Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [linear-approximation]

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

0
votes
0answers
12 views

Approximation factor for Facility location LP problem with square cost function

According to Approximation algorithm By Vijay Vazirani, Consider the following modification to the metric uncapacitated facility location. Define the cost of connecting city $j$ to facility $i$ to be ...
0
votes
0answers
12 views

Null space of an unbounded operator

Consider the following ODE. $$ a(t) \frac{d^2 u}{d t^2} +b(t) u =0 $$ I want to find is general solution, i.e. the space of its solutions without boundary conditions. This is very important because my ...
0
votes
1answer
36 views

Applications of Partial Derivatives on $f(x, y) = \sqrt{\frac{1}{|1-x^2 - y^2|}}$

Applications of Partial Derivatives on $f(x, y) = \sqrt{\frac{1}{|1-x^2 - y^2|}}$ a) Find the domain of the function. I think the domain should be real numbers without a circle with radius 1, and ...
0
votes
1answer
17 views

linear approximation method x=0.1+ln10

I was wondering how you would approach this question: Estimate $e^x$ at $x = ln(10) + 0.1$, using the method of small increments (i.e. the linearisation method). Im not sure what to do i made $f(x) = ...
0
votes
1answer
59 views

How to start this proof? The line through $(a,b)$ and $(c,d)$ is the best linear approximation of any function through those points, as $a\to c$.

I want to prove that the line passing through $(a,b)$ and $(c,d)$ is the best linear approximation of any function passing through those 2 points, as $a$ approaches $c$. I'm not sure how to define "...
0
votes
1answer
44 views

Linear approximation question where f(x) = g'(x) - how to use formula?

I'm a little confused by this question: Lines and things that are linear are relatively boring in mathematics. What if my function f(x) = g'(x). I’m going to ask the same question in a ...
0
votes
0answers
19 views

Function expansion in terms of itself

I am given a function $f(t,x_1,x_2,...x_n)$, $R^{n+1} \rightarrow R$ I know that the small changes in variable $t$ lead to small changes in value of the function, $lim_{\Delta t \to 0}f(t+\Delta t,...
0
votes
3answers
57 views

How to find a function's approximation?

I am having problems with the following question: Use the linear approximation $(1+x)^k\approx 1+kx$ to find an approximation for the function $f(x)$ for values of $x$ near zero $$f(x)=\sqrt[3]{\...
0
votes
3answers
54 views

approximate $\sqrt{x+y}$ [closed]

How can one approximate the expression $$\sqrt{x+y}$$ I think this can achieved by a Taylor expansion but I don't know how.
0
votes
2answers
34 views

How to do this Linear Approximation?

this question has been giving me a little trouble: Use a linear approximation to estimate the number $8.07^{2/3}$ I tried using $f(a)+f'(a)(x-a)$ but the answer I get ($4.02$) is apparently wrong. ...
0
votes
0answers
40 views

Estimating the error of approximation for $\sqrt{x}$ for $|x-1| \leq 0.5$. No.4.5.3 Petrovic

Question 4.5.3 is given below: And the answer of Question 4.5.1 is given below: But I do not know how the condition $|x-1|\leq 0.5$ in Question 4.5.3 will make its solution different from ...
0
votes
1answer
74 views

How to solve linearization to approximate $\sqrt[3]{61}$. to the nearest ten thousandth

I'm not sure what the nearest ten-thousandth place means in the question above. How can I solve this problem step by step?
3
votes
2answers
49 views

Clarification of Notion of a “Good Approximation”

My textbook says the following: $$\lim_{x \to x_0} \dfrac{f(x) - f(x_0) - f'(x_0)(x - x_0)}{x - x_0} = 0$$ Thus, the tangent line $l$ through $(x_0, f(x_0))$ with slope $f'(x_0)$ is close to $...
1
vote
1answer
79 views

Using Differentials to Calculate the Volume of a Square Pyramid

Use differentials to solve the problem: The Louvre Pyramid is a tourist attraction in Europe. It is a square pyramid, with a height of $21 m$, and base of side length $35 m$. The four faces of this ...
0
votes
1answer
34 views

Does the inequality $\frac{p^Tp}{p^TAp} < 2\frac{p^TA^{-1}p}{p^Tp}$ hold?

Consider $A \in \mathbb{R}^{m \times m}$, $b\in\mathbb{R}^m$, $p \in \mathbb{R}^m$ and $x\in\mathbb{R}^m / \{A^{-1}b\}$, were the following properties and relations hold: $\quad A>0$ $\quad A^T=A$ ...
0
votes
1answer
184 views

How to linearize the distance formula?

So I have an equation that contains the distance formula squared. However, I am interested in linearizing this equation. My equation is: Constant/distance squared My distance is ...
2
votes
3answers
60 views

If $f(x_0)+f'(x_0)(x-x_0)$ is the affine approximation to $f$ at $x_0$, is $f'(x_0)(x-x_0)$ the linear approximation?

If $f: \mathbb R \to \mathbb R$ is differentiable at $x_0$, the affine approximation of $f$ at $x_0$ is $f(x_0)+f'(x_0)(x-x_0)$. So, would that mean that $f'(x_0)(x-x_0)$ is the linear approximation?
4
votes
1answer
121 views

Intuition of error in Taylor appoximation and finding error in approximation of a function by a constant function

I am reading up on Taylor approximation of a function and I'm trying to develop the intuition for the remainder, when approximating a function with $n^{th}$ degree polynomial which has a continuous $(...
1
vote
2answers
86 views

Placing $n$ linear functions so that it is best fit to another function in integral norm sense?

Say we want to build a function which is piecewise linear $$f(x) = \sum_{\forall k} (H(x-x_k)-H(x-x_{k+1}))l_k(x)\\l_k(x) = c_{k1}x+c_{k2}$$ And also so that it fits best possibly some function $x\to ...
2
votes
2answers
80 views

What is the linear approximation of $f(x) = \frac1x$ when $x \approx 0$?

I did not find an example when the denominator $x$ approximates to $0$. $f(0) + f'(0)x$ does not work because $f(0)$ would be $+\infty$.
0
votes
3answers
638 views

Linear Approximation and Volume of Cylinder

I was given the question: The volume V of a cylinder is computed using the values 6m for the diameter and 9.8m for the height. Use the linear approximation to estimate the maximum error in V if each ...
1
vote
1answer
83 views

Explanation for the Continuity of the Jacobi Matrix

Let $U \subset \mathbb R^{d}$ be an open set, $x \in U$ and $f: U \to \mathbb R$ partially differentiable, it states: $\partial_{k}f$ is continuous $\forall k \in \{1,...,d\} \iff f$ is ...
1
vote
1answer
15 views

Differentials: to the nearest milimetre question

When it says the side length of a cube was measured to be 20mm to the nearest millimeter. In this case, can I regard the maximum error for the side to be 0.5 mm? to compute the absolute error of the ...
1
vote
0answers
59 views

Linear approximation for an implicit function 2

This is a question related to my previous question but due to wrong formulated my question I would like to re-post and not only edit since it was solved for the first time after I put 200 bounty ...
2
votes
1answer
192 views

Linear approximation for an implicit function

Let $$F(r,\mu,t)=r- \frac{2\mu\left ( \dfrac{e^t(t-1)-1}{t^2} \right )}{(1-2\mu)+2\mu \left ( \dfrac{e^t-1}{t} \right )}=0,$$ I would like to find a linear approximation$$ t(r,\mu) \approx t(0,0) + r\...
0
votes
0answers
26 views

Find a vector $x \in \mathbb{R}^n$ such that $x \neq 0$, $x_1w_1 + \dots + x_nw_n \approx 0$ and $x \approx z$

This is a problem I created to myself but now I can't solve it. Hope you can help me to solve or give some insights. Fix two vectors $z,w \in \mathbb{R}^n$ and consider the Euclidian inner product. I ...
0
votes
1answer
259 views

Product and Quotient Rule proof using linearisation

So I've recently been introduced to the concept of linearization and now I'm beginning to apply this concept to prove certain differenation rules. I've managed to prove the chain rule so far, but I ...
1
vote
2answers
656 views

Taylor approximation of inverse square root

Given the function $f(x)=\sqrt{1+mx+\mathcal{O}(x^2)}$ I am reading that $g(x) = \frac{1}{f(x)}$, the inverse square root, can be computed with first order Taylor approximation and take $g(x) = 1 - \...
3
votes
4answers
91 views

“Good” linear approximation criteria?

I've been told that linear approximation is considered as "good" if it meets the criteria below: $$\lim_{x \to a} \frac{f(x)-f(a)-f'(a)(x-a)}{x-a} = 0$$ As far as I understand, the differentiation of ...
0
votes
0answers
92 views

Geometrical interpretation of derivative?

Statement: being unable to solve the famous derivative's limit at the point $P$ is actually equals to being unable to graph a tangent line to that point $P$. Statement seems to be incomplete. At any ...
0
votes
0answers
24 views

Linear approximation measure?

The problem: given any $F(x)$ and a point $P_1$, and an $\Delta \gt 0$, find an $P_2$ such that $L(x)$ is a straight line crossing both points which meets the requirement: $|F(x) - L(x)| \lt \Delta$ ...
0
votes
1answer
52 views

Complete the matrix through a rank-one approximation.

Complete the matrix through a rank-one approximation: \begin{pmatrix}1&2\\3&?\end{pmatrix} I am new to these types of problems, therefore a detailed explanation would really be appreciated.
1
vote
1answer
1k views

How can I formulate the 3-SAT problem as a 0-1 Linear integer program?

I understand the 3-Sat problem but I do not understand 0-1 Linear Integer Program. I know in a linear integer program I would have an indicator variable $X_i$ that indicates whether a clause is true ...
0
votes
0answers
81 views

The order of accuracy when finding the LTE

When finding the Local Truncation Error (LTE) of a Linear multistep method (LMM), I'm aware how to taylor expand the expression, multiply by $1/H$, and then simplify as shown below: Now my question ...
1
vote
0answers
147 views

Derivative as a linear approximation? [closed]

Is it true, that any derivative of a single-argument function $f(x)$ gives an $A$ coefficient form the line equation: $Ax + b$? Does it mean, that having a derivative of any $f(x)$ I could graph a ...
2
votes
1answer
404 views

Linear Approximation to Estimate Integral

The function $g$ is defined by $g(x)=\int_{-20}^x (f(t))^2dt,$ and we're given $$g(20)=100,f(0)=4,f'(0)=12, f''(0)=20.$$ I am to use a tangent line approximation to estimate $g(0.1)$. I believe I ...
1
vote
1answer
122 views

Taylor Remainder, $\arcsin(x)$

Given that the $0$ degree Taylor polynomial centered at $x=\frac{1}{2}$ for $f(x)=\arcsin(x)$ is given by $T_{0,\frac{1}{2}}(x)=\frac{\pi}{6}$ how would I use this to show that $1.2(x-\frac{1}{2})\leq ...
2
votes
1answer
216 views

Newtons Cooling Law estimate constant

Newton's law of cooling states that $\frac{dT}{dt}= -k(T-T_s)$ if $T(0)=100$, $T(3)=75$ and $T_s=25$ then how would you use linear approximation to estimate the value of k? Can you use the Mean Value ...
3
votes
1answer
59 views

How can I find closest line to $x^2$ in the $C^1[0,1]$ norm

Find all closest lines $p(x)=ax+b$ to $f(x)=x^2$ in the $C^1[0,1]$ norm. Note that the best approximation is not unique Attempt : Let $r(x)=x^2-ax-b$. Then $\|r(x)\|_{C^1}=\max\{|r^{(i)}(x)| : 0 \...
0
votes
1answer
29 views

Does the function $U=\frac{kx}{(x+x_{0})^2}$ reduce to a simple harmonic potential energy function for small oscillations around $x=x_0$?

The question is from a physics problems book, A particle of mass $m$ moves in a potential energy function given by $$U=\frac{kx}{(x+x_{0})^2}$$ where $x$ denotes the position and $x_0$ is a ...
-1
votes
2answers
2k views

When using linear approximation to estimate 1/4.002, what is the x in L(x) = f(a)-f'(a)(x-a)

Sorry in advance for the formatting I've never used this site before. I'm asked to use linear approximation to estimate 1/4.002 I found the derivative = -1/(x^2) and I believe f(x) is 1/4 so a = ...
0
votes
1answer
144 views

A question about linear approximation. [closed]

Since we know that in a good linear approximation, $L(x)=f(a)+f'(a)(x-a).$ But what if $f'(a)$ does not exist? How to prove that if a function has a good linear approximation, then it must be ...
0
votes
0answers
1k views

How to approximate the numbers after finding the linear approximation?

Here is the problem: Find the linear approximation of the function $g(x)= \sqrt[5]{1 + x}$ at a = 0. I found the linear approximation to be g(x) $\approx$ $\frac 15$x + 1 Use it to approximate the ...
2
votes
0answers
27 views

replace function by linear approximation

Lets be $f: \mathbb{R}^2 \longrightarrow \mathbb{R}$ and $g:\mathbb{R} \longrightarrow \mathbb{R}$. If $\epsilon(x)=\alpha x+\beta$ is a linear approximation of $g$ at $x_0 \in \overline{\mathbb{R}}$ (...
0
votes
2answers
58 views

estimate $f(3.8)-f(3)$ with $f(x) = \sqrt{x+1}$ [closed]

We are given $f(x)= \sqrt{x+1}$ and use the Linear Approximation to this function at $a=3$ with $\triangle x = 0.8$ to estimate $f(3.8)-f(3) = \triangle f \approx df$ Can you guys explain each step ...
3
votes
1answer
160 views

Linear approximation of hyperbola vs of circle?

I was trying to do a linear approximation using hyperbolic trig functions and comparing that with a linear approximation to a circle at any given angle $a$. So I found that for the unit circle: $$f(x)...
0
votes
1answer
2k views

Percentage error: linear approximation

I have trouble understanding the question below and I do not really know what linear approximation has to do with this: Determine how accurate should we measure the side of a cube so that the ...
1
vote
0answers
175 views

Calculating critical points of functions

I know how to calculate critical points using the derivative, but on this question I don't even know where to start. HELP! It is the final exam period and you have only math and chemistry exams left. ...
2
votes
2answers
35 views

Why does one need to use $f(1,0)$ as linear approximation, rather than calculating $f(1.1, -0.1)$ directly?

Why does one need to use $f(1,0)$ a linear approximation, rather than calculating $f(1.1, -0.1)$ directly? e.g. when $f(x,y)=xe^{xy}$ (used to be more complicated $e^{xy}+xye^{xy}$) It's easy to ...
3
votes
1answer
1k views

What is the difference between linear approximation and a differential?

From my understanding, linear approximations and differentials both use the tangent line to a function to estimate the value of the function at a point. I understand that their respective equations ...