1. What is Instantaneous Rate of Change?

The instantaneous rate of change (IRC) is the derivative of a function at a specific point, representing how quickly the function's value changes at that exact point. It's the limit of the average rate of change as the interval approaches zero.

2. How Does the Calculator Work?

The calculator uses the fundamental definition of derivative:

Where:

• \( f(x) \) — The input function
• \( x \) — The point at which to calculate the derivative
• \( h \) — The interval approaching zero

Explanation: The calculator numerically approximates the derivative by evaluating the function at points very close to the input point.

3. Importance of IRC Calculation

Details: Instantaneous rate of change is fundamental in physics (velocity, acceleration), economics (marginal costs), biology (growth rates), and many other fields where precise measurement of change is needed.

4. Using the Calculator

Tips: Enter the function using standard mathematical notation (e.g., "x^2 + 3*x - 5") and the point where you want to calculate the derivative. The calculator will approximate the instantaneous rate of change at that point.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between average and instantaneous rate of change?
A: Average rate measures change over an interval, while instantaneous rate measures change at an exact point.

Q2: Can I use trigonometric functions?
A: In a full implementation, yes (e.g., sin(x), cos(x)), but this requires a proper math parser.

Q3: How accurate is the numerical approximation?
A: The accuracy depends on the method used (forward difference, central difference, etc.) and the step size.

Q4: What functions can't be differentiated?
A: Functions that aren't continuous or smooth at the point of interest may not have a defined derivative there.

Q5: Is this the same as slope?
A: Yes, the derivative at a point gives the slope of the tangent line to the function at that point.