The Fundamental Theorem of Calculus is a fundamental concept in calculus that establishes the relationship between the derivative and the integral of a function. It states that if a function is continuous on a closed interval, then the definite integral of the function can be evaluated by finding an antiderivative of the function and evaluating it at the endpoints of the interval.
The Fundamental Theorem of Calculus is significant because it allows us to easily evaluate definite integrals without having to use Riemann sums or other tedious methods. It also connects the two branches of calculus, differentiation and integration, by showing that they are inverse operations of each other.
In calculus, a limit is a mathematical concept that represents the value that a function approaches as the input approaches a certain value. It is often used to describe the behavior of a function near a specific point.
Limits are often used in proofs to show the behavior of a function at a certain point or as the input approaches a certain value. In the context of fundamental theorem and limit proofs, limits are used to establish the connection between the derivative and integral of a function.
Some common techniques used in fundamental theorem and limit proofs include the definition of a limit, the Squeeze Theorem, the Intermediate Value Theorem, and techniques from differential and integral calculus such as the Mean Value Theorem and the Fundamental Theorem of Calculus.