Tag: integral calculus – ii

Questions Related to integral calculus – ii

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 $C'\left(x\right)$

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

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

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