Homework Statement
The notation for a Riemann sum - Ʃ f(x*i)Δx - is very similar to the notation for the integral (the Ʃ becomes ∫, the f(x*i) becomes f(x) and the Δx becomes dx).
\int f(x)dx = \lim_{n \to \infty}\sum_{k=0}^{n} f(x_i) Δx
Is there a way to explicitly define the values on the...
Homework Statement
Generally the derivative has the limit x-- h applied to the whole thing like
$$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$
I'm guessing you can't express it as
$$\frac{\lim_{h\to 0} f(x+h)-f(x)}{\lim_{h\to 0} h}$$
because the quotient rule for limits doesn't hold when...
Absolutely a typo. :) sorry!
You'd be surprised at the quality of the instruction here - for example, one of my teachers didn't know what this
f(x+h)-f(x)
----------
h
lim h--> 0
was.
I understand that the upper limit is x, I just mean that that I would expect the definition of the...
Ok, here's what I meant:
I was looking at ## \int_a^b f(t)dt = \int_0^b f(t)dt - \int_0^a f(t)dt = F(a) - F(b)##
and I've been taught that F(x), or F(a) or so on, in this context, is an indefinite integral. Like this:
$$\int f(x) dx$$
But that didn't make sense to me because I don't...
Homework Statement
I hope this is in the right forum, because this is a question on theory and not related to a specific problem.
I was reading onlne about the Fundamental Theorem of Calculus. On one site the author wrote:
F(x) = \int_{0}^{x} f(t) dt
Later, he wrote:
\int_{a}^{b}...
Thank you both for your answers :)
I still haven't really managed to explain what I'm talking about...I'll have a look at related threads in the forum to see if I can clarify.
I think you misunderstood me, I probably haven't been using the correct terminology.
I dug up my calculus book, and I think what I'm thinking about is a step function.
In the book, it says the following:
"...if s(x) = c for all x in the closed interval [a,b], the ordinate set of s is a...
Homework Statement / The Attempt at a Solution
I know that given f(x)=c, the integral from a≤x≤b is c(b-a) (at least, I hope I know that! :D).
Is the integral the same value if you don't include an endpoint? That is, if you were evaluating f(x) from a≤x<b?
Intuitively I think it both...
I'm familiar with the A-Level system, although I haven't taken Economics.
I think that other than possible A-Levels universities will require in order to do physics/engineering (I imagine Math/Further Math, Physics and Chemistry will more than satisfy any prerequisites, although I haven't...
Jorriss - Thanks. I was worried that you have to do many classical mechanics courses in order to learn quantum mechanics - an intro course and an advanced course doesn't sound as bad as I thought.
Oh, I actually am looking forward to Thermodynamics/Stat. Mechanics :) When I said I was less...
A while back I read that, roughly speaking, these are the main topics that make up the backbone of a physicist's education: classical mechanics, electromagnetism, statistical mechanics/thermodynamics, special relativity, quantum mechanics, and general relativity.
(If that's incorrect, please...
You should probably know that, even if you take "genius" to be a construct that is equivalent to high IQ percentile (the concept is more complex, for instance it encompasses cognitive traits and not merely cognitive capacities), it is incorrect to state that IQ above 140 (or 145, or 135, or...
This I did learn.
I was never explained this, nor was I *formally* explained the definition of the integral as an anti-derivative (so you can see how I don't understand this, for example:
)
I know I have huge gaps in my knowledge. Basically, I don't know how it works in the states, but...
I appreciate your replies - and the bits I understand have been really helpful - but I haven't learned this rigorously enough to understand other parts.
I obviously need to find a good text on the subject because there are simply too many issues here that I haven't learned (I'm in a...
HallsofIvy, could you explain a bit about the undetermined constant? I haven't been taught this rigorously. Why is there always a constant?
Does what chiro talked about imply
\int_{b}^{c} f(x) = \int_{0}^{c} f(x) - \int_{0}^{b} f(x) ?
Sorry for asking question after question, thinking about...
Thanks :)
That's interesting chiro - in what kind of problems would you use a non-zero initial condition?
This sort of leads me to my second question:
What I want to evaluate the integral of cos(x) at x=0. I'd have to set up a definite integral
\int_{a}^{b} cos(x) dx
where both a...
Homework Statement
This isn't a specific problem ;)
Homework Equations
The Attempt at a Solution
My thought process, skip to the last paragraph if you want the question:
I was wondering how you compute an integral at a specific x value, lets say c. Based on the fact that the...
I'm new to this so I'm probably wrong, but here are my two cents:
I'm not sure splitting the limit like that works.
limit (1/2^n-1)/(1/2^n+1) doesn't equal limit (1/2^n)/(1/2^n) + limit (-1/1).
(x+a)/(x+b) doesn't equal x/x + a/b, it equals x/(x+b) + a/(x+b).
(also if your way is correct the...
Thanks, RGV :)
Anyway, here goes with the related issues this made me think of:
1. Is there any way to formally define or quantify "how fast" the limits approach a point? (I realize "how fast" is terribly inexact, I just don't know how to express it better). Does it depend on the...
Thank you everybody so much for all this help and the fantastic explanations! You guys really really helped me out a lot, and I understand things much better now.
This made me think of a few other related issues -can I post them to this thread and continue the discussion or do I need to make...
About what HallsofIvy said about how fast the functions get to a point -
If I take the two expressions we've been talking about, (x-1) and (x/2-1/2) that both equal 0 at x=1, and check their value close to 1, say at x = 1.001 I get:
x-1 = 1.001 - 1 = 0.001.
x/2 -1/2 = 1.001/2 -1/2 =...
OK, now I think I get it.
If I have limit x-> A f(x), and limit x-> A g(x), it's not relevant that at A both limits have the same value, because the values of f(x) as the function approaches A are different from the values of g(x) as g(x) approaches A, and we're looking at values approaching...
Ok, D H, you mean that because the expressions (x-1) and (x/2-1/2) are not equal, even if I prove that in the limit they are equal, I can't cancel them?
(You say they aren't equal. Surely in the limit they are equal? You can evaluate each one separately and show that)
Again, sorry if...
Thank you both for such detailed replies! I understand some ideas much more now.
But I still don't fully understand why I'm not allowed to cancel the two terms in the second limit.
It's like in the first limit I gave, you can cancel the two (x-5)'s because:
a) Even though they approach...