# Tagged Questions

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

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### Test if point is in convex hull of $n$ points

I have $n$ points $x_1,\dots,x_n\in\Bbb R^d$, and I would like to check that some other point $y$ lies in their convex hull. How can I do this in some efficient way? I think that there was an ...
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### Understanding a necessary step in a solution in variational calculus

I'm reviewing calculus of variations using a pdf that I found online (link) and in the example about the minimal surface of revolution, the writer simplified an equation tagged $(3.16)$ as follows: ...
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### how to find the derivate of a function g(x)

It's $g(x)={{x^{2}-1}\over{x^{2}+2}}$ and i have to calculate $g^{13}(0)$. I can't calculate all the derivates so i think to use power series. $g(x)={{x^2\over{x^{2}+2}}-{1\over{x^2+2}}}$ Can i use ...
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### Is the Weierstrass function given in Counterexamples in Analysis a typo?

Let $0 < a < 1$; let $b$ be an odd integer; let $ab > 1 + \frac{3\pi}{2}$; let $f: x \mapsto \sum_{n \geq 0}a^{n}\cos (b^{n}\pi x): \Bbb{R} \to \Bbb{R}$. Then $f$ is everywhere continuous ...
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### Attempting to draw G(x) from G'(x)

We know $G(0) = 0$ Okay, so I have the above graph but I'm having a difficult time translating it into the graph of $G(x)$. What I know so far is that the slope changes abruptly from 0 to 2 at ...
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### Indefinite integral problem: $\int_1^\frac{n+1}{1} \frac{(x - [x])^{[x]}}{[x]} dx$ [duplicate]

$I =\int_1^\frac{n+1}{1} \frac{(x - [x])^{[x]}}{[x]} dx$ my attempt: $I=\int_1^n \frac{(x - [x])^{[x]}{[x]}}\implies\sum_{i=1}^n\int_r^{r+1} \frac{(x-r)^r}{r}dx$ Now by ...
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### What is $x^2$-1 applied n times

For the function $F(x)=x^2-1$. How do I write $F^n(x)$ ($F$ applied $n$ times) in terms of $x$?
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### Find the solution of the differential equation that satisfies ${{dP} \over {dt}} = 8\sqrt {Pt} ,\,P(1) = 5$

Please help. My homework is grading my answer as incorrect, but I can't tell what I did wrong. The second photo is the work of the problem done correctly but with dp/dt=2sqrt(Pt). I based my work off ...
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### Confusion of proving limits

Ok say we were to prove this simple limit: $\lim \limits_{x \to 2}$ $x^2$ For all epsilon > 0, there exist a delta > 0 such that IF $0 < |x-2|< \delta$ THEN $|x^2-4| < \epsilon$ We know ...
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### Trigonometric Substitution on $\frac{1}{x\sqrt{(x^2 +25)}}$

How can I find $$\int\frac{1}{x\sqrt{(x^2 +25)}} \space dx$$ using trigonometric substitution?
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### Improper Integral $\int _ {0}^{1} \frac{1}{\sqrt{(1-x) \sin{x}}} dx$

The question is : Does the following improper integral converges? $$\int _ {0}^{1} \frac{1}{\sqrt{(1-x) \sin{x}}} dx$$ I have tried some approaches but I'm not sure whether it was correct or not. ...
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### Is f(x,y) differentiable at the origin?

My book defines $f(x,y)$ is differentiable at $(a,b)$ when $\lim_{(x,y)\to(a,b)} \frac{R_{1,(a,b)} (x,y)}{\|(x,y)-(a,b)\|}=0$ where $R_{1,(a,b)}(x,y)=f(x,y)-L_{(a,b)} (x,y)$ and ...
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### Laplace transform of the zeroth-order Bessel function [duplicate]

Consider $J_0$ the zeroth order Bessel function. I'm trying to compute the Laplace transform $$\mathcal{L}[J_0](s) = \int_0^\infty J_0(t) e^{-st}dt,$$ but until now I couldn't find a good way to do ...
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### $\sum_{k=n}^{\infty}\left(n-k\right)e^{-\lambda}\frac{\lambda^{k}}{k!}= ?$

Could you please help me. How do I sum the following: $$\sum_{k=n}^{\infty}\left(n-k\right)e^{-\lambda}\frac{\lambda^{k}}{k!}$$ If the summation had started at 0, then it would be simply an ...
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### Can anyone solve this odd integral? [closed]

Can anyone solve this odd integral? $$\int\frac{e^{-50(\frac 1x-1)^2}}{x}\,dx$$ for $x>0$. I couldn't . . .
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### Solving differential equation and obtain expressions for unknowns?

I have the following differential equation $my'' + \beta y' + mg = 0$ , with condition $y(0)=0$. I need to solve the equation and obtain expressions for the unknowns. I have attempted to use the ...
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### what is wrong with this derivation?

We have simple function : $$Y = X^2$$ Writing $X^2$ as : $X^2 = X+X+X+...........+X$ $(X times)$ We can write above equation as : $$Y = X+X+X+X+...........+X$$ Differentiating with respect to X, ...
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### Prove Newton's iteration will diverge for these functions, no matter what real starting point is selected. $f(x)=x^2+1$ and $g(x)=7x^4+3x^2+\pi$.

Prove Newton's iteration will diverge for these functions, no matter what real starting point is selected. $f(x)=x^2+1$ and $g(x)=7x^4+3x^2+\pi$. We know that $f(x)>0$ and $g(x)>0$ for all ...
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### Convert Power Series to function

I tried to solve the attached Power Series, however I can't get to the right answer. I wrote down the correct answer at the top-right of the page. Appreciate your help!
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