Can we back up a bit? I, sort of a newbie, but not really, want to ask about something that some newbie asked, but I can't remember for the life of me where it is.
It's true, if you look at the statement, "d / dx", it's saying that a small change in x changes the function following this by this much. Let the following function be and only be x^2 (and maybe others but having no greater than a power of 2).
By staying with the logic of this statement, what is stopping us from saying that a small change in x can be the increase of one integer under consideration by 1, and that if we were to take the difference of the two results of the function using x and x+1 (for example, 81 - 64), we can say that a small change in x (x + 1) changes the result of the function by 2x + 1, rather than 2x?
81 - 64 = 17
(2)(8) = 16
2x + 1 = (x + 1)^2 - (x)^2
It just happens to be a coincidence that the rate of change can be measured consistently with the second power. Why does this give us 2x + 1 and not 2x like calculus says?
>>7956052
because an increase of "1" is not a small change. Unless you're talking about finite differences, which are not denoted by d/dx anyway.
d/dx does not say how much a function changes by increasing x by a small amount.
d/dx is just a notation.
Look up the definition of a derivative and you will see why it's 2x and not 2x+1
will help you prove it if you have trouble.
>>7956083
>d/dx does not say how much a function changes by increasing x by a small amount.
It doesn't?! Why can't we read it out exactly as the notation indicates? Why then not just use a sign created like the sum of all parts or something, rather than a notation that uses operations and variables?
derivative is the gradient, integral is the area under the graph
>>7956091
>rather than a notation that uses operations and variables?
You will understand why it is useful this way when you do non-trivial integrals.
All of your doubts would be cleared if you just took the time to read a Calculus I book. What is stopping you from doing so?
>>7956099
Attention span. Can you just explain it now?
>>7956124
df(x) / dx = lim h->0 (f(x+h)-f(x))/h
dx^2 / dx = lim h->0 ((x+h)^2-x^2)/h
lim h->0 (2xh+h^2)/h
lim h->0 2x+h
2x
>>7956132
Yes, so you can say that for every small change in x. f(x) changes by 2x... That is literally what this is saying.
>>7956139
Why do you have to assume that a small change has to imply that it "approaches" so that you take the leap forward and assume what would be the case if it was? You're working under an impossible premise. Why not just say that "in a scale where x and x+1 is a distance so inconceivably small that humans cannot understand it, this is the case".
>>7956160
Answer this: Where does that limit comes from?
After you do answer that, you will have answered your own question.
>>7956169
It comes from the assumption that you can take the difference of two different points from an axis and assume it is so small that it is 0. But here's the part that I don't like... You CAN'T ever get 0.
>>7956174
Let me expand on that and say that in a scale where an atom is a universe itself, the assumptions you make to assume that the negligible difference is essentially 0 would be astronomical.
>>7956174
>the limit as x approaches 0 is the same as x = 0
Looks like you will have to back up even further. Drop calculus for now, Mr. 20 IQ, and first study limits.
>>7956177
Isn't that why you drop the h out of the equation of 2x + h? It, being the difference between two points along an axis (being one part of the two components that make up a slope), is so small that you essentially ignore it.
>>7956160
I don't understand the question. The derivative describes the instantaneous change in the function, not a "small change". Another way of thinking about it is that you are finding the slope of the line tangent to the function at a point. This line describes the instantaneous behavior of that point.
>>7956181
I understsand, but you're working under the premise that the distance between two points is so small that you can consider it instantaneous, when realistically, that is absolutely impossible?
>>7956179
No, it's infinitesimally small, which is equivalent to 0.
>>7956179
Yes but you are ignoring all of the algebra going on.
In the limit you have f(x+h) - f(x)
If you h=0 insteado f being the limit as h approaches 0, you would have f(x) - f(x).
You would end up with the undefined expression 0/0
The thing is that algebraically, h exists but numerically it is a really small value. Once you cancel the h that is dividing then you are done with your algebra tricks and can safely and rigorously treat h as a numerical value.
>>7956182
>I understsand, but you're working under the premise that the distance between two points is so small that you can consider it instantaneous, when realistically, that is absolutely impossible?
No, the limit is not a distance between two points, it is what that distance approaches as the two points get closer and closer together. But there is no such thing as two different points that are infinitely close to each other.
>>7956186
>Let me expand on that and say that in a scale where an atom is a universe itself, the assumptions you make to assume that the negligible difference is essentially 0 would be astronomical.
I have no idea what this means, but atoms have no relevance here. We are talking about infinitesimal points on the real number line.
>>7956187
So you're basically taking the word "infinity" and isolating it to a finite point, or a "goal" point?
I don't mean to give the appearance that I'm a know-it-all, but this is one of the faultiest premises in modern times. I wish I would be around in a time where this is understood to be too inaccurate.
>>7956192
Yes, it has much relevance. Please read it over.
The distance of .000000000000000054 nanometers to us is so inconceivably small, but to a scale where an atom to us is the size of our universe, that distance is gargantuan. For that reason, you cannot assume that it is essentially 0, because to that universe, it is not.
>>7956196
>So you're basically taking the word "infinity" and isolating it to a finite point, or a "goal" point?
Huh? Again, I don't understand what you're trying to say. You keep bringing in these philosophical-sounding assumptions about impossibility and infinity when we are talking about a simple, strictly defined mathematical system that doesn't contain these assumptions. If you want to ignore calculus as "inaccurate" then go ahead, but this is a rather meaningless and pointless position as the various real-life applications of calculus seem to be working fine.
>>7956204
>ITT: I haven't taken calculus and I haven't even taken pre-calc limits but I swear on my heart that all of calculus is wrong and I am a genius.
>>7956204
It's not a matter of perspective. Finite amounts will always be finite, no matter how large they "appear". None of what you're saying is mathematical.
>>7956207
But it /does/ contain these assumptions.
It may work for your mechanical understanding of reality, but it does not solve an absolute understand of reality. It confines truth to your empirical perception of it and completely ignores other scales, and other possibilities.
If you want to be a mechanic, then worry nothing at all. This assumption is just fine and the margin of error is too negligible to notice.
>>7956215
OK, what is the margin of error in saying that the derivative of x^2 is 2x?
>>7956213
Then this is why we need philosophy, because it is more importantly to our understanding of reality than mathematics. I refuse to accept your premise as being an absolute with relation to reality.
>>7956220
>two points approaches 0 when it never can
>you can't approach 0
>0.0000001 doesn't exist
>0.00000000001 doesn't exist
>0.00000000000000001 doesn't exist
Yup. I already knew it. You were a Newton tier genius. I will now burn all of my calculus books and subscribe to your own personal brand of bullshit.
>daily reminder that when you say that the limit as h approaches 0 is 2x, that means that it is approaching 2x, not that it is exactly 2x.
>>7956218
I don't think your position is going to produce any understanding of reality, let alone a better one than mathematical physics. Regardless, this thread began with a purely mathematical question, and calculus is absolutely true mathematically. You have not given any mathematical reasons why it's "inaccurate".
>>7956220
>By assuming that the distance between two points approaches 0 when it never can
OK, so what does the distance between two points approach as they get closer together?
>It's lazy.
It's math.
>>7956223
Rather,
>0.0000001 does exist
>0.00000000001 does exist
>0.00000000000000001 does exist
and so does 0.000...1
You will never approach (get close to) 0 because there will almost be another scale ahead of it, and that scale will still think there is an infinitude of "0.000...1"'s to consider to reach 0. It's a cute idea, but you can never "approach" 0 no matter how many 0's you tag on after the decimal.
>>7956228
What you just described is approaching 0 though. If it's impossible to approach 0 then there must be a point at which you can get no closer to zero. But as you just said, you can always get closer and closer to zero. The limit describes this behavior.
>>7956228
Yeah, but you'er working from the premise that the universe is infinite both microscopically and macroscopically. How do you know?
>>7956234
Beautiful. Thank you.
Why do I have a reason to believe that the distance between two points suddenly comes to a halt and you cannot carry that operation any further?
>>7956228
Lets study the function f(x) = x
Lets try to approach f(x) as x approaches 0 from the positive side.
We have 1, then 0.1, then 0.001, etc.
What is happening here? Is there a value that we will never be able to reach without going past 0? Yes there are. You will never -1, -0.1, -0.01
Basically, you will never get smaller than 0, approaching it from the positive side.
Lets now approach it from the negative side. -1,-0.1,-0.01,-0.01, etc.
Now what values can't you reach? You will never reach 0.01,0.1,1 because you would have to go beyond 0.
It is ALMOST like if there was some kind of barrier that didn't let us get through to other values... and that barrier is 0. It is like 0 is LIMITING us from getting past it. It is almost like being infitinitesimally close to 0 will always be similar to being at 0 but never past it, relative to whatever direction you are approaching it from.
HMMMMMMMMMMMMMMM
>>7956234
>is infinite both microscopically and macroscopically. How do you know?
epsilon delta
>>7956239
>Why do I have a reason to believe that the distance between two points suddenly comes to a halt and you cannot carry that operation any further?
Why would I*
Sorry, I really need to watch my English. It is the most important. I promise to proofread up to 10 times before I post.
>>7956249
It doesn't. That's the point. If it came to a halt before reach 0 then that would be the limit, not zero.
>>7956252
But you have to understand that to say "the distance between two points..." means that there is a space between two points, and that no matter how hard you try or think, you can /never/ reach a point where that space is nothing.
>>7956256
Thought experiment for you:
Bring two fingers closer and closer to each other. If your idea is true, my fingers would never converge. overlap, and distance away in their opposite directions. There has to be a point where you cannot get any smaller, and it does in fact reach 0.
>>7956259
Hm, wow. Thanks.
>>7956256
>But you have to understand that to say "the distance between two points..." means that there is a space between two points, and that no matter how hard you try or think, you can /never/ reach a point where that space is nothing.
The distance between two different points can never be 0, but the distance *approaches* zero. Again, the derivative is not the effect of a small change, it is what the effect approaches, i.e. the instantaneous change. And if you just can't wrap your mind around that, just think of it as the slope of the tangent line. What problem do you have with that?
>>7956268
>instantaneous change
Those are oxymorons.
If something is instantaneous, you /cannot/ observe change because to say change means you're comparing it to another point, which is the "distance between two points" I'm trying to get through to you.
For example:
>The sky changed to green
You're comparing the sky to a state that it previously was. So there are two points: when it was green, and when it was blue.
>>7956277
>If something is instantaneous, you /cannot/ observe change because to say change means you're comparing it to another point, which is the "distance between two points" I'm trying to get through to you.
Basically you're just saying "change between points" doesn't mean "instantaneous change", which is what I already said. This is just semantics. Try to respond to the concept, not the name of the concept.
>You're comparing the sky to a state that it previously was. So there are two points: when it was green, and when it was blue.
If you want to measure how fast the color of the sky was changing at a particular point in time, then you would use the derivative. For example, how fast your car is currently going is not a change between two points, it is an instantaneous rate.
>>7956290
I totally understand your concept, but I'm having a difficult time agreeing with it.
I guess going back to Newton's idea which says "An object will continue to travel in a straight line if uninterrupted" implies that it's not taking into consideration the negligible gravity of all the bodies surrounding it. Maybe in an ideal situation where the symmetry of bodies around the [object moving in a "straight line"] allows for it to travel in a straight line, it really doesn't in an entropic universe where everything is random and some may be on one side more than the other.
It is a matter of semantics, and I guess math is all about taking the negligible and assuming it is so small that it doesn't matter.
Is it correct to say that "0" in this case would be the velocity of an object travelling uniformly, and the "0" in an accelerating object would be the velocity it would be travelling at any given point if the force were no longer being applied?
>>7956319
In addition to this, who is also to say that the "uniform acceleration" in space can't be attributed to the gravity in all directions which may "negligibly" cancel each other out?
>>7956277
Let's now look at integrals to see the true beauty of calculus. Let's say you have some curve, and you want to know the area this curve bounds. This is a very common problem right, we want to know the area of lots of shapes like circles and things. Well we know how to calculate the area of rectangles and triangles, so a good way of measuring the area would be to split the shape up into triangles and rectangles. The change between points gives us a nice hypotenuse of a right triangle, sitting on top of a rectangle. But as you can see, we have areas of the shape that aren't included in these rectangles and triangles, and areas of the triangles that go beyond the shape. So these are going to be errors and adding up the areas of the triangles and rectangles will only give us an approximation of the shape's area. The smaller and smaller we make the width of the rectangles, the better the approximation will be, but it will never be perfect. But wait! If the approximation gets better and better, then the *true area* is the limit as the width of the rectangles get smaller and smaller! All we have to do is find this limit, and we can get the true area of the shape! Can we do this? Why yes we can, because the derivative of the curve gives us this limit!
>>7956319
Is it correct to say that "0" in this case would be the position of an object travelling uniformly*
I did not stick to my promise. I am sorry.
>>7956319
>I guess going back to Newton's idea which says "An object will continue to travel in a straight line if uninterrupted" implies that it's not taking into consideration the negligible gravity of all the bodies surrounding it. Maybe in an ideal situation where the symmetry of bodies around the [object moving in a "straight line"] allows for it to travel in a straight line, it really doesn't in an entropic universe where everything is random and some may be on one side more than the other.
I don't see what any of this has to do with what we're talking about. Remember that we are talking about mathematics, empirical physics has no bearing on it. And even if it did, physics uses calculus to describe the real world, so it wouldn't make sense to appeal to physics anyway.
>It is a matter of semantics, and I guess math is all about taking the negligible and assuming it is so small that it doesn't matter.
No, it has nothing to do with negligible values. The derivative is not an approximation, it is the limit of approximations. So it is ideal, it has no error, not even negligible error.
>Is it correct to say that "0" in this case would be the velocity of an object travelling uniformly
An object travelling uniformly would have constant, non-zero velocity and 0 acceleration.
>>7956333
I very much understand what you are saying. So much so, that I could confidently teach it to a classroom. But there's a part of me that doesn't agree with it, and I guess that is what makes us different.
You can go ahead a say "sub 20 IQ piece of shit", and I'll say "Sorry, I don't agree with your premise" while departing from this discussion.
Thank you and take care!
>>7956360
If you can't articulate a coherent reason why you disagree, why continue to disagree? Feelings and intuition don't trump mathematical truth.
>>7956343
The position of an object traveling uniformly would be changing linearly, not zero. If you have the velocity of an object, then the integral of that is the position of the object. A uniformly travelling object has constant velocity v and position vx + C
>>7956369
I've responded to everything you've said, have I not?
>>7956369
>I've already explained my reasoning coherently
Do you just have some sort of brain damage?
Your reasoning is not coherent.
>>7956277
>Those are oxymorons
No.
I think you have trouble understanding what a function is.
