An extension of the Fundamental Theorem of Calculus

In summary, the statement in the book of Introduction to Probability states that F(x) = \int_{-\infty}^{x} f(t)dt, and by the Fundamental Theorem of Calculus, F'(x) = f(x). The question was whether this is a valid application of the FTC, to which the answer is yes. This is because F(x) can be written as the sum of two integrals that follow the FTC for any constant a < x.
  • #1
Castilla
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In a book of Introduction to Probability I found this statement:

" Let be [tex]F(x) = \int_{-\infty}^{x} f(t)dt.[/tex] Then, by the Fundamental Theorem of Calculus, [tex]F'(x) = f(x).[/tex]"

With the minus infinity on the lower limit, it is this a valid aplication of the FTC?

Thanks.
 
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  • #2
Castilla said:
In a book of Introduction to Probability I found this statement:

" Let be [tex]F(x) = \int_{-\infty}^{x} f(t)dt.[/tex] Then, by the Fundamental Theorem of Calculus, [tex]F'(x) = f(x).[/tex]"

With the minus infinity on the lower limit, it is this a valid aplication of the FTC?

Thanks.


[tex]F(x) = \int_{-\infty}^{a} f(t)dt + \int_{a}^{x} f(t)dt[/tex]

for any constant a < x. So the answer is yes.
 
  • #3
Oh god, it was so easy...thanks, Greg.

Castilla
 

1. What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a theorem in calculus that connects the concepts of differentiation and integration. It states that if a function is continuous on a closed interval, then the integral of the function over that interval can be calculated by finding the antiderivative of the function at the endpoints of the interval and subtracting them.

2. What is an extension of the Fundamental Theorem of Calculus?

An extension of the Fundamental Theorem of Calculus is a generalization of the original theorem that applies to a wider range of functions and intervals. It takes into account functions that are not continuous and intervals that are not closed, providing a more powerful tool for solving problems in calculus.

3. How is the extension of the Fundamental Theorem of Calculus different from the original theorem?

The extension of the Fundamental Theorem of Calculus differs from the original theorem in that it allows for the evaluation of integrals over non-continuous functions and non-closed intervals. It also includes techniques such as substitution and integration by parts, which are not necessary in the original theorem.

4. What are some real-world applications of the extension of the Fundamental Theorem of Calculus?

The extension of the Fundamental Theorem of Calculus has many applications in physics, engineering, economics, and other fields. It is used to solve problems involving rates of change, optimization, and area under curves. For example, it can be applied to calculate the work done by a varying force, the volume of a three-dimensional object, or the total revenue of a company.

5. How can I use the extension of the Fundamental Theorem of Calculus in my own work?

If you are a scientist, the extension of the Fundamental Theorem of Calculus can be a valuable tool for solving problems in your field. It can help you analyze data, make predictions, and understand the behavior of complex systems. It is important to have a solid understanding of the original theorem and how it applies to your specific problem before attempting to use the extension.

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