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

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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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24 views

approximating functions via a piecewise combination of linear and constant functions

I am curious if anyone has encountered any literature on approximating functions via a piecewise combination of linear and constant functions. I have seen a couple of papers which use piecewise ...
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40 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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66 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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82 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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59 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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38 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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1answer
171 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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1answer
15 views

Dual form of $L_1$ norm approximation as a linear programming problem

According to my text: Given an overdetermined system, the residual vector is: $$\textbf{r} = \textbf{Ca} - \textbf{f}$$ The $L_1$ norm approximation seeks to minimize the residual r: ...
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15 views

When linearizing nth degree polynomials, is there any advantage in using Taylor series versus taking n derivatives?

If I need to get a linear approximation of a nonlinear function (linearize), for example approximate the values of a nonlinear function with a tangent line about point ...
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38 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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91 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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1answer
57 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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1answer
31 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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46 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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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 ...
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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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31 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 ...
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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 ...
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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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114 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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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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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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16 views

Question about linearising system with second derivative

I need to linearise a system: $\ddot{x}+4\dot{x}^5+(x^2+1)u=0$. The referenced answer is :$\ddot{x}+0+(0+1)u\approx0$. So, the linearly approximated about $x=0$ is: $\ddot{x}=-u$ I can understand ...
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19 views

Using Linear Kalman Filters with a Nonlinear System?

Can you answer these questions I have about using linear Kalman filters and extended Kalman filters with a nonlinear system? 1. Does using a linear Kalman filter mean that I must have a ...
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32 views

Linear independent set of function applied to water waves.

I need to show that a given surface elevation $\zeta(x,y,t)$ defined on a closed region $D(x,y,t): 0<x<L_x,0<y<L_y,0<t<T$ and not periodic on D: $$ ΞΆ(x,y,t) = \sum_{n=1}^{\infty} ...
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22 views

Algorithm for finding linear recurrent approximations to integer sequences

Is there an algorithm for taking a sequence of integers and approximating a first part of it piecewise if need be with pieces like: $$ \text{ if } n = 2k, \\ a_n = a_{n-1} - a_{n-2} + 1 $$ then, ...
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23 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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37 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, ...