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 Theorem Work?

The theorem is expressed mathematically as:

Where:

• \( f'(x) \) — Derivative of the function (rate of change)
• \( a \) — Lower limit of integration
• \( b \) — Upper limit of integration
• \( f(b) - f(a) \) — Net change in the function over [a, b]

Explanation: The integral of the derivative gives the total accumulation of change, which equals the difference in the function's values at the endpoints.

3. Applications of the Net Change Theorem

Details: The theorem has wide applications in physics (displacement from velocity), economics (total cost from marginal cost), biology (population change from growth rate), and other fields where rates of change are known.

4. Using the Calculator

Tips: Enter the derivative function (f'(x)) using standard mathematical notation. Input the lower and upper limits of integration. The calculator will show both the integral representation and the net change result.

5. Frequently Asked Questions (FAQ)

Q1: How is this different from the Fundamental Theorem of Calculus?
A: The Net Change Theorem is essentially Part 2 of the Fundamental Theorem, specifically focusing on computing net change.

Q2: Can this be used for vector-valued functions?
A: Yes, the theorem extends to vector functions, computing net change component-wise.

Q3: What if the derivative isn't continuous?
A: The theorem still holds as long as the derivative is integrable over the interval.

Q4: How is this applied in real-world problems?
A: Common applications include finding distance traveled from velocity, work done from power, or total growth from growth rate.

Q5: 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 (always positive).