Questions tagged [linear-approximation]

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

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48 views

Can someone help me attempt this question? I don't know what equations to propose. Thank You.. [closed]

Propose a nonlinear system of equations that contains at least four variables, and its corresponding initial approximation to solve the equations. The initial approximation should contain both ...
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1answer
59 views

(Why) Do Functions have to be Twice Differentiable to use linear approximation?

According to Wiki on Linear Approximation Given a twice continuously differentiable function $f$ of one real variable, Taylor's theorem for the case $n = 1$ states that $$f(x) = f(a) + f'(a)(x - a) +...
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$\zeta(1 + 2/x)$ has a strange, nearly linear behaviour

I was messing around with some infinite sums in $\ell^p$ spaces and I encountered a strange result: $\zeta\left(1 + \frac{2}{x}\right)$ looks like it is linear in $x$ for $x > 1$! A simple linear ...
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Why is the linear approximation of my function, after I remove negligible terms, more accurate than the linear approximation not removing the terms?

I have the function $ F(x) $ where $ x >> a $ and I have derived two linear approximations of $ F(x) $: $ L_{1}(x) $, where I take that $ \frac{\left(a^{2}-2xa\right)}{x^{2}} $ is way way less ...
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42 views

Taylor approximation of $\phi_1(\lambda) = \frac{1}{\sqrt{\psi(\lambda)}} - \frac{\sigma}{\lambda}$

I am reading this paper. In some point of the analysis the non linear equation $$\phi_1(\lambda) = \frac{1}{\sqrt{\psi(\lambda)}} - \frac{\sigma}{\lambda} \tag{A}$$ is studied, i.e. $(6.7)$ in the ...
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20 views

Number of fixed points in a fixed point iteration function

I'm trying to understand if a fixed point iteration function can have more than 1 fixed points. In theory I tried to find an f(x) such that I could generate an iteration function g(x) in a way that g(...
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3answers
72 views

How exactly is the value of $\frac{1}{y\sqrt{y^2+\frac{l^2}{4}}}$ as $y \gg l$ calculated?

I am having trouble understanding if a particular calculation is or is not a limit calculation. I suspect it is not, but a particular set of notes from an MIT OCW physics course (problem starts on ...
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1answer
40 views

linear approximation of a function at a number

question: Find linear approximation of the function f(x) = √x at x=9. Use it to approximate √9.1 linearisation formula L(x) = f(a) + f'(a)(x-a) In this question, a=9 right? and when we substitute ...
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13 views

Maximise sum of squares eigenvector problem

I'm looking to find the vector $x$ that maximises the following equation: (1) - $\Sigma_i (x^TA_ix)^2$ I have 9200 matrices $A_i$ and each matrix is 160$\times$160. However each matrix only has about ...
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What can be inferred about the original function from the best RMSE linear approximation

This is a very open question. Assume I have a continuous, differentiable function $f$ on a closed interval $[a,b]$. Now I also have a linear approximation of this function $\lambda x+\tau$ that is ...
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29 views

What is the computational complexity (in flops) of nonnegative least square optimization?

Suppose I have a vector $x\in \mathbb{R}^D$ and a matrix $U\in\mathbb{R}^{m\times D}$. I would like to solve the following nonnegative least squares optimization problem: $a = \text{argmin}_{y\in \...
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27 views

Using the tangent line of a value to approximate a value

Good Day, For some reason, my brain is failing to craft a reasonable solution to this equation. Question: Use the tangent line of $f(x) = \sqrt x$ at $a=16$ to approximate $\sqrt 17$ I believe we are ...
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6 views

Should the approximated values automatically be increasing when the experimental data is increasing?

I am trying to approximate groups of discrete data which are monotonically increasing by continuous peicewise linear function. The objective is minimizing the sum of absolute error. Numerically, I ...
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1answer
47 views

Approximating $\ln(1+2x) = 2x$ using linear approximation

The question is as follows: Use the linear approximation formula $$\boxed{f(x+\Delta x) \approx f(x) +f'(x)\Delta x}$$ to show that $\ln(2x+1) \approx 2x$ for small values of $x$. I am having ...
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31 views

Efficiently approximate Ax=b if you can choose values for A

Here is the situation. You have a black box x, whose values are not known to you, but you can choose values for A and generate the corresponding b value using Ax=b. There is some noise in the system ...
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52 views

What is the reason for this step in proving the Borsuk-Ulam Theorem (by triangulation)

I'm reading Using the Borsuk-Ulam Theorem, which presents several proofs of the theorem. Since this is a book for combinatorialists, the first involves a triangulation $\mathsf{T}$ of the $n$-...
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1answer
26 views

Why does the linear approximation of $f \circ g$ near $a$ imply $f$ gets linearized by $g$ for a small enough neighborhood of $g$ near $a$?

I've noticed that, if we input the linear approximation of $g$ near $a$ into $f$, we get a decent approximation (althought non-linear) of the linear approximation of $f \circ g$ near $a$. In order to ...
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34 views

Finite element approximation of the cofactor matrix of the Jacobian

Let $\Omega = (0,1)^d$ and assume there is a tensor product mesh $\mathcal T_h$ of quadrilaterals, of maximum diameter $h$, covering $\overline \Omega$, i.e., $\overline{\Omega}= \bigcup_{K\in\mathcal ...
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These are two similar problems on estimating error; Why does one involve taking a linear approximation, and not the other?

I'm having a hard time understanding where to apply linear approximation. These two problems seem very similar to me, except one does not use linear approximation: Problem 1: Suppose that you measure ...
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2answers
38 views

In this error approximation, why can we "kill" the $\Delta x\cdot \Delta \theta$ term, but not the term with $\Delta \theta$ alone?

While trying to solve a problem I've come to this equation -- where $\Delta y$ is measurement error of $y$: $$\Delta y = \Delta h+\Delta x \cdot \tan(\theta)+x \cdot \sec^2(\theta) \cdot \Delta\theta +...
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1answer
35 views

What is the purpose of taking the linear approximation of this tangent function for estimating the measurement error?

$h$ = distance from ground to eye of a person, who measured the angle to the top of the building with a protractor. $θ$ = from the eye to the top of the building. $x$ = distance of the person from the ...
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49 views

Best linear approximation for x(x-y)

I have a non-linear term in the form of $$x(x-y)$$ I need to replace it with a linear term in the form of $$c_1x-c_2y$$ I have seen the following linear approximation in literature but I need to know ...
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Is this manipulation of the Taylor series valid?

The Taylor series of a function $f(x)$, (assuming differentiability) is: $$\tag{1}f(x)=f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^2+...$$ If I take $x=x+h$ I get: $$\tag{2}f(x+h)=f(a)+f'(a)(x+h-a)+\frac{1}...
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1answer
69 views

Struggles understanding the derivation of the Householder's method

The Wikipedia article on Householder's method for root-finding derives it in their first approach with: I understand this first part. They continue by saying: "The coefficient of degree $d$ has ...
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2answers
115 views

Proving that pressure is approximately equal to $P_0 \left(1 + \frac{k}{2} M \right)$

After conducting a series of experiments, a physicist concluded that the pressure around an object placed in a moving fluid is given by $$P(M) = P_0 \left( 1 + \frac{k - 1}{2}M \right)^{k/(k-1)},$$...
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32 views

Picking a numerical methods for solving stiff ODE systems with fixed time step.

I am currently studying analytical and numerical methods and can't wrap my head around which methods to choose from. The methods that I have learned so far are: Forward Euler Backward Euler Modified ...
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36 views

One-sided Difference Vs Ghost Node Approach.

Can someone tell me the difference between Central Difference and Ghost Node Approach for approximating ODE's? What is the benefit of one over the other, what are the disadvantages of one over the ...
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pertubative approach starting from an approximated probability distribution

I approximate a probability distribution $P_x(x)$ with a $P_x^{app}(x)$, such that $P_x(x)-P_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity. Consider the generic ...
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Name of approximation by linearly adding basis functions and its traits e.g. fourier transfer

Questions What is the name of the operation which approximates a continuous function by a linear combination of basis functions? What kind of traits do they have? Methods of function expansion ...
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179 views

Is the linear approximation of the product of two functions the same as the product of the linear approximations of the two functions?

I have just seen a lecture about linear approximation, in which it was established that when $x$ is near $0$: The linear approximation of $e^x$ is $1+x$ The linear approximation of $(1+x)^r$ is $1+...
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How does linear approximation give an approximation over an entire interval?

In my textbook, there is a statement as follows: I don't understand how "linear approximation gives an approximation over an entire interval". I guess that it may have something to do with ...
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1answer
139 views

Composition of Linear Approximation

Suppose you have $x,y,z\in\mathbb{R}$ where $$y:=f(x)$$ $$z:=g(y)$$ for some smooth functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$. Note that there is $h$ (the composition of $g$ and $f$) such that $$...
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1answer
118 views

Response surface and variables importance

Disclaimer: I'm not expert in math; the solution to this problem may be trivial. I have a process based on a deterministic computer simulation where two continuous input variables $x_1$ and $x_2$ (...
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1answer
27 views

Linear approximation using the generalised formula

I have been reviewing my analysis course and there is something that seems odd to me. My professor has wrote that the linear approximation for a function of two variables is : $ x = x_{0} + h$ $ y =...
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159 views

Average, Instantaneous Change And Linear Approximation

I have these 2 question: A function f(x), is any function, so that $L_{a}(x)=L_{b}(x), a \not= b$. Show $\frac{f(b)-f(a)}{b-a}=\frac{f'(b)+f'(a)}{2}$. Assume that $a \not= b$ and that $\frac{f(b)-f(a)...
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How can I use for the generalized logistic equation a series approximation?

The question is how can I show the following approximation stated in this paper. I have tried to follow , but I am not pretty sure if the following results are correct Generalized logistic growth ...
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158 views

Two dimensional Taylor Series Linear Approximation

Hi so I'm Currently studying Multivariable Calculus and Real Analysis and have came to a wall on this question. I know from reading my notes that I have to find F(x,y), Fx(x,y), Fy(x,y), Fxx(x,y), Fyy(...
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1answer
51 views

Graphic Interpretation of the Linear Approximation for Vector Valued Functions

A linear approximation of a function $f:\mathbb{R} \to \mathbb{R}$ at the point $x=a$ draws a tangent line to $f(x)$ at $(a,f(a))$ and approximates values of $f(x)$ for $x$ near $a$ by using the ...
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1answer
54 views

find linear approximation

given function $f:\mathbb{R^2} \to \mathbb{R^2}$ $f(x,y)=(x^3y^2-y,xy^3-x)$ how do we calculate linear approximation to $(fof)(1+h,1+k)$ for $h,k$ near $0$ i got this $f(x_o+ \triangle x,y_o+\...
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1answer
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How to find the linearization of this function($ x^4+y^3-x^3-y^2+x^2+y-2=0$)? [closed]

This past paper question about linearization for my calculus 1 course has been bothering me. I spent 5 days trying to solve it. Considering the equation: $ x^4+y^3-x^3-y^2+x^2+y-2=0$ We accept and can ...
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Linearization of a non-linear model

Context:-Consider you are holding a plate of cookies in your hand at angle $x$ with force $k$(you may change this at your discretion.The plate puts a force $f$ on your hands. Some people take cookies ...
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Diferential equations's linear approximation

Take for example the well known $$ \ddot{\phi} + \frac{g}{l} \sin{\phi} = 0.$$ Usually, in a physics course they might say that by Taylor's Theorem we can write, for small angles: $$\ddot{\phi} + \...
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72 views

How to provide a linear approximation from the chain rule

If given an equation $$ z=e^{xy} $$ where $$ x = 1+st , y = s^2-t^2 $$ I found $$dz/dt , dz/ds$$ using the chain rule and got the following: $$ dz/ds = e^{xy}(yt+2xs)$$ $$dz/dt = e^{xy}(ys-2xt) $$ ...
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2answers
154 views

L'Hôpital's rule to approximate a function

I came across a simple problem: Show $\sqrt{1+x}\approx 1 + \frac{x}{2}$ holds near $0$. Interesting to me was the way the author solved this using l'Hôpital's rule: Since $\lim_{x\rightarrow 0}\frac{\...
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43 views

Proof of the Formula for Linear Approximation With $n$ Variables?

The linear approximation of $f(\vec{x})$ where $\vec{x}$ has $n$ elements is given by $$f(\vec{x}) \approx f(\vec{a}) + \sum_{j=1}^n \frac{\partial f}{\partial x_j} (\vec{a})(x_j-a_j)$$ for $\vec{a}$ ...
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76 views

A ball of ice melts so that its radius decreases from $5 cm$ to $4.92 cm$. By approximately how much does the volume of the ball decrease?

The following exercise is presented in my book as an example of linear approximation and calculating the rate of change using derivatives. The rate of change for the volume gives $ \approx-25.13 cm^3 $...
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2answers
156 views

In this textbook explanation of needing partial derivatives, how is this partial derivative not an indeterminate form?

$$ f(x,y) = x^\frac{1}{3}y^\frac{1}{3} $$ $$\frac{\partial f}{\partial x}(0,0) = \lim_{x \to 0} \frac{f(h,0)-f(0,0)}{h}= \lim_{x \to 0} \frac{0-0}{h} = 0$$ "and, similarly, $\frac{\partial f}{\...
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22 views

Approximating $\int_{-\infty}^\infty e^{-f(q)/\hbar} dq$

I'm a Physics student (self-studying Zee's Quantum Field Theory in a Nutshell) hoping to get the following mathematical query solved. It is mentioned that to approximate $I=\int_{-\infty}^\infty e^{-f(...
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1answer
49 views

How to construct the matrix of a 2-D system of differential equations

The task is to find and classify the fixed (equilibrium) points of the system : $$ \begin{cases} \dot x = (2x - y) (x - 2) \\ \dot y = xy - 2\end{cases}$$ Solving $$ \begin{cases} \dot x = (2x - y) ...
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1answer
32 views

What is the power series of $1/(D-h(x))$ if $h(x)\ll D$?

I have a problem, which I do not conceptually understand. I need to approximate an arbitrary function $$\frac{1}{D-h(x)}$$ where $h(x)$ is arbitrary, $h(x)\ll D$ and $D$ is a constant. Friends say ...

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