U-Substitution Examples

Business Calculus · Axiom Academy

EXAMPLE U-Substitution Examples Worked examples applying u-substitution to business functions A company's marginal cost is MC(x) = (x^2+1)^2 dollars per unit. Find the cost function if fixed costs are 500. Notice that 2x (derivative of x²+1) is similar to 6x in the numerator The cost function shows how total cost changes with production. The term x^2+1 represents variable costs that decrease as production scales up. (Rate of change of demand with respect to price) A startup's revenue growth rate is R'(t) = 50te^ 0.1t^2 thousand dollars per month. Find R(t) if initial revenue is 200,000. We have 50t in the integral, and du = 0.2t dt, so we need to adjust. The exponential term e^ 0.1t^2 shows accelerating revenue growth, typical of successful startups experiencing viral growth or network effects. (Appears in compound interest calculations) After an advertising campaign ends, sales decline at the rate S'(t) = (4t+1)^2 units per week. Find S(t) if sales were 1000 units/week when t = 0. This model shows sales approaching 800 units/week as t → ∞. The 4t+1 term represents the "boost" from the campaign that gradually fades. Based on the example above, which approach is correct?

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