Fundamental theorem of calculus

The fundamental theorem of calculus is a like between the concept of the derivative of a function and the concept of integral. There are two parts to the theorem.

First part

This part is also known as the first fundamental theorem of calculus.

The theorem

Let f be a continuous real-valued function which is continuous on the closed interval [a,b]. Let F be the function defined, for all x in [a,b], by the following equation:

The theorem

Then, F is a primitive of f on [a,b].

Prerequisite information

  • Have a knowledge of the definition of the primitive.
  • Know the definition of the derivative.
  • Know the mean value theorem.

The proof

We want to find the primitive of F, denoted F. This is why we use the defintion of the derivative to get started:

The proof

We know that f is continuous on [a,b], this is why it is continuous on [x,x+h[x;x+h][a;b]. From the mean value theorem, we know that there exist a point c[x,x+h] such that:

The proof 1

Then, we deduce from the results (1) and (2) that:

The proof 2

When h0cx, this is why we have:

The proof 3

Second part

This part is also known as the second fundamental theorem of calculus. Sometimes it is referred to as the evaluation theorem (in context of integral calculus).

The theorem

Let F be a primitive of f on [a,b]. If f is continuous on the interval [a,b], then:

The theorem

Prerequisite information

  • Know the first part of the fundamental theorem of calculus.

The proof

From the first part of the fundamental theorem of calculus, we have:

The proof

Then, we deduce that:

Then, we deduce that

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