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by a.s.h.
Tue Jan 30, 2018 1:20 pm
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

(cont.) So perhaps you were trying to convey your physical intuition in your posts, but from a mathematical standpoint, I don't see any paradoxes here. (For example with your Riemann sums, you are basically saying that since ∆x → 0, our sum is 0 + 0 + ... 0 = 0. But this is not valid because there a...
by a.s.h.
Tue Jan 30, 2018 1:18 pm
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

Sometimes we can replace a variable by infinity or 0 and have something that makes sense. When we do this, we are using the extended reals , which is basically the real numbers together with +∞ and -∞. For example, for the limit lim n → ∞ (3/n), we can indeed say this is 3/∞ = 0 (i.e. replace n by ∞...
by a.s.h.
Tue Jan 30, 2018 7:45 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

Basically, the definition of saying "the limit of that sum as n → ∞ is 1" is that the sum can be made to stay arbitrarily close to the value 1 whenever n is sufficiently large. The value of n never has to "equal infinity" when using the definition of the limit. Precise formulatio...
by a.s.h.
Tue Jan 30, 2018 7:44 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

I don't think n ever "equals" infinity in these infinite sums. The infinite sums are defined as limits as n → ∞. In the sum I wrote ((1/n) + (1/n) + … + (1/n), where there are n terms in total), when we take the limit as n → ∞, the answer is 1. Taking the limit does not mean we are multipl...
by a.s.h.
Sun Jan 28, 2018 10:28 pm
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

You seem to be saying that since each term of a series goes to 0 as n -> ∞, the entire sum goes to 0 as n -> ∞. Unfortunately, this is not true because the number of terms in the sum is also going to infinity! That is, ∞*0 is an indeterminate form . A simple example of a series where the terms goes ...
by a.s.h.
Thu Feb 09, 2017 7:09 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

Hello, Here are the three last pages of my integral paradox proof. Have a nice day. So in summary, what are you claiming the paradox to be? You wrote an expression for f(x) (the derivative of F(x) := (x^3)/3) but didn't take the limit as Δx → 0. When we do this, we see that f(x) = x^2, and the Riem...
by a.s.h.
Sat Oct 22, 2016 2:57 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

We can't take a regular derivative at the endpoint 3, because the function wouldn't be defined on the right of 3, so we can't take a limit from that side (remember, for a regular limit, h can approach 0 from both sides). But we can still consider a one-sided limit, since the function would be define...
by a.s.h.
Sat Oct 22, 2016 2:03 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

We can only talk about differentiability at an interior point of the domain of the function. So at the endpoint 3, we can't talk about differentiability, but we can talk about one-sided derivatives, as I said in the previous post.
by a.s.h.
Sat Oct 22, 2016 12:52 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

In such cases, we would talk about the one-sided derivative of the function. In the example you've given, we would need to exam whether the function is left differentiable at 3. See for instance this page for definitions and information on one-sided derivatives: https://en.wikipedia.org/wiki/Semi-di...
by a.s.h.
Tue Oct 18, 2016 3:56 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

a.s.h, Yes, we need a slope at some x-value. When we say that a slope (derivative) at, say, x = 1 of some function, we mean a precise value of a slope at x = 1 (not x = 1.01 or less), but the quotient expression defining the slope at x = 1 is undefined at x = 1 in x =1! As I see, it is not persuasi...
by a.s.h.
Tue Oct 18, 2016 3:19 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

Even though p(0) is not defined in your example, the limit as x -> 0 of p(x) is perfectly well-defined. Remember, a slope (derivative) at a particular point is a limit, so as long as the limit is well-defined, the derivative exists.
by a.s.h.
Tue Oct 18, 2016 2:03 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

a.s.h., That's is right, but h cannot be zero. The value of the slope in the situation that you described in your last post can be equal exactly to 2 IF AND ONLY IF h = 0! As far as I understand, we agreed that h (a difference of two x-values) cannot be zero for defining a slope. If h = 0, then it ...
by a.s.h.
Tue Oct 18, 2016 12:21 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

While we can use any arbitrarily small h to get difference quotients, the point is that these difference quotients will approach something, namely, the slope of the tangent (if the function is differentiable). E.g. consider finding the slope at x = 1 for f(x) = x^2. With h = 0.1, the difference quot...
by a.s.h.
Mon Oct 17, 2016 1:31 pm
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

I think I misinterpreted what you were asking. I thought you meant how many numbers are there between 0 and h, for some given h > 0. Now it seems like you're asking how many values in that interval are equal to h. Well the answer to this would be just one: h itself.
by a.s.h.
Mon Oct 17, 2016 5:53 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 116
Views: 9846

Re: Discussion of Integral calculus

How many values of x do you think there could be within the interval, say, from x = 0 to (x + h) = h at x -axis of x,y -coordinate system? There are infinitely many such x-values, and the type of infinity is called an uncountable infinity . This is in a sense bigger than say the total number of who...

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