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1answer
20 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) = ...
1
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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) $ ...
0
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1answer
63 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 ...
2
votes
4answers
61 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 ...
1
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1answer
19 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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2answers
43 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: ...
1
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1answer
30 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?
3
votes
1answer
53 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
13 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 ...
0
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1answer
31 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} ...
0
votes
2answers
21 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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0answers
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 ...
0
votes
1answer
81 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. ...
2
votes
2answers
20 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 ...
0
votes
1answer
16 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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0answers
15 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 ...
0
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1answer
37 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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2answers
52 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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2answers
42 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' ...
0
votes
1answer
24 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 ...
0
votes
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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0answers
20 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 & ...
0
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0answers
42 views

Log-linearization around the steady-state

I am doing log-linearization around the steady-state and currently stuck to the following two problems: $K_{t+1}= \frac{1}{u_{t}}[(a+q_{t})K_{t}-b_{t}]$ ...
1
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2answers
84 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 ...
0
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0answers
37 views

What conditions are required to approximate an unknown function with trapezoidal rule?

What conditions are required to approximate an unknown function with trapezoidal rule? I have been told that function should be linear, what does that mean? Does it mean for their graph to be a ...
1
vote
2answers
50 views

Generalization of linear approximation? [closed]

How is the linear approximation is generalized to the Taylor series? I do not get that concept.
0
votes
1answer
31 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 ...
2
votes
1answer
33 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 ...
1
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0answers
23 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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vote
0answers
77 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
1
vote
1answer
40 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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vote
2answers
42 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 ...
1
vote
2answers
40 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 ...
0
votes
2answers
77 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 ...
1
vote
1answer
51 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 ...
0
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3answers
42 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)}, ...
0
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0answers
48 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 ...
1
vote
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 ...
4
votes
1answer
197 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 ...
1
vote
1answer
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 ...
0
votes
0answers
42 views

Euler's method and its error

A student asked me how to approximate the error of Euler's method, and the book he is using said "If a value of a function is approximated with $n$ steps, the error is proportional to ...
0
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3answers
129 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 ...
1
vote
1answer
59 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 ...
0
votes
1answer
50 views

Use economisation to find linear approximation to x^2-x-1?

I've been given the solution to this question... It uses chebychev, and you get: $1/2(2x^2-1)-2x-1/2$ So the Chebyshev economisation polynomial is $-2T1 -1/2 T0$ I can see the logic in how this ...
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0answers
75 views

Questions About The Linear Algebra Behind Least Square Approximation

I am working on a few linear algebra problems, and I am stuck. I was hoping to get some directions on this site. Before I state my questions, here's the necessary context: Given a matrix $X$ ...
1
vote
1answer
28 views

How do I do Linearization at a point that lies on a curve?

I keep applying the formula to the info given but I keep getting lost/weird answers. Can someone please help me? I know $L(x)=f(a)+f'(a)(x-a)$ question Y(x) satisfies $x^2y^2 + xy = 6$. Point (x,y) ...
1
vote
1answer
2k views

Linear approximation of cos(28 degrees)

Evaluate cos(28 degrees) using linear approximation. I have done this so far, but the answer seems to be incorrect and I can't figure out why. ...
0
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1answer
33 views

Write an equation that approximates this relationship. Y = Seconds of daylight in day, X = a range of days

Disclaimer: This is a project for a math class. (who does math for fun anyways? Jk I actually enjoy math when I understand it, and not so much when I feel lost, but I digress ) The problem roughly ...
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2answers
64 views

Is best approximation from a linear subspace a linear map?

Let $X$ be a strictly convex Banach space, and $Y \subset X$ a closed subspace. Then for any $x \in X$ there exists a unique $y \in Y$ that minimizes the distance to $x$, i.e. a best approximation of ...
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0answers
29 views

Iterative approximation of non-constant values in linear equation

The issue regards an algorithm for iterative approximation of unknown transaction values. For each iteration (each day), we are give the total revenue of all transactions for that day, and we have the ...