Tag: integral calculus – ii

Questions Related to integral calculus – ii

The demand function if $p = 60 + 2D - 10D^2$, the rate of charge in price with respect to demand is ______

business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. $\dfrac{60}{D} + 2 - 10D$

  2. $2 - 20D$

  3. $2 - 40D$

  4. None of these

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Correct Option: B

Given the total cost function, $TC = a+bQ+cQ^2+dQ^3$, Find the Marginal cost.

business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. $b+2cQ+3dQ^2$

  2. $a + b+2cQ+3dQ^2$

  3. $a + 2b+2cQ+3dQ^2$

  4. None

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have $TC=a+bQ+cQ^2+dQ^3$

$\Rightarrow$  $Marginal\,cost = \dfrac{d}{dQ}TC$

$\Rightarrow$  $Marginal\, cost=\dfrac{d}{dQ}(a+bQ+cQ^2+dQ^3)$

$\therefore$    $Marginal\,cost=b+2cQ+3dQ^2$

 The total cost function is $TC = 12x + 2x^2$. Find the $MC$.

business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. $12-4x$

  2. $4x - 12$

  3. $12 + 4x$

  4. None of these

Show answer
Correct Option: C
Explanation:

$MC = \dfrac{\mathrm{d} TC}{\mathrm{d} x}$

$TC = 12x + 2x^{2}$
$\Rightarrow \dfrac{\mathrm{d} TC}{\mathrm{d} x}=12+4x$
$\Rightarrow MC = 12 + 4x$

$y = 48x - 2x^2$
where, $y=$ Total revenue $ $ $.
$x = $ Output
At what output is total revenue a maximum?

business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. $2$

  2. $12$

  3. $48$

  4. $4$

Show answer
Correct Option: B
Explanation:

$\Rightarrow$  We have given, $y=48x-2x^2$

$\Rightarrow$  Differentiate on both sides w.r.t $x$ we get,
$\Rightarrow$  $\dfrac{dy}{dx}=48-4x$          --- ( 1 )
$\Rightarrow$  Let $\dfrac{dy}{dx}=0$
$\Rightarrow$  $48-4x=0$
$\Rightarrow$  $4x=48$
$\Rightarrow$  $x=12$ is a turning point.
$\Rightarrow$  $\dfrac{d^2y}{dx^2}=-4$         [Differentiate ( 1 ) on both sides]
$\Rightarrow$  So, the turning point is a maximum 

The cost, in dollars, of producing $x$ gallons of detergent is given by
$C(x)=350+20x0.08x^2 + 0.0004x^3$
What is a formula for the marginal cost function $C'(x)$

business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. $C(x)=200.16x^2+0.0012x^2$

  2. $C(x)=200.16x+0.0012x^2$

  3. $C(x)=20x0.16x+0.0012x^2$

  4. $C(x)=20x0.16x^2+0.0012x^2$

Show answer
Correct Option: B
Explanation:
$c'(x)=\cfrac{d}{dx}(c(x))$
$=2\times(100+0.8)x+3\times0.0004x^{2}$
$=200.16x+0.0012x^{2}$

If we differentiate the cost function: $y = \dfrac{x^4}4 + 2x^2$, we get

business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. $\dfrac{x^3}4 + 4x$

  2. $4x^3+4x$

  3. $x^3 + 4x$

  4. $4x^3+2x^2$

Show answer
Correct Option: C
Explanation:

Suppose: $y = ax^n$
Then, $dy = nax^{n-1}$
Remember
$\dfrac{x^4}{4} = \dfrac{1}{4}x^4$
So differentiating both terms in the expression for using the formula gives:
$\dfrac{dy}{dx} = x^3+4x$

A ABC firms start producing pens and finds that the production cost of each pen is Rs $10$. and the fixed expenditures of production is Rs 4500. If each pen is sold for Rs 25 , determine cost function.

business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. 4500+10x

  2. 10+4500x

  3. 25+10x

  4. 10+25x

Show answer
Correct Option: A
Explanation:

The cost function equation is expressed as C(x)= FC + V(x), where C equals total production cost, FC is total fixed costs, V is variable cost and x is the number of units.
$4500$ =fixed expenditures
$10x$ = variable cost
C(X)= $4500+10x$

A company sells its product at the rate of Rs. $6$ per unit. The variable costs are estimated to run $25\%$ of the total revenue received. If the fixed costs for the product are Rs. $4500$. Find the break even point.
business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. $1000$

  2. $2000$

  3. $3000$

  4. $3500$

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  Here, price per unit $(p)=Rs.6$

$\Rightarrow$  Total revenue $R(x)=p.x=6x$  where $x$ is the number of unit sold.
$\Rightarrow$  Cost function $C(x)=4500+\dfrac{25}{100}R(x)$

$\Rightarrow$  Cost function $C(x)=4500+\dfrac{25}{100}\times 6x$

$\Rightarrow$   $C(x)=4500+\dfrac{3}{2}x$

$\Rightarrow$   Profit function $P(x)=R(x)-C(x)$

$\Rightarrow$  $P(x)=6x-(4500+\dfrac{3}{2}x)$

$\therefore$    $P(x)=6x-\dfrac{3}{2}x-4500$
$\Rightarrow$   At break even point $P(x)=0$
$\Rightarrow$   $6x-\dfrac{3}{2}x-4500=0$

$\Rightarrow$   $\dfrac{12x-3x}{2}-4500=0$

$\Rightarrow$   $x=\dfrac{9000}{9}=1000$
$\Rightarrow$  Hence, $x=1000$ is break even point.

A company sells its product at the rate of Rs $6$ per unit . The variable costs are estimated to run 25% of the total revenue received. If the fixed costs for the product are Rs $4500$.
Find the total revenue function.

business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. $6x$

  2. $4x$

  3. $4500x$

  4. $6x+4$

Show answer
Correct Option: A
Explanation:
$\Rightarrow$  If x is the number of units of certain product sold at a rate of $Rs.p$ per unit, then the amount derived from the sale of $x$ units of a product is the total revenue. 
$\Rightarrow$  Here, price per unit is $Rs.6$, so $p=6$
$\Rightarrow$  $Total\,revenue\,\,R(x)=p.x$
$\Rightarrow$  $Total\,revenue\,\,R(x)=6x$
A firm $ABC$ starts producing pens and finds that the production cost of each pen is Rs $10$, and the fixed expenditures of production is Rs. $4500$. If each pen is sold for Rs. $25$, find Revenue function
business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. $25x$

  2. $10x$

  3. $4500x$

  4. $4500+10x$

Show answer
Correct Option: A
Explanation:

R(x)= ( price per unit)*(number of units produced or sold)
Revenue function = $25 \times x$ (number of units)
Revenue function =$25x$