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Complex fundamental theorem of calculus Hot Network Questions Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
8. To give you background, I have been studying calculus from Stewart. In the Integrals chapter, he proved the Evaluation Therorem by applying the mean value theorem on the Riemann sum of a continous function f f: ∫b a f(x)dx = F(b) − F(a). ∫ a b f (x) d x = F (b) − F (a). Now when I studying the Fundamental Theorem of Calculus, I think ...
Here is a problem I have been working on recently: Let f: [a, b] → R be continuous, differentiable on [a, b] except at most for a countable number of points, and f′ is Lebesgue integrable, then the fundamental theorem of calculus holds, i.e. ∀x, y ∈ [a, b] we have f(y) = f(x) + ∫y xf ′ (t)dt. The proof I have at the moment is ...
My teacher tells me the following statement: Suppose f ∈ L1Loc(Ω) and ∫Ωfφ = 0, ∀φ ∈ C∞0(Ω) Then f = 0 a.e. on Ω. It is known as fundamental lemma of calculus of variation. My teacher told me it suffices to prove this statement holds for the case f is continuous. But I find it's not easy to deduce the lemma from the case f is ...
As you have written it F(x, y) = ∫ba∫dcf(u, v)dudv indicates that the function F is a constant with zero partial derivatives since the integral on the RHS is a constant (real number) independent of x and y. Assuming that f ∈ C(R) you can apply the fundamental theorem of calculus twice to prove (*). First you must show that G(u, y) = ∫ ...
Your proof would be correct, if you can first prove something like the following: ... First Fundamental ...
Silly question. Can someone show me a nice easy to follow proof on the fundamental theorem of calculus. More specifically, $\displaystyle\int_{a}^{b}f(x)dx = F(b) - F(a)$ I know that by just googling fundamental theorem of calculus, one can get all sorts of answers, but for some odd reason I have a hard time following the arguments.
Proof of fundamental theorem of calculus part 1 Rudin Theorem 6.20. 1. Attempt at proving the first part ...
80. 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). In more detail, chop up the interval [a, b] into tiny pieces: a = x0 <x1 <⋯ <xN = b.
4. Intuitively, this theorem says that "the total change is the sum of all the little changes", and this can be made into a rigorous proof. Here is some intuition. Chop the interval [a, b] up into tiny pieces: a = x0 <x1 <⋯ <xN = b. Let ΔFi be the change in F as its input changes from xi to xi + 1. Of course, F(b) − F(a) = ∑iΔFi.