Fundemental Theorem of Calculus

In summary: I.e., ifF(x) = \int^x f(t)dtthenF'(x) = f(x)and\frac{d}{dx} \int^x f(t)dt = f(x).In summary, the conversation discusses the use of the fundamental theorem of calculus to find the derivative of a continuous function, as well as various notations for expressing this concept. It is noted that while \frac{d}{dx}\int_a^x f(t)dt=f(x) is a commonly used notation, it is not technically correct as the integral does not name a function of x. The notation \int^x f(t)dt is sometimes used to indicate the general anti-derivative
  • #1
Niles
1,866
0

Homework Statement


Hi guys

From the Fundemental Theorem of Calculus we know that if we have a continuous function f : [a,b] -> R and F is the function on (a,b) defined by
[tex]
F(x)=\int_a^xf(t)dt,
[/tex]
then F is differentiable on (a,b) with F'(x)=f(x) for all x in (a,b), i.e.
[tex]
\frac{d}{dx}\int_a^x f(t)dt=f(x).
[/tex]Question: Is it correct also to write
[tex]
\frac{d}{dx}\int f(t)dt=f(x)?
[/tex]

If not, then is there a way of expressing [itex]\frac{d}{dx}\int_a^x f(t)dt=f(x)[/itex] without limits on the integral?
 
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  • #2
Niles said:
Question: Is it correct also to write
[tex]
\frac{d}{dx}\int f(t)dt=f(x)?
[/tex]

When you write that, everyone knows what you mean -- but it's not correct. The reason is that
[tex]\int f(t)\,dt[/tex]
doesn't name a function of [tex]x[/tex], even though some dependence on [tex]x[/tex] is clearly intended. You could think of it as naming a one-parameter family of functions
[tex]F_a(x) = \int_a^x f(t)\,dt[/tex]
where the parameter [tex]a[/tex] is the lower limit of the integral. The parameter shows up in another guise as the customary "constant of integration" [tex]C[/tex], which is connected with the fact that [tex]F_b - F_a[/tex] is the constant function equal to
[tex]\int_a^b f(t)\,dt[/tex].

The functions [tex]F_a[/tex] all differ by a constant, so they have the same derivative, which is [tex]f[/tex] by the fundamental theorem. But the one-parameter family of functions [tex]\{F_a : a \in \mathbb{R}\}[/tex] isn't something you can use the same notation to differentiate.
 
  • #3
Occaisionally, you will see the notation
[tex]\int^x f(t)dt[/tex]
(note the lack of a lower limit) to indicate the general anti-derivative.
 

Related to Fundemental Theorem of Calculus

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a fundamental concept in calculus that links the concepts of differentiation and integration. It states that if a function is continuous on a closed interval and has an antiderivative, then the integral of the function over that interval can be evaluated by using the antiderivative at the endpoints of the interval.

Why is the Fundamental Theorem of Calculus important?

The Fundamental Theorem of Calculus is important because it provides a powerful tool for evaluating definite integrals, which are used to find the total change or accumulation of a quantity. It also allows for the connection between the geometric interpretation of the definite integral as area under a curve and the algebraic interpretation as the antiderivative of a function.

What are the two parts of the Fundamental Theorem of Calculus?

The first part of the Fundamental Theorem of Calculus states that if f(x) is continuous on the closed interval [a,b] and F(x) is any antiderivative of f(x), then the definite integral of f(x) from a to b is equal to F(b) - F(a). The second part of the theorem states that if a function g(x) is defined as the definite integral of f(x) from a to x, then g'(x) = f(x).

How is the Fundamental Theorem of Calculus used in real-life applications?

The Fundamental Theorem of Calculus is used in various real-life applications, such as physics, engineering, economics, and statistics. It allows for the calculation of areas, volumes, and other quantities that are important in these fields. For example, it can be used to calculate the distance traveled by an object given its velocity function, or to find the average value of a function over a given interval.

Is the Fundamental Theorem of Calculus difficult to understand?

The Fundamental Theorem of Calculus can be a difficult concept to understand at first, but with practice and a solid understanding of the basic concepts of calculus, it can be grasped. It is important to build a strong foundation in calculus before attempting to understand the theorem, and to work through examples and practice problems to gain a better understanding of its applications.

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