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

When the price of refrigerator rises from Rs.$2000$ per unit to Rs. $2500$ per unit and in response to this rise is price the quantity supplied increases from $2500$ units to $3500$ units, find out the price elasticity of supply.

  1. $4.5$
  2. $1.5$
  3. $3.5$
  4. $2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$\Rightarrow$  Let $q _1=2500,\,q _2=3500,\,p _1=2000$ and $p _2=2500$.
$\Rightarrow$  $\Delta q=q _2 - q _1=3500-2500=1000$

$\Rightarrow$  $\dfrac{q _2 + q _1}{2}=\dfrac{3500+2500}{2}=3000$

$\Rightarrow$  $\Delta p=p _2 - p _1=2500-2000=500$

$\Rightarrow$  $\dfrac{p _2+p _1}{2}=\dfrac{2500+2000}{2}=2250$

$\Rightarrow$  We know, $e _s=\dfrac{\Delta q}{\Delta p}\times \dfrac{p _2+p _1}{q _2+q _1}$

$\Rightarrow$  $e _s =\dfrac{1000}{3000}  \times \dfrac{2250}{500}$

$\Rightarrow$  $e _s=\dfrac{1}{3}\times \dfrac{4.5}{1}=\dfrac{4.5}{3}=1.5$
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

If both average cost (AC) and marginal cost (MC) are U shaped, then

  1. AC will reach a minimum at a level of output that is less than that at which MC reaches a minimum.

  2. the total cost curve will be a straight line.

  3. AC will reach a minimum at a level of output that is greater than that at which MC reaches a minimum.

  4. both AC and MC will reach a minimum at the same level of output.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The marginal cost curve intersects the average cost curve at the latter's minimum point. Because the marginal cost curve rises after its own minimum, it must cross the average cost curve from below at a point where the average cost is at its minimum, meaning the average cost minimum occurs at a higher output level.

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 of cultivating a square field at the rate of $Rs.\,135$ per hectare is $Rs.\,1215$. The cost of putting a fence around it at the rate of $75\;paise$ per metre would be ...... .

  1. $Rs.\,360$
  2. $Rs.\,810$
  3. $Rs.\,900$
  4. $Rs.\,1800$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

rate of cultivating a square field =Rs.135 per hectare, 
cost of cultivating a square field=Rs.1215
area of square field=cost of cultivating a square field/rate of cultivating a square field
$A=1215/135$
$A=9 hectare$
1 hectare =10000m^2
$Area=a^2$
$90000=a^2$
$a=300$

cost of putting a fence around it at the rate of 75 paise per metre 
Total cost=0.75*4*300Rs
C=900Rs
Answer (C) 900RS

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

If average cost is at a minimum, then

  1. it is equal to marginal cost.

  2. total cost is also at a minimum.

  3. profit is at a maximum.

  4. all of the above are true.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\Rightarrow$  If average cost is at a minimum, then $it\, is\, equal\, to\, marginal \,cost.$

$\Rightarrow$  Average cost is Production cost per unit of output, computed by dividing the total of fixed costs and variable costs by the number of total units produced (total output). Lower average costs are a potent competitive advantage.
$\Rightarrow$ MC indicates the rate at which the total cost of a product changes as the production increases by one unit. However, because fixed costs do not change based on the number of products produced, the marginal cost is influenced only by the variations in the variable costs.
$\Rightarrow$  When average cost is at its minimum, $Marginal\, cost = Average\, cost$. That's because the second marginal cost gets higher than average cost, average cost has to get higher because Marginal cost is greater than average cost.

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

If the amount of taxes paid $(T)$ depends on income $(x)$, how would you use calculus notation to describe the marginal tax rate? If taxes and income are both measured in dollars per year, what are the units of the marginal tax rate?

  1. Same as units of $T$
  2. Same as units of $x$

  3. Unit - less

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

If taxes and income are both measured in dollars per year, then the units of the marginal tax rate will be unit-less.

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

If a consumer daily income rises from Rs. $300$ to Rs.$350$, his purchase of a good $X$ increases from $25$ units per day to $40$ units; find the income elasticity of demand for $X$?

  1. $2$
  2. $1$
  3. $3$
  4. $4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Change in quantity demand $\Delta Q$ = ${Q} _{2} - {Q} _{1}$ = $40 - 25 = 15$
Change in income $\Delta M $ = ${M} _{2} + {M} _{1}$ = $350 -300$ = $50$
$\dfrac{\Delta Q}{\Delta M}$ x $\dfrac {{M} _{2} + {M} _{1}}{{Q} _{2} - {Q} _{1}}$


= $\dfrac{15}{50} \times \dfrac{350+300}{25+40}$

= $\dfrac{3}{10} \times {650}{65}$


= $\dfrac{3}{10} \times {10}$

= $3$
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

Differentiation refers to the process whereby we

  1. Calculate the area underneath a curve

  2. Calculate the intercept of a curve with the horizontal axis

  3. Calculate the gradient to a curve at any point on the curve

  4. Calculate the intercept of a curve with the vertical axis
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Differentiation is the process where the slope/gradient of a curve is calculated at a point lying on it.

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 $C(x)=\dfrac{x^3}{3}+x^2-15x+3$. Find marginal cost function.

  1. $x^2+2x-15$
  2. $x^2-2x-15$
  3. $x^2+2x+15$

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

$\Rightarrow$  We have, $C(x)=\dfrac{x^3}{3}+x^2-15x+3$.

$\Rightarrow$  Marginal cost = $\dfrac{d}{dx}(C)$

$\Rightarrow$  Marginal cost = $\dfrac{d}{dx}(\dfrac{x^3}{3}+x^2-15x+3)$

$\Rightarrow$  Marginal cost = $\dfrac{3\times x^2}{3}+2\times x-15$

$\therefore$    Marginal cost = $x^2+2x-15$

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

When we differentiate an expression with respect to one of a number of independent variables, we are engaged in

  1. Finding definite integrals

  2. Total differentiation

  3. Partial differentiation

  4. Integration
Reveal answer Fill a bubble to check yourself
C Correct answer
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.