Rate of change

Definition

Rate of change quantifies how one quantity changes with respect to another. In calculus, it is often represented as the derivative of a function.

5 Must Know Facts For Your Next Test

1. The rate of change can be found using the definite integral to measure the net change over an interval.
2. It is directly related to the Fundamental Theorem of Calculus, which connects differentiation and integration.
3. In practical applications, it describes phenomena such as velocity, acceleration, and growth rates.
4. The Net Change Theorem states that the integral of a rate of change over an interval gives the net change in the quantity.
5. Understanding how to manipulate and interpret integrals is crucial for solving problems involving rates of change.

Review Questions

• How do you find the net change in a quantity using its rate of change?
• Explain how the Fundamental Theorem of Calculus relates differentiation and integration.
• Provide an example of a real-world scenario where understanding the rate of change is essential.

Related terms

Derivative: A measure of how a function changes as its input changes; represents an instantaneous rate of change.

Integral: A mathematical object that represents accumulation or area under a curve; used to compute total quantities from rates of change.

Net Change Theorem: $$\text{States that } \int_a^b f'(x) dx = f(b) - f(a), \text{ connecting integrals with differences in function values.}$$