Recent questions tagged function

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What is a Lipschitz function and its significance?
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Define and explain the concept of a limit of a function.
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Here is a Pascal equation that yields y as a function of theHighLowRange of the data over a number of days. Can you reverse theequation to yield the HighLowRange as a fu...
69
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What is the conjugate of the complex number \(7+3i\) ?
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What is a Bot?
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To solve the maximization problem of the investor, we need to set up the Lagrangian and find the optimal conditions. Let's consider the following problem:\[\underset{\lef...
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Given a sequence ${\{x_n\}_{n\in\mathbb{N}}}$ in a metric space X, prove the following statements:(a) If $d(x_n,x_{n+1}) < 2^{-n}$ for every $n \in \mathbb{N}$, then ${\{...
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Assume ${\{x_n\}_{n\in\mathbb{N}}}$ is a Cauchy sequence in a metric space X, and there exists a subsequence ${\{x_{n_k}\}_{k\in\mathbb{N}}}$ that converges to $x \in X$...
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Prove that every meromorphic function on $S^2$ is rational.
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Let $L_1, L_2$ be lines in the plane. For which pairs of $L_1, L_2$ do there exists real functions, harmonic on the entire plane, 0 on $L_1 \cup L_2$, but not vanishing i...
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Let $\Omega$ be a region, $K$ a compact subset of $\Omega$, and fix some $z_0 \in \Omega$. Let $u$ be any positive harmonic function. Prove that there exists $\alpha, \be...
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Let $\Omega$ be a region, and $f_n \in \mathcal{H}(\Omega)$ for all $n$. Set $u_n = \Re(f_n)$, and suppose $u_n$ converges uniformly on compact subsets of $\Omega$ and th...
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Suppose $f$ is a complex function on a region $\Omega$, and both $f, f^2$ are harmonic on $\Omega$. Prove that either $f, \overline{f}$ must be holomorphic on $\Omega$.
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Let $u,v$ be real harmonic functions on a plane region $\Omega$. Under what conditions is $uv$ harmonic?Further, show that $u^2$ may not be harmonic on $\Omega$, unless $...
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Let $u$ be a harmonic function on a region $\Omega$. What can we say about the set of points such that $\nabla u = 0$, that is, the set of points where $u_x = u_y = 0$?
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Suppose $\Omega$ is a region, $f_n \in \mathcal{H}(\Omega)$ for $n \geq 1$. Suppose further that none of the $f_n$ has a zero in $\Omega$, and $f_n \to f$ uniformly on co...
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Compute$$\int_0^\infty \frac{dx}{1 + x^n}$$for $n \geq 2$.
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Suppose $f$ is an entire function, and that for every power series:$$ f(z) = \sum_{n=0}^\infty c_n (z - a)^n$$at least one coefficient is 0. Prove that $f$ is polynomial....
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Let $f \in \mathcal{H}(\Omega)$. Under certain conditions on $z, \Gamma$, we have that:$$ f^{(n)}(z) = \frac{n!}{2\pi i} \int_\Gamma \frac{f(\zeta)}{(\zeta - z)^{n+1}} d\...
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There is a region $\Omega$ such that $\text{exp}(\Omega) = D(1,1)$. Show that the exponential function is one-to-one on $\Omega$, but that there are many such $\Omega$. F...
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Suppose that $f$ is an entire function, and$$ | f(z) | \leq A + B|z|^k$$for all $z$, where $A, B, k$ are positive real numbers. Prove that $f$ must be polynomial.
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Suppose $f, g$ are entire functions, and suppose that for all $z \in \mathbb{C}$, that $| f(z) | \leq | g(z)|$. What conclusion can you draw?
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Suppose $x_1,...,x_n \in \mathbb{R}^D$ are data points, and we introduce an outlier $x^o$ with the property that, for some $\delta 0$, $\Vert x_i - x^o \Vert_2 \delta$...
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Let $x_1,...,x_n \in \mathbb{R}^d$. Fix some positive integer $K$. Let $C_1,...,C_K$ be a partition of the data with centroids $\mu_1,...,\mu_K$. Let$$ F(C_1,...,C_k) \s...
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Let $x,y \in \mathbb{R}^{d \times 1}$. Prove that $xy^T \in \mathbb{R}^{d \times d}$ has at most rank 1.
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Recall that the variance of a set of numbers $x_1,...,x_n \in \mathbb{R}$ is defined as $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2$, where we define the mean as $...
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The function: \( v(t) = \frac{5}{2}t^{\frac{3}{2}}(7-t) - t^{\frac{5}{2}} \)Please find its derivative and show me the steps.
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Find (a) all horizontal and (b) all vertical asymptotes of the graph\[y=\frac{|x| \sqrt{4 x^{4}+6 x^{2}+4}}{(2 x-1)^{2}(x+1)}\]
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The number of guests allocated to each wine waiter at a function where various drinks are to be served, is ... people.
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\( f l\left(x_{2}\right)=\frac{-2 c}{b-\sqrt{b^{2}-4 a c}}=\frac{-2.000}{62.10-62.06}=\frac{-2.000}{0.04000}=-50.00 \)
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Solution \( e^{i \pi}+1=0 \)
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Explain \( e^{i \pi}+1=0 \)
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What does Euler's totient function \(\phi (n)\) represent?
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\( \int \frac{x}{x^{2}+4 x+5} d x \)
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\( \int_{0}^{\frac{\pi}{2}} \sqrt{ }(1+\sqrt{ } \sin x) d x \)
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\( \begin{array}{l}200 \% \times 200 \%=? \\ \begin{array}{ll}\text { A. } 40 & \text { C. } 4 \\ \text { B. } 400 \% & \text { D. } 0.4 \%\end{array}\end{array} \)
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if \( 2^{m+n}=32 \) and \( 3^{3 m-2 n}=243 \) find \( m: n= \) ?
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Plot the constant, linear, quadratic, and cubic Taylor polynomials for \( \cos (x) \) computed at \( x_{0}=0 \) over the interval \( [a, b]= \) \( [-\pi / 2, \pi / 2] \)....
122
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Prove the mean value theorem by first reducing to the case \( u(a)=u(b)=0 \) and then using the fact that \( u(x) \) must take on a maximum or minimum value for some poin...
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Suppose that \( u \) is continuous on \( [a, b] \) and differentiable on \( (a, b) \). Then there is a point \( \xi \) in \( (a, b) \) such that\[u(b)-u(a)=u^{\prime}(\xi...
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Write down a definition for \( \lim _{x \rightarrow \bar{x}} u(x)=\infty \).
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Compute, if possible, the limits: \( \lim _{x \rightarrow 0} \sin (x) \) and \( \lim _{x \rightarrow 0} \sin (1 / x) \)
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Solve the first-order linear differential equation \(\frac{dx}{ dy} ​ +y=e ^x \)
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Solve the separable differential equation \(\frac{dx}{ dy} ​ =xy\).
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Use Newton’s Method to determine \(x_2\) for the given function \(f(x)=7x^3-8x+4, x_0=-1\)
73
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Compute the differential of the given function \(u=t^2\cos{2t}\)
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Compute the differential of the given function\( f(x)=3 x^{6}-8 x^{3}+x^{2}-9 x-4 \)
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Find the linear approximation to \( h(y)=\sin (y+1) \) at \( y=0 \). Use the linear approximation to approximate the value of \( \sin (2) \) and \( \sin (15) \). Compare ...
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Find the linear approximation to \( h(y)=\sin (y+1) \) at \( y=0 \). Use the linear approximation to approximate the value of \( \sin (2) \) and \( \sin (15) \). Compare ...
91
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Find a linear approximation to the function \( g(t)=\mathbf{e}^{\sin (t)} \) at \( t=-4 \)
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