Amount of change

Definition

The amount of change refers to the difference in the value of a function as its input changes. It is crucial for understanding and calculating derivatives, which measure how functions change.

5 Must Know Facts For Your Next Test

1. The amount of change can be represented as $f(b) - f(a)$ for a function $f$ over an interval $[a, b]$.
2. In calculus, the derivative represents the instantaneous rate of change of a function at a given point.
3. The limit process used to define derivatives involves examining the amount of change over increasingly smaller intervals.
4. For linear functions, the amount of change is constant and equal to the slope of the line.
5. Understanding the relationship between average rate of change (over an interval) and instantaneous rate of change (at a point) is fundamental.

Review Questions

• How do you compute the amount of change for a given function over an interval?
• What does the derivative tell us about the amount of change at any specific point?
• How do you interpret the average rate of change versus instantaneous rate of change?

Related terms

Derivative: A measure of how a function's output value changes as its input value changes; defined as $\lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}$.

Instantaneous Rate of Change: The rate at which a function is changing at any given point; represented by its derivative.

Average Rate of Change: $\frac{{f(b) - f(a)}}{b - a}$; measures how much a function's value changes on average over an interval $[a, b]$.