34
you are viewing a single comment's thread
view the rest of the comments
view the rest of the comments
this post was submitted on 04 Apr 2024
34 points (100.0% liked)
Actually Useful AI
1989 readers
1 users here now
Welcome! ๐ค
Our community focuses on programming-oriented, hype-free discussion of Artificial Intelligence (AI) topics. We aim to curate content that truly contributes to the understanding and practical application of AI, making it, as the name suggests, "actually useful" for developers and enthusiasts alike.
Be an active member! ๐
We highly value participation in our community. Whether it's asking questions, sharing insights, or sparking new discussions, your engagement helps us all grow.
What can I post? ๐
In general, anything related to AI is acceptable. However, we encourage you to strive for high-quality content.
What is not allowed? ๐ซ
- ๐ Sensationalism: "How I made $1000 in 30 minutes using ChatGPT - the answer will surprise you!"
- โป๏ธ Recycled Content: "Ultimate ChatGPT Prompting Guide" that is the 10,000th variation on "As a (role), explain (thing) in (style)"
- ๐ฎ Blogspam: Anything the mods consider crypto/AI bro success porn sigma grindset blogspam
General Rules ๐
Members are expected to engage in on-topic discussions, and exhibit mature, respectful behavior. Those who fail to uphold these standards may find their posts or comments removed, with repeat offenders potentially facing a permanent ban.
While we appreciate focus, a little humor and off-topic banter, when tasteful and relevant, can also add flavor to our discussions.
Related Communities ๐
General
- !Artificial@kbin.social
- !artificial_intel@lemmy.ml
- !singularity@lemmy.fmhy.ml
- !ai@kbin.social
- !ArtificialIntelligence@kbin.social
- !aihorde@lemmy.dbzer0.com
Chat
Image
Open Source
Please message @sisyphean@programming.dev if you would like us to add a community to this list.
Icon base by Lord Berandas under CC BY 3.0 with modifications to add a gradient
founded 1 year ago
MODERATORS
You are correct. I apologize for the confusion. The correct statement should be:
As max(norm(P)) -> 0, the Riemann sum S(P, {x_i*}) approaches the definite integral:
lim (max(norm(P)) -> 0) S(P, {x_i*}) = โซ(a, b) f(t) dt
@Mistral@lemmings.world Okay. Now can you rewrite the proof in language a five-year-old would understand?
Of course! Let's imagine a long pathway, like a hallway, that goes from a door at one end (point A) to another door at the other end (point B). We want to find out the total distance of this hallway.
Now, let's pretend we don't know the exact distance, but we can find the distance between some points along the hallway. We can do this by placing markers on the floor, like stickers, and measuring the distance between them. We can call these markers "partitions" and the distance between them "widths."
When we make these partitions closer and closer to each other, our estimate of the total distance of the hallway will get better and better. In the end, we want to find out the exact total distance of the hallway, which is what we call the "definite integral."
So, imagine we have a friend who can tell us the height of the hallway at any point. We can think of the height as the "function" we want to integrate. Now, we can make a "Riemann sum" by adding up the height of the hallway at each partition and multiply it by the width of that partition.
As we make more and more partitions and the widths of the partitions get smaller and smaller, our Riemann sum will get closer and closer to the actual total distance of the hallway, which is the definite integral we're trying to find. And that's how we can find the definite integral by using partitions and Riemann sums!
@Mistral@lemmings.world That's great! Can you rewrite the proof so a dog can understand it?
Sure, let's give it a try! Imagine you're a dog in a big park, and there's a long pathway (like a hallway) that goes from one end of the park to the other end. Your human friend wants to find out the total distance of this pathway, which we can think of as the "definite integral."
Now, your human friend is not sure about the exact distance of the pathway, but they can find the distance between some points along the pathway. They can do this by putting markers (let's call them "partitions") on the ground and measuring the distance between them. We can call the distance between the markers "widths."
Your human friend then wants to estimate the total distance of the pathway. They can do this by making the partitions closer and closer to each other, which will make their estimate better and better.
Now, imagine your human friend also knows the height of the pathway at any point. We can think of the height as the "function" we want to integrate. Your human friend can make a "Riemann sum" by adding up the height of the pathway at each partition and multiplying it by the width of that partition.
As your human friend makes more and more partitions and the widths of the partitions get smaller and smaller, their Riemann sum will get closer and closer to the actual total distance of the pathway, which is the definite integral they're trying to find. And that's how your human friend can find the definite integral by using partitions and Riemann sums!
Welp my brain is sufficiently melted. 10/10.
In all seriousness, this is really cool!