Proving the second fundamental theorem of calculus?

That is why you are having trouble.In summary, the conversation discusses how to show that ##Dx\int f(u)du = f(x)##, where the integral is evaluated from a to x. The hint suggests using a Taylor expansion of f(u) around x, but the issue is that the final answer should only be f(x) without the f(a) term. The problem arises because the integral is a function of x, not u.
  • #1
Vitani11
275
3

Homework Statement


Show that Dx∫f(u)du = f(x) Where the integral is evaluated from a to x. (Hint: Do Taylor expansion of f(u) around x).

Homework Equations


None

The Attempt at a Solution


I have

... = Dx(F(u)+C) = Dx(F(x-a)+C) = dxF(x) - dxF(a) = f(x)-f(a). My problem is that it should be only f(x), not f(x) - f(a). I did a taylor expansion of f(u) around x and I'm not sure how that is supposed to help me...
 
Physics news on Phys.org
  • #2
I know that for an integral F(a) (where a is not a variable) the derivative of the integral would be 0, and that's why it would not be included in the final answer, but I don't know how to show that.
 
  • #3
Vitani11 said:
Dx(F(u)+C)
You are starting with ##\int_a^xf(u)du##, so that is a function of x (or maybe of a and x). It is not a function of u.
 

FAQ: Proving the second fundamental theorem of calculus?

What is the Second Fundamental Theorem of Calculus?

The Second Fundamental Theorem of Calculus is a theorem in calculus that states that if a function is continuous on a closed interval and has a derivative at every point in the interval, then the integral of the function over that interval can be evaluated using the antiderivative of the function at the endpoints of the interval.

How is the Second Fundamental Theorem of Calculus proven?

The Second Fundamental Theorem of Calculus can be proven using the first fundamental theorem of calculus, which states that the integral of a function can be computed by finding any antiderivative of the function and evaluating it at the endpoints of the interval. By using this theorem and the properties of integrals and derivatives, the Second Fundamental Theorem of Calculus can be derived and proven.

What is the significance of the Second Fundamental Theorem of Calculus?

The Second Fundamental Theorem of Calculus is significant because it provides a powerful tool for evaluating integrals. It allows for the evaluation of integrals without having to use the limit definition of the integral, making calculations much simpler and more efficient. It also connects the concepts of derivatives and integrals, providing a deeper understanding of these fundamental concepts in calculus.

Are there any conditions for the Second Fundamental Theorem of Calculus to be applicable?

Yes, there are two main conditions for the Second Fundamental Theorem of Calculus to be applicable. First, the function must be continuous on the closed interval. Second, the function must have a derivative at every point in the interval. If these conditions are met, then the theorem can be applied to evaluate the integral of the function over the given interval.

What are the practical applications of the Second Fundamental Theorem of Calculus?

The Second Fundamental Theorem of Calculus has many practical applications, especially in physics and engineering. It can be used to calculate areas, volumes, and other physical quantities that are represented by integrals. It is also used in optimization problems, where it allows for the determination of optimal solutions by finding the points where the derivative of a function is equal to zero.

Similar threads

Differentiate the given integral
Replies
15
Views
1K
Help me prove integral answer over infinitesimal interval
Replies
9
Views
1K
Application of the Fundamental Theorem of Calculus (cosmological red-shift)
Replies
1
Views
1K
Proofs about the second-order linear differential equation?
Replies
21
Views
1K
Finding the Wrong Answer with Stokes' Theorem
Replies
4
Views
1K
Prove limit comparison test for Integrals
Replies
2
Views
1K
Attempting to prove the Intermediate Value Theorem
Replies
26
Views
2K
How to apply the fundamental theorem to partial derivatives?
Replies
2
Views
3K
Intermediate Value Theorem Problem on a String
Replies
14
Views
2K
Solving the Minimization Operator Problem
Replies
19
Views
2K

Hot Threads

Recent Insights

Back
Top