Given the total cost function of x units of a commodity as C x = 1 3 x 3 + 3 x 2 - 16 x + 2 . Find the average cost function.

Given the total cost function of x units of a commodity as, C x = 1 3 x 3 + 3 x 2 - 16 x + 2 . Average cost function A C = Total cost Number of units ⇒ A C = 1 3 x 3 + 3 x 2 - 16 x + 2 x ⇒ A C = 1 3 x 2 + 3 x - 16 + 2 x Hence, average cost function is 1 3 x 2 + 3 x - 16 + 2 x .

The total cost function for a production is given by C ( x ) = 3 4 x 2 - 7 x + 27 . Find the number of units produced for which M . C . = A . C . ( M . C . = Marginal Cost and A . C . = Average Cost.)

Total cost function for a production is given by C ( x ) = 3 4 x 2 - 7 x + 27 And, Marginal cost M . C . = d d x C x = d d x 3 4 x 2 - 7 x + 27 = 3 4 d d x x 2 - 7 d d x x + d d x 27 = 3 4 2 x - 7 + 0 ∵ d d x x n = n x n - 1 , n ∈ R = 3 2 x - 7 . . . . . i Now, we know that Average cost A . C . = Total cost Number of units of a good produced ⇒ A . C . = 3 4 x 2 - 7 x + 27 x . . . . i i Now, according to the question, using i , and i i , we have M . C . = A . C . ⇒ 3 2 x - 7 = 3 4 x 2 - 7 x + 27 x ⇒ x 3 2 x - 7 = 3 4 x 2 - 7 x + 27 ⇒ 3 2 x 2 - 7 x = 3 4 x 2 - 7 x + 27 ⇒ 3 4 x 2 = 27 ⇒ x 2 = 27 3 × 4 = 36 ⇒ x = 6 Hence, the number of units produced is 6 .

Given the total cost function of x units of a commodity as C x = 1 3 x 3 + 3 x 2 - 16 x + 2 . Find the marginal cost function.

Given, C x = 1 3 x 3 + 3 x 2 - 16 x + 2 . The marginal cost, denoted by M C , is defined as the rate of change of C ( x ) with respect to x , i.e., M C x = d C d x . ⇒ M C x = d 1 3 x 3 + 3 x 2 - 16 x + 2 d x ⇒ M C x = 1 3 d x 3 d x + 3 d x 2 d x - 16 d x d x + d 2 d x Using d x n d x = n x n - 1 ⇒ M C x = 1 3 × 3 x 2 + 6 x - 16 + 0 ⇒ M C x = x 2 + 6 x - 16 Therefore, the marginal cost function is x 2 + 6 x - 16 .

A manufacturer's marginal cost function is 500 2 x + 25 . Find the cost involved to increase production from 100 units to 300 units.

Marginal cost is the derivative of the cost function. ∴ C ( x ) = ∫ M ( x ) dx = ∫ 500 2 x + 25 dx Let 2 x + 25 = t ⟹ 2 dx = dt ∴ C ( x ) = ∫ 250 t dt C ( x ) = 500 ( t ) + c = 500 ( 2 x + 25 ) + c C ( 100 ) = 500 ( 225 ) + c = 7500 + c C ( 300 ) = 500 ( 625 ) + c = 12500 + c C ( 300 ) - C ( 100 ) = 5000 Therefore, cost involved to increase production from 100 units to 300 units is 5000 .

The average cost function for x units of a commodity is A C = 1 3 x 2 + 3 x - 7 + 16 x . Show that the marginal average cost is given by x ( M C ) - C ( x ) x 2 .

Given, average cost function A C = 1 3 x 2 + 3 x - 7 + 16 x Average cost = Total cost x Hence, total cost C ( x ) = ( 1 3 x 2 + 3 x - 7 + 16 x ) × x = 1 3 x 3 + 3 x 2 - 7 x + 16 Marginal cost function M C = d d x ( C ( x ) ) M C = d d x ( 1 3 x 3 + 3 x 2 - 7 x + 16 ) ⇒ M C = x 2 + 6 x - 7 Marginal average cost = d d x ( A C ) = d d x ( x 2 3 + 3 x - 7 + 16 x ) = 2 3 x + 3 - 16 x 2 x ( M C ) - C ( x ) x 2 = x ( x 2 + 6 x - 7 ) - ( x 3 3 + 3 x 2 - 7 x + 16 ) x 2 = x 3 + 6 x 2 - 7 x - x 3 3 - 3 x 2 + 7 x - 16 x 2 = 2 x 3 3 + 3 x 2 - 16 x 2 = 2 3 x + 3 - 16 x 2 Hence, marginal average cost = x ( M C ) - C ( x ) x 2

The average cost function A C for a commodity is given by A C = x + 5 + 36 x in terms of output x . Find the total cost and the marginal cost as the functions of x .

Given, A C = x + 5 + 36 x Total cost = Average cost × Number of output units Total cost C ( x ) = ( x + 5 + 36 x ) x = x 2 + 5 x + 36 Marginal cost is rate of change of total cost. Marginal cost M C = d d x ( C ) = d d x ( x 2 + 5 x + 6 ) = 2 x + 5 Hence, total cost is x 2 + 5 x + 36 and 2 x + 5 .

The demand x of a commodity in terms of price p is given by x = 1 9 ( 245 - 7 p ) . The average cost is R s . 20 per unit. Find the cost function in terms of p .

Average cost = R s 20 Average cost = Total cost Number of units Total cost = Average cost × Number of units 20 × 1 9 ( 245 - 7 p ) = C ( x ) C ( x ) = 20 9 ( 245 - 7 p ) Hence, cost function in terms of p is 20 9 ( 245 - 7 p ) .

The average cost function AC for a commodity is given by A C = x + 5 + 36 x in terms of output x. Find the output for which AC increases.

Given, average cost function A C = x + 5 + 36 x To check output for which AC increases, we will differentiate A C d d x ( A C ) = d d x ( x + 5 + 36 x ) = 1 - 36 x 2 For A C to be increasing, d d x ( A C ) > 0 1 - 36 x 2 > 0 1 > 36 x 2 x 2 > 36 x > 6 Hence, for x > 6 , A C increases.

EASY

Earn 100

The cost of manufacturing $x$ units of a commodity is $x^{2} + 4 x$ . Find the output where marginal cost is equal to average cost.

Important Questions on Application of Calculus

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

The average cost function of a good is

2 Q + 6 + \frac{13}{Q}

where

Q

is the quantity produced. The approx. cost at

Q = 15

is ________

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

Which of the following curve is never U shape

MEDIUM

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

For

15

units of output, average cost was

Rs . 250

end fixed cost per unit of output was

125

, then the average variable cost per at

15

units?

MEDIUM

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

Average fixed cost for producing

8

units of output was

60

, what is the average Fixed cost at

6

units of output

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

Given the total cost function of

x

units of a commodity as

C (x) = \frac{1}{3} x^{3} + 3 x^{2} - 16 x + 2

. Find the average cost function.

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

If variable cost of

5

units of output is 100 and fixed cost is

40

. Find average cost at

5

units of output.

MEDIUM

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

A firm producing 7 units of output has an average total cost of

Rs . 150

and has to pay

Rs . 350

to its fixed factors of production whether it produces or not. How much of the average total cost is made up of variable costs?

MEDIUM

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

The total cost function for a production is given by

C (x) = \frac{3}{4} x^{2} - 7 x + 27

. Find the number of units produced for which

M . C . = A . C .

(

M . C . =

Marginal Cost and

A . C . =

Average Cost.)

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

The cost function of a product is given by

C (x) = \frac{x^{3}}{3} - 45 x^{2} - 900 x + 36

where

x

is the number of units produced. How many units should be produced to minimise the marginal cost?

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

The marginal cost function of $x$ units of a product is given by $MC = 3 x^{2} - 10 x + 3$ . The cost of producing one unit is $\mp 7$ . Find the total cost function and average cost function.

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

Given the total cost function of

x

units of a commodity as

C (x) = \frac{1}{3} x^{3} + 3 x^{2} - 16 x + 2

. Find the marginal cost function.

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

A manufacturer's marginal cost function is

\frac{500}{\sqrt{2 x + 25}}

. Find the cost involved to increase production from

100

units to

300

units.

MEDIUM

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

The total cost

C (x)

associated with the production of

x

units of an item is given by

C (x) = 0 \cdot 005 x^{3} - 0 \cdot 02 x^{2} + 30 x + 5000

. Find the marginal cost when

3

units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

A publishing house finds that the cost directly attributed to each book is

₹ 30

and that the fixed costs are

₹ 15000

. If each book can be sold for

₹ 45

, then determine the cost function.

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

The total cost of producing and marketing x units of commodity is given by

C = 2 x + \frac{1}{5} x^{2} + 5 e

. Find the fixed cost.

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

The average cost function for $x$ units of a commodity is $A C = \frac{1}{3} x^{2} + 3 x - 7 + \frac{16}{x}$ . Show that the marginal average cost is given by $\frac{x (M C) - C (x)}{x^{2}}$ .

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

The average cost function

A C

for a commodity is given by

A C = x + 5 + \frac{36}{x}

in terms of output

x

. Find the total cost and the marginal cost as the functions of

x

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

The demand

x

of a commodity in terms of price

p

is given by

x = \frac{1}{9} (245 - 7 p)

. The average cost is

R s . 20

per unit. Find the cost function in terms of

p

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

Cost of producing

75

units of a commodity is

275

and cost of producing

150

units is

300

. Assuming that TC is linear, find the cost function.

EASY

Mathematics>Integral Calculus>Application of Calculus>Average and Marginal Cost

The average cost function AC for a commodity is given by

A C = x + 5 + \frac{36}{x}

in terms of output x. Find the output for which AC increases.

The cost of manufacturing x units of a commodity is x2+4x. Find the output where marginal cost is equal to average cost.

Important Questions on Application of Calculus

Learn and Prepare for any exam you want !

The cost of manufacturing $x$ units of a commodity is $x^{2} + 4 x$ . Find the output where marginal cost is equal to average cost.