Optimizing Profits with Integral Calculus?

📌 CONCEPT: To maximize profit, we need to find the optimal number of units to produce by determining the critical point of the profit function using integral calculus.

📐 RULE / FORMULA: The profit function P(x) = revenue - cost, and we use the formula for critical points: P'(x) = 0 to find the maximum profit.

💡 WORKED EXAMPLE: Let's consider a company with revenue R(x) = 200x - 1000 and cost C(x) = 50x + 500. The profit function P(x) = R(x) - C(x) = (200x - 1000) - (50x + 500) = 150x - 1500. To find the critical point, we take the derivative P'(x) = 150 and equate it to zero. Solving for x, we get x = 10.

⚠️ COMMON MISTAKE: Students often forget to check the second derivative to confirm if the critical point corresponds to a maximum, minimum, or point of inflection.