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

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5
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1answer
249 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 ...
3
votes
1answer
58 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 ...
3
votes
0answers
94 views

Reason for the name “Ring of dual numbers”

The ring of dual numbers over a field $k$ is defined as the quotient $$k[\varepsilon]/\varepsilon^2.$$ I was reading this question with an interesting answer about some of their basic properties ...
3
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0answers
85 views

Question about linearization

Given a data matrix $D\in\mathbb{R}^{N \times N}$ Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$, if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j ...
2
votes
4answers
122 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 ...
2
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2answers
77 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) ...
2
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2answers
67 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 ...
2
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2answers
29 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 ...
2
votes
1answer
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 ...
2
votes
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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vote
2answers
3k views

Linear approximation to ln(x) at x = 1, then estimate ln(1.08)

I know that the derivative of $\ln(x)$, or log of whatever base (x) = $(1/x)$ *the original function. If x is a more complicated expression, then the derivative would be $(x'/x)*f(x)$. If I knew the ...
1
vote
1answer
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 ...
1
vote
2answers
58 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
44 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 ...
1
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2answers
84 views

Linear approximation formulas for $x$ not near $0$

I am following MIT's calculus videos and I have noticed that when dealing with linear approximations, the professor calculates a set of approximation "formulas" for $x$ near $0$ like $1+x$ for $e^x$ , ...
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2answers
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' ...
1
vote
1answer
47 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 ...
1
vote
1answer
4k 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. ...
1
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2answers
128 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. ...
1
vote
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 $v$ ...
1
vote
1answer
103 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?
1
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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 ...
1
vote
1answer
68 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 ...
1
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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 ...
1
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1answer
30 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
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1answer
36 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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0answers
30 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 ...
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) $ ...
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0answers
27 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 ...
1
vote
2answers
132 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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0answers
26 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
111 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
164 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 ...
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0answers
83 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 $X$ ...
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0answers
33 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 ...
1
vote
1answer
128 views

Use least squares to estimate coefficients in a linear system containing noise

Problem I have a linear system of the form, $y=Ax+v$, where $v$ is noise. I need to use least squares to estimate the coefficients of the matrix $A$. Atempt I made the assumption that the error, $v$, ...
0
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3answers
200 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 ...
0
votes
3answers
3k views

Linear approximation to $y = \sqrt{1-x}$ at $x=0$, then approximate $\sqrt{0.9}$ and $\sqrt{0.99}$

How do I find this? I know that the derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$. Here, I would plug in $(1-x)$ instead of $x$. When $x = 0$, the slope would evaluate to $\dfrac{1}{2}$. I got ...
0
votes
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 = ...
0
votes
2answers
26 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 ...
0
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2answers
303 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. ...
0
votes
2answers
160 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 ...
0
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1answer
38 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
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2answers
199 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 ...
0
votes
1answer
144 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 ...
0
votes
1answer
25 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 ...
0
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1answer
32 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 ...
0
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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) = ...
0
votes
2answers
64 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: ...
0
votes
1answer
20 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 ...