Total Profit Over a Period

Business Calculus · Axiom Academy

EXAMPLE Total Profit Over a Period Calculating accumulated profit using integration A tech startup's profit rate (in thousands of dollars per month) is modeled by: Find the total profit during the first year (12 months). Total profit = Integral of profit rate over time The startup accumulated 186,000 in profit during year one. Notice profit rate was negative early on (losses), but accelerating growth (t² term) eventually dominated. 2 Seasonal Business: Ice Cream Shop An ice cream shop's monthly profit rate follows a seasonal pattern: where t = 0 is January. Find total annual profit. The sine terms cancel over a complete cycle (12 months)! Despite huge seasonal swings (from - 2,000 to + 8,000/month), the annual profit is simply 12 × 3,000 = 36,000. The seasonal fluctuation averages out! Two business ventures have different profit rate models: Option A: P'_A(t) = 10 + 2t (steady growth) Option B: P'_B(t) = 15e^ -0.1t (high initial, decaying) Even though Option B starts with a higher profit rate ( 15k vs 10k), Option A's steady growth eventually wins. This illustrates the importance of sustainable growth over short-term gains. A new product has profit rate: P'(t) = -20 + 8t - 0.5t^2 (thousands/month) When does cumulative profit become positive (break even)? Factoring or numerical methods give T ≈ 5.3 months

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