What does the Fundamental Theorem of Calculus, Part 1 state?
If $g(x) = \int_{a}^{x} f(t) dt$, then $g'(x) = f(x)$.
What does the Fundamental Theorem of Calculus, Part 2 state?
If $F(x)$ is an antiderivative of $f(x)$, then $\int_{a}^{b} f(x) dx = F(b) - F(a)$.
How do you evaluate $\int_{a}^{b} f(x) dx$ using FTOC Part 2?
1. Find the antiderivative F(x) of f(x). 2. Evaluate F(b) and F(a). 3. Subtract F(a) from F(b): F(b) - F(a).
How do you find $g'(x)$ if $g(x) = \int_{a}^{x} f(t) dt$?
Apply FTOC Part 1: $g'(x) = f(x)$.
How do you handle a constant of integration when using FTOC Part 2?
The constant of integration cancels out when evaluating F(b) - F(a), so it's not necessary to include it.
How to find $g'(x)$ if $g(x) = \int_{a}^{h(x)} f(t) dt$?
Apply FTOC Part 1 and the chain rule: $g'(x) = f(h(x)) * h'(x)$
Define definite integral.
An integral with upper and lower limits, resulting in a numerical value.
What is an antiderivative?
A function whose derivative is the original function.
Define the Fundamental Theorem of Calculus.
A theorem that connects the derivative and the integral, stating that differentiation and integration are inverse processes.
