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

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0
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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
3answers
84 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
votes
0answers
25 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
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: ...
2
votes
0answers
25 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 ...
0
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2answers
89 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.
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2answers
32 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 ...
0
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2answers
418 views

How to prove the convexity of $f$ if the strict epigraph of $f$ is convex

I have trouble to prove the equivalence of the following two definitions of convex function. For convenience, I list them as follows: Def-A. Let $f : I\to \mathbb{R}$ be a function, where $I$ is (and ...
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2answers
66 views

How to differentiate $\ln(a^x)$?

Can someone give me the process to differentiate this (with respect to $x$)? $$ \ln(a^x) $$
2
votes
1answer
5k views

Showing properties of a surge function

I am working on the question below and I am getting stuck. Consider the surge function $y=axe^{-bx}$ with $a$ and $b$ positive constants. (a) Find the local maxima, local minima, and ...
-1
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2answers
73 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 = ...
2
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4answers
55 views

Proving convergence or divergence of series: Tips and Tricks

I currently write an article where I collect some tips for students for proving the convergence or divergence of series. What tips and tricks do you know or use or teach? Remark: I will add some ...
2
votes
1answer
40 views

Fixed point, bounded derivative

Let $p\in\mathbb{N}$. Let $f:I\to\mathbb{R}$ differentiable in the closed interval $I$ (bounded or not), with $f(I) \subset I$, and let $g = f\circ f\circ \cdots \circ f = f^p$, where $\circ$ means ...
0
votes
1answer
40 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
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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 ...
0
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3answers
42 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 ...
2
votes
0answers
28 views

Integral of familly of curves

Let $f_n(t):=t^{n+m+1/3}e^{-(t^{-n}+t^{-m}+t^2)}$, where $n,m\geq 1$. I have been having difficulty calculating this ingetral on $\mathbb{R}$. Please help, thanks.
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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 ?
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2answers
96 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 ...
7
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2answers
205 views

Find $\lim_{n \to \infty} n \int_0^1 (\cos x - \sin x)^n dx$

Find: $$\lim_{n \to \infty} n \int_0^1 (\cos x - \sin x)^n dx$$ This is one of the problems i have to solve so that i could join college. I tried using integration by parts, i tried using ...
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)...
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1answer
42 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 ...
5
votes
1answer
35 views

How to prove the parallel projection of an ellipsoid is an ellipse?

Take the following ellipsoid in implicit form as an example: $$x^2 + 2 y^2 + 3 z^2 + x y + y z - 2 xz = 5$$ which shows: The parallel projection of the ellipsoid onto $xoy$ coordinate plane can ...
0
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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: $$\int_0^\infty dx \int_0^\infty dy ~\vec\nabla\cdot\vec F=\oint_C ds~\vec n\cdot\vec F$$...
0
votes
2answers
41 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 ...
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{...
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2answers
46 views

Differentiation under the integral sign in $R^3$

I'm trying to take derivative from an integral. I know about the Reynolds transport theorem, but I do not know how to obtain the unit normal and the velocity. I'm going to take the derivate from the ...
0
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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
25 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\...
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0answers
35 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 ...
7
votes
2answers
230 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?
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3answers
34 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 ...
2
votes
1answer
30 views

Calculating the Volume of a Cylindrical Shell

Hey I'm trying to solve this problem and I'm stuck. The integral I'm using to solve this problem is the integral from $0$ to $4$ of $2{\pi}y(27 - (y + 2)^3)$ but the answer isn't correct. Can anyone ...
2
votes
5answers
143 views

Definite integral of $\sqrt{\frac{1}{\cos^2(x)}}$

I've got problems with this integral: $$\int_0^{\frac{\pi}{4}} \sqrt{\frac{1}{\cos^2(x)}} \, \mathrm{d}x$$ First I substitute $x=2\arctan(x)$ but this leads nowhere. Any hints for solving?
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5answers
47 views

Determine values $a$,$b$, and $c$ such that the graph of $y = ax^2 + bx + c$ has a relative maximum at $(3, 12)$ and crosses the $y$-axis at $(0,1)$.

I understand to find $c$ we incorporate the point at the $y$-axis $(0,1)$ into the question which gives us $c=1$ but I can't seem to get the correct numbers for $a$ and $b$. If you could help that ...
29
votes
1answer
567 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}xdx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
0
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1answer
42 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 ...
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1answer
603 views

Finding closed form of a series where the first term is $n=0$

"The figure below shows the quantity of the drug atenolol in the blood as a function of time, with the first dose at time $t = 0$. Suppose atenolol is taken in $75$ mg doses once a day to lower blood ...
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votes
1answer
33 views

Please explain the answer to this polar-cartesian problem

If $r>0$ and $0< \theta <2\pi$, then the polar coordinates of the Cartesian point $(2, [-2 * 3^{1/2}] )$ are ___________. The answer is $(4, 5\pi/3).$ I don't understand how they came up ...
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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 ...
3
votes
3answers
53 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 ...
3
votes
1answer
64 views

Solving $ \frac {dy}{dx} = \sqrt{y} x\cos(x) $ with $y(0) = 1$

I was helping someone work this problem out for an online course and I thought it'd be pretty easy since it's a first order separable DE. I ended up with $$ y = \frac {(x\sin(x) + \cos(x) + 1)^2}{4} ...
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0answers
43 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 $$y(x)=\int_{0}^{x} h(t) \, \mathrm{d}t$$ ...
0
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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}{...
17
votes
2answers
433 views

Conjectured closed form for $\sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n+\frac{1}{\sqrt{2}}}$

I was trying to find closed form generalizations of the following well known hyperbolic secant sum $$ \sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n}=\frac{\left\{\Gamma\left(\frac{1}{4}\right)\right\}^2}{...
1
vote
1answer
34 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 ...
6
votes
2answers
478 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 $...
4
votes
2answers
151 views

Did I count the limit of this function correctly: $\lim_{x\to1}\frac{x-1}{x^n-1}$?

Given: $\lim_{x\rightarrow 1}\frac{x-1}{x^{n}-1}, x \in \mathbb{R}$ Because you cannot really get the limit with the current given function, I have used L'Hôpitals rule. $f(x) = x-1$ $f'(x) = 1$ $...
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 ...