Tag: integral calculus – ii

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Questions Related to integral calculus – ii

Multiple choice 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

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

  1. $\dfrac{60}{D} + 2 - 10D$
  2. $2 - 20D$
  3. $2 - 40D$

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

The rate of change of price with respect to demand is the derivative dp/dD. Differentiating p = 60 + 2D - 10D² term by term gives dp/dD = 0 + 2 - 20D = 2 - 20D. Option A incorrectly divides 60 by D, and Option C has an incorrect coefficient.

Multiple choice 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

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

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

  4. None
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

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

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

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

Multiple choice 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

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

  1. $12-4x$
  2. $4x - 12$
  3. $12 + 4x$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

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

Multiple choice 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

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

  1. $2$
  2. $12$
  3. $48$
  4. $4$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 

Multiple choice 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

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 $C'(x)$

  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. $C(x)=20x0.16x^2+0.0012x^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$c'(x)=\cfrac{d}{dx}(c(x))$
$=2\times(100+0.8)x+3\times0.0004x^{2}$
$=200.16x+0.0012x^{2}$
Multiple choice 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

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

  1. $\dfrac{x^3}4 + 4x$
  2. $4x^3+4x$
  3. $x^3 + 4x$
  4. $4x^3+2x^2$
Reveal answer Fill a bubble to check yourself
C Correct answer
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:
$\dfrac{dy}{dx} = x^3+4x$

Multiple choice 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

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.

  1. 4500+10x

  2. 10+4500x

  3. 25+10x

  4. 10+25x
Reveal answer Fill a bubble to check yourself
A Correct answer
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)= $4500+10x$

Multiple choice 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
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.
  1. $1000$
  2. $2000$
  3. $3000$
  4. $3500$
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice 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

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.

  1. $6x$
  2. $4x$
  3. $4500x$
  4. $6x+4$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $p=6$
$\Rightarrow$  $Total\,revenue\,\,R(x)=p.x$
$\Rightarrow$  $Total\,revenue\,\,R(x)=6x$
Multiple choice 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
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
  1. $25x$
  2. $10x$
  3. $4500x$
  4. $4500+10x$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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