# Tagged Questions

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

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### 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.
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### 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....
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### 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.
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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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### 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 ...
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### 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 $$y(x)=\int_{0}^{x} h(t) \, \mathrm{d}t$$ ...
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### 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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### 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 ...
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### 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 ...
I have read the following question : Non-centered Gaussian moments where it is stated that : $$E|X|^p = \sigma^p 2^{p/2} \frac{\Gamma \left(\frac{p+1}{2}\right)}{\sqrt{\pi}} {}_1 F_1 \left(-\frac{1}{... 1answer 31 views ### Derivation of the Euler Lagrange Equation I'm self studying a little bit of physics at the moment and for that I needed the derivation of the Euler Lagrange Equation. I understand everything but for a little step in the proof, maybe someone ... 0answers 21 views ### Koshliakov-Voronoi formulas reading the papers https://cmup.fc.up.pt/main/sites/default/files/publications/vorklcmup.pdf i read that$$-\sum_{k=1}^\infty \sigma_2(k) K_0(2 \pi k x)=\sum_{k=1}^\infty \frac{\sigma_2(k) \left(x^3 ...
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 ...