Given the total cost function for x units of a commodity as C ( x ) = 1 3 x 3 + x 2 - 8 x + 5 , find the marginal cost function.

Given, cost function C ( x ) = 1 3 x 3 + x 2 - 8 x + 5 Marginal cost is rate of change of cost. Marginal cost function M C = d d x ( 1 3 x 3 + x 2 - 8 x + 5 ) M C = x 2 + 2 x - 8 Hence, marginal cost is x 2 + 2 x - 8 .

Given the total cost function for x units of a commodity as C ( x ) = 1 3 x 3 + x 2 - 8 x + 5 . Find average cost function.

Given, cost function C ( x ) = 1 3 x 3 + x 2 - 8 x + 5 Average cost function A C = Total cost Number of units A C = 1 3 x 3 + x 2 - 8 x + 5 x A C = 1 3 x 2 + x - 8 + 5 x Hence, average cost function is 1 3 x 2 + x - 8 + 5 x .

Given the total cost function for x units of a commodity as C ( x ) = 1 3 x 3 + x 2 - 8 x + 5 , find slope of average cost function.

Given, cost function C ( x ) = 1 3 x 3 + x 2 - 8 x + 5 Average cost function A C = Total cost Number of units A C = 1 3 x 3 + x 2 - 8 x + 5 x A C = 1 3 x 2 + x - 8 + 5 x Slope of average cost function = d d x ( 1 3 x 2 + x - 8 + 5 x ) = 2 3 x + 1 - 5 x 2 Hence, Slope of average cost function is 2 3 x + 1 - 5 x 2 .

A product can be manufactured at a total cost C ( x ) = x 2 100 + 100 x + 40 , where x is the number of units produced. The price at which each unit can be sold is given by P ( x ) = 200 - x 400 . Determine the production level x at which the profit is maximum. What is the price per unit and total profit at this level of production?

Given, cost function C ( x ) = x 2 100 + 100 x + 40 Price P ( x ) = 200 - x 400 Revenue function R ( x ) = P ( x ) × x = ( 200 - x 400 ) × x = 200 x - x 2 400 Profit G ( x ) = R ( x ) - C ( x ) = 200 x - x 2 400 - x 2 100 - 100 x - 40 = 100 x - 5 x 2 400 - 40 For maxima/minima d d x G ( x ) = 0 d d x G ( x ) = d d x ( 100 x - 5 x 2 400 - 40 ) 0 = 100 - x 40 x = 4000 d 2 d x 2 G ( x ) = - 1 40 < 0 Hence, x = 4000 is point of maxima. P ( 4000 ) = 200 - 4000 400 = 190 G ( 4000 ) = 100 ( 4000 ) - 5 ( 4000 ) 2 400 - 40 = 199960 Hence, price per unit is ₹ 190 and profit is ₹ 199960 .

E M B I B E

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Application of Calculus in Commerce and Economics>EXERCISE>Q 15

EASY

12th ICSE

IMPORTANT

Earn 100

A company finds that total revenue from sale of $q$ units of its product is $2 q$ rupees whereas the cost is $500 + \frac{1}{2} {(\frac{q}{20})}^{2}$ . Find the rate of change of profit when $q = 500$ .

Important Questions on Application of Calculus in Commerce and Economics

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Application of Calculus in Commerce and Economics>EXERCISE>Q 16

Given the total cost function for

x

units of a commodity as

C (x) = \frac{1}{3} x^{3} + x^{2} - 8 x + 5

, find the marginal cost function.

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Application of Calculus in Commerce and Economics>EXERCISE>Q 16

Given the total cost function for

x

units of a commodity as

C (x) = \frac{1}{3} x^{3} + x^{2} - 8 x + 5

. Find average cost function.

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Application of Calculus in Commerce and Economics>EXERCISE>Q 16

Given the total cost function for

x

units of a commodity as

C (x) = \frac{1}{3} x^{3} + x^{2} - 8 x + 5

, find slope of average cost function.

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Application of Calculus in Commerce and Economics>EXERCISE>Q 17

If the cost function of a production unit is

C (x) = 40 - 20 x + 10 x^{2}

, find the average cost function and the output x where average cost is minimum. Verify that the average cost is equal to the marginal cost at this level of output.

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Application of Calculus in Commerce and Economics>EXERCISE>Q 18

The demand function of a monopolist is