# Tagged Questions

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

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### How to convert liters to quarts by linear approximation? [closed]

One liter is 1.0567 quarts. How do you use linear approximation to estimate how many liters are in one quart?
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### Jacobian matrix and Taylor expansion

Let $\mathbf{W}(\alpha)$ be a matrix which depends to parameter $\alpha$ and let $\mathbf{f}$ be a vector. I want to approximate $\mathbf{W}(\alpha+\Delta \alpha)\mathbf{f}$ using Taylor expansion. ...
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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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
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 ...
### 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 ...