Tagged Questions

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Big-O Notation for remainder terms in Taylor expansion

The Big-O notation is commonly used in Taylor expansions of the form $$f(x+\epsilon)=f(x)+\epsilon f'(x)+O(\epsilon^2)$$ to say that the remainder term grows at least quadratic around $\epsilon=0$. ...
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Partial derivatives + Taylor's Formula in several variables

Given a function $f(x) = (x_1+...+x_n)^k$, how do we show that $$D_1^{j_1}\cdots D_n^{j_n}f(x) = k!$$ if $j_1+...+j_n = k$?
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Third degree Taylor polynomial in two variables

How does one find the third-degree Taylor polynomial of $f(x,y) = (x+y)^3$ at the points $(0,0)$ and $(1,1)$? Many thanks
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Differentiating second order term of Taylor polynomial (multivariable)

I am trying to derive Newton step in an iterative optimization. I know the step is: $$\Delta x=-H^{-1}g$$ where H is Hessian and $g$ is gradient of a vector function $f(x)$ at $x$. I also know the ...
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Find the 3rd degree Taylor polynomial about the origin of $$f(x,y)=\sin (x)\ln(1+y)$$ So I used this formula to calculate it ...
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First order multivariate approximation

To demonstrate that $\nabla\!_{\hat{\boldsymbol u}}\,f(\boldsymbol{x}) \equiv \left \langle \hat{\boldsymbol u}, \nabla f(\boldsymbol{x}) \right \rangle$ I plug a first order expansion of ...
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How Hessian feature detector works?

I know about Harris corner detector, and I understand the basic idea of its second moment matrix, $$M = \left[ \begin{array}{cc} I_x^2 & I_xI_y \\ I_xI_y & I_y^2 \end{array} \right]$$, edges ...
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Bounding a continuously differentiable function using Taylor given the function is bounded by the norm of x

Suppose $0 < r < 1$ and that $f \colon B_1(0) \to \mathbb R$ is continuously differentiable. If there is an $\alpha > 0$ s.t. $|f(x)<\Vert x\Vert^\alpha$ for all $x \in B_r(0)$, prove ...
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Exact expansion of functions

Prove that for any twice differentiable function $f: {R}^n \to R$, $f(y) = f(x) + \nabla f(x)^T (y-x)+ \frac{1}{2} (y-x)^T \nabla^2f(z)(y-x)$, for some $z$ on the line segment $[x, y]$. Note that ...
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taylor's formula in $\mathbb{R}^n$ as exponential of derivative operator

I hope this question is not to vague. There is some relationship between the formal operator $$e^\frac{d}{dx} = 1 + \frac{d}{dx} + \frac{1}{2!}\frac{d^2}{dx^2} + \ldots$$ and taylor's formula. Is ...
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$f(y) \leq f(x)+\nabla f(x)\cdot (y-x)$ and $f(x)\geq 0$ implies that $f$ is constant.

Here is the question. Suppose that $f: \mathbb R^n \rightarrow \mathbb R$ has two derivatives and the associated hessian matrix is negative semidefinite on all of $\mathbb R^n$. Show that for any ...
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