Search found 104 matches

by a.s.h.
Thu Feb 09, 2017 7:09 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 106
Views: 4057

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: 106
Views: 4057

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: 106
Views: 4057

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: 106
Views: 4057

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: 106
Views: 4057

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: 106
Views: 4057

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: 106
Views: 4057

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: 106
Views: 4057

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: 106
Views: 4057

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: 106
Views: 4057

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...
by a.s.h.
Sat Oct 15, 2016 1:24 pm
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 106
Views: 4057

Re: Discussion of Integral calculus?

Dear a.s.h. and Joe, I think that some wrong impression that I do not understand your understanding of the limits could appear. Of course, I know all these conceptions more or less well. Here, I offer somehow another view and try to prove it. It would be interesting to know which exactly understand...
by a.s.h.
Fri Oct 14, 2016 4:50 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 106
Views: 4057

Re: Fundamental paradox of Integral calculus?

If the difference between the x-values was 0, we'd get 0/0 for the difference quotient, which is indeterminate. So we instead look at a limit as the difference between the x-values (h) tends to 0.
by a.s.h.
Wed Oct 12, 2016 4:30 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 106
Views: 4057

Re: Fundamental paradox of Integral calculus?

We cannot use the limit quotient rule here, since it results in a zero denominator. The quotient rule (i.e. the rule lim_{x->a} (A(x)/B(x)) = (lim_{x->a} (A(x)))/(lim_{x->a} (B(x)))) can only be used for limits where the denominator does not have 0 as limit, i.e. where lim_{x->a} (B(x)) ≠ 0.
by a.s.h.
Wed Oct 12, 2016 4:10 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 106
Views: 4057

Re: Fundamental paradox of Integral calculus?

a.s.h., Do you suppose that the limits rules for functions can be applied for h in the definition of the derivative? Thank you. Which rules are you referring to? Remember, we can't move the limit into the numerator and denominator, because this results in 0/0, and so the limit quotient rule doesn't...
by a.s.h.
Wed Oct 12, 2016 3:58 am
Forum: Physics Questions
Topic: Discussion of Integral calculus
Replies: 106
Views: 4057

Re: Fundamental paradox of Integral calculus?

Boris Lagutin wrote:a.s.h.,

Do you think that h is a variable or a function in the definition of the derivative (limit): F'(x) = (F(x+h) - F(x)) / h , as h goes to zero?

Thank you.


It's a dummy variable for a limit.

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