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 .

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 .

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 .

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 .

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The total cost function $C (x)$ of a company is given as $C (x) = 1000 + 25 x + 2 x^{2}$ where $x$ is the output then determine marginal cost when $x = 15 .$

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Important Questions on Application of Derivatives

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

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>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

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?

EASY

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

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 ________

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

Let

f (x + y) = f (x) f (y)

for all

x

and

y .

Then what is

f^{'} (5)

equal to

[

where

f^{'} (x)

is the derivative of

f (x)] ?

EASY

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

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>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

If variable cost of

5

units of output is 100 and fixed cost is

40

. Find average cost at

5

units of output.

EASY

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

Which of the following curve is never U shape

HARD

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is

{t a n}^{- 1} (\frac{1}{2}) .

Water is poured into it at a constant rate of

5 cubic m / \min .

Then the rate

(

m / \min),

at which the level of water is rising at the instant when the depth of water in the tank is

10 m;

is:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

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>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

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.

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

A man

6^{'}

tall moves away from a source of light

20^{'}

above the ground level, his rate of walking being

4 m / h .

At what rate is the tip of his shadow moving?

EASY

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

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.

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

If the volume of spherical ball is increasing at the rate of

4 π

{cm}^{3} / \sec

then the rate of change of its surface area when the volume is

288 π {cm}^{3}

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

Average fixed cost for producing

8

units of output was

60

, what is the average Fixed cost at

6

units of output

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

If the volume of a spherical ball is increasing at the rate of

4 π cc / \sec

then the rate of increase of its radius

(in cm / \sec),

when the volume is

288 π cc

EASY

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

2

m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate

25 c m / s e c

, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is

1

m above the ground is:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

A spherical iron ball of radius

10 c m

is coated with a layer of ice of uniform thickness that melts at a rate of

50 c m^{3} / m i n .

When the thickness of the ice is

5 c m,

then the rate at which the thickness

(

c m / m i n)

of the ice decreases, is :

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

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>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

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>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

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.

The total cost function C(x) of a company is given as C(x)=1000+25x+2x2 where x is the output then determine marginal cost when x=15.

Important Questions on Application of Derivatives

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The total cost function $C (x)$ of a company is given as $C (x) = 1000 + 25 x + 2 x^{2}$ where $x$ is the output then determine marginal cost when $x = 15 .$