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

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447 views

How to find probability distribution function given the Moment Generating Function

After searching, I found two questions like mine, but didn't see my answer to my question. Finding a probability distribution given the moment generating function Finding probability using ...
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43 views

Does convergence of $(a_{5n})$ and $(a_{n+1} - a_n)\rightarrow0$ imply convergence of $a_n$

As I wrote in the title: Does convergence of $(a_{5n})$ and the fact that $(a_{n+1} - a_n)\rightarrow0$ imply convergence of $a_n$. I understand that convergence of $(a_{5n})$ means that $(a_n)$can ...
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0answers
49 views

Proof of l'Hôpitals rule, g'(x)=0

l'Hôpitals rule for limits where $\mathbf{x\to 0}$. Let $f$ and $g$ be continuously differentiable in the neighborhood of $x=0$. Also, let the functions and their derivatives be defined in the ...
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43 views

Convergence of a integral

The question is: exists a natural number $n \geq 2$such that $$ \displaystyle\int_{0}^{+ \infty} \displaystyle\frac{\ln r}{(1 + r^2)^{n}} r dr< \infty ?$$ I am trying to do this : i know that ...
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45 views

Why does the moment of approximation matter for the end result?

I am trying to wrap my head around the reason why the moment of approximation matters for the end result of my analysis. As an example, let's take an equation for which we can still find the full ...
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851 views

Finding the upper and lower Riemann-sums of trigonometric functions

I am asked to find Riemann-sums for the function $f(x) = \cos x, x \in [0, 2\pi], n = 4 \rightarrow \Delta x = \frac{\pi}{2}$ I was able to get the correct answers by intuition, but I have some ...
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81 views

Is my calculation of $\frac{\partial}{\partial x}\frac{x+y}{\sqrt{y^2-x^2}}$ correct?

Is my calculation of the partial derivative (with respect to $x$) of the function $$f(x,y)=\frac{x+y}{\sqrt{y^2-x^2}}$$ correct? ...
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47 views

An Integral which when evaluated leads to a better bound on a familiar constant. What is the bound and the constant?

An Integral which when evaluated leads to a better bound on a familiar constant. What is the bound and the constant? $$ \frac{1}{3164}\int_0^1 \frac{x^8(1-x)^8(25 + 816x^2)}{1+x^2} \ dx $$
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42 views

Finding the angle between vectors

What is the difference between $cos(\theta) = \frac{(v \cdot u) }{( \|v\| \|u\| )}$ and $cos(\theta) = \frac{|(v\cdot u)|}{( \|v\| \|u\| )}$ I noticed that my textbook uses simply the value of the ...
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168 views

Text with alternative definition of “derivative”?

Instantaneous rates of change are conventionally defined as limits of difference quotients. Rates of things moving at constant speed are definable without delicate issues. If I pass someone moving ...
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82 views

Working with projection of areas?

I was recently solving a physics problem which had to do with the momentum imparted by a photon beam to a perfectly absorbing sphere and a perfectly reflecting one. Considering the former and Putting ...
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86 views

When should one use the two-point compactification of $\mathbb R$?

The real line $\mathbb R$ has a one-point compactification $\mathbb R\cup\{\infty\}$, where this "$\infty$" is at both ends of the line, so that the compactification is topologically a circle. It ...
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119 views

Multivariable calculus: optimizing for shortest path along a curvy plane?

I want to write a computer program which can help me spend the least amount of energy and time walking between locations on my university campus. My campus is very hilly, and it is also extremely hot ...
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56 views

partial derivative notation question

I'm reading a book called Correlated Data Analysis, Analytics, and Applications and I simply don't understand some notation. The author says, in chapter 2, page 26: A unit deviance is called ...
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99 views

Physical Significance of normalised and non-normalised function?

$\dfrac {\sin(x)}{x}= \rm{sinc}(x)$ unnormalized \rm{sinc} function And for the normalized sinc we have: $\dfrac {\sin(\pi x)}{\pi x}=\rm{sinc}(x)$ normalized \rm{sinc} function Is there any ...
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38 views

Attach term to solution of PDE(perturbation theory)

Currently I am struggeling with the following problem: Actually I have a found a solution to the PDE $\Delta \Phi(r,\theta)=f(r,\theta)$ and now I want to include a small extra term given by ...
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82 views

How to find function satisfying a known sum over it?

I've got the following (simplified somewhat) situation; $f(x) = A(e^{-x/B} - e^{-x/C}) = \sum\limits_{n=x}^{x+f(x - 1)D} p(n)$ Is there a way to find $p(n)$? Math is not my specialization and ...
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65 views

Is this answer sufficient to prove? The question is related to second partial derivatives.

Is this answer sufficient to prove ? Does there exist a notation mistake or else? Problem Suppose that the functions $\varphi: \mathbb R \rightarrow \mathbb R$ and $\psi: \mathbb R \rightarrow ...
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113 views

Spivak Calculus on Manifolds Exercise 2-9

I'm kind of stumped on this exercise in two spots. First I'll state the problem: Two functions $f,g : \mathbb{R} \to \mathbb{R}$ are equal up to $n$th order at $a$ if $$\lim_{h \to 0} \frac{f(a + ...
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68 views

Integration by Parts for PDEs

I'm reading a paper on PDEs in preparation for some research. In it, integrals like this appear repreatedly: $$ I(x,t) = \int_{|y|>1} e^{-i\alpha xy} \frac{e^{i\beta y^2}}{(1+y^2)^m} dy. $$ Here ...
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131 views

Transversality condition equation

I'm somewhat baffled: I have a problem in calculus of variations: $$ \int_0^T \!(x-\dot x^2)dt,\qquad x(0)=0,\qquad x(T)=T^2-2. $$ Let $ F(t,x, \dot x) =x-\dot x^2. $ I calculate all the ...
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56 views

Limit of sequence, sequence is formed by the root of an equation.

$$f_n:[0, \infty) \to \Bbb R, \ f_n(x) \ = x^n-nx-1, \ \forall n \ge2$$ $$f_n(x)=0$$ Has one root, $$X_n \in (1, \infty)$$ Calculate $$\displaystyle \lim_{n \to \infty} X_n$$ $$f_n(X_n)= ...
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36 views

Laplacian of a generalized Whitehead potential

I'm trying to calculate the Laplacian $\triangle\Phi_W^{IJ}$ of the following "generalized Whitehead potential": $$ \Phi_W^{IJ}(\vec{x}) = ...
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124 views

For what values of x does it converge?

How can i study the character of the following series? $$ \sum_{n=1}^\infty\,\,\frac{x^n(\sin1\cdot\sin2\cdot ...\cdot\sin n)^2}{(1+x\cos^21)\cdot(1+x\cos^22)\cdot...\cdot(1+x\cos^2n)},\qquad ...
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45 views

Need $f:[0,\infty)\to[0,\infty)$ such that $f$ is not convex but $f(x^p)$ is for $p>1$.

To be more specific I need to find an $f:[0,+\infty)\to[0,+\infty)$ which satisfies the following: (somewhat trivial stuff) The function $f$ is continuous, nondecreasing, there exists $k>0$ such ...
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163 views

Sketching the graph of a function

How would I sketch the graph of the following function. Sketch the graph of a function satisfying the following conditions $f(1)=2$ $f(-1)=f(3)=-1$ $f'(1)=0$ $f'(x)>0$ for all $x<1$ ...
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278 views

How to calculate the asymptotic expansion of $\sum \sqrt{k}$?

Denote $u_n:=\sum_{k=1}^n \sqrt{k}$. We can easily see that $$ k^{1/2} = \frac{2}{3} (k^{3/2} - (k-1)^{3/2}) + O(k^{-1/2}),$$ hence $\sum_1^n \sqrt{k} = \frac{2}{3}n^{3/2} + O(n^{1/2})$, because ...
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43 views

bound for $a,b,c$ in $\mid ax^2+bx+c \mid \leq 1\;\forall x\in \left[0,1\right]$

If $\mid ax^2+bx+c \mid \leq 1\;\forall x\in \left[0,1\right]$ and $a,b,c\in \mathbb{R}$. Then Prove that $|a|\leq 8\;\;,\mid b \mid \leq 8$ and $\mid c \mid \leq 1$ My Try:: Put $x = 0$ in ...
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119 views

How can I find a maximal inscribed ellipsoid to a *concave* set of points, in 3D?

I have a set of points which describe the surface of an irregular, natural (i.e., occurs in nature) object. This point set is not necessarily convex, and contains occasional indentations so parts of ...
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134 views

Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$

Being inspired by this post, I've wondered if the infinite series below may be expressed as an intregral. I'm very curious about that. $$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$ ...
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200 views

Second derivative of a vector field

I wonder how to treat the "second derivative" of a vector field. For example, imagine we have a vector field $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$. Then we evaluate the derivative at two points ...
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73 views

Did I set up this Newtons Method problem correctly?

First off I want to say that this is homework. So please don't give me the answer. I'm just looking for guidance and confirmation of whether I have the right idea. Please forgive me as I don't know ...
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225 views

Heaviside unit step- and delta function

The following question is right from the book: Show that $$ H(x-x_i) = \int_{-\infty}^x \delta(x_0-x_i)dx_0\, $$ satisfies $$ H(x-x_i) \equiv \begin{cases} 0 & x < x_i \\ 1 ...
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82 views

The Limit of the Following Derivative

Suppose you have two functions $F$ and $G$ with the following properties. $G(0)=F(0)=0, G'>0, G''<0, F'>0, F''<0 $ and also $\lim_{x\to0} F'(x)=\infty, \lim_{x\to\infty} F'(x)=0, ...
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165 views

evaluate the sum of the following series

I have tried for a long time on this: $$\sum_{n=1}^{\infty}\ln\left(1+\frac{1}{n}\right)\ln\left(1+\frac{1}{2n}\right)\ln\left(\frac{1}{2n+1}\right)$$ Even I have no idea whether it could be ...
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271 views

Minimizing a functional definite integral

I have a definite integral defined by $$T\left(G\left(g\right)\right)=\int_{g_{1}}^{g_{2}}G(g)\mathrm{d}g$$ where $G$ is a continuous function of a variable $g$, and $g_{1}$ and $g_{2}$ are known ...
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132 views

Continuity criteria for Radon-Nikodym derivative

I have been looking for results or theorems which give me regularity conditions of the Radon-Nikodym derivative, but I have not found any :( For instance, we know that if $\nu\ll\mu$ then there ...
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152 views

Cauchy criterion

We know that the series $\sum_{n=1}^\infty \dfrac{1}{n(n+1)}$ coverge to 1. I want use Cauchy criterion to show the series converges as an exercise. Is the following proof correct? Given $\epsilon ...
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809 views

Good introductory book for matrix calculus

Hi I am an electronics graduate and working on image processing for the past one year...I have a basic exposure to linear algebra(thanks to Gilbert Strang..!!!). Now I am facing problems with matrix ...
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73 views

Monotonicity of a discrete function

Let $k_1+k_2=k$, where $k, k_1$ are all positive integers with $k_1 \ge 1$. Also let $K=\min\{k_1, \lfloor k_2/9 \rfloor +1\}$. Define $g(x)=\max\{1 \le i \le k: \lfloor i/9 \rfloor +1=x\}, x=1, ...
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247 views

system of implicit nonlinear differential equations

Here I have a system of nonlinear differential equations: $ (M+2m)\ddot{x} + m(l_1 \ddot{\theta}_1\cos\theta_1 - l_1\dot{\theta}_1^2\sin\theta_1) + ...
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179 views

Divergence Theorem to prove equality of integrals

I'm trying to wrap my head around this problem - the interplay between $\nabla$ and $\Delta$ is doing my head in. It says to use the divergence theorem. Prove that $$\int_\Omega u \cdot \Delta v\, ...
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217 views

Global optimum of sum of convex functions

Take two real differentiable convex functions, $f_1$ and $f_2$, defined on the unit interval $[0; 1]$. I want to find the global optimum of: $\min_{x \in [0;1]} af_1(x)+bf_2(x)$, for given $a, b \in ...
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303 views

Calculate the integral $xF(x) \, dx$, between $0$ and $1$

Where, $F(x) = \int^x_0 t\cosh(t^4)\,dt$ This is a 2 part question: (a) calculate $\int^1_0 xF(x)\,dx$ (the answer will involve $F(1)$) (b) $ \lim_{x\to0}\frac{F(x)}{x^2}$ For part (a) I am ...
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260 views

differentiable square root of nonnegative smooth function

Suppose I have a smooth function $f(x):\mathbb{R}\to\mathbb{R}\geq 0.$ Does there always exist a differentiable $g(x):\mathbb{R}\to\mathbb{R}$ with $g(x)^2 = f(x)$? If so, clearly $g(x) = ...
3
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0answers
282 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 ...
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209 views

Properties of the lemniscate functions as meromorphic functions on $\mathbb{C}$

We consider the following function. $$u(x) = \int_{0}^{x} \frac{dt}{\sqrt{1 - t^4}}$$ $u(x)$ is defined on $[-1, 1]$. Since $u'(x) = \frac{1}{\sqrt{1 - x^4}} > 0$ on $(-1, 1)$, $u(x)$ is strctly ...
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271 views

Is there an example of using L'Hospital's Rule on a product where it doesn't work?

I was reading that, when trying to solve something like: $$\lim_{x\to\infty} f(x)g(x)$$ I can rewrite is as: $$\lim_{x\to\infty} \frac{f(x)}{\frac{1}{g(x)}}$$ and use L'Hospital's Rule to solve. ...
3
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0answers
152 views

Evaluating $\int \limits_{a}^{\infty} \frac{\exp\left(-ax\right)}{\log(x)\left(c+x\right)^2}dx$

I have the following integral $$\int \limits_{a}^{\infty} \frac{\exp\left(-ax\right)}{\log(x)\left(c+x\right)^2} dx$$ that I do not know how to evaluate. Could you please give me a hint? Thanks in ...
3
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228 views

Sign of derivative of a complicated function

EDIT (for bounty): Consider the differential equation $G(p;x,\lambda)p \left[1-\lambda-x(1+\lambda)\right] + x(1+\lambda)p + (1-x)(1-\lambda) \int_{p}^{1} z G'(z;x,\lambda) dz - (1-\lambda) = 0$, ...