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**The cost function and average cost function from marginal cost function : **If C is the cost of producing an output x then marginal cost dC function, mc = dc/dx Using integration as reverse process of differentiation we obtain,

**Cost function, **

where k is the constant of integration which can be evaluated if the fixed cost is known. If the fixed cost is not known, then k = 0.

**Average cost fucntion, **

**Example 12 The marginal cost function of manufacturing x units of a commodity is 6 + 10x - 6x ^{2} . Find the total cost and average cost, given that the total cost of producing 1 unit is 15. **

Solution : Given that,

Given, when x = 1,C = 15

15= 6 + 5 - 2 + k

⇒ k = 6

∴ Total Cost function, C = 6x + 5x 2- 2x 3 + 6

**Example 13 The marginal cost function of manufacturing x units of a commodity is 3x ^{2} - 2x + 8. If there is no fixed cost find the total cost and average cost functions.**

Solution : Given that,

Solution : Given that,

** The revenue function and demand function from marginal revenue function**

If R is the total revenue function when the output is x, then marginal revenue MR = dx/dR . Integrating with respect to ‘x’ we get

where ‘k’ is the constant of integration which can be evaluated under given conditions. If the total revenue R = 0, when x = 0

**Example 15 If the marginal revenue for a commodity is MR = 9 - 6x ^{2 }+ 2x, find the total revenue and demand function. **

Solution : Given that, MR = 9 - 6x 2 + 2x

For the marginal revenue function MR = 3 - 2x - x 2 , find the revenue function and demand function.

Solution : Given that

MR = 3 - 2x - x

since R = 0, when x = 0, k = 0

If the marginal revenue for a commodity is

Solution : Given that,

when no product is sold, revenue is zero. when x = 0, R = 0.

**The demand function when the elasticity of demand is given**

We know that, Elasticity of demand

This equation yields the demand function ‘ p’ as a function of ‘x ’.

The revenue function can be found out by using the relation, R = px .

**Example 18 The elasticity of demand with respect to price p for a commodity is demand x- 5/x , x > 5 when the demand is ‘x’. Find the if the price is 2 when demand is 7. Also find function the revenue function**

Solution : Given that, Elasticity of demand

Integrating both sides,

when p = 2, x = 7, k = 4

∴ The demand function is,

The elasticity of demand with respect to price for a commodity is a constant and is equal to 2. Find the demand function and hence the total revenue function, given that when the price is 1, the demand is 4.

Solution : Given that, Elasticity of demand, η

Integrating both sides,

Given, when x = 4, p = 1

From (1) we get k = 4

∴ (1) ⇒ xp2 = 4 or p^{2} = 4/x

**Example 20 Revenue R = px = 2 x The marginal cost and marginal revenue with respect to a commodity of a firm are given by C' (x) = 4 + 0.08x and R'(x) = 12. Find the total profit, given that the total cost at zero output is zero.**

Solution :

Given that,

MC = 4 + 0.08x

But given when x = 0,C = 0

∴ (1)⇒ 0 = 0 + 0 + k

∴ k = 0

∴ C(x) = 4x + 0.04x 2 ---------(2)

Revenue = 0 when x = 0.

∴ k = 0

∴ R (x) = 12x ---------(3)

Total profit function, P(x) = R(x) - C (x)

= 12x - 4x - 0.04x

= 8x - 0.04x

**Example 21 The marginal revenue function (in thousands of rupees) of a c o m mo d ity is 7 + e ^{-0.05x }where x is the number of units sold. Find the total revenue from the sale of 100 units ( e ^{-5} = 0.0067)**

Solution : Given that,

Marginal revenue, R'(x) = 7 + e ^{- 0.05x }

∴ Total revenue from sale of 100 units is

Revenue, R = Rs.7,19,866.

**Example 22 The marginal cost C'(x) and marginal revenue R'(x) are x given by C'(x) = 20 +x/20 and R'(x) = 30 The fixed cost is Rs. 200. Determine the maximum profit. Solution :**

When quantity produced is zero, the fixed cost is Rs. 200. i.e. when x = 0, C = 200,

⇒ k = 200

The revenue, R'(x) = 30

When no product is sold, revenue = 0 --------(2)

i.e., when x = 0, R = 0

∴ Revenue, R(x) = 30x

Profit, P = Total revenue - Total cost

∴ Profit is maximum when x = 200

**Example 23 A company determines that the marginal cost of producing x units is C'(x) = 10.6x. The fixed cost is Rs. 50. The selling price per unit is Rs. 5. Find (i) Total cost function (ii) Total revenue function (iii) Profit function. **

Solution : Given,

= 5.3x^{2} + k --------(1)

Given fixed cost = Rs. 50

(i.e.) when x = 0, C = 50 ∴ k = 50

Hence Cost function, C = 5.3x 2 + 50

(ii) Total revenue = number of units sold x price per unit Let x be the number of units sold. Given that selling price per unit is Rs. 5.

∴ Revenue R(x) = 5x.

(iii) Profit, P= Total revenue - Total cost

= 5x - (5.3x 2 + 50)

= 5x - 5.3x 2 - 50.

**Example 24 Determine the cost of producing 3000 units of commodity if the marginal cost in rupees per unit is C'(x) **

Solution : Given, Marginal cost, C'(x)** **

When x = 3000,

Cost of production, C(x) = Rs.9000

Solution :

The cost of producing 10 incremental units after 15 units have been produced