# 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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### 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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### 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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### 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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### How to approximate Heaviside function by polynomial

I have a Heaviside smooth function that defined as $$H_{\epsilon}=\frac {1}{2} [1+\frac {2}{\pi} \arctan(\frac {x}{\epsilon})]$$ I want to use polynominal to approximate the Heaviside function. ...
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### 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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### 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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### 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 know ...
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 ...
### 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 \$\...