this post was submitted on 04 Apr 2024
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Actually Useful AI

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The Mistral bot here on Lemmy now remembers context (lemmings.world)
submitted 2 years ago by rikudou@lemmings.world to c/auai@programming.dev
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Iโ€™ve updated @Mistral@lemmings.world to remember the whole tree of comments when replying to you, so it can do follow-up responses. You still have to mention it on every message, though. Example in the comments.

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[โ€“] PM_ME_VINTAGE_30S@lemmy.sdf.org 1 points 2 years ago (1 children)

@Mistral@lemmings.world Prove the two Fundamental Theorems of Calculus.

[โ€“] Mistral@lemmings.world 1 points 2 years ago

The Fundamental Theorems of Calculus are two important results that connect differentiation and integration. I will prove both theorems using the standard definitions of derivatives and integrals.

First, let's recall the definitions:

  1. Derivative: If f(x) is a function, then the derivative of f(x) with respect to x, denoted as f'(x), is the limit:

f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]

  1. Integral: If f(x) is a function and [a, b] is a closed interval, then the definite integral of f(x) over [a, b], denoted as โˆซ(a to b) f(x) dx, is the limit:

โˆซ(a to b) f(x) dx = lim (n -> โˆž) [ฮฃ (from i = 1 to n) f(x_i) * (x_i - x_(i-1))]

where x_0 = a, x_n = b, and x_i are points in the interval [a, b] such that x_(i-1) < x_i.

Now, let's prove the two Fundamental Theorems of Calculus:

Fundamental Theorem of Calculus, Part 1 (Differentiation):

If f(x) is continuous on [a, b], and g(x) is its antiderivative, then g'(x) = f(x).

Proof:

By definition, we need to show that the limit:

g'(x) = lim (h -> 0) [g(x + h) - g(x)] / h

exists and equals f(x) for all x in [a, b].

Since g(x) is an antiderivative of f(x), we have:

g(x + h) - g(x) = โˆซ(x to x+h) f(t) dt

Now, we can use the definition of the integral to rewrite this as:

g(x + h) - g(x) = lim (n ->

[โ€“] ExperimentalGuy@programming.dev -1 points 2 years ago

@Mistral@lemmings.world where in the world is Carmen San Diego?