1. What is the Net Change Theorem?

The Net Change Theorem states that the integral of a rate of change gives the net change over that interval. It connects differential and integral calculus, showing how accumulation relates to rates.

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
• \( f(b) - f(a) \) — Net change in the quantity

Explanation: The theorem shows that integrating a rate of change over an interval gives the total change in the original quantity.

3. Importance of Net Change Theorem

Details: This theorem is fundamental in physics (displacement from velocity), economics (total cost from marginal cost), and biology (population change from growth rate).

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

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

Q2: Can this be used for non-continuous functions?
A: The theorem applies to integrable functions, which may have finite discontinuities but must be bounded.

Q3: How is this related to the Fundamental Theorem of Calculus?
A: The Net Change Theorem is essentially Part 2 of the Fundamental Theorem of Calculus in practical application.

Q4: What are common applications of this theorem?
A: Calculating distance traveled from velocity, work done from power, or total profit from marginal profit.

Q5: What if I don't know the derivative function?
A: You need the rate function (derivative) to use this theorem. If you have the original function, just compute f(b)-f(a) directly.