# Tagged Questions

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### Finding $f(x)$.

If $$f(x)=1+x+x^2+\displaystyle\int_{0}^{x}e^k f(x-k) dk$$ then how do we find the function $f(x)$? Is there a way to solve it, with or without arriving at a differential equation? This a homework ...
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### Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$\delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
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### Divergent of a vector field on a sequence of spheres

I'm studying for my exams and I found this problem in the book "Advanced Calculus", written by Friedman: "Consider a sequence of spheres $S_n$ in $\mathbb{R}^3$ with center $P_n$ and radius $r_n$, ...
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### Find the Equation of the Tangent Line (Real-Analysis)

Let $f: \mathbb R \rightarrow \mathbb R$ be defined by $f(s) = {1 \over \pi} \int^s_0 e^{-t^2} dt$. Find the equation of the line tangent to the graph of $y = f^{-1}(x)$ at $x=0$. I left out the ...
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### Complex Roots and calculations

roots of the equation $z^6 =1-\sqrt3 i$ are $$z_1,z_2,z_3,z_4,z_5,z_6$$ calculate:$$|z_1|^3 +|z_2|^3+|z_3|^3+|z_4|^3+|z_5|^3+|z_6|^3$$ also calculate: $$z_1^6 +z_2^6+z_3^6+z_4^6+z_5^6+z_6^6$$ ...
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### Taylor series of the function $f(x) = (1+x) ^{\frac{1}{x}}$

Good night!! I got this problem and I'd like to find all the mistakes in my statement. This is my prblem. Find the binomial coefficients of $f(x) = (1+x)^{\alpha}$, with $\alpha \in \mathbb{R}$, and ...
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### Study of a function

I have this function $\displaystyle g(s)=\frac{s^{2-\sigma}}{1+s^2}, ~\text{for all} ~s\in \mathbb{R}$ , i need to find the interval of $\sigma$ and the maximum of the function $g$. I calculate the ...
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### Weak solution Boundary Value problem

I have to prove that the following problem $$(P) \begin{cases} -u''-u=1\,\,\,\,\,\,\,\,\text{if}\,\,\, x\in(0,\pi)\\ u(0)=u(\pi)=0 \end{cases}$$ doesn't admit weak solutions. I'm proceeding by ...
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### a question about multivariable integral!

If $\lfloor x \rfloor$ denotes the greatest integer in $x$, evaluate the integral$$\iint_{R} \lfloor x+y \rfloor ~ \mathrm{d}x~ \mathrm{d}y$$where $R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\}$. This ...
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### When is $\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$

An elementary question on Riemann - Integration: Under what conditions on $f$ is the following true: $$\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$$ If $f$ is bounded in $[a,b]$, then this is ...
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I have the equation$\begin{cases} x'(t)=x(t)+y(t) \\y'(t)= \mu y^2(t)+x(t)\end{cases}$ Cauchy problem $\begin{cases} x(0)= 1 + \mu \\y(0)=-2\end{cases}$ . I must calculate $\frac{\partial ... 0answers 39 views ### problem of computing limit The problem is to prove the following for$n \geq 3$$$u(0)=\frac{1}{n\alpha (n) r^{n-1}}\int_{\partial B(0,r)} g dS +\frac{1}{n(n-2)\alpha (n)} \int_{B(0,r)} (\frac{1}{|x|^{n-2}} - ... 1answer 27 views ### Pointwise and uniform convergence of sequence of functions Let (f_n) be a sequence of continuous functions on \mathbb R. If (f_n) converges to f pointwise on \mathbb R, then$$\lim\limits_{n\to ... 0answers 25 views ### Find Jordan decomposition constructed by function. $$F(x) =\left\{ \begin{array}{l l} 0 && x \leq 0\\ 1-x && 0 \leq x \leq 1\\ 2x-4 && 1 < x \leq 2 \\ 1 && 2 < x \end{array} \right.$$ ... 2answers 35 views ### Substitution of an implicit variable I wasn't sure how to title this question: I want to manipulate the integral $$I(a,b) = \int_0^{\frac{\pi}{2}} \frac{d \phi}{\sqrt{a^2\cos^2 \phi + b^2 \sin^2 \phi}}$$ with this subsitution: $$\sin ... 0answers 25 views ### calculate the sup of the max of 3 functions Let a function be the variable, how the calculate the following expression?$$\inf_{c(t) \in C[-1,0]} \max \{ \max_{-1 \leq t \leq 0} |c(t)| , \max_{0 \leq t \leq 1} | \int_{0}^{t} c(v-1) +1 dv +c ... 2answers 54 views ### Convergence radius of power series is infinite Which function is given by a power series whose convergence radius is infinite? $$A. \ \ \ e^{-\frac{1}{x^2}}$$ $$B. \ \ \ \sin{\left(\frac{1}{x}\right)}$$ $$C. \ \ \ ... 0answers 20 views ### Identity proof vector calculus I encounter massive problems tackling the following math problem. Especially the notation in line #2 with the \frac{\partial u}{\partial x^2} and in line #3 the U:]0,R[\times\mathbb R\ , ... 1answer 119 views ### Convergence of \sum_{n=1}^{\infty}\frac{1}{3^n\ \sin(n)} Does this series converge? Root test and ratio test are inconclusive. 3answers 58 views ### Let f:(0,1) \rightarrow \mathbb R be continous. Suppose that |f(x) - f(y)| \leq |\sin x - \sin y | for all x,y\in (0,1). Then f is discontinuous at least one point in (0,1) f is continuous everywhere on (0,1), but not uniformly continuous on (0,1) f is uniformly continuous on (0,1) \lim_{x \rightarrow 0^+} ... 3answers 52 views ### Show that series converges Show that if \{ p_n \} is a Cauchy sequence, then it has a subsequence \{ p_{n_k}\} such that the series \sum_{k=1}^\infty b_k converges, where b_k = d(p_{n_k}, \, p_{n_{k+1}}) . My ... 1answer 40 views ### Maximization of Function with two restrictions. Maximize$$f(x,y,z)=xy+z^2,$$while$2x-y=0$and$x+z=0$. Lagrange doesnt seem to work. 1answer 49 views ### Show that series in Cauchy Sequence Let$a_n = d(p_n, p_n+1)$for$n = 1, 2,\cdots $. Show that if the series$\displaystyle \sum^{∞}_{n=1} a_n$converges, then$\{p_n\}$is a Cauchy sequence. My Approach: I thought of using the ... 1answer 18 views ### Let$K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\}$and$S := K_1(0) \setminus \mathbb{Q}^2$. Is M path connected? The Assignment: Let$K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\}$and$S := K_1(0) \setminus \mathbb{Q}^2$. Is S path connected? Explain your answer. I don't think S is path-connected since ... 1answer 48 views ### proof of coarea formula for n dimensional hypersurface in$R^nf:R^n \rightarrow R$be continuous and summable. please give the proof for these formulas$\int_{R^n}f dx = \int_0^\infty(\int_{\partial B(x_0,r)}fdS)dr\frac{d}{dr}\int_{ ...
Let $x_1> 1$ and let $x_{n+1} := 2 - \displaystyle\frac{1}{x{_n}}$ for $n \in \mathbb{ N}$. Show that $(x_n)$ is bounded and monotone. Find the limit. I am confused on how to show that the ...
### Find $f(t)$ such that $f\left(\dfrac{t^2}{2-t}\right) = -5t+4$.
Find $f(t)$ such that $f\left(\dfrac{t^2}{2-t}\right) = -5t+4$. I don't really know how to approach this problem. Would you give any hint on how to start?