Curve Sketching for Business

Business Calculus · Axiom Academy

EXAMPLE Curve Sketching for Business Functions Complete analysis of a profit function using calculus Problem: Tech Startup Profit Analysis A tech startup's monthly profit function (in thousands of dollars) based on advertising spending x (in thousands) is: Goal: Sketch the profit curve and identify all key features to guide business decisions. Domain: Since x represents advertising spending, x 0 With no advertising, the company loses 10,000/month. They break even at 1,000 and 10,000 ad spend. Find where the profit is increasing or decreasing: Set P'(x) = 0 to find critical points: P(5) = -(5)^2 + 10(5) - 10 = -25 + 50 - 10 = 15 Maximum profit of 15,000 occurs at 5,000 ad spend Find the second derivative to determine concavity: The curve is concave down everywhere This confirms x = 5 is a maximum (not a minimum) There are no inflection points The negative concavity shows diminishing returns on advertising . Each additional dollar spent on ads produces less additional profit than the previous dollar. Optimal Ad Spend: 5,000/month for maximum profit Profitable Range: Between 1,000 and 10,000 Warning: Spending more than 10,000 on ads will cause losses Key Insight: Diminishing returns kick in immediately - every additional dollar is less effective You've mastered the complete curve sketching process for business applications. Based on the example above, which approach is correct?

This is the written version of the interactive lesson above. See the full Business Calculus course.