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Marginal income and marginal cost - class-XII

Description: marginal income and marginal cost
Number of Questions: 26
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Fill in the blanks:
___________INR 50 notes make 300 rupees

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. 3

  2. 4

  3. 6

  4. 8

Show answer
Correct Option: C
Explanation:

$6$ INR $50$ notes make $300 rupees$ as:

$6\times 50=300$ 

Marginal cost is defined as the instantaneous rate of change of total cost at any level of output.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. True

  2. False

Show answer
Correct Option: A

A man spends $Rs\ 1800$ per month on an average for the first four months and $Rs\ 2000$ per month for the next $8$ months and saves $Rs\ 5600$ a year. What is his average monthly income?

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $Rs\ 2400$

  2. $Rs\ 2000$

  3. $Rs\ 1800$

  4. $Rs\ 2500$

Show answer
Correct Option: A
Explanation:

Total exp in 4 months $ = 4 \times 1800 = 7200$

Total exp in 8 months $ = 8 \times 2000 = 16000$
total exp $ = 7200 + 16000$
$=23200$
savings = 5600
total income = 23200 + 5600
=28800
$Avearage\,\,income = \dfrac{{28800}}{{12}}$
$=2400/month.$

When the price of refrigerator rises from Rs.$2000$ per unit to Rs. $2500$ per unit and in response to this rise is price the quantity supplied increases from $2500$ units to $3500$ units, find out the price elasticity of supply.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $4.5$

  2. $1.5$

  3. $3.5$

  4. $2$

Show answer
Correct Option: B
Explanation:
$\Rightarrow$  Let $q _1=2500,\,q _2=3500,\,p _1=2000$ and $p _2=2500$.
$\Rightarrow$  $\Delta q=q _2 - q _1=3500-2500=1000$

$\Rightarrow$  $\dfrac{q _2 + q _1}{2}=\dfrac{3500+2500}{2}=3000$

$\Rightarrow$  $\Delta p=p _2 - p _1=2500-2000=500$

$\Rightarrow$  $\dfrac{p _2+p _1}{2}=\dfrac{2500+2000}{2}=2250$

$\Rightarrow$  We know, $e _s=\dfrac{\Delta q}{\Delta p}\times \dfrac{p _2+p _1}{q _2+q _1}$

$\Rightarrow$  $e _s =\dfrac{1000}{3000}  \times \dfrac{2250}{500}$

$\Rightarrow$  $e _s=\dfrac{1}{3}\times \dfrac{4.5}{1}=\dfrac{4.5}{3}=1.5$

If both average cost (AC) and marginal cost (MC) are U shaped, then

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. AC will reach a minimum at a level of output that is less than that at which MC reaches a minimum.

  2. the total cost curve will be a straight line.

  3. AC will reach a minimum at a level of output that is greater than that at which MC reaches a minimum.

  4. both AC and MC will reach a minimum at the same level of output.

Show answer
Correct Option: C

The cost of cultivating a square field at the rate of $Rs.\,135$ per hectare is $Rs.\,1215$. The cost of putting a fence around it at the rate of $75\;paise$ per metre would be ...... .

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $Rs.\,360$

  2. $Rs.\,810$

  3. $Rs.\,900$

  4. $Rs.\,1800$

Show answer
Correct Option: C
Explanation:

rate of cultivating a square field =Rs.135 per hectare, 
cost of cultivating a square field=Rs.1215
area of square field=cost of cultivating a square field/rate of cultivating a square field
$A=1215/135$
$A=9 hectare$
1 hectare =10000m^2
$Area=a^2$
$90000=a^2$
$a=300$

cost of putting a fence around it at the rate of 75 paise per metre 
Total cost=0.75*4*300Rs
C=900Rs
Answer (C) 900RS

If average cost is at a minimum, then

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. it is equal to marginal cost.

  2. total cost is also at a minimum.

  3. profit is at a maximum.

  4. all of the above are true.

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  If average cost is at a minimum, then $it\, is\, equal\, to\, marginal \,cost.$

$\Rightarrow$  Average cost is Production cost per unit of output, computed by dividing the total of fixed costs and variable costs by the number of total units produced (total output). Lower average costs are a potent competitive advantage.
$\Rightarrow$ MC indicates the rate at which the total cost of a product changes as the production increases by one unit. However, because fixed costs do not change based on the number of products produced, the marginal cost is influenced only by the variations in the variable costs.
$\Rightarrow$  When average cost is at its minimum, $Marginal\, cost = Average\, cost$. That's because the second marginal cost gets higher than average cost, average cost has to get higher because Marginal cost is greater than average cost.

If the amount of taxes paid $(T)$ depends on income $(x)$, how would you use calculus notation to describe the marginal tax rate? If taxes and income are both measured in dollars per year, what are the units of the marginal tax rate?

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. Same as units of $T$

  2. Same as units of $x$

  3. Unit - less

  4. None of these

Show answer
Correct Option: C
Explanation:

If taxes and income are both measured in dollars per year, then the units of the marginal tax rate will be unit-less.

If a consumer daily income rises from Rs. $300$ to Rs.$350$, his purchase of a good $X$ increases from $25$ units per day to $40$ units; find the income elasticity of demand for $X$?

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $2$

  2. $1$

  3. $3$

  4. $4$

Show answer
Correct Option: C
Explanation:

Change in quantity demand $\Delta Q$ = ${Q} _{2} - {Q} _{1}$ = $40 - 25 = 15$
Change in income $\Delta M $ = ${M} _{2} + {M} _{1}$ = $350 -300$ = $50$
$\dfrac{\Delta Q}{\Delta M}$ x $\dfrac {{M} _{2} + {M} _{1}}{{Q} _{2} - {Q} _{1}}$


= $\dfrac{15}{50} \times \dfrac{350+300}{25+40}$

= $\dfrac{3}{10} \times {650}{65}$


= $\dfrac{3}{10} \times {10}$

= $3$

Differentiation refers to the process whereby we

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. Calculate the area underneath a curve

  2. Calculate the intercept of a curve with the horizontal axis

  3. Calculate the gradient to a curve at any point on the curve

  4. Calculate the intercept of a curve with the vertical axis

Show answer
Correct Option: C
Explanation:

Differentiation is the process where the slope/gradient of a curve is calculated at a point lying on it.

The cost function for x units of a commodity is given by $C(x)=\dfrac{x^3}{3}+x^2-15x+3$. Find marginal cost function.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $x^2+2x-15$

  2. $x^2-2x-15$

  3. $x^2+2x+15$

  4. None of these

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have, $C(x)=\dfrac{x^3}{3}+x^2-15x+3$.

$\Rightarrow$  Marginal cost = $\dfrac{d}{dx}(C)$

$\Rightarrow$  Marginal cost = $\dfrac{d}{dx}(\dfrac{x^3}{3}+x^2-15x+3)$

$\Rightarrow$  Marginal cost = $\dfrac{3\times x^2}{3}+2\times x-15$

$\therefore$    Marginal cost = $x^2+2x-15$

If a function $p = 50 3x$, find $TR$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $50x-3x^2$

  2. $50x+3x^2$

  3. $50+3x^2$

  4. $50-3x^2$

Show answer
Correct Option: A
Explanation:

Total revenue$(TR)=px\ \implies (50-3x).x\ \implies 50x-3x^2$

When we differentiate an expression with respect to one of a number of independent variables, we are engaged in

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. Finding definite integrals

  2. Total differentiation

  3. Partial differentiation

  4. Integration

Show answer
Correct Option: C
Explanation:

A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant.

Hence, C is correct.

The cost function of a firm $C(x)=2x^2-4x+5$. Find the average cost when $x=2$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $\dfrac{5}{2}$

  2. $\dfrac{1}{4}$

  3. $\dfrac{3}{4}$

  4. $\dfrac{5}{7}$

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have, $C(x)=2x^2-4x+5.$


$\Rightarrow$   Average cost = $\dfrac{C(x)}{x}$


$\Rightarrow$    Average cost = $\dfrac{2x^2-4x+5}{x}$

$\Rightarrow$    Average cost = $2x-4+\dfrac{5}{x}$

$\Rightarrow$    Now, substitute value of $x=2$.
$\Rightarrow$    Average cost = $2(2)-4+\dfrac{5}{2}=\dfrac{5}{2}$

The cost function of a firm $C(x)=2x^2-4x+5$. Find the average cost when $x=10$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $16.5$

  2. $15.5$

  3. $12.5$

  4. None of these

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have, $C(x)=2x^2-4x+5$.

$\Rightarrow$  Average cost = $\dfrac{C(x)}{x}$

$\Rightarrow$  Average cost = $\dfrac{2x^2-4x+5}{x}$

$\Rightarrow$  Average cost = $2x-4+\dfrac{5}{x}$

$\Rightarrow$  Substitute value of $x=10$.
$\Rightarrow$  Average cost = $2\times 10-4+\dfrac{5}{10}=20-4+0.5=16.5$
$\therefore$  $ Average\, cost = 16.5$

The cost function of a firm $C(x)=4x^2-x+70$. Find the marginal cost when $x=3$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. 23

  2. 24

  3. 25

  4. 26

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have, $C(x)=4x^2-x+70$.


$\Rightarrow$  Marginal cost = $\dfrac{d}{dx}C(x)$


$\Rightarrow$  Marginal cost = $\dfrac{d}{dx}(4x^2-x+70)$

$\Rightarrow$  Marginal cost = $2\times 4x-1=8x-1$
$\Rightarrow$  Substitute value of $x=3$,
$\Rightarrow$  Marginal cost = $8\times3-1=23$
$\therefore$    $Marginal\, cost = 23$.

The cost function of a firm $C(x)=2x^2-4x+5$. Find the marginal cost when $x=10$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. 34

  2. 35

  3. 36

  4. None of these

Show answer
Correct Option: C
Explanation:

$\Rightarrow$   We have, $C(x)=2x^2-4x+5$.

$\Rightarrow$   Marginal cost = $\dfrac{d}{dx}C(x)$
$\Rightarrow$   Marginal cost = $\dfrac{d}{dx}(2x^2-4x+5)$
$\Rightarrow$   Marginal cost = $4x-4$
$\Rightarrow$   Now substitute value of $x=10$.
$\Rightarrow$   Marginal cost = $4(10)-4=40-4=36.$

The cost function of a firm $C(x)=2x^2-4x+5$. Find the marginal cost when $x=2$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. 4

  2. 5

  3. 6

  4. 7

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have, $C(x)=2x^2-4x+5$.


$\Rightarrow$  Marginal cost = $\dfrac{d}{dx}C(x)$.


$\Rightarrow$  Marginal cost = $\dfrac{d}{dx}(2x^2-4x+5)$

$\Rightarrow$  Marginal cost = $4x-4$
$\Rightarrow$  Substitute value of $x=2$.
$\Rightarrow$  Marginal cost = $4\times 2-4=8-4=4$
$\therefore$    $Marginal\,cost=4.$

The cost function of a firm $C(x)=3x^2-2x+3$. Find the marginal cost when $x=3$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. 19

  2. 18

  3. 16

  4. 17

Show answer
Correct Option: C
Explanation:

$\Rightarrow$   We have, $C(x)=3x^2-2x+3$

$\Rightarrow$   Marginal cost = $\dfrac{d}{dx}C(x)$

$\Rightarrow$   Marginal cost = $\dfrac{d}{dx}(3x^2-2x+3)$

$\Rightarrow$   Marginal cost = $2\times 3x-2=6x-2$.
$\Rightarrow$   Now, substitute value of $x=3$,
$\Rightarrow$   Marginal cost = $6\times 3-2=18-2=16$

The cost function for x units of a commodity is given by $C(x)=3x^3-6x+5$. Find marginal cost function , when $x=2$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $6$

  2. $4$

  3. $2$

  4. None of these

Show answer
Correct Option: D
Explanation:
The derivative of the cost function $C(x)$ is called marginal cost with notation:

$C'(x)=  \dfrac{dC}{dx} $

$C'(x)=  9\times x^{2} -6 $

Putting the value of x as $2$

We get

$C'(x)=  9\times 2^{2} -6 $

$C'(x)=  30 $

$Marginal\space cost =36$

The cost function of a firm $C(x)=4x^2-x+70$. Find the average cost when $x=3$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $\dfrac{104}{3}$

  2. $\dfrac{103}{3}$

  3. $\dfrac{105}{3}$

  4. $\dfrac{103}{4}$

Show answer
Correct Option: B
Explanation:

$\Rightarrow$   We have, $C(x)=4x^2-x+70$.

$\Rightarrow$   Average cost = $\dfrac{C(x)}{x}$
$\Rightarrow$   Average cost = $\dfrac{4x^2-x+70}{x}$
$\Rightarrow$   Average cost = $4x-1+\dfrac{70}{x}$
$\Rightarrow$   Substitute value of $x=3$.
$\Rightarrow$  Average cost = $12-1+\dfrac{70}{3}=\dfrac{36-3+70}{3}=\dfrac{103}{3}$
$\therefore$     $Average\, cost=\dfrac{103}{3}$

The demand function of a monopolist is given by $p=1500-2x-x^2$. Find the marginal revenue when $x=10$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $1170$

  2. $1160$

  3. $1150$

  4. None of these

Show answer
Correct Option: B
Explanation:

$\Rightarrow$  We have, $p=1500-2x-x^2$

$\Rightarrow$  Revenue Function   $R=p\times x$
$\therefore$       $R=1500x-2x^2-x^3$.
$\Rightarrow$   Marginal revenue = $\dfrac{d}{dx}R$

$\Rightarrow$   Marginal revenue = $\dfrac{d}{dx}(1500x-2x^2-x^3)$ 

$\Rightarrow$   Marginal revenue = $1500-4x-3x^2$
$\Rightarrow$   Now, substitute $x=10$.
$\Rightarrow$   Marginal revenue = $1500-2(100)-3(100)^2=1160$
$\therefore$   Marginal revenue is $1160$.

Given the marginal cost function $\dfrac{2x}{3}+3-\dfrac{16}{x^2}$, find  average cost function.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $\dfrac{1}{3}x^2+3x-7+\dfrac{16}{x}$

  2. $\dfrac{1}{2}x^2+3x-7+\dfrac{16}{x}$

  3. $\dfrac{1}{4}x^2+3x-7+\dfrac{16}{x}$

  4. None of these

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have $MC=\dfrac{2x}{3}+3-\dfrac{16}{x^2}$


$\Rightarrow$  $Average\,\,cost=\int (MC)dx$ 


$\Rightarrow$   $Average\,\,cost=\int (\dfrac{2x}{3}+3-\dfrac{16}{x^2})dx$


$\therefore$   $Average\,\,cost=\dfrac{2x^2}{2\times 3}+3x-7+\dfrac{16}{x}$

$\Rightarrow$   $Average\,\,cost=\dfrac{1}{3}x^2+3x-7+\dfrac{16}{x}$

The cost function of a firm $C(x)=3x^2-2x+3$. Find the average cost when $x=3$.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. 8

  2. 9

  3. 10

  4. 12

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have, $C(x)=3x^2-2x+3$

$\Rightarrow$  Average cost = $\dfrac{C(x)}{x}$

$\Rightarrow$  Average cost = $\dfrac{3x^2-2x+3}{x}$

$\Rightarrow$  Average cost = $3x-2+\dfrac{3}{x}$.

$\Rightarrow$  Substitute value of $x=3$,
$\Rightarrow$  Average cost = $3\times 3-2+\dfrac{3}{3}=9-2+1=8$
$\therefore$    $Average\, cost\,=8$. 

If the demanding Law is given by $q = \dfrac{20}{p+1}$, find the elasticity of demand with respect to price at the point when $p = 3.$

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $\dfrac43$

  2. $-\dfrac34$

  3. $\dfrac23$

  4. $-\dfrac32$

Show answer
Correct Option: B
Explanation:
Elasticity of demand $=\cfrac{\cfrac{dq}{q}}{\cfrac{dp}{p}}=-\cfrac{p}{(p+1)}$
When $p=3$
Elasticity of demand $=-\cfrac{3}{4}$

If the total cost function for a manufacturer is given by $C =\dfrac{5x^2}{\sqrt(x^2+3)}+5000$, find marginal cost function.

business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

  1. $\dfrac{3x(x^2+6)}{(x^2+3)^{(3/2)}}$

  2. $\dfrac{4x(x^2+6)}{(x^2+3)^{(3/2)}}$

  3. $\dfrac{5x(x^2+6)}{(x^2+3)^{(3/2)}}$

  4. None of these

Show answer
Correct Option: C
Explanation:

Given cost function $C\left(x\right)=\dfrac{5x^2}{\sqrt{\left(x^2+3\right)}}+5000$


Marginal cost function is given by $C'\left(x\right)$

$C'\left(x\right)=\dfrac{\left(\sqrt{x^2+3}\right)\left(10x\right)-\dfrac{5x^2}{2\sqrt{x^2+3}}\times\left({2x+10}\right)}{\left(\sqrt{x^2+3}\right)^2}$

$\dfrac{d}{dx}\left(\dfrac{u}{v}\right)=\dfrac{v\dfrac{du}{dx}-u\dfrac{dv}{dx}}{v^2}$

$C'\left(x\right)=\dfrac{2\left(x^2+3\right)\left(10x\right)-5x^2\left(2x\right)}{2\left(\sqrt{x^2+3}\right)^3}=\dfrac{5x\left(x^2+6\right)}{\left(x^2+3\right)^\tfrac{3}{2}}$

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