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is there a good (linear) approximation for the following function: $$(1+f(x))^T f(x) \over (1+ f(x))^T -1 $$ where $f(x)$ is a function with the variable $x$ and $T$ is a time index. I already tried a Taylor-approximation but I'm not really satisfied with my solution. Although I only used it up to the first derivative because the second derivative for $f(x)$ is pretty nasty. I'm only looking for a general concept. Thanks

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    $\begingroup$ That's what first-order Taylor approximation is: the best linear approximation around a chosen point. It uses the derivative and value at that point to find the tangent line to the graph (or plate, etc. for multivariable functions). If you used the second derivative as well, you would get a quadratic approximation. $\endgroup$ Mar 22, 2017 at 16:10
  • $\begingroup$ Thank you for your answer. And is there another method to get a linear approximation? $\endgroup$
    – PAS
    Mar 22, 2017 at 16:12
  • $\begingroup$ If you want the approximation to be as accurate as possible as x approaches a chosen point $a$, no. If you have some other criterion for how the function is approximated, there could be other methods. For example, I could be say that $g(x)=0$ is a "linear approximation" to $f(x) = sin(x)$ if my criterion is "the error is never more than 1", and I wouldn't get that (silly) answer with a first-order Taylor approximation anywhere.There's also complications with non-analytic functions. But essentially, no. You may wish to read this short page: math.brown.edu/UTRA/linapprox.html $\endgroup$ Mar 22, 2017 at 16:34

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