Net change theorem

Definition

The Net Change Theorem states that the integral of a rate of change function over an interval gives the net change in the quantity over that interval. It is essentially the Fundamental Theorem of Calculus applied to real-world problems involving rates of change.

5 Must Know Facts For Your Next Test

1. The Net Change Theorem can be expressed as $\int_a^b F'(x) \, dx = F(b) - F(a)$.
2. It connects derivatives (rates of change) and integrals (accumulated quantities).
3. The theorem is useful for calculating total distance, displacement, or accumulated quantity.
4. It applies to any context where you have a rate function and need the total change over an interval.
5. Understanding initial values and boundary conditions is crucial when applying this theorem.

Review Questions

• What does the Net Change Theorem calculate when applied to a velocity function?
• How would you express the Net Change Theorem in integral form?
• In what types of real-world problems is the Net Change Theorem particularly useful?

Related terms

Fundamental Theorem of Calculus:

States that differentiation and integration are inverse processes; specifically, $\int_a^b f'(x) \, dx = f(b) - f(a)$ links antiderivatives with definite integrals.

Definite Integral:

The integral of a function over a specific interval $[a, b]$, representing the accumulated quantity or area under the curve between $a$ and $b$.

Rate of Change:

A measure of how a quantity changes with respect to another variable, often represented as a derivative in calculus.