Advanced Placement Calculus AB covering limits, derivatives, and integrals.
The Fundamental Theorem of Calculus connects differentiation and integration, showing they are inverse processes.
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.
This theorem makes evaluating definite integrals much faster—just find an antiderivative!
If you know your velocity at every moment, integrating gives total distance. If you know total change, differentiating gives the rate.
\[\int_a^b f(x),dx = F(b) - F(a)\]
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 \( F'(x) = \cos x \).
The Fundamental Theorem links finding areas and rates of change, making calculus powerful.