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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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1answer
18 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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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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1answer
22 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) ...
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1answer
146 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. ...
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1answer
24 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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1answer
28 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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19 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 ...
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2answers
63 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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0answers
26 views

Correct terminology for polylines, their segments, knots, etc.

Background: piecewise-linear continuous functions $f(x_k)=y_k$ with fixed set of knots $x_k$ with restrictions on the angles between adjacent segments. The translator who dealt with my paper, ...
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2answers
92 views

Regression analysis on temperature/sensor data

Looking for a solution to what I thought should be an easy problem, but has me running in circles somehow... I'm working with two sets of data: he first set is raw values from a sensor ...
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61 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 ...
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2answers
791 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 ...
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3answers
979 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 ...
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1answer
64 views

Which number is larger if f'(x) is a differentiable increasing function for all x?

Suppose is a differentiable increasing function for all x. Which number is the larger and why? or ? I believe f(x) must be concave up everywhere since the derivative is increasing, but I am not ...
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78 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 ...
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1answer
93 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, $v$, ...
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1answer
316 views

Relative error; absolute error divided by real value or approx value?

I have the following function; $$f(x) = -\frac{x}{2x + 4}\cdot v_r \Rightarrow \hat{f}(x) = -\frac{x}{4}\cdot v_r$$ Because $x$ is very small we can approximate $f(x)$ to $\hat{f}(x)$. Now the ...