For questions about linear approximations, $f(x) \approx f(a)+f'(a)(x-a)$ for $x$ around $a$.

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Jacobian matrix and Taylor expansion

Let $\mathbf{W}(\alpha)$ be a matrix which depends to parameter $\alpha$ and let $\mathbf{f}$ be a vector. I want to approximate $\mathbf{W}(\alpha+\Delta \alpha)\mathbf{f}$ using Taylor expansion. ...
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30 views

Dual form of $L_1$ norm approximation as a linear programming problem

According to my text: Given an overdetermined system, the residual vector is: $$\textbf{r} = \textbf{Ca} - \textbf{f}$$ The $L_1$ norm approximation seeks to minimize the residual r: $$\text{...
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27 views

When linearizing nth degree polynomials, is there any advantage in using Taylor series versus taking n derivatives?

If I need to get a linear approximation of a nonlinear function (linearize), for example approximate the values of a nonlinear function with a tangent line about point ...
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16 views

Question about linearising system with second derivative

I need to linearise a system: $\ddot{x}+4\dot{x}^5+(x^2+1)u=0$. The referenced answer is :$\ddot{x}+0+(0+1)u\approx0$. So, the linearly approximated about $x=0$ is: $\ddot{x}=-u$ I can understand ...
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19 views

Using Linear Kalman Filters with a Nonlinear System?

Can you answer these questions I have about using linear Kalman filters and extended Kalman filters with a nonlinear system? 1. Does using a linear Kalman filter mean that I must have a time-...
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27 views

percentage decrease of the edge of an icecube

I have a question that is asking to find an approximation for the percentage the the edge length of an ice cube will decrease if the cube loses six percent of its volume. The question instructs us to ...
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1answer
38 views

Linear Approximation for functions with first derivative as $0$

Linear approximation around a point through Taylor series requires the first order derivative to be non-zero unless you want to get only the value at that point. However this is only true when you are ...
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21 views

Implementing Recursive descent algorithm for PWL Approximation

I am currently trying to linearise a convex function at hand (an M/M/1 curve) using piecewise linear functions. Since I wanted the approximation error to be as low as possible, I searched for some ...
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25 views

approximating functions via a piecewise combination of linear and constant functions

I am curious if anyone has encountered any literature on approximating functions via a piecewise combination of linear and constant functions. I have seen a couple of papers which use piecewise ...
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78 views

Linear approximation to the product: $\prod_{k=0}^r\left(1+\frac12\left(\frac{\frac12+k+1}{\frac12+k}-\frac{\frac12+k}{\frac12+k+1}\right)\right)$

I have come upon with the next expression: \begin{equation} P_r=\prod_{k=0}^r \left(1+\frac{1}{2}\left(\frac{\frac{1}{2}+k+1}{\frac{1}{2}+k} -\frac{\frac{1}{2}+k}{\frac{1}{2}+k+1}\right)\right) \end{...
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32 views

Linear independent set of function applied to water waves.

I need to show that a given surface elevation $\zeta(x,y,t)$ defined on a closed region $D(x,y,t): 0<x<L_x,0<y<L_y,0<t<T$ and not periodic on D: $$ ζ(x,y,t) = \sum_{n=1}^{\infty} ...
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24 views

Algorithm for finding linear recurrent approximations to integer sequences

Is there an algorithm for taking a sequence of integers and approximating a first part of it piecewise if need be with pieces like: $$ \text{ if } n = 2k, \\ a_n = a_{n-1} - a_{n-2} + 1 $$ then, ...
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31 views

Defining the differentiability of a multivariable function (if/then)

I'm trying to understand differentiability for multivariable functions and am thoroughly confused by the introduction (and the direction of implications in a certain definition) I'm given the ...
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44 views

Finding a two-variable function that is distinct from another on every open disk, with specifics.

Consider the two-variable function $$f(x, y) = \sin(x) + \cos(x) + y^2.$$ Find a two-variable function $g(x, y)$ that is distinct from $f(x, y)$ on every open disk which contains the point $(1, 2)$ ...
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35 views

Finding $w_1,w_2,b$ such that $w_2 \sigma(w_1 x + b) \approx x$

Let $\sigma(z) = 1/(1+e^{-z})$. How can I find $w_1, w_2, b$ such that $w_2 \sigma(w_1 x + b) \approx x$ for $x \in [0,1]$? The hint provided was to rewrite $x = \frac{1}{2}+\Delta$, assume $w_1$ is ...
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71 views

Using linearization to calculate the thickness of a layer of paint on a spherical ball

The volume of a sphere with radius $r$ is given by the formula $V(r) = \frac{4 \pi}{3} r^3$. a) If $a$ is a given fixed value for $r$, write the formula for the linearization of the volume function $...
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170 views

% Error of Linear Approximations: Example Problem

I received the following question on my exam and got it right, although it was entirely a guess and I had absolutely no idea how to approach it. Any help with the logic or steps behind this would be ...
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1answer
26 views

Minimizing sums of values versus minimizing cubes of sums.

I am attempting to find the best path from start to finish from a set of points. Say that one path has costs $x_1,x_2,...,x_n$ and $y_1,y_2,...,y_n$ associated with it. I am attempting to find the ...
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1answer
40 views

$f(x) = e^x$ and $a = 1$. Find the linear approximation $L(x)$

I a little confused on this question and I feel I shouldn't be. So, I take the derivative of f(x) which is $f'(x)=e^x$ Next I plug in the point $a = 1$, which then gives me the slope $2.71$ Knowing $...
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35 views

How to find the point after which a discrete function follows a linear and steady trend

I have many discrete functions that follow the same trend. An example of discrete function is shown in the figure below. At each step, represented on x-axis, we reduce a given area, represented on ...
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45 views

linear approximation of $f(x)$

Let $y=f(x)=(x_1^2+2x_2, x_1x_2-3x_1)$ Is the linear approximation just $f(y)=f(x)+A(y-x)$ whenever $y$ is approximately near $x$? I know that if I calculate the Jacobian matrix, I can get that $Df=\...
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135 views

Using linear approximation for a two variable function to estimate $0.999^{10}(1 + \sin(0.01))$

I am trying to evaluate $$0.999^{10}(1 + \sin(0.01))$$ using linear approximation for a function with two variables, but I am a little confused as to how to do that, as I don't have any x or y terms. ...
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24 views

Let $l(x)$ be the linear approximation of $f(x) = x^{2/5}$ at $a = 32$. Approximation?

I'm still a bit confused on how to figure out linear approximations. What are the basic steps to solving a problem like this? Thanks so much! Let $l(x)$ be the linear approximation of $f(x) = x^{2/5}$...
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1answer
31 views

Rewriting approximated terms

The following data are inferred from a presentation slide, so I do not much info. Using linear approximation and log rules $\sqrt x $ can be rewritten as $\frac{x+1}{2}$, where $(1 \leq x \lt 2) $ ....
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93 views

Finding the closest function describing a “magnetic line” (given magnetic readings)

I'm collecting data from a smartphone magnetometer while I move a magnet along a straight line (a slider). I am collecting the values of the magnetic field strength along the three axes. I would like ...
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4answers
141 views

The Significance of Linear Approximation

I want to know what makes linear approximation so important (or useful). What I am aware of in my current state of limited understanding is that linear approximation is one of the applications of a ...
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1answer
26 views

Matrix Approximation with outer difference

Given a skew-symmetric matrix $A$ what is the best approximation by an outer difference of a vector. Approximation norm could be either Frobenius or Euclidean. The outer difference $D$ of vector $v$ ...
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65 views

Proving Lower bounds on an Approximately Linear Function

We are looking for a lower bound on the function, $\frac{1.31}{e^{\frac{1.31}{x+1}} - 1}$ for $x \geq 2$. This function seems to behave linearly. We believe that the following statement holds: $$\...
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1answer
115 views

Implicit differentiation and linear approximations

Consider the implicit function $$(w(x)+1)e^{w(x)}=x.$$ I need to approximate $w(1.1)$ using the fact that $w(1)=0$. Could you give me any hints?
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59 views

Minimizing Area by Approximation

Suppose I have an increasing step function $E_c$ given by $$E_c(\phi) = \sum_{i=1}^n E_i \theta(\phi - \phi_i),$$ where $\theta$ is the Heaviside step function and $E_i$, $\phi$, and $\phi_i$ are all ...
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1answer
41 views

Basic Linear Approximation

I have come across the need to quickly perform linear approximations, for example I ran across this simplification provided r << d (I think maybe it should be r >> d). $2(r + d)^{-2} - r^{-2} -...
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22 views

Differentials to find approximate values

I'm asked to solve the following without a calculator: $80^{3/4}$ I only know that $f(x+dx) \approx f(x) + dy$ I then proceed to find $dy$, it should follow that if $f(x) = x^{3/4}$, then $dy = \...
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Newton's method $f(x) = \frac{1}{2} x^8 + \frac{8}{5} x ^5+ 3 x +10$ near the point $x = 3.$

$\displaystyle f(x) = \frac{1}{2} x^8 + \frac{8}{5} x ^5+ 3 x +10$ near the point $x = 3.$ Use $x_1 = 3$ as the initial approximation. Find the next two approximations, $x_2$ and $x_3$, to four ...
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368 views

How to approximate Heaviside function by polynomial

I have a Heaviside smooth function that defined as $$H_{\epsilon}=\frac {1}{2} [1+\frac {2}{\pi} \arctan(\frac {x}{\epsilon})]$$ I want to use polynominal to approximate the Heaviside function. ...
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31 views

Quadratic/ Cubic/ etc approximations without the Taylor series

It's easy to convince someone that the linear approximation is the best line which approximates a function at a point because everyone learns early that the derivative of a function is just the slope ...
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1answer
21 views

error in approximation a monotonic function in L^1

I am trying to solve this problem, but I am getting an incorrect solution. Here is the problem and my approach: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize $\int_0^...
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57 views

Linear approximation to find partial derivatives

If the equations $f(x, y, u, v) = 0$ and $g(x, y, u, v) = 0$ can be solved for $u$ and $v$ as differentiable functions of $x$ and $y$, compute their first partial derivatives. Pretty lost on this one....
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222 views

Linear approximation with two variables

The problem I have is this: Use suitable linear approximation to find the approximate values for given functions at the points indicated: $f(x, y) = xe^{y+x^2}$ at $(2.05, -3.92)$ I know how to do ...
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46 views

Linear Approximation of a quantity

How do i proceed estimating this quantity using Linear Approximation? $$\dfrac{1}{\sqrt{95}}-\dfrac{1}{\sqrt{99}}$$ My understanding is that I need to decide what the function is, find a 'nice' ...
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31 views

Second Order Approximation for a Polynomial

if I have an expression: $L=\frac{12a^3d^3-4wa^3d^2+16a^2d^2-4wa^2d+6ad+1}{12a^3d^3-4wa^3d^2-4a^2wd+16a^2d^2+7ad-aw+1}$ what is the second order approximation in $\frac{d}{w}$? I know that $(\frac{d}...
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40 views

Messing with the linear approximation… I don't get Taylor's formula??

Alright, so I was messing around with the linear approximation for a function; I wanted to see if I could get the formula for the 2nd degree approximation from it. I went about it like this: $$ f(x + ...
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204 views

Find all planes which are tangent to a surface

I'm given the surface $z=1-x^2-y^2$ and must find all planes tangent to the surface and contain the line passing through the points $(1, 0, 2)$ and $(0, 2, 2).$ I know how to calculate tangent planes ...
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69 views

How is Taylor expansion a generalization of linear approximation? [closed]

The concept of derivative is related to linear approximation of a function: $$f(x)\approx f(a)+f'(a)(x-a)$$ I was told that this linear approximation is generalized by the Taylor series. What does ...
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35 views

Linear approximation to a system in the neighbourhood of the origin?

What would a linear approximation to the following system near the origin be? $${dx \over dt}=-y-x(x^2+y^2), {dy \over dt}=x-y(x²+y²)$$ I have no idea how to find this... I'm looking at this as an ...
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1answer
42 views

Approximation of $f\in C[a,b]$ by functions constant on intervals of length $(b-a)/2^n$

I read (p. 405 here) that a continuous function on $[a,b]$ can be arbitrarily approximated according to the distance of space $L^2[a,b]$ defined by $d(f,g):=\|f-g\|_2=\int_{[a,b]}|f-g|^2d\mu$ with $\...
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Linear approximation with different modifiers

The given function was $$f(x)=ln(\frac{2}{x})$$ and I had to compute the linear approximation at x = 2. I obtained the answer of $$L(x)=-\frac{1}{2}(x-2)$$ I am then supposed to use that ...
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116 views

Expanding in powers of $\epsilon$ and big O notation

I do not understand how to approach (D.1) equation Where did that big O notation come from?Is it using taylor series and linear approximation? Thanks in advance
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49 views

Linear approximation, new volume versus volume change

Would it be correct to say, supposing that we are dealing with the volume of a ball (just to exemplify), that: $f(a+\Delta x)\approx L(a+\Delta x)=f(a)+f'(a)\Delta x$ is an approximation of the new ...
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59 views

Using linear approximation to approximate $\sqrt{81.3}$

Use linear approximaiton to approximate $\sqrt{81.3}$ as follows: Let $f(x)=\sqrt{x}$. The equation of the tangent line to $f(x)$ at $x=81$ can be written in the form $y=mx+b$ where $m$ is:____ and ...
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45 views

Finding the linear approximation of $\frac{1}{\sqrt{2-x}}$ at $x=0$

The linear approximation at $x=0$ to $\dfrac{1}{\sqrt{2-x}}$ is $A+Bx$, where $A$ is: _____, and where $B$ is: ______. I don't understand what this question is asking and how to solve it. I know ...