Calculus & Analysis Marathon & Questions (1 Viewer)

seanieg89 · Jul 2, 2016

Re: First Year Uni Calculus Marathon

Oh my bad, you are totally right. Take g to be the constant function g(t)=b. Then (fog) is identically f(b), which need not be L. This is also a counterexample to my other claim, but my original point about the well defined-ness of the limit of that trigonometric expression at t=0 stands.

InteGrand · Jul 2, 2016

Re: First Year Uni Calculus Marathon

seanieg89 said:
Oh my bad, you are totally right. Take g to be the constant function g(t)=b. Then (fog) is identically f(b), which need not be L. This is also a counterexample to my other claim, but my original point about the well defined-ness of the limit of that trigonometric expression at t=0 stands.

OK. So essentially the limit of that trig. expression, defined with domain D being what you said (basically can just take it to be some punctured neighbourhood of 0 with all points where sin(1/t) = 0 also removed, i.e. remove all the numbers of the form 1/(n*pi)) as t -> 0 will still be 2 (same as the limit of the expression with the change of variables x = sin(1/t)), because for any epsilon > 0, there'll exist a punctured neighbourhood of 0 such that for any t in this punctured neighbourhood and also D (i.e. in the intersection with D, which is indeed clearly non-empty as you explained), the trig. expression will be within epsilon of 2 (i.e. |f(t) – 2| < epsilon)? And is the reason this is the case basically from the change of variables, which is valid to use in this case?

Also, when you said counterexample to your other claim in that quoted post, which claim was this?

seanieg89 · Jul 3, 2016

Re: First Year Uni Calculus Marathon

InteGrand said:
OK. So essentially the limit of that trig. expression, defined with domain D being what you said (basically can just take it to be some punctured neighbourhood of 0 with all points where sin(1/t) = 0 also removed, i.e. remove all the numbers of the form 1/(n*pi)) as t -> 0 will still be 2 (same as the limit of the expression with the change of variables x = sin(1/t)), because for any epsilon > 0, there'll exist a punctured neighbourhood of 0 such that for any t in this punctured neighbourhood and also D (i.e. in the intersection with D, which is indeed clearly non-empty as you explained), the trig. expression will be within epsilon of 2 (i.e. |f(t) – 2| < epsilon)?

Also, when you said counterexample to your other claim in that quoted post, which claim was this?

Yes.

And the reason things are okay here is that the point x=0 is not in the domain of the function f(x)=((1+x)^2-1)/x, so the countable family t near 0 for which g(t)=0 are domain holes (*) of (fog) rather than points at which we could have exceptional values of (fog)(t).

If you look at my counterexample a couple of posts ago, you will notice the key thing it exploits is that the limit definition works with punctured neighbourhoods, so I chose my g to map everything to the puncture. I.e. knowledge of lim_{x->p} f(x) tells us nothing about the behaviour of (fog) if g(x)=p always.

(*) These removable singularities are so rarely thought of as domain holes, we usually fill them in immediately.

If we gave f its natural continuous extension to all of R, and did the same for g. f(0)=2, g(0)=0. Then things work out because the composition of two continuous functions is continuous.

seanieg89 · Jul 3, 2016

Re: First Year Uni Calculus Marathon

InteGrand said:
Also, when you said counterexample to your other claim in that quoted post, which claim was this?

"And yes, I think you can just use this change of variables, in that I claim the following:

Suppose f is a function defined on a set D in R^n, take n=1 if you like.
Suppose E is another subset of R^n and g:E->R^n is another function.

Then fog is a function defined on the g-preimage of D. (If this set is empty then it doesn't really make sense to call g a change of variables lol.)

Now if lim_{x->q} g(x)=p and lim_{x->p} f(x)=L for some q in E, p in D, L in R, then lim_{x->q} (fog)(x)=L."

^ Untrue as stated there.

InteGrand · Jul 3, 2016

Re: First Year Uni Calculus Marathon

seanieg89 said:
"And yes, I think you can just use this change of variables, in that I claim the following:

Suppose f is a function defined on a set D in R^n, take n=1 if you like.
Suppose E is another subset of R^n and g:E->R^n is another function.

Then fog is a function defined on the g-preimage of D. (If this set is empty then it doesn't really make sense to call g a change of variables lol.)

Now if lim_{x->q} g(x)=p and lim_{x->p} f(x)=L for some q in E, p in D, L in R, then lim_{x->q} (fog)(x)=L."

^ Untrue as stated there.

Oh, I thought this was the first claim you were referring to (i.e. not the "other" one). So the one without the "other" was the one about the well-defined limit (which was of course true)?

seanieg89 · Jul 3, 2016

Re: First Year Uni Calculus Marathon

Nomenclature for claims is getting confusing

.

It is indeed true that the limit of f(g(t)) as t->0 is well defined.

InteGrand · Jul 3, 2016

Re: First Year Uni Calculus Marathon

seanieg89 said:
Nomenclature for claims is getting confusing .

It is indeed true that the limit of f(g(t)) as t->0 is well defined.

Ah, OK.

seanieg89 · Jul 3, 2016

Re: First Year Uni Calculus Marathon

And more generally, that if f: D->R, we can make sense of lim_{x->p} f(x) for any p in the closure of D, not just the interior of D.

Of course these limits might not always exist.

Paradoxica · Jul 5, 2016

Re: First Year Uni Calculus Marathon

$$\huge$ ^{\displaystyle \lim_{n \to \infty}} \quad \frac{\displaystyle \sum_{k=1}^n {k^n}}{n^n}\\\\$You are given:$\\\\ \lim_{n \to \infty} \left(1+\frac{z}{n}\right)^n \equiv e^z$

seanieg89 · Jul 5, 2016

Re: First Year Uni Calculus Marathon

Paradoxica said:
$$\huge$ ^{\displaystyle \lim_{n \to \infty}} \quad \frac{\displaystyle \sum_{k=1}^n {k^n}}{n^n}\\\\$You are given:$\\\\ \lim_{n \to \infty} \left(1+\frac{z}{n}\right)^n \equiv e^z$

$S_n=\sum_{k=0}^n (1-\frac{k}{n})^n=\sum_{k=0}^\infty u_{n,k}$

where

$u_{n,k}=0$

for k>n, and

$u_{n,k}=(1-\frac{k}{n})^n$

otherwise.

As $n\rightarrow \infty$ , $u_{n,k}\nearrow e^{-k}$ , the monotonicity of this convergence following from basic calculus / other methods if you prefer.

Hence the monotone convergence theorem applies and

$\lim_{n\rightarrow\infty} S_n= \lim_{n\rightarrow\infty} \left(\sum_{k=0}^\infty u_{n,k}\right)=\sum_{k=0}^\infty \left(\lim_{n\rightarrow\infty} u_{n,k}\right)=\sum_{k=0}^\infty e^{-k}=\frac{e}{e-1}.$

InteGrand · Jul 5, 2016

Re: First Year Uni Calculus Marathon

$\noindent Let $a<b$, $a,b$ real. Let $f\colon \left[a,b\right] \to \mathbb{R}$ be a \emph{continuous} function that satisfies $f(x) \geq 0$ for all $x\in \left[a,b\right]$. Suppose $\int _a ^b f(x)\text{ d}x=0$. Show $f(x)\equiv 0$.$

seanieg89 · Jul 8, 2016

Re: First Year Uni Calculus Marathon

Here is an exercise to exhibit a couple of quirks of high dimensional solids. (Not as directly related to first year calculus courses as most things in this thread, but it is first year level stuff, and it is in the calculus/analysis ballpark.)

a) By generalising MX2 methods of volume calculation, find an expression for the volume of the n-dimensional ball of radius r in terms of the Gamma function

$\Gamma(s+1)=\int_0^\infty x^se^{-x}\, dx.$

b) How does this quantity behave asymptotically as n->inf? Interpret this as a comparative statement about n-dimensional balls and n-dimensional cubes.

c) What happens to |B(r-d)|/|B(r)| as n-> inf? Here B(r) denotes the n-dimensional ball of radius r, and d < r is fixed. Interpret this result as a statement about the asymptotic concentration of mass in high dimensional balls.

d) Show that the the limiting behaviour in (c) can occur even if d(n) depends on n and monotonically decreases to zero. For which power rates of decay d(n)=n^(-p) will this happen?

e) Repeat c) and d), this time for strips about the equator. That is, what can we say about the limiting behaviour of |{x in B(r): |x_n| < d}|/|B(r)|? What does this say about the asymptotic concentration of mass in high dimensional balls? Can we take d(n)->0 and still have the same behaviour? At what power rate can d tend to zero with us having the same limiting behaviour?

dan964 · Jul 8, 2016

Re: First Year Uni Calculus Marathon

seanieg89 said:
Here is an exercise to exhibit a couple of quirks of high dimensional solids. (Not as directly related to first year calculus courses as most things in this thread, but it is first year level stuff, and it is in the calculus/analysis ballpark.)

a) By generalising MX2 methods of volume calculation, find an expression for the volume of the n-dimensional ball of radius r in terms of the Gamma function

$\Gamma(s+1)=\int_0^\infty x^se^{-x}\, dx.$

b) How does this quantity behave asymptotically as n->inf? Interpret this as a comparative statement about n-dimensional balls and n-dimensional cubes.

c) What happens to |B(r-d)|/|B(r)| as n-> inf? Here B(r) denotes the n-dimensional ball of radius r, and d < r is fixed. Interpret this result as a statement about the asymptotic concentration of mass in high dimensional balls.

d) Show that the the limiting behaviour in (c) can occur even if d(n) depends on n and monotonically decreases to zero. For which power rates of decay d(n)=n^(-p) will this happen?

e) Repeat c) and d), this time for strips about the equator. That is, what can we say about the limiting behaviour of |{x in B(r): |x_n| < d}|/|B(r)|? What does this say about the asymptotic concentration of mass in high dimensional balls? Can we take d(n)->0 and still have the same behaviour? At what power rate can d tend to zero with us having the same limiting behaviour?

what definition of the gamma function are you assume. Also is there a difference between $s$ and $n$ ?

seanieg89 · Jul 8, 2016

Re: First Year Uni Calculus Marathon

dan964 said:
what definition of the gamma function are you assume.

The one written in the question.

dan964 · Jul 8, 2016

Re: First Year Uni Calculus Marathon

seanieg89 said:
The one written in the question.

that makes sense.

seanieg89 · Jul 8, 2016

Re: First Year Uni Calculus Marathon

dan964 said:
what definition of the gamma function are you assume. Also is there a difference between $s$ and $n$ ?

s is a variable used only in the definition of the Gamma function, n refers to dimension.

Paradoxica · Jul 9, 2016

Re: First Year Uni Calculus Marathon

Easy exercise. Leave your answer completely in terms of the Gamma function, in as simple a form you can muster.

$\int_0^\infty \left(\sqrt{x^4+1}-x^2\right)\,$d$x$

Paradoxica · Jul 9, 2016

First Year Uni Calculus Marathon

Solve the differential equation without initial boundaries:

$y \ddot{y} = k \dot{y}^2$

seanieg89 · Jul 12, 2016

Re: First Year Uni Calculus Marathon

seanieg89 said:
Related:

$Prove that $\lim_{n\rightarrow \infty }\left(1+\frac{x}{n}\right)^n$ and $\lim_{n\rightarrow \infty }\left(\sum_{k=0}^n \frac{x^k}{k!}\right)$ both exist and coincide for all real numbers $x$.$

seanieg89 said:
$A series $S_n:=\sum_{k=1}^n x_k$ converges as $n\rightarrow \infty$ iff for every $\epsilon >0$ there exists a $N>0$ such that for every $m\geq n\geq N$ we have $|S_m-S_n|<\epsilon.$ (You can prove this if you like, but you don't have to. It just comes from the completeness of the reals.)\\ \\ Using this criterion or otherwise, show that for any $\theta$, the series $\sum_{k=1}^n e^{ik\theta}g(k)$ converges, where $g(k)$ is a sequence of positive real numbers monotonically decreasing to zero.\\ \\ (The case $\theta=\pi$ is the hopefully familiar alternating series test.)$

seanieg89 said:
Here is a question related to a basic tool in the analysis I do.

$$For positive $\lambda$ let $I(\lambda):=\int_\mathbb{R} a(x)e^{i\lambda \phi(x)}\, dx$\\ \\where $a,\phi$ are smooth and $a$ is zero outside of a bounded interval $[-M,M]$.\\ \\ Expressions like this come up a lot in analysis, and we often want to understand how this expression behaves as $\lambda\rightarrow\infty.$ The general intuition is that it gets small because increasing $\lambda$ makes the integrand more oscillatory, so it is small at more values of $x$.\\ \\ 1. Suppose $\phi'(x)\neq 0$ on $[-M,M]$. Show that for each $n> 0$, we can find a positive constant $c_n$ such that $|I(\lambda)|\leq c_n\lambda^{-n}.$ (i.e $I(\lambda)$ decays faster than any polynomial in $\lambda$.)\\ \\ 2. Let $\phi(x)=x^2$. Find an asymptotic expansion for $I(\lambda)$.\\ \\ 3. Suppose $\phi(x)$ has exactly one stationary point in $[-M,M]$ and has nonzero second derivative there. Find an asymptotic expransion for $I(\lambda)$. (What if it has a finite number of such stationary points?)$

$Note: An asympotic expansion in this sense is an increasing real sequence $\rho_j\rightarrow \infty$ and a sequence of reals $b_j$ such that $\lambda^{\rho_{j+1}}(I(\lambda)-\sum_{j=1}^N b_j \lambda^{-\rho_j})$ is bounded for each positive integer $N$. Think of Taylor series but exponents in such an expansion need not be integers, and the formal series $\sum_{j=1}^\infty b_j \lambda^{-\rho_j}$ need not converge to be a valid asymptotic expansion.$

Bumping some unanswered questions of mine. The third is a bit harder than the first two.

seanieg89 · Jul 12, 2016

Re: First Year Uni Calculus Marathon

And here's a new one simpler than all three of the above.

$Suppose there exist positive constants $C,\rho$ such that $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the inequality $|f(x)-f(y)|\leq C|x-y|^{\rho}$ for all $x,y\in \mathbb{R}$.\\ \\ a) For which values of $\rho,C$ is $f$ necessarily continuous?\\ \\ b) For which values of $\rho,C$ is $f$ necessarily differentiable? \\ \\ c) You will notice that this condition on $f$ is global, whilst the definitions of continuity and differentiability are local. (i.e. to check if a function is continuous/differentiable at a point p, we only need to know what it does in an arbitrarily small neighbourhood of p). Can you come up with a local version of this property?\\ \\ d) Relate the property in c) to differentiability.$

Calculus & Analysis Marathon & Questions (1 Viewer)

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