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

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lineariztion of a nonlinear system

Consider a system described by the following nonlinear differential equation $\ddot{y} \ln{x} + \dot{x} (\dot{y})^{3/2} + x^2y = 1$. Let $x_0$ and $y_0$ be positive real numbers. Linearize the ...
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2answers
39 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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34 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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26 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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85 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
23 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
36 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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1answer
29 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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1answer
43 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 ...
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102 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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1answer
23 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) = ...
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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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88 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
103 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
25 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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62 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
80 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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57 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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0answers
19 views

How do I linearize an equation in discrete “z” space

I have a discretized transfer function, shown below: $$\frac{az^{-1}(e^{-ak}-e^{-bk})}{{a e^{-a k-b k}}z^{-3} - {b e^{-a k-b k}}z^{-3} - {a e^{-b k}}z^{-2} - {a e^{-a k-b k}}z^{-2} + {b ...
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1answer
37 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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2answers
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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24 views

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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213 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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2answers
27 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
19 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 ...
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27 views

linear approximation of two variables function

Let $\displaystyle f(x,y)=\frac{\sin(x+y)}{\sin(x)}$. Find linear approximation of $f$ near $\displaystyle \left(\frac{\pi}{2},0\right)$. My try: The linear approximation of $f$ is ...
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1answer
52 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 ...
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132 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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45 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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27 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 ...
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1answer
39 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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25 views

Deriving Matlab Algorithms for Linear Approximation (LU factorization)

This is Question 3 from here For part 2 I got the matrix L and U to be (using LU factorization): $L=\left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ -1/2 & 1 & 0 & 0 & ...
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117 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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65 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
39 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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25 views

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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0answers
98 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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1answer
47 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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51 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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41 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 ...
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138 views

Proof that the best linear approximation to $f(x)$ near $a$ is given by the linear function $L_{a}(x) = f(a) + f '(a)(x-a)$

The title basically says everything. The formula for linear approximation appears to be right intuitively but is there a proof for it? Secondly, is there also a proof why to put the $\frac{1}{2}$ in ...
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1answer
64 views

Estimate value using Lagrange's MVT

Estimate the value of $51^{1/2}$ using Lagrange's MVT. Answer both in terms of inequalities and approximately estimated value. My method: Let $f(x)=x^{1/2}$ defined in $[49,51]$ and ...
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43 views

For what values of $r$, $x^r$ has infinite slope at $x=0$?

I'm learning calculus form MIT OCW 18.01SC. In session 23 (it's about linear approximation), prof computes linear approximation near $0$ of some basic functions. $$\sin{x}, \cos{x}, e^x, \ln{(1+x)}, ...
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67 views

solve equation involving digamma function

I have the following equations that I need to solve. $$ \psi(\alpha)-\psi(\alpha+\beta)=X_0 \\ \psi(\beta)-\psi(\alpha+\beta)=Y_0 $$ $X_0$ and $Y_0$ are known constants. Is there a way to atleast ...
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1answer
46 views

Arc Length in two dimensions by integration

I'm really at the end of my wits on this problem. Basically I'm trying to find arc length. The vector-valued function is: $R=\langle t,\sqrt{t}\rangle$ and $t\ge0$. We're looking for the length of ...
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233 views

Computer Algebra Systems for Experimental Mathematics (especially Integer Relations with PSLQ)

I would like to use a computer algebra system to do some experimental mathematics, particularly Integer Relation problems using the PSLQ algorithm. I know that Maple has a PSLQ implementation, but ...
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26 views

Most accurate linear approximation for two lines

Consider two lines defined by: $$\begin{aligned}y_1 &= m_1 x + b_1\\y_2 &= m_2 x + b_2\end{aligned}$$ where for the sake of argument, the domain of both lines is the same and everything is ...
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184 views

Where did the linear approximation/linearization formula come from?

Where did the linear approximation/linearization formula come from? I understand that it takes root in the point-slope form and slope intercept form of a linear equation, but I don't understand where ...
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126 views

Determine the values of x for which the linear approximation is accurate to within 0.1.

So I've got a function: $$\frac{1}{(1+2x)^4}$$ with its linear approximation: $$1-8x$$ For all values of $x$ where the linear approximation is accurate within $0.1$, then surely we subtract the ...