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

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1
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2answers
55 views

Find the exact critical numbers for $f(x) = 3x - \sqrt{x}$

I found the derivative of the function which I believe is $3-\frac{1}{2\sqrt{x}}$ but I am not sure how to find the $x$ value for the critical number.
0
votes
3answers
63 views

Understanding Big O

If $f(x)=O(x^2)$ as $x \rightarrow 0$ and $f(x)$ is continuous at $x=0$, what does this tell us? Can we assume $f(0)=0$? Is $f(x)$ differentiable at $x=0$? I am having trouble understanding this stuff....
-1
votes
2answers
22 views

How to demostrate Sub-set from flat is a open set

The set C={(x,y);2< x^2 + y^2 < 4} How can we defined this set as open using the definition.
-1
votes
1answer
33 views

help with real analysis [on hold]

Let $S \subset \mathbb{R}$ be nonempty. Show that if $u= \sup S$, then for every number $n$ belong to $\mathbb{N}$ the number $u -\frac{1}{n}$ is not an upper bound of $S$, but the number $u + \frac{1}...
2
votes
2answers
55 views

Summing a series of integrals

I asked this question on Mathoverflow, but it was off-topic there (though it is related to my research...) and I was told to ask it here. I have a series of integrals I would like to sum, but I don't ...
0
votes
1answer
40 views

Path length of Gaussian

I am trying to find the path length of a Gaussian $f(x)=e^{-x^2/a^2}$ from $x=0$ to some positive point $x_0$. I've tried this by integrating the differential length, $ds^2=dx^2+dy^2$, but getting ...
0
votes
1answer
14 views

How to minimize $(p_1^2 + (1-p_1)^2)^n$ where $p_1 = 1-(1-(k/n))^N$

Consider $S_{n,N,k} = (p_1^2 + (1-p_1)^2)^n$ where $p_1 = 1-(1-(k/n))^N$. If we fix $N$ and $n$, how do we find a $k$ which minimizes $S_{n,N,k}$? We assume that $1 \leq k < N$ if that makes a ...
6
votes
3answers
294 views

Limit of the sequence $\lim_{n\rightarrow \infty}n\left ( 1-\sqrt{1-\frac{5}{n}} \right )$, strange result

$\lim_{n\rightarrow \infty}n\left ( 1-\sqrt{1-\frac{5}{n}} \right )$ $\lim_{n\rightarrow \infty} n *\lim_{n\rightarrow \infty}\left ( 1-\sqrt{1-\frac{5}{n}} \right ) = \infty * \left ( 1-\sqrt{1-0} \...
2
votes
2answers
49 views

Limit and convergence/divergence of an integral

I was working on a problem concerning the function $$f(x) = \frac{x^2}{\ln(x)^\sqrt{x}}$$ asking for the value of $$\lim_{x \to \infty}f(x)$$ and for the convergence/divergence of $$\int_2^\infty f(x) ...
0
votes
2answers
23 views

Differential $\mathrm{d}^2f$ of implicit function $F(x,y,z)=xyz-x-y-z=0$

Determine the differential $\mathrm{d}^2f$ of the implicit function defined as $z=f(x,y)$: $$F(x,y,z)=xyz-x-y-z=0$$ So in fact of the implicit function I have to use the implicit function ...
0
votes
1answer
20 views

Gradient of a maximum

How do you compute the gradient of a function that involves a maximum? For example, I have the function: $$ f(\vec{t}) = v(1-\exp(-\lambda\cdot \max(t_0,t_1)))$$ With $v$ and $\lambda$ constant, for ...
0
votes
2answers
20 views

How would I use the difference quotient on this logarithmic function?

This is no homework, it's for exam practice. Show that $\lim_{x\rightarrow 0}\frac{1}{x}ln(1+ax) = a$ where $a \in \mathbb{R}\setminus \left \{ 0 \right \}$ is chosen definitely / fixed (...
4
votes
3answers
389 views

Integration by parts: is this legitimate way of using?

Is it legitimate to write $$\int_0^a\mathrm{d}x\,f(x)g(x)=\left[f(x)\int_0^x\mathrm{d}x\,g(x)\right]_0^a-\int_0^a\mathrm{d}x\,\frac{\mathrm{d}f(x)}{\mathrm{d}x}\int_0^x\mathrm{d}x\,g(x)$$ Thanks.
0
votes
0answers
46 views

analytical solution to equation

I am trying to solve the following equation by $x$. (put sole $x$ on one side of the equation), but I am not sure if there is a analytical solution to this problem: $$\large -2\ln\left( \frac{(1-p)^{...
1
vote
1answer
39 views

Doubly infinite matrices $A=(a_{i,j})_{i,j=\infty}^{\infty}$

Let $A=(a_{i,j})_{i,j=\infty}^{\infty}$, where $$ \|A\|:=\sum_{r=-\infty}^{\infty}\sup_{j}|a_{j,j+r}|<\infty. $$ I want to show that for all matrices $\|AB\|\leq\|A\|\|B\|$. I obverse that $$ (AB)...
0
votes
0answers
18 views

Koshliakov-Voronoi formulasII

reading the papers https://cmup.fc.up.pt/main/sites/default/files/publications/vorklcmup.pdf i read that $$\sum _{k=1}^{\infty } \sigma _0(k) K_0(\pi k x)=\sum _{k=1}^{\infty } \frac{\sigma _0(k) \...
1
vote
1answer
69 views

Find $\int^{2}_{0}f(x)\cos(\pi x)dx=?$

I would appreciate if somebody could help me with the following problem. Q:$f(x)$ satisfy 1,2,3 $f'(x)=f'(1-x)$ $f(x)f(1-x)=\sin(\pi x)$ $f(0)+f(1/2)=1$ Find $\int^{2}_{0}f(x)\cos(\pi x)dx=?$ I ...
2
votes
0answers
24 views

Problem in understanding a statement on finding the velocity field of fluid.

I was reading The equations of motion of The Flow of Dry Water in Lectures on Physics: Vol II by Feynman; here he is explaining the equation of motion of incompressible fluid where $\bf v$ is the ...
1
vote
1answer
21 views

Need know all ways to show function is continuous, convergent and differentiable [on hold]

Please tell me all ways to show / proof that a function is continuous, convergent and differentiable. continuous: show that function is differentiable if yes then it is continuous also convergent: ...
0
votes
3answers
79 views

A paradox in differential calculus

Say I have a function $f=f(x,y)$ where $x,y$ are independent variables. Now, it is given that $p=x+y$. It can be shown that, since $x,y$ are independent, we get $$\frac{\partial p}{\partial x}=\frac{...
1
vote
2answers
46 views

I find correct limit of the sin cos function?

This is no homeworks I only do for learn. $$\lim_{x\rightarrow \pi}\frac{\sin^{2}x}{1+\cos x}$$ I use l'Hôpital rule because no idea where limit go for both. Top is called $g(x)$ and bottom is $...
1
vote
2answers
31 views

Limit of functions - always for both sides (+-) necessary?

I'm very confused when I read some pages on the internet about limits (for functions). Let's say I got any function f(x) given and someone tells me to find the limit (towards 3 or $\infty$ or ...
1
vote
0answers
52 views

if $f''$ is continuous then $f''(x_0)=0$ where $x_0$ is an inflection point

I came across the following statement and I am trying to get some intuition why is it right. "if $f''$ is continuous then $f''(x_0)=0$ where $x_0$ is an inflection point, take a note: the opposite is ...
-1
votes
2answers
72 views

Reciprocal of the sum of powers of $1/x$ [duplicate]

Incidentally, I found $$\frac{1}{\sum_{n=1} \frac{1}{x^{n}}} = (x-1)$$ where $x\ge 2$. Please direct me to how others have developed the relationship. My computer cannot compute more than X = ...
0
votes
2answers
88 views

Why does $\sum_{n=1}^{\infty} \frac{2^n+1}{5^n+1}$ converge?

Why does $\sum_{n=1}^{\infty} \frac{2^n+1}{5^n+1}$ converge? I've tried by using the ratio test but I don't get so far, I'm a little lost with it. Any help will be really aprecciated.
0
votes
1answer
39 views

Graph the following function $f(x) = \frac{x + 3}{x^2 -4}$

I have been able to calculate the Intercepts, Asymptotes, Local Max/Min but when finding the inflection points, I can't seem to factor my derivative if you could kindly help that would be great. ...
0
votes
0answers
27 views

Dirichlet Kernel function: Can you please derive the solution for $n=0$ on the Cosine equation? [on hold]

Below is the link for the intro of Dirichlet function: http://i.stack.imgur.com/ydWab.png I came across an equation $$\frac{1}{2} + \cos{t} + \cos{2t} + \dots + \cos{nt} = \frac{\sin((2n+1)\frac{t}{2}...
0
votes
1answer
31 views

Finding a delta for the greatest integer function given an epsilon = 1/2

I'm having trouble with the following problem. Given the standard greatest integer function $\lfloor x \rfloor = int(x)$ where $ \lfloor x \rfloor $ returns the greatest integer less than or equal to ...
-1
votes
0answers
56 views

Evaluate $\int_0^\frac12 \frac{\sin(\pi x)}{(x+1)(x+2)} dx$ [on hold]

$f(x) = \int_0^\frac12 \frac{\sin(\pi x)}{(x+1)(x+2)} dx$ Could not solve the problem. Can anyone help me ?
0
votes
3answers
40 views

General chain rule help/ derivatives help.

I've been thinking too much about the chain rule and I've got myself in a muddle: Suppose $y=f(g(x))$, we can easily show that $\frac {dy}{dx} = f'(g(x))\cdot g'(x)$. I would ask please that ...
0
votes
1answer
28 views

Finding $f(x)$ in $\cos^2(x)f(x)=x^2-2\int_1^x \sin(t)\cos(t)f(t) \, \mathrm{d}t$

I need to find a valid $f(x)$ such that: $$\cos^2(x)f(x)=x^2-2\int_1^x \sin(t)\cos(t)f(t) \, \mathrm{d}t$$ I can apply the FToC and I get: $$(2\cos(x)-\sin(x)f(x))+(\cos^2 x f'(x))=2x\sin(x)\cos(x)...
0
votes
0answers
9 views

Divergence Theorem: Conditions for the boundary integration to vanish?

Consider the Divergence Theorem for example in two dimensions, in the upper right quadrant of Euclidean space: $$\int_0^\infty dx \int_0^\infty dy ~\vec\nabla\cdot\vec F=\oint_C ds~\vec n\cdot\vec F$$...
1
vote
1answer
40 views

Derivative of a fraction

I want to derivate: $$f(x)=\frac{x^2-\frac{1}{3}}{x^3}$$ I apply the table formula: $$Dx\frac{f(x)}{g(x)}=\frac{f′(x)g(x)−f(x)g′(x)}{g(x)^2}$$ But i always get a wrong result. My result is: $$\frac{...
0
votes
2answers
40 views

Converting 'velocity with respect to distance' to 'distance with respect to time'

If I have a formula for velocity with respect to distance, like: $73 (km / s / megaparsec)$ And I want to convert it to a formula for velocity (or any of its derivatives or its integral) with ...
0
votes
1answer
41 views

Maximum and minimum of a composition

I have : $$f(x)=g(3x^2)+xg(x)$$ I know:$$$$ $f(0)$ is a critical point, $g(0)=0$, $$$$ $g'(x)\ne0$, $\exists g''(x)$ I also know $g$ function is a strictly decreasing function. How can I ...
0
votes
0answers
18 views

Show that the vector path is regular?

Let $\{\tilde{i}, \tilde{j}, \tilde{k}\}$ be the standard basis of vectors for IR3. If the path $\tilde{x}$ : IR → IR3 is defined by $$\tilde{x}= \cos 4t\ \tilde{i}+ \sin 4t\ \tilde{j}+ 2t^2\ \tilde{k}...
0
votes
1answer
23 views

PDE with a condition

Considering the heat equation, $$\frac{du}{dt}=\frac{d^2u}{dx^2}$$ if $$u(x,t)=t^{\alpha}\phi(\xi)$$ with $$\xi=x/\sqrt{t} \enspace then \enspace \phi \enspace satisfies \enspace \alpha\phi-(1/2)\xi\...
0
votes
0answers
33 views

Maclaurin expasion of $\sin(x)/(1-4x^2)$

I have to expand this function $f(x)=\dfrac{\sin(x)}{1-4x^2}$ around $x=0$ and then find tis radius of convergence. I expand $\sin(x)$ on series, but i dont know how to use $1/(1-4x^2)$, it is the ...
1
vote
3answers
33 views

Given a function, how can one tell if it doesn't have a limit at $x=a$ due to a discontinuity?

For example, if you have the $$\lim_{x \to 2} \frac{1}{x-2},$$ the limits approaching from the positive and negative are different. You can tell because the $x-2$ becomes $0$ and the entire binomial ...
5
votes
4answers
117 views

Not understanding derivative of a matrix-matrix product.

I am trying to figure out a the derivative of a matrix-matrix multiplication, but to no avail. This document seems to show me the answer, but I am having a hard time parsing it and understanding it. ...
0
votes
1answer
41 views

What is the area of triangle ABC?

Verbatim my Math test- Consider a polynomial $y=P(x)$ of the least degree passing through $A(-1,1)$ and whose graph has two points of inflexion $B(1,2)$, and $C$ with abscissa 0, at which, the curve ...
1
vote
4answers
162 views

How to show $ \lim_{(x,y) \to (0,0)} \frac{3x^2y^2}{x^4+y^4}=\frac{3}{2} $ using the $\epsilon$-$\delta$ notation.

I need to prove that: $$ \lim_{(x,y) \to (0,0)} \frac{3x^2y^2}{x^4+y^4}=\frac{3}{2} $$ using the $\epsilon$-$\delta$ notation. I have tried everything I could think of to make the expression into a ...
0
votes
0answers
42 views

Solving this ODE 1

Trouble solving this ODE : $$\frac{d^2y}{dx^2}=\int_{-\infty}^{x^2/2} e^{x-t^2/2} \, \mathrm{d}t$$ $$x>0,\, y(0)=0,\, \frac{dy}{dx}(0)=0$$ in the form $$y(x)=\int_{0}^{x} h(t) \, \mathrm{d}t$$ ...
7
votes
2answers
210 views

Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured

By trial and error I have found numerically $$\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}=\frac{1}{6}-\frac{1}{2\pi}$$ how can this result be derived analytically?
0
votes
2answers
93 views

Another (perhaps tricky) integral.

While solving my Math paper, I came across this integral, and I can't see any way to solve it. At least, any easy way. The integral is- $$ \int{x^{2} \over 1 + x^{5}}\,\mathrm{d}x $$ I'm not even ...
3
votes
3answers
51 views

How-to proof this integral

I saw it in the Hurwitz zeta function , $$ \int_0^ty^{p-1}\left(1-e^{-zy}\right)dy=\frac{t^p}{p}+e^{-tz}\sum_{k=0}^{p-1}k!\binom{p-1}{k}\frac{t^{p-1-k}}{z^{k+1}}-\frac{(p-1)!}{z^p}$$ And I was not ...
0
votes
0answers
16 views

Non integer, non-centered Gaussian moments

I have read the following question : Non-centered Gaussian moments where it is stated that : $$E|X|^p = \sigma^p 2^{p/2} \frac{\Gamma \left(\frac{p+1}{2}\right)}{\sqrt{\pi}} {}_1 F_1 \left(-\frac{1}{...
1
vote
1answer
31 views

Derivation of the Euler Lagrange Equation

I'm self studying a little bit of physics at the moment and for that I needed the derivation of the Euler Lagrange Equation. I understand everything but for a little step in the proof, maybe someone ...
0
votes
0answers
21 views

Koshliakov-Voronoi formulas

reading the papers https://cmup.fc.up.pt/main/sites/default/files/publications/vorklcmup.pdf i read that $$-\sum_{k=1}^\infty \sigma_2(k) K_0(2 \pi k x)=\sum_{k=1}^\infty \frac{\sigma_2(k) \left(x^3 ...
0
votes
2answers
45 views

related rates- rate a man's shadow changes as he walks past a lamp post (is a fixed distance away from it)

A $186$ cm man walks past a light mounted $5$ m up on the wall of a building, walking at $2\ m/s$ parallel to the wall on a path that is $2$ m from the wall. At what rate is the length of his shadow ...