Net change theorem

Definition

The Net Change Theorem states that the integral of a rate of change over an interval gives the net change in the quantity over that interval. Mathematically, it is expressed as $\int_{a}^{b} F'(x) \, dx = F(b) - F(a)$.

5 Must Know Facts For Your Next Test

1. The Net Change Theorem links differentiation and integration by showing how integrals can be used to calculate total accumulation.
2. It is often applied in physical contexts, such as calculating displacement from velocity or total consumption from a rate of consumption.
3. The theorem requires that the function being integrated is continuous on the interval $[a, b]$.
4. If the function $F(x)$ represents a quantity, then $F'(x)$ represents its rate of change.
5. Understanding definite integrals is crucial for applying the Net Change Theorem correctly.

Review Questions

• How does the Net Change Theorem connect differentiation and integration?
• What conditions must be met for the Net Change Theorem to be applicable?
• Provide an example where you would use the Net Change Theorem in a real-world scenario.

Related terms

Definite Integral: A definite integral computes the accumulation of quantities and has limits of integration, written as $\int_{a}^{b} f(x) \, dx$.

Fundamental Theorem of Calculus: This theorem links differentiation and integration, stating that differentiation and integration are inverse processes.

Rate of Change: A measure of how a quantity changes with respect to another variable; often represented as a derivative.