Calculus I · Unit 3A · lesson
The Net Change Theorem
Learn The Net Change Theorem through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
The Fundamental Theorem of CalculusWhat this section is building
Learn The Net Change Theorem through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Differentiating a running total recovers its current integrand, while evaluating an accumulated total subtracts antiderivative endpoint values.
Separate FTC Part I, FTC Part II, net change, and variable-bound chain-rule tasks before manipulating notation.
Forgetting a chain-rule factor at a variable bound, reversing endpoint subtraction, or adding +C to a definite value.
Learning objectives
Translate a contextual rate integral into final amount minus initial amount.
The Net Change Theorem
Rates become measurable changes
If is the rate at which a quantity changes, then . This net-change statement is the Fundamental Theorem written in applied language. It tells us that integrating a rate over time recovers the actual change in the underlying quantity, whether that quantity is position, volume, charge, population, energy, or cost.
The word "net" matters whenever the rate can be negative. Inflow and outflow, charging and discharging, or forward and backward motion contribute with opposite signs. To recover the final amount, add the net change to the initial amount. To recover total activity without cancellation, a different model may be needed, often involving absolute value or separate positive and negative contributions.
If is a rate of change, then
This is FTC Part II interpreted in context.
Charge flowing through a circuit
Current , measured in amperes, is charge flow in coulombs per second. If
for , then the net charge transferred is
The units make the interpretation unavoidable: amperes times seconds equals coulombs.
u3a-net-change-01A population changes at rate people/year for . What is the net population change?
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Integrate the rate over the time interval.
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