1. What is the Net Change Theorem?

The Net Change Theorem states that the integral of a rate of change (derivative) over an interval gives the net change in the original function over that interval. It connects differential and integral calculus.

2. How Does the Calculator Work?

The calculator uses the Net Change Theorem:

Where:

• \( f'(x) \) — Derivative function (rate of change)
• \( a \) — Lower limit of integration
• \( b \) — Upper limit of integration
• \( f(x) \) — Antiderivative of \( f'(x) \)

Explanation: The theorem shows that the definite integral of a rate of change gives the total change in the quantity over the interval.

3. Importance of Net Change

Details: The Net Change Theorem is fundamental in physics (displacement from velocity), economics (total cost from marginal cost), and other applications where we know the rate of change and want to find the total change.

4. Using the Calculator

Tips: Enter the derivative function in proper mathematical notation, and the integration limits. The calculator will find the antiderivative and compute the net change.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between net change and total change?
A: Net change accounts for direction (can be negative), while total change sums absolute values.

Q2: Can this be used for piecewise functions?
A: Yes, but you need to integrate each piece separately and sum the results.

Q3: What if the antiderivative can't be found?
A: Numerical integration methods can be used to approximate the integral.

Q4: How is this related to the Fundamental Theorem of Calculus?
A: The Net Change Theorem is essentially Part 2 of the Fundamental Theorem.

Q5: What are common applications?
A: Calculating distance from velocity, work from force, population growth from growth rate, etc.