AP Calculus AB
Advanced Placement Calculus AB covering limits, derivatives, and integrals.
Fundamental Theorem of Calculus
Bridging Derivatives and Integrals
The Fundamental Theorem of Calculus connects differentiation and integration, showing they are inverse processes.
Two Main Parts
Part 1: If \( F(x) \) is an antiderivative of \( f(x) \), then the integral of \( f(x) \) from \( a \) to \( b \) is \( F(b) - F(a) \).
Part 2: If you define a function by an integral, its derivative is the original function inside the integral.
Why Is This Important?
This theorem makes evaluating definite integrals much faster—just find an antiderivative!
Applications
- Calculating areas and accumulated quantities.
- Relating instantaneous rates to total change.
Example in Action
If you know your velocity at every moment, integrating gives total distance. If you know total change, differentiating gives the rate.
Key Formula
\[\int_a^b f(x),dx = F(b) - F(a)\]
Examples
To find the area under \( f(x) = 3x^2 \) from 1 to 2, compute \( [x^3]_1^2 = 8 - 1 = 7 \).
If \( F(x) = \int_0^x \cos t,dt \), then \( F'(x) = \cos x \).
In a Nutshell
The Fundamental Theorem links finding areas and rates of change, making calculus powerful.