# Tagged Questions

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

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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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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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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### 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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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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### 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 ...
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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 ...
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