Application of Calculus (Commerce) • Topic 1 of 2

Cost, Revenue & Profit Functions

Commerce mathematics models a business using functions of the output level $x$ (the number of units produced or sold).

The three core functions

  • Total cost $C(x)$ = fixed cost + variable cost.
  • Average cost $\text{AC}=\dfrac{C(x)}{x}$ — the cost per unit.
  • Revenue $R(x)=p\cdot x$, where $p$ is the price per unit (the demand function links $p$ and $x$).
  • Profit $P(x)=R(x)-C(x)$.

Break-even point

The break-even output is where the firm neither profits nor loses: $R(x)=C(x)$, equivalently $P(x)=0$. Below it the firm runs a loss; above it, a profit.

Solved Examples
1
Worked Example
If $C(x)=2x+50$ and price per unit is ₹7, find the profit function.
Solution

$R(x)=7x$, so $P(x)=R(x)-C(x)=7x-(2x+50)=5x-50.$

2
Worked Example
For the firm above, find the break-even output.
Solution

Set $P(x)=0$: $5x-50=0\Rightarrow x=10$ units.

3
Worked Example
If $C(x)=x^2+10x+100$, find the average cost at $x=10$.
Solution

$\text{AC}=\dfrac{C(10)}{10}=\dfrac{100+100+100}{10}=\dfrac{300}{10}=30$ (in ₹ per unit).

4
Worked Example
If demand is $p=20-x$, write the revenue function.
Solution

$R(x)=p\cdot x=(20-x)x=20x-x^2.$

Key Points

  • $C(x)=$ fixed $+$ variable cost; average cost $=\dfrac{C(x)}{x}$.
  • Revenue $R(x)=p\,x$; profit $P(x)=R(x)-C(x)$.
  • Break-even: $R(x)=C(x)$, i.e. $P(x)=0$.
  • Use the demand function to express $p$ in terms of $x$.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Profit function is:
Explanation: Profit = revenue minus cost.
Q2.Break-even occurs when:
Explanation: Profit is zero at break-even.
Q3.If $C(x)=2x+50$ and $R(x)=7x$, break-even output is:
Explanation: $5x-50=0\Rightarrow x=10$.
Q4.Average cost is:
Explanation: Cost per unit $=C(x)/x$.
Practice more MCQs → 📝 Assignment · 35 marks