Total Revenue from Marginal

Business Calculus · Axiom Academy

EXAMPLE Total Revenue from Marginal Revenue Recovering revenue functions through integration 1 Tech Startup: Software Licenses A software company finds that their marginal revenue function for enterprise licenses is: where x is the number of licenses sold. The -0.5x² term shows diminishing marginal revenue - each additional license brings in less revenue than the previous one (volume discounts or market saturation). Maximum revenue occurs when MR = 0, i.e., when x = 400 licenses. 2 E-commerce: Subscription Service An online streaming service has marginal revenue: where x is subscribers in thousands. At 10,000 subscribers (x=10), monthly revenue is 85,000. Check: R(10) = -0.05(100) + 10(10) + (-10) = -5 + 100 - 10 = 85 ✓ The constant C = -10 represents the baseline adjustment. This could account for initial setup costs or free trial periods that reduce effective revenue when subscriber counts are low. A company knows MR(x) = 50 - 0.4x and wants to find the demand (price) function. Since R(x) = p · x, we have: p = x First find R(x), then divide by x to get the demand function p(x). The demand function p = 50 - 0.2x is linear. At x = 0, maximum price is 50. For each unit increase in quantity, price must drop by 0.20 to maintain demand. Unlike cost functions (which have fixed costs), revenue functions always satisfy R(0) = 0 because selling zero units generates zero revenue. This means the constant of integration is always 0. Based on the example above, which approach is correct?

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