Calculator Input
Use a demand price curve and a cost curve. The calculator compares critical points and endpoints.
Formula Used
Demand price: P(q) = a + bq + cq²
Revenue: R(q) = q × P(q)
Cost: C(q) = F + vq + wq² + zq³
Profit: π(q) = R(q) − C(q)
First derivative: π′(q) = MR − MC
Maximum test: Solve π′(q) = 0, then compare endpoints and second derivative.
Example Data Table
The table below uses sample values from the default setup.
| Quantity | Price | Revenue | Cost | Profit |
|---|---|---|---|---|
| 0 | 120.00 | 0.00 | 800.00 | -800.00 |
| 20 | 90.00 | 1800.00 | 1360.00 | 440.00 |
| 30 | 75.00 | 2250.00 | 1685.00 | 565.00 |
| 40 | 60.00 | 2400.00 | 2040.00 | 360.00 |
| 60 | 30.00 | 1800.00 | 2840.00 | -1040.00 |
How to Use This Calculator
- Enter the demand price function coefficients.
- Enter fixed, variable, quadratic, and cubic cost values.
- Set the allowed production quantity range.
- Choose whole units if production must be counted in units.
- Press the calculate button.
- Review maximum profit, quantity, price, revenue, and cost.
- Use CSV or PDF buttons to save the result.
Maximum Profit and Calculus
Why Profit Needs a Function
A business earns profit when revenue is greater than cost. Calculus helps when revenue and cost change with quantity. A fixed price is easy to inspect. A changing price needs a curve. This calculator uses a demand price curve. It also uses a flexible cost curve. These two curves create a profit function.
The Role of Marginal Values
Marginal revenue shows the extra revenue from one more unit. Marginal cost shows the extra cost from one more unit. Profit usually improves while marginal revenue is above marginal cost. Profit usually falls after marginal cost becomes higher. The best point often appears where marginal revenue equals marginal cost.
Why Endpoints Matter
A derivative finds internal critical points. It does not always prove the best allowed output. Production limits can change the answer. A company may have a minimum order size. It may also have capacity limits. For that reason, the calculator compares derivative roots with both endpoints.
Second Derivative Check
The second derivative explains curve shape. A negative value suggests a local maximum. A positive value suggests a local minimum. A zero value needs extra comparison. This tool avoids a weak conclusion by checking every valid candidate.
Practical Planning
The result gives the best quantity, price, revenue, cost, and profit. It also estimates break even points. These points show where profit changes from loss to gain, or from gain to loss. Use the output for planning. Adjust the coefficients when market conditions change. Small changes in demand can move the best quantity. Cost inflation can also reduce the profit peak. Recalculate before pricing, purchasing, or capacity decisions.
FAQs
1. What does this calculator maximize?
It maximizes profit over the quantity range you enter. It compares critical points from calculus with both endpoints.
2. What is the main calculus rule used?
The calculator solves where the first derivative of profit equals zero. That means marginal revenue equals marginal cost.
3. Why does it check endpoints?
A restricted range may make an endpoint better than an internal critical point. Endpoint checking prevents a misleading answer.
4. Can I use linear demand only?
Yes. Set the quadratic demand value to zero. Then the price curve becomes a simple linear demand equation.
5. What does fixed cost do?
Fixed cost lowers profit by the same amount at every quantity. It affects profit and break even points.
6. What if quantity must be whole units?
Select the whole unit option. The calculator checks nearby whole quantities around each calculus solution.
7. What is a break even point?
It is a quantity where profit equals zero. Revenue and total cost are equal at that quantity.
8. Can I download the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable summary.