Example Solved Problems with Answer, Solution, Formula , 12th Business Maths and Statistics : Chapter 3 : Integral Calculus - II

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

12th Business Maths and Statistics : Chapter 3 : Integral Calculus - II : Integration: Cost functions from marginal cost functions | Example Solved Problems with Answer, Solution, Formula

**Related Topics **

Privacy Policy, Terms and Conditions, DMCA Policy and Compliant

Copyright Â© 2018-2023 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.