Questions tagged [linear-approximation]

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

170 questions
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...