Example
Zoe's lemonade stand satisfies a demand characterized by:
$$ P = 10 - 2 Q $$Her revenue is
$$ R \left( Q \right) = PQ = \left( 10 - 2 Q \right) Q = 10 Q - 2 Q^2 $$The marginal revenue is the derivative of the revenue `R \left( Q \right)`:
$$ MR \left( Q \right) = \frac{d R \left( Q \right)}{d Q} = 10 - 2 \times 2 Q = 10 - 4 Q $$Question
The inverse demand for lemonade is `P = 68 - 7Q`.
What is Zoe's marginal revenue?
First, her revenue is
$$ R \left( Q \right) = PQ = (68 - 7Q) Q = 68 Q - 7Q^2 $$So her marginal revenue is
$$ MR \left( Q \right) = \frac{d R \left( Q \right)}{d Q} = 68 - 7 \times 2 Q = 68 - 14Q $$