Tag: business mathematics and statistics

Questions Related to business mathematics and statistics

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 $1160$.