Tag: applications of calculus

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Questions Related to applications of calculus

Multiple choice 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

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

  1. $\dfrac{5}{2}$
  2. $\dfrac{1}{4}$
  3. $\dfrac{3}{4}$
  4. $\dfrac{5}{7}$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 = $2(2)-4+\dfrac{5}{2}=\dfrac{5}{2}$

Multiple choice 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

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

  1. $16.5$
  2. $15.5$
  3. $12.5$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
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$

Multiple choice 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

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

  1. 23

  2. 24

  3. 25

  4. 26
Reveal answer Fill a bubble to check yourself
A Correct answer
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 = $8\times3-1=23$
$\therefore$    $Marginal\, cost = 23$.

Multiple choice 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

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

  1. 34

  2. 35

  3. 36

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
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 = $4(10)-4=40-4=36.$

Multiple choice 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

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

  1. 4

  2. 5

  3. 6

  4. 7
Reveal answer Fill a bubble to check yourself
A Correct answer
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 = $4\times 2-4=8-4=4$
$\therefore$    $Marginal\,cost=4.$

Multiple choice 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

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

  1. 19

  2. 18

  3. 16

  4. 17
Reveal answer Fill a bubble to check yourself
C Correct answer
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 = $6\times 3-2=18-2=16$

Multiple choice 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

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

  1. $6$
  2. $4$
  3. $2$

  4. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
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 $

$Marginal\space cost =36$
Multiple choice 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

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

  1. $\dfrac{104}{3}$
  2. $\dfrac{103}{3}$
  3. $\dfrac{105}{3}$
  4. $\dfrac{103}{4}$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 = $12-1+\dfrac{70}{3}=\dfrac{36-3+70}{3}=\dfrac{103}{3}$
$\therefore$     $Average\, cost=\dfrac{103}{3}$

Multiple choice 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

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

  1. $1170$
  2. $1160$
  3. $1150$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $1160$.

Multiple choice 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

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

  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
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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


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


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


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

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