What does the second fundamental theorem of calculus state?

The second part of the fundamental theorem of calculus tells us that to find the definite integral of a function ƒ from 𝘢 to 𝘣, we need to take an antiderivative of ƒ, call it 𝘍, and calculate 𝘍(𝘣)-𝘍(𝘢).

Why is the fundamental theorem of calculus Part 1 and 2 useful for computing integrals?

The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. See Note. The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula.

What does the fundamental theorem of calculus give us?

The fundamental theorem of calculus establishes the relationship between the derivative and the integral. It just says that the rate of change of the area under the curve up to a point x, equals the height of the area at that point. This theorem helps us to find definite integrals.

Why is the fundamental theorem of calculus useful for computing integrals?

There is a reason it is called the Fundamental Theorem of Calculus. Not only does it establish a relationship between integration and differentiation, but also it guarantees that any integrable function has an antiderivative. Specifically, it guarantees that any continuous function has an antiderivative.

Which two important concepts are connected by the fundamental theorem of calculus?

The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially inverses of one another.

Why does the fundamental theorem of calculus make intuitive sense?

Intuitively, the fundamental theorem of calculus states that “the total change is the sum of all the little changes”. f′(x)dx is a tiny change in the value of f. You add up all these tiny changes to get the total change f(b)−f(a).

What is the second fundamental theorem of welfare economics?

The Second Fundamental Theorem of Welfare Economics states that if every consumer has convex preferences and every firm has a convex production set then any Pareto-efficient allocation can be decentralized as a competitive equilibrium.