For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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4
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158 views

How many points does one need for an epsilon-net

Does anyone know, how many points does one need to have an $\varepsilon$-net on a unit sphere sitting in the three-dimensional Euclidean space? Thanks!
4
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145 views

When are sums and integrals “identical” in form?

In the answer to this question Eric Näslund showed that logarithms can be written as the following limit of a sum: $$\displaystyle \log(x) = \lim_{k\to \infty } \, \sum\limits_{n=k}^{x k} ...
4
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415 views

Lebesgue Integration fundamental questions

My question involves the definition of the Lebesgue integral. Most colloquial definitions I've read follow (2), in that f*(t) is the "length" of one of the horizontal rectangles and dt is the ...
4
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0answers
217 views

Re: Rain droplets falling on a table

This questions is almost exactly similiar to the the following question, with an extra condition : Rain droplets falling on a table Suppose you have a circular table of radius R. This table has ...
4
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451 views

Fastest convergence Series which approximates function

The question is the following: Is there any proof that shows that the Taylor series of an analytical function is the series with the fastest convergence to that function? The motivation to this ...
4
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0answers
273 views

How can I prove this inequality?

I have a pretty nasty looking function $$\sigma (t,y) = \sqrt{\frac{\sum_{i=1}^N \lambda_i \sigma_i \exp \left (-\frac{1}{2 t \sigma_i^2}\left(\ln{\frac{y}{S_0}} - \left(r - \frac{\sigma_i^2}{2} ...
4
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0answers
213 views

Understanding surface area of a revolution/length of curve

I don't quite understand why the formula to find the surface area of a revolution is what it is: $$A = 2\pi \int_a^b x\ \sqrt{1 + \left(\frac{\text{d}y}{\text{d}x}\right)^2}\ \text{d} x.$$ I ...
4
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427 views

Find roots of sum of sinusoids

Given this function and an initial point, find the next root: $$ \begin{align} f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - ...
4
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166 views

Help with removing singularities involving $ \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}$

This post can be thought of as the prototype proof and the motivation for the question posted here Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion ...
4
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0answers
116 views

Why is this change of variables true?

I have an equation $F_{xx}+yF_{yy}+{1\over 2}F_y=0$ defined on $y<0$. I found that the characteristics are $\alpha={2\over 3}(-y)^{3\over 2}-x,\,\,\,\,\,\beta={2\over 3}(-y)^{3\over 2}+x$ and that ...
4
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1k views

Integral of the derivative of a function of bounded variation

Let $f\colon [a,b] \to \mathbb R$ be of bounded variation. Must it be the case that $|\int_a ^b f' (x) |\leq |TV(f)|$, where $TV(f)$ is the total variation of $f$ over $[a,b]$? If so, how can one ...
4
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264 views

Computing complex principal value integral - sgn-function?

I currently face a less appealing integral which emerged computing the expectation of some random variable. It reads as (omitting all unnecessary constants except $\alpha\in(0,1)$) $$ PV ...
4
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140 views

Tricky integral $\int_{a}^{b}\frac{\gamma d \gamma}{\gamma + \phi_{1}(\mu)-e^{-\frac{\phi_{2}(\mu)}{\gamma}}}$

in this integral $a=\psi_{1}(\mu), \ b=\psi_{2} (\mu)$. I expanded the function in Taylor series (3 terms) around ($\gamma= \frac{b}{2}$), numerically (for varioud values of $\mu$, and other constants ...
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39 views
+50

solution uniqueness of an algebraic system

$A(v),B(v),C(v)$ are positive, convexly decreasing functions on $\mathbb{R_+}$; $x$ is a random variable that obeys distribution F; Function $v(a,b,x)$ is implicitly defined as ...
3
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54 views

What is the “$\cdot$” in this: $\left\langle \nabla f,\:\cdot\right\rangle$

I've read the Wikipedia page on exterior derivatives, and it states the following: \begin{align} df&=\sum_{i=1}^n\frac{\partial f}{\partial x_i}dx_i=\left\langle \nabla ...
3
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48 views

Proving a strictly decreasing sequence which tends to zero is positive

Suppose $(a_n)$ is a strictly decreasing sequence such that $a_n\underset{n\to\infty}{\rightarrow}0$. I'm asked to prove that $(a_n)$ is positive. My approach: suppose there is a negative element ...
3
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63 views

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) $

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) = ?$ By integrating I am getting $$y = \ln (x+1)+C$$ I am stuck somewhat as it looks tricky from here. Any help ? Thanks!
3
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60 views

About the closed form for $\lim_{y\to +\infty}\left(-\frac{2}{\pi}\log(1+y)+\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx\right)$

Recently, when facing a baby Rudin's exercise, I proved that: $$ \int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx = \frac{2}{\pi}\log(1+y)+O(1) $$ holds by integration by parts. Now I wonder if ...
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780 views

A difficult integral (expectation of the function of a random variable)

For $H>L$ , $p,q,\alpha,\beta>0$, and B(.,.) the beta functon, trying to solve this integral: $$\mathbb{E}(X)=\frac{\alpha H }{\beta B(p,q)}\int_0^H \frac{x \left(\frac{-H \log ...
3
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26 views

Convergence range of $\sum_{n=0}^{\infty}\frac {(-1)^n3n}{(x-4)^n}$

Find the range of convergence of $\displaystyle\sum_{n=0}^{\infty}\frac {(-1)^n3n}{(x-4)^n}$ Finding the radius first, setting $t=x-4$, $R=1/|((-1)^n3n)^{1/n}|=1$ So $\mid \frac 1 ...
3
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18 views

Spivak Calculus Chapter II Exercise 22 :: Arithmetic mean and geometric mean

the exercise says: If $a_1,\ldots,a_n\ge0$ then the A-M is $A_n=\frac{a_1+\cdots+a_n}{n}$ and G-M is $G_n=\sqrt[n]{a_1a_2\cdots a_n}$ we would like to show that $G_n\le A_n$ (1) suppose that ...
3
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104 views

Visualising surface integrals

For a current problem I am working on, I have run into angular surface integrals, i.e. the differential solid angle $\text{d}\Omega$. Specifically the surface integrals are defined by ...
3
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0answers
1k views

A difficult integral

For $\gamma>0,\delta>0$, trying to evaluate this integral: $$ I=\int_0^H\frac{e^{i t x} \log\left(\frac{H}{H-x}\right) ^{\frac{1}{\gamma }-1} \left(\left(\frac{k}{H \log ...
3
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0answers
47 views

How can we show that the functions are differentiable?

Show that the following functions $$f(x, y)=\frac{xy}{\sqrt{x^2+y^2}} \\ f(x, y)=\frac{x^2y}{x^4+y^2}$$ are differentiable at each point of the domain. Determine which of them is $C^1$. $$$$ The ...
3
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77 views

Prove a simple relation involving the function $f(x)=\sum_{n=1}^{\infty}\frac{\sin(n x)}{n^2}$

Prove that the following identity holds: $$ 16f\left(\frac{\pi}{4}\right) + 16f\left(\frac{3\pi}{4}\right)= 27 f(\alpha) -9 f(2\alpha) + f(3\alpha)$$ where $\alpha = 2\arctan\left(\sqrt{2}\right)$
3
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0answers
38 views

Chain rule for the curl of a vector-valued function

I am looking for a vector expression for the curl of a composite vector-valued function. In other words $$ \nabla\times\mathbf{A}(\mathbf{B}) = ? $$ In indicial notation, this can be ...
3
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0answers
21 views

Equivalent Definition Using Limit of Continuity

Definition : Note that $f: {\bf R}\rightarrow {\bf R}$ is continuous at $x_0$ if $$ \lim_{x\rightarrow x_0} f(x)= f(x_0)$$ And $f$ is continuous on $(a,b)$ if $f$ is continuous at any $x_0\in ...
3
votes
0answers
43 views

Multivariable Calculus - Stokes' theorem and conservative fields - showing that a vector field does not have a potential on a domain.

I understand that the curl of a vector field $\textbf{F}$ being zero means that the field is conservative if its domain is simply connected. This was demonstrated in part a where I showed that the ...
3
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0answers
94 views

Graphs of interesting integrals of the form: $\int \sin^a(x^a)\cos^a(x^a)$

Here are a few graphs of the form:- $$\int \sin^a(x^a)\cos^a(x^a)dx$$ Where $a$ is an even, positive integer. $a = 2$ $a = 4$ $a = 6$ Now, a few graphs of the form:- $$\int ...
3
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51 views

Is there an explicit expression for this integral?

Considering $$ I(z,a)=\int_{0}^{\infty}\frac{r}{z+r^{2\beta}}\sin{(ar)}dr,~~~a>0, $$ where $0<\beta<\frac{3}{2}$, $z=e^{i\theta}$. My question is that can we compute this integral explicitly ...
3
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0answers
41 views

Analytical solution for volume when a plane cuts a hemi-sphere

I need to find the analytical solution when the plane $ P: z = grad\cdot y + z_{cut} $ cuts the hemi-sphere $ S: x^2 + y^2 + z^2 = r^2;\:y \leq 0 $. I constructed two 3D images in MatLab of the ...
3
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0answers
47 views

Prove that there is an interval $[a, b] ⊆ I$ such that $f$ coincides with a polynomial there.

Let $f ∈ C^{\infty}$ satisfy the following property: To each $x ∈ I=[0,1]$ there corresponds a finite integer $N_x$ such that $f^n(x) = 0 $for all $n ≥ N_x.$ Prove that there is an interval $[a, b] ⊆ ...
3
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0answers
60 views

Help me to figure out this integral

$$\int_{0}^{\infty} \frac{x^3\sin\left(\frac{1}{2}\pi x\right)}{e^{2\pi\sqrt{x}} - 1}~dx$$ I've been thinking a long time ,but I have no idea how to do it.
3
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0answers
62 views

Understanding why $\int_{-a}^a \sin^{100}x\,\mathrm dx = \frac{99}{100} \int_{-a}^a \sin^{98}x\,\mathrm dx$

In an answer here on Math.SE it is claimed that $$\int_{-a}^a \sin^{100}x\,\mathrm dx = \frac{99}{100} \int_{-a}^a \sin^{98}x\,\mathrm dx$$ but I don't understand how it could be. Since $$ ...
3
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0answers
58 views

Prove that a sequence $a_n$ converges if and only if $a_n^3$ converges

I want to prove that $A\ sequence\ a_n\ converges\ \longleftrightarrow\ a_n^3\ converges$ If $a_n$ converges, then by arithmetics, $a_n^3$ converges. Now let $a_n^3$ converge to a real $L$. ...
3
votes
0answers
87 views

Definite integral involving arctangents

Compute the integral: $$ \int_1^2 \frac{\arctan x}{\arctan(x-1) -\arctan(x-2) }\,\mathrm{d}x. $$ It is an exercice given in Calculus book and I've tried a lot of methods (Wolfram was the first) to ...
3
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0answers
33 views

Continuity of third derivative in extremum test

Consider the following standard real analysis textbook theorem: Let $I$ be a perfect interval, $f\colon I \to \mathbb{R}$ be $C^3$ (i.e. three times differentiable and $f'''$ continuous). If $x_0 ...
3
votes
0answers
44 views

Center of Mass of objects with infinite length?

Suppose you have $f(x) = \frac{\sin(x)}{x}$ And you have that shape, find the center of mass of $f(x)$ in $x \in (-\infty, \infty)$ Is it possible considering $f(x)$ is an even function?
3
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0answers
57 views

Using the Intermediate Value Theorem to prove a statement about an equation true

I want to prove this statement true by using the IVF: For any real number $b > 2$, the equation $2^x = bx$ has a solution. Here are some questions I need help with answering: Define a function ...
3
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0answers
68 views

how to calculate this line integral $\int_{0}^{2\pi} (16\sin^2 3t +16\cos^2 4t)\sqrt{(144\cos^2 3t +256\sin^2 4t)}dt$

I am working on a line integral to calculate the amount of chocolate to cover a pretzel. the density of the pretzel is given by this formula $\lambda=3(x^2+y^2)$ and the parameter equation of a ...
3
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0answers
39 views

Multi-variable function is undefined at every point, then limit still may exist

The following question was posed; If a multi-variable function $f(x,y)$ was undefined at every point on a curve, then could a limit exist of a point $(x_0, y_0)$ on this curve for this function? i.e ...
3
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0answers
31 views

Verification of $\frac{d}{dt}(\hat{v}\cdot\hat{v})=0$

I would like to verify if the following is actually true: $$\frac{d}{dt}(\hat{v}\cdot\hat{v}) = \frac{d}{dt}\left|\left|\hat{v}\right|\right|^2 = 0$$ My thought on this is that since $\hat{v}$ is a ...
3
votes
0answers
79 views

How to find the minimum $m$ for a given $n$ in this inequality?

For a given $n \in \Bbb N$, how do you find the minimum $m \in \Bbb N$ which satisfies the inequality below? $$3^{3^{3^{3^{\unicode{x22F0}^{3}}}}} (m \text{ times}) > ...
3
votes
0answers
46 views

How prove this indentity with $\sum_{0\le i\le l,0\le j\le m,0\le k\le n}\frac{f(a_{i},b_{j},c_{k})}{N(a_{i},b_{j},c_{k})}=1$

Question: Let $$f(x,y,z)=x^ly^mz^n+\sum_{s=1}^{t}A_{s}x^{p_{s}}y^{q_{s}}z^{r_{s}}$$ where $l,m,n$ is postive integers,and $A_{s}(1\le s\le t)$is real numbers,$p_{s},q_{s},r_{s}(1\le s\le t)$ ...
3
votes
0answers
63 views

Differentation uder the integral sign

Let $F(x)=\int_{\sin x}^{\cos x} e^{x\sqrt{1-y^2}} \, dy $. My task is to calculate $F'(x)$. My idea is to use http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign and I get: ...
3
votes
0answers
56 views

Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
3
votes
0answers
73 views

Do mathematicians consider functional integration to be good mathematics? If so, what is it?

I am trying to learn about path integration in physics and, from what I understand, this means learning to do functional integrals. (Note: I know what a functional is and I also have done functional ...
3
votes
0answers
107 views

Prove that $\left\vert\prod_{k=1}^{n}{\sin (k)}\right\vert\leq\prod_{k=1}^{n-1}{\sin \left(\frac{k\pi}{n}\right)}$

Prove that $$\left\vert\prod_{k=1}^{n}{\sin (k)}\right\vert\leq\prod_{k=1}^{n-1}{\sin \left(\frac{k\pi}{n}\right)}\quad\forall n\in\mathbb{N}\backslash\{1\}.$$ Please show all passages and what ...
3
votes
0answers
27 views

Correct integral to compute volume of a solid

I want to compute the volume obtained by rotating the bounded region of $y=3-x^2$, $y=2x$, $x \leq 0$ around the $y$-axis. I want to use the cylindrical shell method, so my integral is: ...
3
votes
0answers
103 views

Integrate $\int\tan (x)\sin (ax)dx$

$$\int\tan (x)\sin (ax)dx$$ If $a$ integer number I used the wolfram.Alpha.com site to give me the following How can I use this result if I need to compute some values by using a program in ...