Net Change Theorem

Business Calculus · Axiom Academy

Connecting rates of change to total change The Net Change Theorem is a direct consequence of the Fundamental Theorem of Calculus, stated in a way that highlights its practical applications. The integral of a rate of change equals the total (net) change If you know how fast something is changing (the rate), you can find how much it changed (the total) by integrating the rate over the time interval. Integrate marginal cost to find total variable cost Integrate marginal revenue to find total revenue Integrate profit rate to find accumulated profit Integrate production rate minus sales rate A company's cash flow rate is modeled by: where t is months. If cash reserves are 100,000 at t = 0, find cash reserves after 12 months. The company accumulated 168,000 in additional cash over 12 months, bringing total reserves to 268,000. The increasing cash flow rate (3t² term) shows improving business performance over time. A region's employment is changing at the rate: (thousands of jobs per year, t in years) Find the net change in employment from t = 0 to t = 4. The result of 12,000 jobs is the net change. At some point the rate became negative (jobs were lost), but overall, there was a net gain. To find total jobs created vs. total jobs lost, we'd need to split the integral where E'(t) = 0. So the rate is negative (job losses) for t > 3. Net Change: _a^b f'(x)\,dx — Can be positive, negative, or zero Total Change: _a^b |f'(x)|\,dx — Always positive, measures total movement

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