1. What is Rate of Change?

The rate of change (ROC) measures how much a quantity (the function) changes with respect to change in another quantity (the independent variable). It's a fundamental concept in calculus and real-world applications.

2. How Does the Calculator Work?

The calculator uses the rate of change formula:

Where:

• \( f(b) \) — Function value at point b
• \( f(a) \) — Function value at point a
• \( b \) — Second point value
• \( a \) — First point value

Explanation: The formula calculates the average rate of change of a function between two points, which represents the slope of the secant line between these points.

3. Importance of Rate of Change

Details: Rate of change is crucial in physics (velocity, acceleration), economics (marginal costs), biology (growth rates), and many other fields. It's the foundation for differential calculus.

4. Using the Calculator

Tips: Enter the function values at points a and b, and the corresponding x-values (a and b). Ensure (b - a) is not zero to avoid division by zero.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between average and instantaneous rate of change?
A: Average ROC measures change over an interval (this calculator), while instantaneous ROC (derivative) measures change at a single point.

Q2: Can this be used for non-linear functions?
A: Yes, but it gives the average rate over the interval, not the rate at any specific point.

Q3: What does a negative rate of change indicate?
A: It means the function is decreasing over that interval (output decreases as input increases).

Q4: How is this related to slope?
A: For linear functions, ROC equals the slope. For non-linear functions, it's the slope of the secant line.

Q5: What are common units for ROC?
A: Units are "output units per input unit" (e.g., m/s for position over time).