Tag: integral calculus – ii

Questions Related to integral calculus – ii

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. $\dfrac{5}{7}$

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


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


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


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

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