Common Misunderstandings About the Fundamental Theorem of Calculus (FTC)

When students study AP Calculus AB, they often have some mix-ups about the Fundamental Theorem of Calculus. Here are a few of the most common mistakes:

1. Confusing Areas
   Students sometimes think that $F(b) - F(a)$ tells them the area under the curve between points $a$ and $b$. But it really shows the net area. This means it can be positive (above the x-axis) or negative (below the x-axis).

2. Misunderstanding Derivatives
   Many students believe that $F'(x)$, which is the derivative of the integral, is just the same as the original function. However, it’s important to remember that this is based on limits.

3. Constant Functions Confusion
   It’s easy to misunderstand how to find the area under constant functions. For instance, if you have $f(x) = k$ (which is a constant), the area from $a$ to $b$ is actually $k(b-a)$. It's not zero!

Understanding and clearing up these common mistakes helps grasp how integrals and derivatives work together.