1. What is the Net Change Theorem?

The Net Change Theorem is a fundamental concept in calculus that connects the integral of a rate of change to the net change in the quantity itself. It states that the integral of a derivative over an interval equals the net change in the original function over that interval.

2. How Does the Calculator Work?

The calculator uses the Net Change Theorem:

Where:

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

Explanation: The theorem provides a powerful connection between differentiation and integration, showing how the integral of a rate of change gives the total change in the quantity.

3. Importance of Net Change Calculation

Details: The Net Change Theorem is crucial for solving problems involving accumulation, such as calculating total distance traveled from velocity, total growth from growth rate, or total cost from marginal cost.

4. Using the Calculator

Tips: Enter the derivative function in proper mathematical notation, and specify the interval limits. The calculator will compute the net change over the specified interval.

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 all changes without regard to direction.

Q2: Can this be used for piecewise functions?
A: Yes, but you need to break the integral into parts where the derivative is continuous.

Q3: What if I don't know the derivative function?
A: You must have the rate of change function (derivative) to use this theorem directly.

Q4: 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.

Q5: Can this handle multivariable functions?
A: No, this calculator is for single-variable calculus only.