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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
31 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
39 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
39 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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1answer
17 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
36 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
14 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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12 views

Log-linearization around the steady-state

I am doing log-linearization around the steady-state and currently stuck to the following two problems: $K_{t+1}= \frac{1}{u_{t}}[(a+q_{t})K_{t}-b_{t}]$ ...
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2answers
74 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
23 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 ...
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2answers
47 views

Generalization of linear approximation? [closed]

How is the linear approximation is generalized to the Taylor series? I do not get that concept.
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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 ...
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1answer
32 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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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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0answers
65 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
38 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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2answers
40 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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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 ...
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2answers
57 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
46 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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3answers
40 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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41 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
44 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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1answer
136 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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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 ...
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0answers
41 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 ...
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3answers
102 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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1answer
52 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
45 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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68 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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1answer
27 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
1k 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
29 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
54 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
27 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
78 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
31 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
212 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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0answers
82 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
2k 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
2k 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
139 views

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

Suppose $f'(x)$ is a differentiable and increasing function for all $x$. Which number is the largest and why? $f(4+\Delta x)$ or $f(4)+f'(4)\Delta x$? I believe $f(x)$ must be concave up everywhere ...
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1answer
113 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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0answers
81 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
525 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 ...