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

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1answer
103 views

Find the product $\prod_{n=1}^{\infty} (1+x^{2n-1}) $

Find the product $\prod_{n=1}^{\infty} (1+x^{2n-1}) $ , for example in case $|x| \le 1$
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1answer
282 views

How to find the delta of a graph with a limit that approaches infinity?

I thought that in order to find $\delta$, given $\epsilon$, you would need to first subtract epsilon from the limit. How would you do that if the limit is infinity? The exact problem is $(2x+4)^{-1}$ ...
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0answers
44 views

Taylor expansion of an integral in spherical co-ordinates

I've some difficulty deriving this equation from jackson electrodynamics (The equation after 1.30) $\nabla^2 \Phi_a\left({\textbf{x}}\right)=-\frac{1}{\epsilon_0}\int_{0}^{R} ...
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0answers
68 views

Line and surface integrals $R^{3} $

So i actually missed the class where this material was covered so plaese bear with me if my understanding is not so good. one of the problems in my textbook is as follow's. Prove the following ...
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0answers
178 views

Tough integration with change of variables and switch to polar coordinates

I was given this question in class and I was just wondering if I am on the right track… Evaluate: $$I=\iint\left(1-\frac{x^2}{a^2} -\frac{y^2}{b^2} \right)^{3/2} dxdy $$ over the region enclosed by ...
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2answers
54 views

In search for the domain in which the inequality holds

I wrote this simple inequality and raised the question what is the maximal domain such that the inequality holds, and the inequality is: $\dfrac{1}{x+y}>{\dfrac{1}{x}}+{\dfrac{1}{y}}$. ...
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0answers
46 views

Question on Mc Carthy's nowhere differentiable function

Mc Carthy's function is a simple example of a nowhere differentiable but everwhere continuous function. Its construction here contains very little detail and I have some questions (highlighted in ...
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0answers
50 views

evaluation of $\int \ln \left(1+2m\cos x+2m^2\right)dx$

How can i calculate $\displaystyle \int \ln \left(1+2m\cos x+2m^2\right)dx$ My try:: Let $\displaystyle I(m) = \int \ln \left(1+2m\cos x+2m^2\right)dx$ dIfferentiate both side w.r.to $x$ ...
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1answer
38 views

Sequence of continued powers defined recursively

I have a series ${a_n}$ defined recursively by $a_1 = b$ and $a_{n+1} = b^{a_n}$, with $b \ge 1$ and $n \in \mathbb{Z} $. I am trying to show that ${a_n}$ is bounded above if $1 \le b < 3^{1/3}$, ...
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0answers
29 views

Prove limit without using operator norm

prove $$\lim_{h \to 0}\frac{f(\mathbf{x}+h)-f(\mathbf{x})-\langle\nabla f(\mathbf{x})h,h\rangle-1/2\langle\nabla^{2}f(\mathbf{x})h,h\rangle}{\|h\|^{2}}=0$$ where function $f:\Bbb{R}^{n} \to \Bbb{R}$ ...
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1answer
48 views

The length of the boundary

Please I need your help if you can . I was asked yo determine the length of the boundary which formed by the hyperbola $x^2 -81y^2=9$ and the lines $y=-2$ and $y=1$ and given the parameter ...
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3answers
239 views

composition of continuous and uniformly continuous functions

I have a fundamental question. Let $\,f:X\rightarrow Y$ be continuous and $\,g:Y\rightarrow Z$ be uniformly continuous functions. Can $\,g\circ f$ be uniformly continuous?
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0answers
138 views

Difference between distance between two points and metric

if i have a line element given e.g. $ds^2=\frac{dx^2+dy^2}{2y} $ is it then always possible to derive a distance between two points in this metric? and how would one determine the length of a curve if ...
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0answers
37 views

Approximation theorem

I am looking for some theorem that gives me that each curve $x(t)=(x_1(t),x_2(t))$ that is continously differentiable and has $\dot{x_1}(t)\ge 0$ can be approximated by continously differentiable ...
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2answers
74 views

Exact differential equations. Test to tel if its exact not valid, am I doing something wrong?

I got this differential equation: $$(y^{3} + \cos t)'y = 2 + y \sin t,\text{ where }y(0) = -1$$ Tried to check for $dM/dY = dH/dY$ but I cant seem to get them alike. So what would the next step be ...
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0answers
61 views

Differentiation help

I recently got some lecture slides, but needed a little help understanding the maths behind them. (equations) (Working and Answer) Basically, I don't understand how to get from step (4) to (5). ...
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0answers
83 views

How do I find $\sum_{n=1}^\infty \frac{1}{n^n}$ [duplicate]

I stumbled across this problem to find the result of the following expression: $$\sum_{n=1}^\infty \frac{1}{n^n}$$ but I don't know how to approach it. It was suggested to me that I try this: ...
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0answers
58 views

How to evaluate $\int_0^1\frac{\ln(1+x)}{1+x^2}dx$ [duplicate]

How to evaluate : $$\int_0^1\frac{\ln (1+x)}{1+x^2}\text{d}x$$
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1answer
84 views

Taylor polynomials expansion with substitution

I am working on some practice exercises on Taylor Polynomial and came across this problem: Find the third order Taylor polynomial of $f(x,y)=x + \cos(\pi y) + x\log(y)$ based at $a=(3,1).$ In the ...
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1answer
4k views

How to find average rate of change

How would I find the average rate of change over $8$ minutes, of a person that runs at a rate of $v(t)=x\sin(x^2-7x)$ ft/min? I missed when this was taught and I have no clue on how to do it. Help is ...
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1answer
47 views

question about a divergant series does not make sense

I saw a picture in my textbook. I didn't read anything about it, but I got an idea. see :http://i.stack.imgur.com/anf8N.jpg Let the square's side be L : $$ \begin{align} ...
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0answers
36 views

Why does the surface area integral need the arc length differential but the volume doesn't? [duplicate]

When calculating the surface area of a revolution you need to use the arc length differential $$\sqrt{1 + y'^2}$$ but you don't need to use that when calculating the volume. Why is that? Thanks!
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1answer
95 views

Can the following equation be solved for the unknowns?

$$f(x) = x^3 + bx^2 + c$$ A tangent line passes through the point $(-2,6)$. Solve for $b$ and $c$. The above question was on our test, and our teacher insisted it was solvable. Any help is ...
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1answer
39 views

Specify conditions for $\alpha$ so that the iteration $x_{n+1} = x_n - \alpha f(x_n)$ converges to root of f.

Specify conditions on $\alpha$ so that the iterative process $x_{n+1} = x_n - \alpha f(x_n)$ converges to root of f if started with $x_0$ close to the root. It is suggested that the proof should ...
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1answer
846 views

Is the Sigmoid Function a Probability Distribution?

This could be a stupid question but, since sigmoid function maps values between $-\infty$ and $\infty$ to values between 0 and 1, I thought it could be a probability distribution. However when I take ...
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1answer
18 views

What values does the function $Z(y)$ have at various interval?

When $y\leq0$; $H(y)=0$ When $y>0$; $H(y)=e^{-\dfrac1y}$ What values does the function $Z(y)$ have at various interval? Where $Z(y)=H(1-y)(1+y)$ Please show this!
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0answers
146 views

How to take the limit of some integral?

$$ f\left( x^{\prime },t+\varepsilon \right) = \int_{-\infty }^\infty dx\int_{-i\infty }^{i\infty } \frac{d\tilde{x}}{2\pi i} \left(1+\varepsilon \left[ \tilde{x}D_{1}\left( x,t\right) ...
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1answer
59 views

Area of a 3D surface

I need to compute the area of a $3D$ sphere centered on $0;0;0$ and the book I'm following says: "If a curve $y = f (x)$ from $y = a$ to $y = b$ is revolved around the $x$ axis, the surface area of ...
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3answers
247 views

3D - derivative of a point's function, is it the tangent?

If I have (for instance) this formula which associates a $(x,y,z)$ point $p$ to each $u,v$ couple (on a 2D surface in 3D): $p=f(u,v)=(u^2+v^2+4,2uv,u^2−v^2) $ and I calculate the $\frac{\partial ...
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2answers
281 views

Dirac delta function

1)Prove that the dirac delta function property: $$ x\delta'(x)=-\delta(x)$$ 2)and : $$\int_{-\infty}^\infty \delta'(x)f(x)dx=-f'(0) \ $$
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0answers
50 views

Trouble proving/disproving the existence of a particular derivative.

I am having trouble with the following question: Quesiton: Define $f$ such that $$ f(x) = \begin{cases} \cos\left(\frac{1}{x}\right) &\mbox{ if }x \ne 0 \\ 0 &\mbox{ if } x = 0 \end{cases} ...
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0answers
113 views

Proof of an inequality problem

Wise men or women over the world!! I badly ask you to help me. Let $N$ and $B$ be two positive integers such that $1\le B\le \frac{N}{2}$ and $N=ug$ (for convenience, assume that $N$ is even.) For ...
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0answers
159 views

Evaluating the Gaussian integral $ \int_{-\infty}^{1}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}dx $ with the substitution $x=\frac{1}{t}$

I got $dx=-t^{-2}dt$, giving me $$ \int_{0}^{1}-\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2t^{2}}}t^{-2}dt $$ I can already tell that the above equation can't be right since it is negative everywhere. It ...
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1answer
969 views

How do I find the y double prime by implicit differentiation?

I am to find y double prime at $x = 0$ given $$xy + 7e^y = 7e$$ I am confused as to how to use the information of $x = 0$. The only $x$ term in the equation is $xy$ which after implicit ...
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1answer
55 views

Finding a tangent using a point that is undefined for the function

$f(x) = x\ln(a^2x^2), a > 0$ A tangent to the derivative of the function goes through $(0, 0)$. The task is find the tangent's intersection point with the derivative and the function of the ...
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0answers
37 views

Limit of $\epsilon t-f(t)$

Let $f(t)$ be continuously differentiable, increasing, positive and unbounded in $[0,\infty)$. Assume that: $\lim_{t \to \infty} t-f(t) =\infty$; For any $\epsilon >0$, the function $\epsilon ...
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0answers
93 views

Prove $\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$

$$\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$$ The above is an identity frequently used in ...
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1answer
50 views

Can you use indefinite integration to prove equivalence of two functions?

Is it always the case that if: $$ \int f(x) dx = F_1(x) + C $$ and $$ \int f(x) dx = F_2(x) + C $$ then $$ F_1(x) = F_2(x) $$ and why? Is it a legitimate way to prove the equivalence of two ...
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2answers
39 views

Boundedness and min-max

On This Page, We consider the function $$f(x)=\frac{1}{1+x^2}$$ on the interval $[0,\infty)$ The first point argued is that $[0,\infty)$ is not bounded. The next point is minima does not occur. Is ...
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0answers
29 views

Evaluating a simple sum bound

I'm trying to evaluate and prove a simple statement but It seems really raw/bad solution. I would like to advise with you if this is the right way because It is really getting more complicated than It ...
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0answers
191 views

How to find $\kappa$ to minimize integral $I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) \mathrm{d}x$

I am trying to find such value $\kappa \in (0,1)$ that would minimize the integral \begin{equation} \begin{aligned} I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) ...
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2answers
2k views

Find the equation of the tangent to the curve with exponential function

The question is as follows: Find the equation of the tangent to the curve $y = xe^{2x}$ at the point $(\frac{1}{2}, \frac{e}{2})$. Now I figured out that $\frac{dy}{dx} = e^{2x}(2x+1)$, and that ...
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0answers
52 views

Convergence of integrals over divergent parts

I'm wondering if it is possible for an integral which diverges in the limits $1$ to $\infty$ to converge in the limits from $0$ to $\infty$. And if so: how could I find this out? For example $$ ...
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0answers
27 views

Quick question on the simplification of digamma series

How to simplify : $$\sum\limits_{k=1}^{\infty }{\frac{{{\left( -1 \right)}^{k-1}}}{k}\left( \psi \left( \frac{k}{2\left( 2+\sqrt{3} \right)}+1 \right)-\psi \left( \frac{k}{2\left( 2+\sqrt{3} ...
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0answers
66 views

Proving that this function must be even (II)

Suppose $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is continuous. Also let $\mathbf{x}=(x_1,\ldots,x_d)\in\mathbb{R}^d$. I'd like to prove the following: If ...
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1answer
682 views

General Solution to Quasilinear PDE using Method of Characteristics

This is a homework that I'm having a bit of trouble with. I posted it previously but there was a typo in my original post. Since I received an answer for the incorrect problem it was suggested that ...
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0answers
144 views

Convergence of the reciprocal of a function whose derivative tends to infinity

If $$\lim_{x\to\infty} f'(x) = \infty$$ prove that $$\int_1^\infty \frac{1}{f(x)}\neq\infty$$ if $f'(x) \geq 1$ and $f(x) \geq1$ for all values of $x$. I'm thinking I can find a way to write it in big ...
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0answers
52 views

A simple question about a limit

I have the following data: 1) $u(t),u′(t)$ and $u′′(t)$ are bounded 2) $u''(t)+cu'(t)+f(u(t))=0$ where $c>0$ and $f$ is continuous 3) $\int_{0}^{+\infty}{u'(t)^2}<\infty$ I want to prove ...
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0answers
58 views

Implicit function/taylor expansion

So i have the following relation: $$x\cdot g(y(x)) = 1$$ Taking the logarithm of this equation yields $$\log x + \log g(y) = 0 $$ I work with the following assumptions: $(x,y(x)) = (1,0)$ satisfies ...
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1answer
198 views

Finding level curves of this function

$$x^{2}+2ky+y^{2}=u(x,y,k)$$ If $k$ is made constant, what are the level curves of $u$? How do I go about doing this?